[ { "id": "C1-E01", "tex": "\\begin{gathered}\\begin{aligned}\n \\dot R&=dR/dt,\\qquad \\ddot R=d\\dot R/dt,\\\\\n \\rho R_0\\ddot R(0)&=p_b(0)-p_\\infty(0)-2\\sigma/R_0-4\\mu U_0/R_0-\\tfrac32\\rho U_0^2,\\\\\n U_0=0\\quad&\\Longrightarrow\\quad\n \\ddot R(0)=\\frac{p_b(0)-p_\\infty(0)-2\\sigma/R_0}{\\rho R_0}.\n \\end{aligned}\\\\[6pt]\\begin{gathered}R(0)=R_0>0\\\\ \\dot R(0)=U_0\\\\ u(r,t)\\to0,\\quad p(r,t)\\to p_\\infty(t)\\quad(r\\to\\infty)\\end{gathered}\\end{gathered}", "symbols_en": "$R$ is the cavity radius (m); $t$ is time (s), with 0 the chosen initial instant; $R_0$ is its prescribed positive initial value (m); $\\dot R$ is wall speed and $U_0$ its initial value (m s⁻¹); $r$ is radial position (m); $u$ is radial liquid speed (m s⁻¹); $p$ is liquid pressure (Pa); $p_\\infty$ is prescribed far-field pressure (Pa). The arrow denotes the far-field limit. $\\rho>0$ is carrier density (kg m⁻³); $p_b$ is bubble pressure (Pa); $\\sigma$ is constant surface tension (N m⁻¹); $\\mu$ is viscosity (Pa s); $\\ddot R$ is acceleration (m s⁻²). All pressures share an absolute reference.", "symbols_zh": "$R$ 为空腔半径（m）；$t$ 为时间（s），0 为选定的初始时刻；$R_0$ 为给定的正初始半径（m）；$\\dot R$ 为壁面速度，$U_0$ 为其初值（m s⁻¹）；$r$ 为径向位置（m）；$u$ 为液体径向速度（m s⁻¹）；$p$ 为液体压力（Pa）；$p_\\infty$ 为给定远场压力（Pa）。箭头表示远场极限。 $\\rho>0$ 为载液密度（kg m⁻³）；$p_b$ 为气泡压力（Pa）；$\\sigma$ 为常数表面张力（N m⁻¹）；$\\mu$ 为黏度（Pa s）；$\\ddot R$ 为加速度（m s⁻²）。全部压力采用同一绝对参考值。", "kind": "Prescribed initial and boundary conditions", "kind_zh": "给定的初始条件与边界条件", "diagram": { "type": "sphere", "labels": [ "R(t)", "R₀ > 0", "Ṙ(0) = U₀", "r → ∞", "u → 0", "p → p∞", "t = 0" ], "notes": [ "Radial direction points from bubble center into carrier", "Far-field pressure is an input, not a laser calculation" ] }, "caption_en": "The evolving spherical boundary and the prescribed far-field state.", "caption_zh": "运动球形边界与给定的远场状态。" }, { "id": "C1-E02", "tex": "\\begin{gathered}\\begin{aligned}\n \\frac{d}{dt}\\left[\\rho\\frac{4\\pi}{3}((r+\\delta r)^3-r^3)\\right]&=0,\\\\\n 4\\pi\\rho r^2u(r,t)-4\\pi\\rho(r+\\delta r)^2u(r+\\delta r,t)&=0,\\\\\n \\lim_{\\delta r\\to0}\\frac{(r+\\delta r)^2u(r+\\delta r,t)-r^2u(r,t)}{\\delta r}&=\\partial_r(r^2u)=0,\\\\\n 0&=\\frac{1}{r^2}\\frac{\\partial(r^2u)}{\\partial r},\\\\\n \\frac{\\partial(r^2u)}{\\partial r}&=0\n \\quad\\Longrightarrow\\quad r^2u=C(t),\\\\\n u(R,t)&=\\dot R\\quad\\Longrightarrow\\quad C(t)=R^2\\dot R,\\\\\n \\frac{u(2R,t)}{\\dot R}&=\\frac{R^2}{(2R)^2}=\\frac14\\quad(\\dot R\\ne0).\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\frac{1}{r^2}\\frac{\\partial(r^2u)}{\\partial r}=0\\ \\Longrightarrow\\ r^2u=C(t)\\\\ u(R,t)=\\dot R\\ \\Longrightarrow\\ C(t)=R^2\\dot R\\\\ u(r,t)=\\frac{R^2\\dot R}{r^2}\\end{gathered}\\end{gathered}", "symbols_en": "$r\\ge R(t)>0$ is radial position (m); $R$ is cavity radius (m); $t$ is time (s); $u$ is radial carrier velocity (m s⁻¹); $\\dot R$ is wall speed (m s⁻¹); $C(t)$ is the radius-independent flux factor (m³ s⁻¹), not a material constant; $\\partial/\\partial r$ differentiates at fixed time. The implication arrows indicate integration and use of the interface condition. $C(t)$ is the radial volume-flux factor (m³ s⁻¹); $\\rho>0$ is constant liquid density (kg m⁻³); $\\delta r>0$ is fixed shell thickness (m); $\\pi$ is the circle constant; $\\partial_r$ holds time fixed; the fixed shell stays liquid; the liquid domain is $r\\ge R(t)>0$.", "symbols_zh": "$r\\ge R(t)>0$ 为径向位置（m）；$R$ 为空腔半径（m）；$t$ 为时间（s）；$u$ 为载液径向速度（m s⁻¹）；$\\dot R$ 为壁面速度（m s⁻¹）；$C(t)$ 为与径向位置无关的流量因子（m³ s⁻¹），并非材料常数；$\\partial/\\partial r$ 表示固定时间下的径向求导。箭头表示积分及应用界面条件。 $C(t)$ 为径向体积通量因子（m³ s⁻¹）；$\\rho>0$ 为常数液体密度（kg m⁻³）；$\\delta r>0$ 为固定球壳厚度（m）；$\\pi$ 为圆周率；$\\partial_r$ 保持时间固定；固定球壳始终属于液体；液体区域为 $r\\ge R(t)>0$。", "kind": "Exact continuity within the spherical model", "kind_zh": "球形模型内的精确连续性关系", "diagram": { "type": "sphere", "labels": [ "r ≥ R", "u(r,t)", "Ṙ", "C(t) = R²Ṙ", "t", "r²u = constant" ], "notes": [ "Concentric surfaces carry the same volume flux", "At r = 2R the speed is Ṙ/4" ] }, "caption_en": "Inverse-square velocity decay places the inertia in the surrounding carrier.", "caption_zh": "速度按距离平方反比衰减，惯性来自周围载液。" }, { "id": "C1-E03", "tex": "\\begin{gathered}\\begin{aligned}\n \\boldsymbol u&=u(r,t)\\boldsymbol e_r,\\qquad\\nabla\\times\\boldsymbol u=\\boldsymbol0,\\\\\n \\partial_r\\phi&=u=R^2\\dot R/r^2,\\\\\n \\phi(r,t)&=-R^2\\dot R/r+g(t),\\\\\n \\phi(\\infty,t)&=0\\quad\\Longrightarrow\\quad g(t)=0,\\\\\n \\int_\\infty^r s^{-2}\\,ds&=[-s^{-1}]_\\infty^r=-1/r-0.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\phi(r,t)=\\int_\\infty^r u(s,t)\\,ds=R^2\\dot R\\int_\\infty^r s^{-2}\\,ds=-\\frac{R^2\\dot R}{r}\\\\ \\frac{\\partial\\phi}{\\partial r}=u\\end{gathered}\\end{gathered}", "symbols_en": "$\\phi$ is velocity potential (m² s⁻¹); $r$ is liquid radial position and $R$ cavity radius (m); $t$ is time (s); $s$ is a dummy radial integration coordinate (m); $u$ is radial velocity (m s⁻¹); $\\dot R$ is wall speed (m s⁻¹); the integral is an improper integral convergent for $r\\ge R>0$; $\\partial/\\partial r$ is the fixed-time spatial derivative. $\\boldsymbol u$ is velocity (m s⁻¹), $\\boldsymbol e_r$ the outward radial unit vector, $\\nabla\\times$ the curl (inverse-length differentiation), $\\boldsymbol0$ the zero vector; $g(t)$ is a potential gauge (m² s⁻¹).", "symbols_zh": "$\\phi$ 为速度势（m² s⁻¹）；$r$ 为液体径向位置，$R$ 为空腔半径（m）；$t$ 为时间（s）；$s$ 为径向积分哑变量（m）；$u$ 为径向速度（m s⁻¹）；$\\dot R$ 为壁面速度（m s⁻¹）；该广义积分在 $r\\ge R>0$ 时收敛；$\\partial/\\partial r$ 为固定时间的空间导数。 $\\boldsymbol u$ 为速度（m s⁻¹），$\\boldsymbol e_r$ 为向外径向单位矢量，$\\nabla\\times$ 为旋度算子（逆长度微分），$\\boldsymbol0$ 为零矢量；$g(t)$ 为速度势规范函数（m² s⁻¹）。", "kind": "Derived velocity potential", "kind_zh": "推导得到的速度势", "diagram": { "type": "sphere", "labels": [ "φ(r,t)", "r", "R", "Ṙ", "s: integration radius", "u = ∂rφ", "φ∞ = 0", "t" ], "notes": [ "Differentiate spatially to recover the radial flow" ] }, "caption_en": "The potential reference is fixed at infinity before differentiation.", "caption_zh": "在求导之前，先固定无穷远处的速度势参考值。" }, { "id": "C1-E04", "tex": "\\begin{gathered}\\begin{aligned}\n \\frac{d(R^2\\dot R)}{dt}&=2R\\dot R\\dot R+R^2\\ddot R,\\\\\n \\frac{d\\phi(R(t),t)}{dt}&=\\left.\\partial_t\\phi\\right|_{r=R}\n +\\dot R\\left.\\partial_r\\phi\\right|_{r=R},\\\\\n &=-(2\\dot R^2+R\\ddot R)+\\dot R\\dot R\\\\\n &=-\\dot R^2-R\\ddot R.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\left.\\frac{\\partial\\phi}{\\partial t}\\right|_r=-\\frac{2R\\dot R^2+R^2\\ddot R}{r}\\\\ \\left.\\frac{\\partial\\phi}{\\partial t}\\right|_{r=R}=-(2\\dot R^2+R\\ddot R)\\\\ \\frac{d\\phi(R(t),t)}{dt}=-\\dot R^2-R\\ddot R\\end{gathered}\\end{gathered}", "symbols_en": "$\\phi$ is velocity potential (m² s⁻¹); $t$ is time (s); $r$ is fixed radial position, $R(t)>0$ is moving cavity radius (m); $\\dot R$ and $\\ddot R$ are wall speed (m s⁻¹) and acceleration (m s⁻²); vertical bars specify where the derivative is taken or evaluated; $d/dt$ follows the moving wall value. Each derivative has units m² s⁻². ", "symbols_zh": "$\\phi$ 为速度势（m² s⁻¹）；$t$ 为时间（s）；$r$ 为固定径向位置，$R(t)>0$ 为运动空腔半径（m）；$\\dot R$ 与 $\\ddot R$ 分别为壁面速度（m s⁻¹）和加速度（m s⁻²）；竖线表示求导或取值条件；$d/dt$ 跟随运动壁面的取值。各导数的单位均为 m² s⁻²。 ", "kind": "Exact Eulerian and moving-point derivatives", "kind_zh": "精确的欧拉导数与运动点导数", "diagram": { "type": "sphere", "labels": [ "Fixed r", "Moving R(t)", "φ", "Ṙ", "R̈", "∂tφ at fixed r", "dφ(R,t)/dt", "t" ], "notes": [ "Take the partial derivative before setting r = R", "The moving-point derivative differs by Ṙ²" ] }, "caption_en": "A fixed liquid observation point and a moving interface point have different time derivatives.", "caption_zh": "固定液体观察点与运动界面点对应不同的时间导数。" }, { "id": "C1-E05", "tex": "\\begin{gathered}\\begin{aligned}\n \\nabla^2\\boldsymbol u&=\\nabla(\\nabla\\cdot\\boldsymbol u)-\\nabla\\times(\\nabla\\times\\boldsymbol u)=\\boldsymbol0,\\\\\n \\partial_t\\boldsymbol u+(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u&=-\\nabla p/\\rho,\\\\\n \\sum_{j=1}^3u_j\\partial_j u_i&=\\sum_{j=1}^3u_j\\partial_i u_j\n =\\partial_i\\left(\\tfrac12\\sum_{j=1}^3u_j^2\\right)\\quad(i=1,2,3),\\\\\n \\boldsymbol u=\\nabla\\phi,\\quad(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u\n &=\\nabla(|\\boldsymbol u|^2/2),\\\\\n \\nabla\\left(\\partial_t\\phi+|\\nabla\\phi|^2/2+p/\\rho\\right)&=\\boldsymbol0,\\\\\n \\partial_t\\phi+|\\nabla\\phi|^2/2+p/\\rho&=B_0(t),\\\\\n \\phi_\\infty=0,\\quad\\boldsymbol u_\\infty=\\boldsymbol0\n &\\quad\\Longrightarrow\\quad B_0(t)=p_\\infty(t)/\\rho,\\\\\n \\partial_t\\phi+\\tfrac12|\\nabla\\phi|^2+p/\\rho&=p_\\infty/\\rho,\\\\\n (p_l-p_\\infty)/\\rho&=-\\partial_t\\phi|_{r=R}-\\tfrac12\\dot R^2.\n \\end{aligned}\\\\[6pt]\\begin{aligned}\n (p_l-p_\\infty)/\\rho&=-\\partial_t\\phi|_{r=R}-u(R,t)^2/2\\\\\n &=2\\dot R^2+R\\ddot R-\\dot R^2/2\\\\\n &=R\\ddot R+\\tfrac32\\dot R^2.\n \\end{aligned}\\end{gathered}", "symbols_en": "$p_l$ is liquid pressure at the wall and $p_\\infty$ far-field pressure (Pa); $\\rho>0$ is carrier density (kg m⁻³); $\\phi$ is velocity potential (m² s⁻¹); $t$ is time (s); $r$ is radial position and $R>0$ cavity radius (m); $u$ is radial velocity and $\\dot R$ wall speed (m s⁻¹); $\\ddot R$ is wall acceleration (m s⁻²); $\\partial_t$ is taken at fixed position before wall evaluation. $\\boldsymbol u=\\nabla\\phi$ is liquid velocity (m s⁻¹); $\\nabla$, $\\nabla\\cdot$, $\\nabla\\times$, $\\nabla^2$ denote gradient, divergence, curl and vector Laplacian; $u_i,u_j$ are Cartesian velocity components (m s⁻¹), $i,j\\in\\{1,2,3\\}$; $\\partial_i,\\partial_j$ are coordinate derivatives (m⁻¹); $\\sum$ adds over the indicated index; zero vorticity implies $\\partial_j u_i=\\partial_i u_j$. $|\\ |$ is Euclidean magnitude; $B_0(t)$ is the Bernoulli integration function (m² s⁻²); subscript $\\infty$ denotes the far-field limit; $p$ is local liquid pressure (Pa). Smooth fields permit exchanging derivatives.", "symbols_zh": "$p_l$ 为壁面液体压力，$p_\\infty$ 为远场压力（Pa）；$\\rho>0$ 为载液密度（kg m⁻³）；$\\phi$ 为速度势（m² s⁻¹）；$t$ 为时间（s）；$r$ 为径向位置，$R>0$ 为空腔半径（m）；$u$ 为径向速度，$\\dot R$ 为壁面速度（m s⁻¹）；$\\ddot R$ 为壁面加速度（m s⁻²）；$\\partial_t$ 必须先在固定位置求导，再于壁面取值。 $\\boldsymbol u=\\nabla\\phi$ 为液体速度（m s⁻¹）；$\\nabla$、$\\nabla\\cdot$、$\\nabla\\times$、$\\nabla^2$ 分别表示梯度、散度、旋度及矢量拉普拉斯；$u_i,u_j$ 为笛卡尔速度分量（m s⁻¹），$i,j\\in\\{1,2,3\\}$；$\\partial_i,\\partial_j$ 为坐标导数（m⁻¹）；$\\sum$ 对指定指标求和；无旋条件给出 $\\partial_j u_i=\\partial_i u_j$。$|\\ |$ 为欧氏模长；$B_0(t)$ 为伯努利积分函数（m² s⁻²）；下标 $\\infty$ 表示远场极限；$p$ 为局部液体压力（Pa）。场足够光滑时允许交换导数。", "kind": "Derived inertial pressure", "kind_zh": "推导得到的惯性压力", "diagram": { "type": "sphere", "labels": [ "pl at r = R", "p∞", "ρ", "φ", "u(R,t) = Ṙ", "R̈", "r", "t" ], "notes": [ "R R̈ and 3Ṙ²/2 each have units m²/s²" ] }, "caption_en": "Liquid inertia relates the interface pressure to wall acceleration and squared speed.", "caption_zh": "液体惯性把界面压力与壁面加速度及速度平方联系起来。" }, { "id": "C1-E06", "tex": "\\begin{gathered}\\begin{aligned}\n \\boldsymbol n&=-\\boldsymbol e_r,\\qquad\n \\nabla_s\\cdot\\boldsymbol n=\\nabla\\cdot\\boldsymbol n\n =\\frac1{r^2}\\partial_r(-r^2)=-2/r,\\\\\n \\kappa&=(\\nabla_s\\cdot\\boldsymbol n)|_{r=R}=-2/R,\\\\\n p_l-p_b&=\\sigma\\kappa+2\\mu\\partial_r u|_{r=R},\\\\\n \\partial_r(R^2\\dot R r^{-2})&=-2R^2\\dot R r^{-3},\\\\\n 2\\mu\\partial_r u|_{r=R}&=2\\mu(-2\\dot R/R)=-4\\mu\\dot R/R.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\left.\\frac{\\partial u}{\\partial r}\\right|_{r=R}=\\left.-\\frac{2R^2\\dot R}{r^3}\\right|_{r=R}=-\\frac{2\\dot R}{R}\\\\ p_l=p_b-\\frac{2\\sigma}{R}+2\\mu\\left.\\frac{\\partial u}{\\partial r}\\right|_{r=R}=p_b-\\frac{2\\sigma}{R}-\\frac{4\\mu\\dot R}{R}\\end{gathered}\\end{gathered}", "symbols_en": "$u$ is radial velocity and $\\dot R$ wall speed (m s⁻¹); $r$ is radial position and $R>0$ cavity radius (m); $p_l,p_b$ are liquid-side and uniform bubble pressures (Pa); $\\sigma\\ge0$ is surface tension (N m⁻¹); $\\mu\\ge0$ is carrier viscosity (Pa s); $\\partial_r u$ is radial strain rate (s⁻¹); wall evaluation is performed at $r=R$. $\\boldsymbol n$ is the liquid-to-gas unit normal; $\\boldsymbol e_r$ is the outward radial unit vector; $\\kappa$ is signed curvature (m⁻¹); $\\nabla_s\\cdot$ is surface divergence and $\\nabla\\cdot$ volume divergence. Extend the unit normal radially without radial variation, so its normal derivative is zero and both divergences agree at the sphere.", "symbols_zh": "$u$ 为径向速度，$\\dot R$ 为壁面速度（m s⁻¹）；$r$ 为径向位置，$R>0$ 为空腔半径（m）；$p_l,p_b$ 分别为液体侧与空间均匀的气泡压力（Pa）；$\\sigma\\ge0$ 为表面张力（N m⁻¹）；$\\mu\\ge0$ 为载液黏度（Pa s）；$\\partial_r u$ 为径向应变率（s⁻¹）；壁面取值在 $r=R$ 处进行。 $\\boldsymbol n$ 为液体指向气体的单位法向；$\\boldsymbol e_r$ 为向外径向单位矢量；$\\kappa$ 为带符号曲率（m⁻¹）；$\\nabla_s\\cdot$ 为表面散度，$\\nabla\\cdot$ 为体散度。将单位法向沿径向作不变延拓，其法向导数为零，因此两种散度在球面一致。", "kind": "Newtonian stress and capillary boundary condition", "kind_zh": "牛顿流体应力与毛细边界条件", "diagram": { "type": "sphere", "labels": [ "pb inside", "pl outside", "R", "r", "u", "Ṙ", "2σ/R", "4μṘ/R", "∂ru = −2Ṙ/R" ], "notes": [ "Viscous stress changes sign when the wall reverses" ] }, "caption_en": "Capillary and viscous stresses act at the interface even when the radial bulk viscous force is zero.", "caption_zh": "即使径向体内黏性力为零，毛细与黏性应力仍作用于界面。" }, { "id": "C1-E07", "tex": "\\begin{gathered}\\begin{aligned}\n \\rho(R\\ddot R+\\tfrac32\\dot R^2)&=p_l-p_\\infty,\\\\\n p_l-p_\\infty&=(p_b-2\\sigma/R-4\\mu\\dot R/R)-p_\\infty,\\\\\n \\ddot R&=\\frac{p_b-p_\\infty-2\\sigma/R-4\\mu\\dot R/R}{\\rho R}\n -\\frac{3\\dot R^2}{2R},\\\\\n \\dot R=\\ddot R=0&\\quad\\Longrightarrow\\quad p_b-p_\\infty=2\\sigma/R.\n \\end{aligned}\\\\[6pt]\\rho\\left(R\\ddot R+\\frac32\\dot R^2\\right)=p_b-p_\\infty-\\frac{2\\sigma}{R}-\\frac{4\\mu\\dot R}{R}\\end{gathered}", "symbols_en": "$\\rho>0$ is carrier density (kg m⁻³); $R>0$ is cavity radius (m); $\\dot R,\\ddot R$ are wall speed (m s⁻¹) and acceleration (m s⁻²); $p_b,p_\\infty$ are bubble and far-field pressures (Pa); $\\sigma$ is constant surface tension (N m⁻¹); $\\mu$ is constant carrier viscosity (Pa s). Every term is a pressure. ", "symbols_zh": "$\\rho>0$ 为载液密度（kg m⁻³）；$R>0$ 为空腔半径（m）；$\\dot R,\\ddot R$ 分别为壁面速度（m s⁻¹）和加速度（m s⁻²）；$p_b,p_\\infty$ 为气泡与远场压力（Pa）；$\\sigma$ 为常数表面张力（N m⁻¹）；$\\mu$ 为常数载液黏度（Pa s）。各项均具有压力单位。 ", "kind": "Rayleigh–Plesset under the declared assumptions", "kind_zh": "所述假设下的 Rayleigh–Plesset 方程", "diagram": { "type": "sphere", "labels": [ "ρ carrier", "R", "Ṙ", "R̈", "pb", "p∞", "2σ/R", "4μṘ/R" ], "notes": [ "Outward is positive", "The equation predicts radius, not jet shape" ] }, "caption_en": "The spherical pressure–inertia balance has a scalar radial state.", "caption_zh": "球形压力—惯性平衡只具有标量径向状态。" }, { "id": "C1-E08", "tex": "\\begin{gathered}\\begin{aligned}\n p_iV_b&=n_i\\mathcal R T_b\\quad (i\\in\\{v,\\mathrm{PFC};v,w;g\\}),\\\\\n p_bV_b&=\\left(n_{v,\\mathrm{PFC}}+n_{v,w}+n_g\\right)\\mathcal R T_b,\\\\\n p_b&=\\frac{n_{v,\\mathrm{PFC}}\\mathcal R T_b}{V_b}\n +\\frac{n_{v,w}\\mathcal R T_b}{V_b}+\\frac{n_g\\mathcal R T_b}{V_b}.\n \\end{aligned}\\\\[6pt]p_b=p_{v,\\mathrm{PFC}}+p_{v,w}+p_g\\end{gathered}", "symbols_en": "$p_b$ is uniform total bubble pressure (Pa); $p_{v,\\mathrm{PFC}}$ is the PFC-vapor partial pressure, $p_{v,w}$ water-vapor partial pressure, and $p_g$ permanent-gas partial pressure (all Pa). Subscript $v$ labels vapor, PFC its chemical family, $w$ water, and $g$ noncondensable gas. Additivity assumes a gas-mixture pressure description with the stated partial pressures. $i$ labels PFC vapor, water vapor or permanent gas; $p_i$ is component partial pressure (Pa); $n_i$ is component amount (mol); $V_b>0$ is bubble volume (m³); $\\mathcal R$ is the universal gas constant (J mol⁻¹ K⁻¹); $T_b>0$ is common gas temperature (K).", "symbols_zh": "$p_b$ 为空间均匀的气泡总压力（Pa）；$p_{v,\\mathrm{PFC}}$ 为 PFC 蒸气分压，$p_{v,w}$ 为水蒸气分压，$p_g$ 为非凝结气体分压（均为 Pa）。下标 $v$ 表示蒸气，PFC 表示该化学物质族，$w$ 表示水，$g$ 表示非凝结气体。相加关系采用具有上述分压定义的气体混合物压力描述。 $i$ 标记 PFC 蒸气、水蒸气或非凝结气体；$p_i$ 为组分分压（Pa）；$n_i$ 为组分物质的量（mol）；$V_b>0$ 为气泡体积（m³）；$\\mathcal R$ 为通用气体常数（J mol⁻¹ K⁻¹）；$T_b>0$ 为共同气体温度（K）。", "kind": "Pressure composition requiring thermodynamic closure", "kind_zh": "需要热力学闭合的压力组成关系", "diagram": { "type": "phase", "labels": [ "pb", "pv,PFC", "pv,w", "pg", "Finite PFC liquid inventory" ], "notes": [ "Partial pressures need mass and energy histories", "No fixed-mass gas law is automatically a vaporization law" ] }, "caption_en": "The mechanical input comes from the evolving contents, not from radius alone.", "caption_zh": "力学输入来自不断变化的气泡内容物，不能仅由半径决定。" }, { "id": "C1-E09", "tex": "\\begin{gathered}\\begin{aligned}\n dV&=r^2\\sin\\theta\\,d\\theta\\,d\\psi\\,dr,\\\\\n \\int_0^{2\\pi}\\!d\\psi\\int_0^\\pi\\!\\sin\\theta\\,d\\theta&=2\\pi[-\\cos\\theta]_0^\\pi=4\\pi,\\\\\n u^2&=R^4\\dot R^2/r^4,\\qquad r^2/r^4=r^{-2},\\\\\n \\int_R^b r^{-2}\\,dr&=-1/b+1/R\\ \\longrightarrow\\ 1/R\\quad(b\\to\\infty),\\\\\n 2\\pi\\rho R^4\\dot R^2/R&=2\\pi\\rho R^3\\dot R^2.\n \\end{aligned}\\\\[6pt]\\begin{aligned}K_l&=\\int_R^\\infty\\frac12\\rho u(r,t)^2\\,4\\pi r^2\\,dr\\\\&=2\\pi\\rho R^4\\dot R^2\\int_R^\\infty r^{-2}\\,dr\\\\&=2\\pi\\rho R^4\\dot R^2\\left[-\\frac1r\\right]_R^\\infty=2\\pi\\rho R^3\\dot R^2.\\end{aligned}\\end{gathered}", "symbols_en": "$K_l$ is exterior liquid kinetic energy (J); $\\rho$ is carrier density (kg m⁻³); $u$ is radial liquid speed and $\\dot R$ wall speed (m s⁻¹); $t$ is time (s); $R>0$ is cavity radius, $r$ the integration radius (m); $\\pi$ is the circle constant; $4\\pi r^2dr$ is the spherical volume element (m³); endpoint brackets mean upper minus lower evaluation. $dV$ is shell volume (m³); $\\theta\\in[0,\\pi]$ and $\\psi\\in[0,2\\pi]$ are polar and azimuthal angles (dimensionless radians); $b>R$ is a finite outer cutoff (m).", "symbols_zh": "$K_l$ 为外部液体动能（J）；$\\rho$ 为载液密度（kg m⁻³）；$u$ 为径向液体速度，$\\dot R$ 为壁面速度（m s⁻¹）；$t$ 为时间（s）；$R>0$ 为空腔半径，$r$ 为积分半径（m）；$\\pi$ 为圆周率；$4\\pi r^2dr$ 为球形体积元（m³）；端点方括号表示上限取值减下限取值。 $dV$ 为体积元（m³）；$\\theta\\in[0,\\pi]$ 与 $\\psi\\in[0,2\\pi]$ 为极角与方位角（无量纲弧度）；$b>R$ 为有限外截断半径（m）。", "kind": "Exact kinetic-energy integral in the spherical model", "kind_zh": "球形模型内的精确动能积分", "diagram": { "type": "energy", "labels": [ "Kl", "ρ", "u(r,t)", "R", "Ṙ", "r: exterior shells", "4πr²dr", "t" ], "notes": [ "Integrate liquid shells outside the cavity", "The radial field decays enough for finite kinetic energy" ] }, "caption_en": "Exterior shells contribute finite kinetic energy even in an infinite domain.", "caption_zh": "即使外域无限，外部液壳贡献的总动能仍有限。" }, { "id": "C1-E10", "tex": "\\begin{gathered}\\begin{aligned}\n \\frac{d(R^3\\dot R^2)}{dt}&=\\frac{dR^3}{dt}\\dot R^2+R^3\\frac{d\\dot R^2}{dt},\\\\\n \\frac{dR^3}{dt}&=3R^2\\dot R,\\qquad\\frac{d\\dot R^2}{dt}=2\\dot R\\ddot R,\\\\\n 2\\pi\\rho(3R^2\\dot R^3+2R^3\\dot R\\ddot R)\n &=4\\pi\\rho R^2\\dot R(\\tfrac32\\dot R^2+R\\ddot R),\\\\\n \\frac{d(4\\pi\\sigma R^2)}{dt}&=4\\pi\\sigma(2R\\dot R).\n \\end{aligned}\\\\[6pt]\\begin{aligned}\\frac{dK_l}{dt}&=2\\pi\\rho\\left(3R^2\\dot R^3+2R^3\\dot R\\ddot R\\right)=4\\pi R^2\\dot R\\,\\rho\\left(R\\ddot R+\\frac32\\dot R^2\\right),\\\\V_b&=\\frac{4\\pi R^3}{3},\\qquad \\dot V_b=4\\pi R^2\\dot R,\\\\E_\\sigma&=4\\pi\\sigma R^2,\\qquad \\frac{dE_\\sigma}{dt}=8\\pi\\sigma R\\dot R.\\end{aligned}\\end{gathered}", "symbols_en": "$K_l,E_\\sigma$ are liquid kinetic and interface energies (J); $t$ is time (s); $\\rho$ is carrier density (kg m⁻³); $R$ is radius (m); $\\dot R,\\ddot R$ are speed (m s⁻¹) and acceleration (m s⁻²); $V_b$ is cavity volume (m³), $\\dot V_b$ its time derivative (m³ s⁻¹); $\\sigma$ is constant surface tension (N m⁻¹); $\\pi$ is the circle constant. Energy derivatives are powers (W). ", "symbols_zh": "$K_l,E_\\sigma$ 分别为液体动能与界面能（J）；$t$ 为时间（s）；$\\rho$ 为载液密度（kg m⁻³）；$R$ 为半径（m）；$\\dot R,\\ddot R$ 分别为速度（m s⁻¹）与加速度（m s⁻²）；$V_b$ 为空腔体积（m³），$\\dot V_b$ 为其时间导数（m³ s⁻¹）；$\\sigma$ 为常数表面张力（N m⁻¹）；$\\pi$ 为圆周率。能量的时间导数为功率（W）。 ", "kind": "Exact product-rule derivatives", "kind_zh": "乘积求导法则的精确应用", "diagram": { "type": "energy", "labels": [ "Kl", "Eσ", "Vb", "V̇b", "R", "Ṙ", "R̈", "ρ", "σ", "t" ], "notes": [ "Surface energy is stored at the spherical boundary", "Volume work changes both liquid speed and effective inertia" ] }, "caption_en": "Volume work couples kinetic storage, interface storage and dissipation.", "caption_zh": "体积功耦合动能储存、界面能储存与耗散。" }, { "id": "C1-E11", "tex": "\\begin{gathered}\\begin{aligned}\n \\frac{dK_l}{dt}&=(p_b-p_\\infty-2\\sigma/R-4\\mu\\dot R/R)(4\\pi R^2\\dot R),\\\\\n (-2\\sigma/R)(4\\pi R^2\\dot R)&=-8\\pi\\sigma R\\dot R=-dE_\\sigma/dt,\\\\\n (-4\\mu\\dot R/R)(4\\pi R^2\\dot R)&=-16\\pi\\mu R\\dot R^2,\\\\\n \\frac{dK_l}{dt}+\\frac{dE_\\sigma}{dt}\n &=(p_b-p_\\infty)\\dot V_b-16\\pi\\mu R\\dot R^2.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\frac{d}{dt}(K_l+E_\\sigma)=(p_b-p_\\infty)\\dot V_b-16\\pi\\mu R\\dot R^2\\\\ 16\\pi\\mu R\\dot R^2\\ge0\\end{gathered}\\end{gathered}", "symbols_en": "$t$ is time (s); $K_l,E_\\sigma$ are kinetic and interface energies (J); $p_b,p_\\infty$ are bubble and far-field pressures (Pa); $\\dot V_b$ is cavity volume rate (m³ s⁻¹); $\\mu\\ge0$ is carrier viscosity (Pa s); $R>0$ is cavity radius (m); $\\dot R$ is wall speed (m s⁻¹); $\\pi$ is the circle constant. The nonnegative term is dissipated power (W). ", "symbols_zh": "$t$ 为时间（s）；$K_l,E_\\sigma$ 为动能与界面能（J）；$p_b,p_\\infty$ 为气泡与远场压力（Pa）；$\\dot V_b$ 为空腔体积变化率（m³ s⁻¹）；$\\mu\\ge0$ 为载液黏度（Pa s）；$R>0$ 为空腔半径（m）；$\\dot R$ 为壁面速度（m s⁻¹）；$\\pi$ 为圆周率。非负项是耗散功率（W）。 ", "kind": "Derived mechanical-energy balance", "kind_zh": "推导得到的机械能平衡", "diagram": { "type": "energy", "labels": [ "Pressure work (pb − p∞)V̇b", "Kl", "Eσ", "16πμR Ṙ² ≥ 0", "pb", "p∞", "R", "Ṙ", "t" ], "notes": [ "Expansion or collapse may supply positive pressure work", "Viscosity always removes mechanical energy" ] }, "caption_en": "The pressure source supplies mechanical work; focusing cannot create extra energy.", "caption_zh": "压力源提供机械功；聚焦不能产生额外能量。" }, { "id": "C1-E12", "tex": "\\begin{gathered}\\begin{aligned}\n p_b&=p_v,\\quad \\mu=\\sigma=0,\\quad p_v-p_\\infty=-\\Delta p_c,\\\\\n \\rho(R\\ddot R+\\tfrac32\\dot R^2)&=-\\Delta p_c,\\\\\n R_{\\max}\\ddot R(0)&=-\\Delta p_c/\\rho,\\\\\n \\ddot R(0)&=-\\frac{\\Delta p_c}{\\rho R_{\\max}}<0.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\Delta p_c=p_\\infty-p_v>0\\\\ R(0)=R_{\\max},\\quad \\dot R(0)=0\\\\ R\\ddot R+\\frac32\\dot R^2=-\\frac{\\Delta p_c}{\\rho}\\end{gathered}\\end{gathered}", "symbols_en": "$\\Delta p_c>0$ is constant collapse-driving pressure (Pa); $p_\\infty,p_v$ are constant far-field and vapor pressures (Pa); $t=0$ is maximum-radius time (s); $R>0$ and $R_{\\max}>0$ are instantaneous and maximum radii (m); $\\dot R,\\ddot R$ are speed (m s⁻¹) and acceleration (m s⁻²); $\\rho>0$ is carrier density (kg m⁻³). $\\mu$ is viscosity (Pa s) and $\\sigma$ surface tension (N m⁻¹), both set to zero in this control; $p_b$ is bubble pressure (Pa).", "symbols_zh": "$\\Delta p_c>0$ 为常数塌缩驱动压差（Pa）；$p_\\infty,p_v$ 为常数远场压力与蒸气压（Pa）；$t=0$ 为最大半径时刻（s）；$R>0$ 与 $R_{\\max}>0$ 分别为当前与最大半径（m）；$\\dot R,\\ddot R$ 为速度（m s⁻¹）与加速度（m s⁻²）；$\\rho>0$ 为载液密度（kg m⁻³）。 $\\mu$ 为黏度（Pa s），$\\sigma$ 为表面张力（N m⁻¹），在此对照中均置零；$p_b$ 为气泡压力（Pa）。", "kind": "Idealized collapse initial-value problem", "kind_zh": "理想化塌缩的初值问题", "diagram": { "type": "sphere", "labels": [ "Rmax at t = 0", "Ṙ(0) = 0", "R(t)", "R̈", "p∞", "pv", "Δpc > 0", "ρ" ], "notes": [ "Constant pressure deficit initiates inward acceleration" ] }, "caption_en": "The ideal control begins at rest and excludes the mechanisms that arrest real collapse.", "caption_zh": "理想对照从静止开始，并排除了阻止实际塌缩的机制。" }, { "id": "C1-E13", "tex": "\\begin{gathered}\\begin{aligned}\n R(\\tfrac12 y')+\\tfrac32 y&=-\\Delta p_c/\\rho,\\\\\n y'+3y/R&=-2\\Delta p_c/(\\rho R),\\\\\n I'/I&=3/R,\\quad I(R_{\\max})=1,\\\\\n \\ln I(R)-\\ln I(R_{\\max})&=\\int_{R_{\\max}}^R\\frac{3}{s}\\,ds\n =3[\\ln(s/R_{\\max})]_{R_{\\max}}^R=3\\ln(R/R_{\\max}),\\\\\n \\ln I(R)&=3\\ln(R/R_{\\max}),\\quad\n I(R)=\\exp(3\\ln(R/R_{\\max}))=(R/R_{\\max})^3,\\\\\n (R^3y)'&=R^3y'+3R^2y.\n \\end{aligned}\\\\[6pt]\\begin{aligned}y(R)&=\\dot R^2,\\qquad \\frac{dy}{dt}=\\frac{dy}{dR}\\dot R=2\\dot R\\ddot R\\quad\\Longrightarrow\\quad\\ddot R=\\frac12\\frac{dy}{dR},\\\\\\frac{dy}{dR}+\\frac{3y}{R}&=-\\frac{2\\Delta p_c}{\\rho R},\\\\\\frac{d(R^3y)}{dR}&=R^3\\frac{dy}{dR}+3R^2y=-\\frac{2\\Delta p_c}{\\rho}R^2.\\end{aligned}\\end{gathered}", "symbols_en": "$y(R)$ is squared wall speed (m² s⁻²); $R\\in(0,R_{\\max})$ is decreasing radius (m); $t$ is time (s); $\\dot R\\ne0$ and $\\ddot R$ are speed (m s⁻¹) and acceleration (m s⁻²); $\\Delta p_c$ is positive pressure difference (Pa); $\\rho$ is carrier density (kg m⁻³); $d/dR$ and $d/dt$ are ordinary derivatives on the monotonic branch. Radius powers are the integrating factor, not derivatives. $y'=dy/dR$ has units m s⁻²; $I>0$ is a dimensionless integrating factor normalized by $I(R_{\\max})=1$, $I'=dI/dR$ (m⁻¹); $s>0$ is a dummy radius (m); $\\ln$ and $\\exp$ are natural logarithm and exponential with dimensionless arguments.", "symbols_zh": "$y(R)$ 为壁面速度平方（m² s⁻²）；$R\\in(0,R_{\\max})$ 为递减半径（m）；$t$ 为时间（s）；$\\dot R\\ne0$ 与 $\\ddot R$ 为速度（m s⁻¹）与加速度（m s⁻²）；$\\Delta p_c$ 为正压差（Pa）；$\\rho$ 为载液密度（kg m⁻³）；$d/dR$ 与 $d/dt$ 为单调分支上的普通导数。半径幂构成积分因子，并非导数。 $y'=dy/dR$ 的单位为 m s⁻²；$I>0$ 为无量纲积分因子，采用 $I(R_{\\max})=1$ 归一化，$I'=dI/dR$（m⁻¹）；$s>0$ 为积分哑半径（m）；$\\ln$ 与 $\\exp$ 为自然对数和指数，自变量无量纲。", "kind": "Exact chain rule and integrating factor", "kind_zh": "精确的链式法则与积分因子", "diagram": { "type": "sphere", "labels": [ "y(R) = Ṙ²", "R decreases", "Ṙ ≠ 0", "R̈", "Δpc", "ρ", "R³y", "t", "Rmax endpoint" ], "notes": [ "Cancel speed only away from the initial turning point" ] }, "caption_en": "The monotonic collapse branch allows radius to replace time in the first integration.", "caption_zh": "单调塌缩分支允许在第一次积分中以半径取代时间。" }, { "id": "C1-E14", "tex": "\\begin{gathered}\\begin{aligned}\n \\int_{R_{\\max}}^R\\frac{d(s^3y)}{ds}\\,ds\n &=-\\frac{2\\Delta p_c}{\\rho}\\left[\\frac{s^3}{3}\\right]_{R_{\\max}}^R,\\\\\n R^3y(R)&=\\frac{2\\Delta p_c}{3\\rho}(R_{\\max}^3-R^3),\\\\\n y(R)&=\\frac{2\\Delta p_c}{3\\rho}\\left(\\frac{R_{\\max}^3}{R^3}-1\\right)\\ge0,\\\\\n \\dot R&=\\pm\\sqrt{y(R)},\\qquad\\dot R<0\\ \\text{during collapse}.\n \\end{aligned}\\\\[6pt]\\begin{aligned}R^3y(R)-R_{\\max}^3y(R_{\\max})&=-\\frac{2\\Delta p_c}{\\rho}\\int_{R_{\\max}}^R s^2\\,ds=-\\frac{2\\Delta p_c}{3\\rho}(R^3-R_{\\max}^3),\\\\y(R_{\\max})&=0,\\qquad \\dot R=-\\sqrt{\\frac{2\\Delta p_c}{3\\rho}\\left[\\left(\\frac{R_{\\max}}{R}\\right)^3-1\\right]}.\\end{aligned}\\end{gathered}", "symbols_en": "$R\\in(0,R_{\\max}]$ and $R_{\\max}$ are current and maximum radii (m); $y$ is squared speed (m² s⁻²); $s$ is dummy radius (m); $\\Delta p_c>0$ is collapse pressure (Pa); $\\rho>0$ is carrier density (kg m⁻³); $\\dot R\\le0$ is inward wall velocity (m s⁻¹). The square root denotes the nonnegative root; its prefixed minus sign selects collapse. ", "symbols_zh": "$R\\in(0,R_{\\max}]$ 与 $R_{\\max}$ 分别为当前与最大半径（m）；$y$ 为速度平方（m² s⁻²）；$s$ 为积分哑半径（m）；$\\Delta p_c>0$ 为塌缩压差（Pa）；$\\rho>0$ 为载液密度（kg m⁻³）；$\\dot R\\le0$ 为向内壁面速度（m s⁻¹）。根号表示非负平方根；前置负号选择塌缩分支。 ", "kind": "Integrated speed with initial constant and branch", "kind_zh": "含初始积分常数与分支选择的速度积分解", "diagram": { "type": "sphere", "labels": [ "Rmax", "R", "s: integration radius", "y = Ṙ²", "Ṙ < 0", "Δpc", "ρ" ], "notes": [ "Speed is zero at R = Rmax", "Zero radius is an excluded model endpoint" ] }, "caption_en": "The initial condition fixes the integration constant and the inward sign fixes the branch.", "caption_zh": "初值确定积分常数，向内的符号确定解分支。" }, { "id": "C1-E15", "tex": "\\begin{gathered}\\begin{aligned}\n dt&=dR/\\dot R,\\qquad\n \\frac1{\\dot R}=-\\sqrt{\\frac{3\\rho}{2\\Delta p_c}}\\frac1{\\sqrt{(R_{\\max}/R)^3-1}},\\\\\n (R_{\\max}/R)^3-1&=x^{-3}-1=(1-x^3)/x^3,\\\\\n \\frac{dR}{\\sqrt{(R_{\\max}/R)^3-1}}\n &=R_{\\max}\\frac{x^{3/2}}{\\sqrt{1-x^3}}\\,dx,\\\\\n -\\int_1^0\\frac{x^{3/2}}{\\sqrt{1-x^3}}\\,dx\n &=\\int_0^1\\frac{x^{3/2}}{\\sqrt{1-x^3}}\\,dx.\n \\end{aligned}\\\\[6pt]\\begin{aligned}t_c&=\\int_{R_{\\max}}^0\\frac{dR}{\\dot R}=\\sqrt{\\frac{3\\rho}{2\\Delta p_c}}\\int_0^{R_{\\max}}\\frac{dR}{\\sqrt{(R_{\\max}/R)^3-1}},\\\\x&=\\frac{R}{R_{\\max}},\\qquad dR=R_{\\max}dx,\\\\t_c&=R_{\\max}\\sqrt{\\frac{\\rho}{\\Delta p_c}}\\ C_t,\\qquad C_t=\\sqrt{\\frac32}\\int_0^1\\frac{x^{3/2}}{\\sqrt{1-x^3}}\\,dx.\\end{aligned}\\end{gathered}", "symbols_en": "$t_c$ is ideal collapse time (s); $R$ is radius and $R_{\\max}$ its maximum (m); $\\dot R<0$ is inward velocity (m s⁻¹); $\\rho$ is density (kg m⁻³); $\\Delta p_c>0$ is pressure difference (Pa); $x=R/R_{\\max}\\in[0,1]$ is dimensionless radius; $C_t$ is the dimensionless time coefficient; definite integrals follow the collapse branch. $dx$ and $dR$ denote integration measures. ", "symbols_zh": "$t_c$ 为理想塌缩时间（s）；$R$ 为半径，$R_{\\max}$ 为其最大值（m）；$\\dot R<0$ 为向内速度（m s⁻¹）；$\\rho$ 为密度（kg m⁻³）；$\\Delta p_c>0$ 为压差（Pa）；$x=R/R_{\\max}\\in[0,1]$ 为无量纲半径；$C_t$ 为无量纲时间系数；定积分沿塌缩分支进行。$dx$ 与 $dR$ 表示积分测度。 ", "kind": "Exact separation and nondimensionalization", "kind_zh": "精确的变量分离与无量纲化", "diagram": { "type": "timescale", "labels": [ "tc", "Rmax", "R", "Ṙ < 0", "ρ", "Δpc", "x = R/Rmax", "Ct" ], "notes": [ "At x = 0 the integrand behaves as x³ᐟ²", "At x = 1 it behaves as (1 − x)⁻¹ᐟ²" ] }, "caption_en": "The endpoint singularity is integrable, so the model has a finite collapse time.", "caption_zh": "端点奇异性可积，因此该模型具有有限塌缩时间。" }, { "id": "C1-E16", "tex": "\\begin{gathered}\\begin{aligned}\n x^{3/2}&=(q^{1/3})^{3/2}=q^{1/2},\\\\\n x^{3/2}\\,dx&=q^{1/2}\\tfrac13q^{-2/3}\\,dq=\\tfrac13q^{-1/6}\\,dq,\\\\\n \\sqrt{3/2}/3&=1/\\sqrt6,\\quad a-1=-1/6\\Rightarrow a=5/6,\\\\\n b-1&=-1/2\\Rightarrow b=1/2,\\\\\n C_t&=B(5/6,1/2)/\\sqrt6=0.9146813565\\ldots .\n \\end{aligned}\\\\[6pt]\\begin{aligned}q&=x^3,\\quad x=q^{1/3},\\quad dx=\\frac13q^{-2/3}\\,dq,\\\\C_t&=\\frac{1}{\\sqrt6}\\int_0^1q^{-1/6}(1-q)^{-1/2}\\,dq=\\frac{1}{\\sqrt6}B\\left(\\frac56,\\frac12\\right)=0.9146813565\\ldots,\\\\B(a,b)&=\\int_0^1q^{a-1}(1-q)^{b-1}\\,dq\\quad(a>0,b>0).\\end{aligned}\\end{gathered}", "symbols_en": "$x$ and $q=x^3$ are dimensionless radius and substitution variables on [0,1]; $C_t$ is the dimensionless collapse-time coefficient; $B(a,b)$ is the beta function, defined by the displayed convergent integral; $a,b$ are positive dimensionless arguments; $dx,dq$ are integration measures. The numeral 6 inside the root is a dimensionless constant; the ellipsis indicates decimal continuation. ", "symbols_zh": "$x$ 与 $q=x^3$ 分别为区间 [0,1] 上的无量纲半径与换元变量；$C_t$ 为无量纲塌缩时间系数；$B(a,b)$ 为贝塔函数，由公式中的收敛积分定义；$a,b$ 为正的无量纲参数；$dx,dq$ 为积分测度。根号中的数字 6 为无量纲常数；省略号表示后续小数。 ", "kind": "Evaluated beta-function coefficient", "kind_zh": "通过贝塔函数计算的系数", "diagram": { "type": "timescale", "labels": [ "x ∈ [0,1]", "q = x³", "dx = q⁻²ᐟ³dq/3", "Ct = 0.914681…", "B(a,b)", "a = 5/6", "b = 1/2" ], "notes": [ "The coefficient comes from the complete radius trajectory, not a material fit" ] }, "caption_en": "The change of variables exposes the beta integral that fixes the numerical coefficient.", "caption_zh": "换元揭示贝塔积分，从而确定数值系数。" }, { "id": "C1-E17", "tex": "\\begin{gathered}\\begin{aligned}\n (2\\pi\\rho)\\frac{2\\Delta p_c}{3\\rho}&=4\\pi\\Delta p_c/3,\\\\\n R^3\\left(\\frac{R_{\\max}^3}{R^3}-1\\right)&=R_{\\max}^3-R^3,\\\\\n K_l&=\\int_{V_{\\max}}^{V_b}(-\\Delta p_c)\\,dV\n =\\Delta p_c(V_{\\max}-V_b),\\\\\n \\frac{dK_l}{dt}&=-\\Delta p_c\\,4\\pi R^2\\dot R>0\\quad(\\dot R<0),\\\\\n K_l/E_B&=(R_{\\max}^3-R^3)/R_{\\max}^3=1-(R/R_{\\max})^3.\n \\end{aligned}\\\\[6pt]\\begin{aligned}K_l(R)&=2\\pi\\rho R^3\\frac{2\\Delta p_c}{3\\rho}\\left[\\left(\\frac{R_{\\max}}R\\right)^3-1\\right]=\\frac{4\\pi}{3}\\Delta p_c(R_{\\max}^3-R^3),\\\\E_B&=\\Delta p_cV_{\\max}=\\frac{4\\pi}{3}\\Delta p_cR_{\\max}^3,\\qquad \\frac{K_l}{E_B}=1-\\left(\\frac R{R_{\\max}}\\right)^3.\\end{aligned}\\end{gathered}", "symbols_en": "$K_l$ is carrier kinetic energy, $E_B$ the maximum-volume pressure-work scale (J); $R\\in(0,R_{\\max}]$ and $R_{\\max}$ are radii (m); $\\rho$ is density (kg m⁻³); $\\Delta p_c$ is constant positive pressure difference (Pa); $V_{\\max}$ is maximum cavity volume (m³); $\\pi$ is the circle constant. $K_l/E_B$ is dimensionless and this control has zero viscous and capillary losses. $V_b=4\\pi R^3/3$ is cavity volume (m³); $V$ is the dummy volume variable (m³); $t$ is time (s), $\\dot R$ wall speed (m s⁻¹).", "symbols_zh": "$K_l$ 为载液动能，$E_B$ 为最大体积的压力功尺度（J）；$R\\in(0,R_{\\max}]$ 与 $R_{\\max}$ 为半径（m）；$\\rho$ 为密度（kg m⁻³）；$\\Delta p_c$ 为常数正压差（Pa）；$V_{\\max}$ 为最大空腔体积（m³）；$\\pi$ 为圆周率。$K_l/E_B$ 无量纲，此对照模型的黏性与毛细损失为零。 $V_b=4\\pi R^3/3$ 为空腔体积（m³）；$V$ 为积分哑体积（m³）；$t$ 为时间（s），$\\dot R$ 为壁面速度（m s⁻¹）。", "kind": "Independent pressure-work conservation check", "kind_zh": "独立的压力功守恒检验", "diagram": { "type": "energy", "labels": [ "Kl(R)", "EB", "Δpc", "Vmax", "Rmax", "R", "ρ", "Kl/EB = 1 − (R/Rmax)³" ], "notes": [ "Finite energy survives the excluded zero-radius speed singularity" ] }, "caption_en": "The Rayleigh speed solution and pressure-work balance agree exactly.", "caption_zh": "瑞利速度解与压力功平衡严格一致。" }, { "id": "C1-E18", "tex": "\\begin{gathered}\\begin{aligned}\n R_{\\max}&=30\\ \\mu\\mathrm m=30\\times10^{-6}\\ \\mathrm m,\\quad\n \\Delta p_c=100\\ \\mathrm{kPa}=10^5\\ \\mathrm{Pa},\\\\\n \\sqrt{\\rho/\\Delta p_c}&=\\sqrt{1000/10^5}=0.1\\ \\mathrm{s\\,m^{-1}},\\\\\n R_{\\max}^3&=27\\times10^{-15}\\ \\mathrm{m^3},\\\\\n t_c&=0.9146813565(30\\times10^{-6})(0.1)\\ \\mathrm s\n =2.7440440695\\times10^{-6}\\ \\mathrm s,\\\\\n E_B&=(4\\pi/3)(10^5)(27\\times10^{-15})\\ \\mathrm J\n =1.1309733553\\times10^{-8}\\ \\mathrm J,\\\\\n (R_{\\max}/R)^3-1&=2^3-1=7,\\quad K_l/E_B=1-(1/2)^3=7/8.\n \\end{aligned}\\\\[6pt]\\begin{aligned}t_c&=0.9146813565(30\\times10^{-6})\\sqrt{\\frac{1000}{100\\times10^3}}\\ \\mathrm s=2.74404\\ \\mu\\mathrm s,\\\\E_B&=\\frac{4\\pi}{3}(100\\times10^3)(30\\times10^{-6})^3\\ \\mathrm J=11.3097\\ \\mathrm{nJ},\\\\\\dot R\\big|_{R=R_{\\max}/2}&=-\\sqrt{\\frac{2(100\\times10^3)}{3(1000)}(8-1)}\\ \\mathrm{m\\,s^{-1}}=-21.6025\\ \\mathrm{m\\,s^{-1}},\\\\K_l\\big|_{R=R_{\\max}/2}&=\\frac78E_B=9.89602\\ \\mathrm{nJ}.\\end{aligned}\\end{gathered}", "symbols_en": "$t_c$ is collapse time (s), $E_B$ pressure-work scale (J), $K_l$ carrier kinetic energy (J), $R$ current radius and $R_{\\max}=30\\times10^{-6}$ m maximum radius; $\\dot R$ is signed wall speed (m s⁻¹). Substituted inputs are $\\rho=1000$ kg m⁻³ and $\\Delta p_c=100\\times10^3$ Pa. $\\pi$ is the circle constant; s, J, μs and nJ mean seconds, joules, microseconds and nanojoules; the vertical bar specifies half-radius evaluation. ", "symbols_zh": "$t_c$ 为塌缩时间（s），$E_B$ 为压力功尺度（J），$K_l$ 为载液动能（J），$R$ 为当前半径，$R_{\\max}=30\\times10^{-6}$ m 为最大半径；$\\dot R$ 为带符号壁面速度（m s⁻¹）。代入输入为 $\\rho=1000$ kg m⁻³ 与 $\\Delta p_c=100\\times10^3$ Pa。$\\pi$ 为圆周率；s、J、μs 与 nJ 分别为秒、焦耳、微秒与纳焦耳；竖线表示在一半半径处取值。 ", "kind": "Declared teaching calculation", "kind_zh": "采用明确给定参数的教学计算", "diagram": { "type": "sphere", "labels": [ "Rmax = 30 μm", "R = 15 μm", "ρ = 1000 kg/m³", "Δpc = 100 kPa", "tc = 2.74404 μs", "EB = 11.3097 nJ", "Ṙ = −21.6025 m/s", "Kl = 9.89602 nJ" ], "notes": [ "These values are a spherical control, not a PFC performance prediction" ] }, "caption_en": "The original numerical control is preserved with its energy and branch checks.", "caption_zh": "保留原始数值对照，并补充能量与解分支检验。" }, { "id": "C1-E19", "tex": "\\begin{gathered}\\begin{aligned}\n x&=R/R_{\\max},\\qquad M_w^2=\\frac{2\\Delta p_c}{3\\rho c^2}(x^{-3}-1),\\\\\n M_w=M_*&\\quad\\Longrightarrow\\quad\n \\frac{3\\rho c^2M_*^2}{2\\Delta p_c}=x_M^{-3}-1,\\\\\n x_M^{-3}&=1+\\frac{3\\rho c^2M_*^2}{2\\Delta p_c},\\\\\n x_M&=\\left[1+\\frac{3(1000)(1500)^2(0.1)^2}{2(10^5)}\\right]^{-1/3}\\\\\n &=338.5^{-1/3}=0.1434874,\\\\\n R_M&=(30\\ \\mu\\mathrm m)(0.1434874)=4.30462\\ \\mu\\mathrm m.\n \\end{aligned}\\\\[6pt]\\begin{gathered}M_w=\\frac{|\\dot R|}{c}\\\\ x_M=\\left[1+\\frac{3\\rho c^2M_*^2}{2\\Delta p_c}\\right]^{-1/3}\\\\ M_*=0.1,\\ c=1500\\ \\mathrm{m\\,s^{-1}}\\ \\Longrightarrow\\ x_M=0.143487\\end{gathered}\\end{gathered}", "symbols_en": "$M_w$ is wall Mach number (dimensionless); $\\dot R$ is wall speed (m s⁻¹); absolute bars denote magnitude; $c>0$ is stipulated carrier sound speed (m s⁻¹); $M_*>0$ is a selected diagnostic Mach value; $x_M=R_M/R_{\\max}$ is the radius fraction when $M_w=M_*$; $R_M,R_{\\max}$ are radii (m); $\\rho$ is carrier density (kg m⁻³); $\\Delta p_c$ is positive collapse pressure (Pa). The number uses the previous teaching density and pressure. $x$ is normalized radius (dimensionless); $R_M=x_MR_{\\max}$ is diagnostic radius (m); numerical inputs are the previous teaching density and pressure, with $c=1500$ m s⁻¹ and chosen $M_*=0.1$.", "symbols_zh": "$M_w$ 为壁面马赫数（无量纲）；$\\dot R$ 为壁面速度（m s⁻¹）；绝对值符号表示大小；$c>0$ 为给定载液声速（m s⁻¹）；$M_*>0$ 为选择的诊断马赫值；$x_M=R_M/R_{\\max}$ 为 $M_w=M_*$ 时的半径比例；$R_M,R_{\\max}$ 为半径（m）；$\\rho$ 为载液密度（kg m⁻³）；$\\Delta p_c$ 为正塌缩压差（Pa）。数值使用前述教学密度与压差。 $x$ 为无量纲半径；$R_M=x_MR_{\\max}$ 为诊断半径（m）；数值输入采用前述教学密度和压差，并取 $c=1500$ m s⁻¹、选定 $M_*=0.1$。", "kind": "Ideal-trajectory diagnostic, not a physical limit", "kind_zh": "理想轨迹的诊断指标，而非物理极限", "diagram": { "type": "timescale", "labels": [ "Mw = |Ṙ|/c", "Ṙ", "c = 1500 m/s", "M* = 0.1", "xM = RM/Rmax", "ρ", "Δpc", "RM ≈ 4.305 μm" ], "notes": [ "The selected threshold flags omitted physics; it does not stop a real cavity" ] }, "caption_en": "A Mach diagnostic exposes the shrinking validity range of incompressible collapse.", "caption_zh": "马赫数诊断揭示不可压缩塌缩模型的适用范围不断缩小。" }, { "id": "C1-E20", "tex": "\\begin{gathered}\\begin{aligned}\n q(\\boldsymbol x,t)&=p(\\boldsymbol x,t)-p_{\\mathrm{ref}}(t),\\\\\n \\Pi(\\boldsymbol x)&=\\lim_{N\\to\\infty}\\sum_{k=1}^Nq(\\boldsymbol x,t_k)\\Delta t,\n \\quad\\Delta t=(t_1-t_0)/N,\\\\\n \\boldsymbol J_p&=-\\int_{t_0}^{t_1}\\int_{S(t)}q\\boldsymbol n\\,dA\\,dt,\\\\\n S(t)=S\\quad&\\Longrightarrow\\quad\\boldsymbol J_p=-\\int_S\\Pi\\boldsymbol n\\,dA,\\\\\n [\\boldsymbol J_p]&=\\mathrm{Pa\\,s\\,m^2}=\\mathrm{N\\,s},\\qquad[\\Pi]=\\mathrm{Pa\\,s}.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\Pi(\\boldsymbol x)=\\int_{t_0}^{t_1}\\left[p(\\boldsymbol x,t)-p_{\\mathrm{ref}}(t)\\right]dt\\\\ \\tau=t_1-t_0>0\\\\ [\\Pi]=\\mathrm{Pa\\,s}\\end{gathered}\\end{gathered}", "symbols_en": "$\\boldsymbol x$ is fixed liquid position (m); $\\Pi$ is pressure impulse (Pa s); $p$ is local pressure and $p_{\\mathrm{ref}}$ a spatially uniform reference (Pa); $t$ is time and $t_0,t_1$ event endpoints (s); $\\tau$ is event duration (s). Square brackets on $\\Pi$ denote units, not endpoint evaluation. The integral is at fixed position in a liquid domain whose displacement will be assessed below. $q$ is gauge pressure (Pa); $N$ is a positive integer subdivision count; $k$ labels a time sample $t_k$ (s); $\\Delta t$ is time-step width (s); $\\boldsymbol J_p$ is gauge-pressure mechanical impulse on a liquid control volume (N s); $S(t)$ is its surface, $\\boldsymbol n$ its outward unit normal, and $dA$ an area element (m²); brackets denote units. Fixed-surface reduction requires fixed orientation and all sampled points remaining liquid.", "symbols_zh": "$\\boldsymbol x$ 为固定液体位置（m）；$\\Pi$ 为压力冲量（Pa s）；$p$ 为局部压力，$p_{\\mathrm{ref}}$ 为空间均匀参考压力（Pa）；$t$ 为时间，$t_0,t_1$ 为事件端点（s）；$\\tau$ 为持续时间（s）。$\\Pi$ 的方括号表示单位，而非端点取值。积分在固定位置进行，液体区域的位移将在下文评估。 $q$ 为表压（Pa）；$N$ 为正整数分段数；$k$ 标记采样时间 $t_k$（s）；$\\Delta t$ 为时间段宽度（s）；$\\boldsymbol J_p$ 为作用于液体控制体的表压机械冲量（N s）；$S(t)$ 为其表面，$\\boldsymbol n$ 为向外单位法向，$dA$ 为面积元（m²）；方括号表示单位。固定表面约化要求方向固定且全部采样点始终位于液体中。", "kind": "Definition of local pressure impulse", "kind_zh": "局部压力冲量的定义", "diagram": { "type": "impulse", "labels": [ "Π(x)", "p(x,t) − pref(t)", "x: position", "t₀", "t₁", "τ = t₁ − t₀", "Pa s" ], "notes": [ "Shade the time integral of pressure at a selected position" ] }, "caption_en": "Impulse combines duration and amplitude at each liquid position.", "caption_zh": "压力冲量在每个液体位置组合压力幅值与持续时间。" }, { "id": "C1-E21", "tex": "\\begin{gathered}\\begin{aligned}\n \\partial_t\\boldsymbol u&=-\\nabla p/\\rho-(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u+\\nu\\nabla^2\\boldsymbol u,\\\\\n \\int_{t_0}^{t_1}\\partial_t\\boldsymbol u\\,dt\n &=\\boldsymbol u(\\boldsymbol x,t_1)-\\boldsymbol u(\\boldsymbol x,t_0)=\\Delta\\boldsymbol u,\\\\\n \\int_{t_0}^{t_1}\\nabla p\\,dt\n &=\\nabla\\int_{t_0}^{t_1}(p-p_{\\mathrm{ref}})\\,dt\n +\\int_{t_0}^{t_1}\\underbrace{\\nabla p_{\\mathrm{ref}}}_{=0}\\,dt=\\nabla\\Pi,\\\\\n \\int_{t_0}^{t_1}\\nu\\nabla^2\\boldsymbol u\\,dt\n &=\\nu\\int_{t_0}^{t_1}\\nabla^2\\boldsymbol u\\,dt.\n \\end{aligned}\\\\[6pt]\\begin{aligned}\n \\Delta\\boldsymbol u&=-\\nabla\\Pi/\\rho\n -\\int_{t_0}^{t_1}(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u\\,dt\\\\\n &\\quad+\\nu\\int_{t_0}^{t_1}\\nabla^2\\boldsymbol u\\,dt,\\\\\n \\Delta\\boldsymbol u&=\\boldsymbol u(\\boldsymbol x,t_1)-\\boldsymbol u(\\boldsymbol x,t_0),\\\\\n \\nu&=\\mu/\\rho.\n \\end{aligned}\\end{gathered}", "symbols_en": "$\\Delta\\boldsymbol u$ is fixed-position velocity change and $\\boldsymbol u$ velocity (m s⁻¹); $\\Pi$ is pressure impulse (Pa s); $\\rho$ is constant density (kg m⁻³); $\\mu$ is viscosity (Pa s); $\\nu$ is kinematic viscosity (m² s⁻¹); $t_0,t_1,t$ are times (s); $\\boldsymbol x$ is fixed position (m); $\\nabla$ is spatial gradient (m⁻¹), $\\nabla^2$ the vector Laplacian (m⁻²); the dot is a true vector contraction in convective acceleration, not an equation separator. ", "symbols_zh": "$\\Delta\\boldsymbol u$ 为固定位置速度变化，$\\boldsymbol u$ 为速度（m s⁻¹）；$\\Pi$ 为压力冲量（Pa s）；$\\rho$ 为常数密度（kg m⁻³）；$\\mu$ 为黏度（Pa s）；$\\nu$ 为运动黏度（m² s⁻¹）；$t_0,t_1,t$ 为时间（s）；$\\boldsymbol x$ 为固定位置（m）；$\\nabla$ 为空间梯度（m⁻¹），$\\nabla^2$ 为矢量拉普拉斯（m⁻²）；圆点是对流加速度中的真正矢量缩并，并非公式分隔符。 ", "kind": "Exact fixed-position integrated incompressible momentum", "kind_zh": "固定位置处不可压缩动量方程的精确时间积分", "diagram": { "type": "impulse", "labels": [ "Δu", "−∇Π/ρ", "u(x,t)", "ρ", "μ", "ν = μ/ρ", "t₀ → t₁", "Convective impulse", "Viscous impulse" ], "notes": [ "Pressure-gradient impulse shares momentum change with two residual terms" ] }, "caption_en": "The short-event approximation must justify removing convective and viscous impulses.", "caption_zh": "短事件近似必须说明为何可以删除对流与黏性冲量。" }, { "id": "C1-E22", "tex": "\\begin{gathered}\\begin{aligned}\n \\left|\\int_{t_0}^{t_1}(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u\\,dt\\right|\n &\\sim U^2\\tau/L,\\qquad (U^2\\tau/L)/U=U\\tau/L,\\\\\n \\left|\\nu\\int_{t_0}^{t_1}\\nabla^2\\boldsymbol u\\,dt\\right|\n &\\sim\\nu U\\tau/L^2,\\qquad(\\nu U\\tau/L^2)/U=\\nu\\tau/L^2,\\\\\n \\nabla\\cdot\\Delta\\boldsymbol u&=\\nabla\\cdot\\boldsymbol u(t_1)-\\nabla\\cdot\\boldsymbol u(t_0)=0-0=0,\\\\\n 0&=-\\nabla\\cdot\\nabla\\Pi/\\rho=-\\nabla^2\\Pi/\\rho.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\frac{U\\tau}{L}\\ll1\\\\ \\frac{\\nu\\tau}{L^2}\\ll1\\\\ \\Delta\\boldsymbol u\\simeq-\\frac{\\nabla\\Pi}{\\rho}\\\\ 0=\\nabla\\cdot\\Delta\\boldsymbol u\\simeq-\\frac{\\nabla^2\\Pi}{\\rho}\\ \\Longrightarrow\\ \\nabla^2\\Pi=0\\end{gathered}\\end{gathered}", "symbols_en": "$U>0$ is a characteristic liquid speed (m s⁻¹); $\\tau$ is impulse duration (s); $L>0$ is liquid variation length (m); $\\nu$ is kinematic viscosity (m² s⁻¹); $\\Delta\\boldsymbol u$ is velocity change (m s⁻¹); $\\Pi$ is pressure impulse (Pa s); $\\rho$ is density (kg m⁻³); $\\nabla\\cdot$ denotes divergence and $\\nabla^2$ Laplacian; $\\ll$ means asymptotically small, and $\\simeq$ marks the reduced approximation. The last Laplace equation belongs to that reduced model. $\\sim$ denotes a scale estimate rather than an equality or rigorous error bound; $t_0,t_1$ are fixed event endpoints (s); $\\boldsymbol u$ is velocity (m s⁻¹).", "symbols_zh": "$U>0$ 为特征液体速度（m s⁻¹）；$\\tau$ 为冲量持续时间（s）；$L>0$ 为液体变化长度（m）；$\\nu$ 为运动黏度（m² s⁻¹）；$\\Delta\\boldsymbol u$ 为速度变化（m s⁻¹）；$\\Pi$ 为压力冲量（Pa s）；$\\rho$ 为密度（kg m⁻³）；$\\nabla\\cdot$ 表示散度，$\\nabla^2$ 表示拉普拉斯算子；$\\ll$ 表示渐近很小，$\\simeq$ 表示约化近似。最后的拉普拉斯方程属于该约化模型。 $\\sim$ 表示尺度估计，并非等式或严格误差界；$t_0,t_1$ 为固定事件端点（s）；$\\boldsymbol u$ 为速度（m s⁻¹）。", "kind": "Short-event reduced pressure-impulse model", "kind_zh": "短时事件的简化压力冲量模型", "diagram": { "type": "impulse", "labels": [ "U", "τ", "L", "ν", "Π(x)", "Δu ≈ −∇Π/ρ", "∇²Π = 0", "ρ" ], "notes": [ "Freeze the domain only while displacement and residual impulses remain small" ] }, "caption_en": "The velocity depends on a spatial impulse gradient constrained by boundary geometry.", "caption_zh": "速度取决于受边界几何约束的空间压力冲量梯度。" }, { "id": "C1-E23", "tex": "\\begin{gathered}\\begin{aligned}\n d^2\\Pi/dz^2=0&\\Rightarrow d\\Pi/dz=A_1,\\\\\n \\Pi(z)-\\Pi(0)&=\\int_0^z A_1\\,ds=A_1z,\\\\\n \\Pi(0)=\\Pi_0&\\Rightarrow A_0=\\Pi_0,\\\\\n 0=\\Pi(L)&=A_1L+\\Pi_0\\Rightarrow A_1=-\\Pi_0/L,\\\\\n \\Delta u_z&=-(-\\Pi_0/L)/\\rho=\\Pi_0/(\\rho L).\n \\end{aligned}\\\\[6pt]\\begin{aligned}\\frac{d^2\\Pi}{dz^2}&=0\\quad\\Longrightarrow\\quad\\Pi=A_1z+A_0,\\\\\\Pi(0)&=\\Pi_0,\\quad\\Pi(L)=0\\quad\\Longrightarrow\\quad A_0=\\Pi_0,\\quad A_1=-\\frac{\\Pi_0}{L},\\\\\\Pi(z)&=\\Pi_0\\left(1-\\frac zL\\right),\\qquad \\Delta u_z=-\\frac1\\rho\\frac{d\\Pi}{dz}=\\frac{\\Pi_0}{\\rho L}.\\end{aligned}\\end{gathered}", "symbols_en": "$z\\in[0,L]$ is axial position and $L>0$ column length (m); $\\Pi(z)$ and $\\Pi_0>0$ are local and driven-end impulses (Pa s); $A_0$ is an integration constant (Pa s), $A_1$ a gradient constant (Pa s m⁻¹); $\\rho>0$ is carrier density (kg m⁻³); $\\Delta u_z$ is positive axial speed change (m s⁻¹); $d/dz$ is an axial derivative. Subscript $z$ labels the component and 0 the driven endpoint. $s$ is a dummy axial integration coordinate (m). The uniform-speed column is an inviscid core benchmark; no-slip wall layers are omitted.", "symbols_zh": "$z\\in[0,L]$ 为轴向位置，$L>0$ 为液柱长度（m）；$\\Pi(z)$ 与 $\\Pi_0>0$ 为局部与受驱动端冲量（Pa s）；$A_0$ 为积分常数（Pa s），$A_1$ 为梯度常数（Pa s m⁻¹）；$\\rho>0$ 为载液密度（kg m⁻³）；$\\Delta u_z$ 为正轴向速度变化（m s⁻¹）；$d/dz$ 为轴向导数。下标 $z$ 表示分量，0 表示受驱动端点。 $s$ 为轴向积分哑坐标（m）。匀速液柱是无黏核心流基准，省略无滑移壁面层。", "kind": "Exact straight-column solution of the reduced model", "kind_zh": "简化模型中直液柱的精确解", "diagram": { "type": "impulse", "labels": [ "z = 0: Π₀", "z = L: Π = 0", "L", "Π(z)", "A₀ = Π₀", "A₁ = −Π₀/L", "ρ", "Δuz → outlet" ], "notes": [ "The fixed side walls do not change normal velocity" ] }, "caption_en": "A linear spatial impulse drop accelerates liquid toward the outlet.", "caption_zh": "线性的空间冲量下降使液体向出口加速。" }, { "id": "C1-E24", "tex": "\\begin{gathered}\\begin{aligned}\n \\Delta p&=6.0\\times10^5\\ \\mathrm{Pa},\\quad\\tau=5.0\\times10^{-7}\\ \\mathrm s,\n \\quad L=10^{-4}\\ \\mathrm m,\\\\\n \\Pi_0&=(6.0\\times10^5)(5.0\\times10^{-7})=0.30\\ \\mathrm{Pa\\,s},\\\\\n \\rho L&=1000(10^{-4})=0.1\\ \\mathrm{kg\\,m^{-2}},\\quad U=0.30/0.1=3.0\\ \\mathrm{m\\,s^{-1}},\\\\\n U\\tau/L&=3(5\\times10^{-7})/10^{-4}=0.015,\\\\\n \\nu\\tau/L^2&=(10^{-6})(5\\times10^{-7})/10^{-8}=5\\times10^{-5},\\\\\n t_a&=10^{-4}/1500=6.66667\\times10^{-8}\\ \\mathrm s,\\quad \\tau/t_a=7.5.\n \\end{aligned}\\\\[6pt]\\begin{aligned}\\Pi_0&=\\Delta p\\tau=(0.60\\times10^6)(0.50\\times10^{-6})\\ \\mathrm{Pa\\,s}=0.30\\ \\mathrm{Pa\\,s},\\\\U&=\\frac{\\Pi_0}{\\rho L}=3.0\\ \\mathrm{m\\,s^{-1}},\\qquad t_a=\\frac Lc=0.066667\\ \\mu\\mathrm s,\\\\\\frac{U\\tau}{L}&=0.015,\\qquad\\frac{\\nu\\tau}{L^2}=5.0\\times10^{-5},\\qquad\\frac{\\tau}{t_a}=7.5.\\end{aligned}\\end{gathered}", "symbols_en": "$\\Pi_0$ is driven-end impulse (Pa s); $\\Delta p$ is constant applied column pressure difference (Pa); $\\tau$ is pulse duration (s); $U$ is column speed increment (m s⁻¹); $\\rho=1000$ kg m⁻³ is density; $L=100\\times10^{-6}$ m is length; $c=1500$ m s⁻¹ is sound speed; $t_a$ is acoustic crossing time (s); $\\nu=\\mu/\\rho=10^{-6}$ m² s⁻¹ with $\\mu=0.001$ Pa s. The three ratios are dimensionless; μs means microseconds. ", "symbols_zh": "$\\Pi_0$ 为受驱动端冲量（Pa s）；$\\Delta p$ 为施加的常数液柱压差（Pa）；$\\tau$ 为脉冲持续时间（s）；$U$ 为液柱速度增量（m s⁻¹）；$\\rho=1000$ kg m⁻³ 为密度；$L=100\\times10^{-6}$ m 为长度；$c=1500$ m s⁻¹ 为声速；$t_a$ 为声传播时间（s）；$\\nu=\\mu/\\rho=10^{-6}$ m² s⁻¹，且 $\\mu=0.001$ Pa s。三个比值均无量纲；μs 表示微秒。 ", "kind": "Declared impulse calculation and applicability diagnostics", "kind_zh": "采用给定参数的冲量计算与适用性诊断", "diagram": { "type": "impulse", "labels": [ "Δp = 0.60 MPa", "τ = 0.50 μs", "Π₀ = 0.30 Pa s", "ρ", "L = 100 μm", "U = 3.0 m/s", "c", "ta = L/c", "ν" ], "notes": [ "Small bulk residual scales support the ideal column approximation", "Many sound crossings are not a guarantee of correct pressure geometry" ] }, "caption_en": "Pressure impulse and acoustic time have distinct definitions, dimensions and roles.", "caption_zh": "压力冲量与声传播时间具有不同的定义、量纲与作用。" }, { "id": "C1-E25", "tex": "\\begin{gathered}\\begin{aligned}\n \\boldsymbol e_r&=(\\sin\\theta\\cos\\psi,\\sin\\theta\\sin\\psi,\\cos\\theta),\\quad\n d\\Omega=\\sin\\theta\\,d\\theta\\,d\\psi,\\\\\n \\int_0^{2\\pi}\\cos\\psi\\,d\\psi&=[\\sin\\psi]_0^{2\\pi}=0,\\\\\n \\int_0^{2\\pi}\\sin\\psi\\,d\\psi&=[-\\cos\\psi]_0^{2\\pi}=0,\\\\\n \\int_0^\\pi\\cos\\theta\\sin\\theta\\,d\\theta&=[\\tfrac12\\sin^2\\theta]_0^\\pi=0,\\\\\n \\int_{S^2}\\boldsymbol e_r\\,d\\Omega&=(0,0,0).\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\gamma=\\frac h{R_{\\max}}\\\\ \\int_{S^2}\\boldsymbol e_r\\,d\\Omega=\\boldsymbol0\\end{gathered}\\end{gathered}", "symbols_en": "$\\gamma$ is dimensionless stand-off; $h$ is bubble-center-to-boundary distance (m); $R_{\\max}$ is maximum radius (m); $S^2$ denotes all directions on the unit sphere; $\\boldsymbol e_r$ is the outward radial unit vector; $d\\Omega$ is the dimensionless solid-angle measure; $\\boldsymbol0$ is the zero vector. The angular identity expresses directional cancellation under perfect spherical symmetry; it is not a boundary-jet prediction. $\\theta\\in[0,\\pi]$ and $\\psi\\in[0,2\\pi]$ are dimensionless polar and azimuthal angles; tuple entries are Cartesian vector components.", "symbols_zh": "$\\gamma$ 为无量纲离壁比；$h$ 为气泡中心到边界的距离（m）；$R_{\\max}$ 为最大半径（m）；$S^2$ 表示单位球面上的全部方向；$\\boldsymbol e_r$ 为向外径向单位矢量；$d\\Omega$ 为无量纲立体角测度；$\\boldsymbol0$ 为零矢量。角积分恒等式表示完全球对称下方向相互抵消，并非边界射流预测。 $\\theta\\in[0,\\pi]$ 与 $\\psi\\in[0,2\\pi]$ 为无量纲极角和方位角；有序三元组为笛卡尔矢量分量。", "kind": "Geometry definition and exact symmetry identity", "kind_zh": "几何定义与精确的对称性恒等式", "diagram": { "type": "sphere", "labels": [ "h: center to boundary", "Rmax", "γ = h/Rmax", "er", "S²", "dΩ", "Opposite directions cancel" ], "notes": [ "The relevant boundary type and curvature must also be specified" ] }, "caption_en": "Stand-off locates a boundary; perfect spherical symmetry selects no outgoing direction.", "caption_zh": "离壁比定位边界；完全球对称不能选出向外喷射方向。" }, { "id": "C1-E26", "tex": "\\begin{gathered}\\begin{aligned}\n \\partial_t\\rho+\\nabla\\cdot(\\rho\\boldsymbol u)&=0\n \\quad\\xrightarrow{\\rho\\ \\mathrm{constant}}\\quad\\nabla\\cdot\\boldsymbol u=0,\\\\\n \\boldsymbol T&=-p\\boldsymbol I+\\mu[\\nabla\\boldsymbol u+(\\nabla\\boldsymbol u)^{\\mathsf T}],\\\\\n \\rho D\\boldsymbol u/Dt&=\\nabla\\cdot\\boldsymbol T,\\\\\n \\nabla\\cdot\\boldsymbol T&=-\\nabla p+\\mu[\\nabla^2\\boldsymbol u+\\nabla(\\nabla\\cdot\\boldsymbol u)]\n =-\\nabla p+\\mu\\nabla^2\\boldsymbol u,\\\\\n D\\boldsymbol u/Dt&=\\partial_t\\boldsymbol u+(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\nabla\\cdot\\boldsymbol u=0\\\\ \\rho\\left(\\frac{\\partial\\boldsymbol u}{\\partial t}+(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u\\right)=-\\nabla p+\\mu\\nabla^2\\boldsymbol u\\end{gathered}\\end{gathered}", "symbols_en": "$\\boldsymbol u(\\boldsymbol x,t)$ is liquid velocity (m s⁻¹); $\\boldsymbol x$ is spatial position (m); $t$ is time (s); $\\rho$ is constant carrier density (kg m⁻³); $p$ is liquid pressure (Pa); $\\mu$ is constant viscosity (Pa s); $\\nabla\\cdot,\\nabla,\\nabla^2$ denote divergence, gradient and vector Laplacian; $\\partial_t$ is the fixed-position derivative. The dot in convection is vector contraction. The momentum equation has force-per-volume units (N m⁻³). $\\boldsymbol T$ is Cauchy stress (Pa), $\\boldsymbol I$ the dimensionless identity tensor, superscript $\\mathsf T$ transpose, $D/Dt$ the material derivative; body forces are omitted. The density derivative is displayed before imposing constant density.", "symbols_zh": "$\\boldsymbol u(\\boldsymbol x,t)$ 为液体速度（m s⁻¹）；$\\boldsymbol x$ 为空间位置（m）；$t$ 为时间（s）；$\\rho$ 为常数载液密度（kg m⁻³）；$p$ 为液体压力（Pa）；$\\mu$ 为常数黏度（Pa s）；$\\nabla\\cdot,\\nabla,\\nabla^2$ 分别为散度、梯度与矢量拉普拉斯；$\\partial_t$ 为固定位置导数。对流中的圆点为矢量缩并。动量方程各项单位为单位体积力（N m⁻³）。 $\\boldsymbol T$ 为柯西应力（Pa），$\\boldsymbol I$ 为无量纲单位张量，上标 $\\mathsf T$ 表示转置，$D/Dt$ 为物质导数；忽略体力。密度导数写在施加常密度条件之前。", "kind": "Bulk conservation and Newtonian constitutive assumption", "kind_zh": "体相守恒律与牛顿流体本构假设", "diagram": { "type": "jet", "labels": [ "Liquid domain x", "u(x,t)", "p(x,t)", "ρ", "μ", "t", "Moving Γ", "Outlet and wall" ], "notes": [ "Resolve the spatial velocity and pressure, not only one radius" ] }, "caption_en": "Bulk momentum and volume conservation supply the spatial field needed for jet formation.", "caption_zh": "体内动量与体积守恒给出射流形成所需的空间场。" }, { "id": "C1-E27", "tex": "\\begin{gathered}\\begin{aligned}\n F(\\boldsymbol x,t)=0,\\quad\\boldsymbol n&=\\nabla F/|\\nabla F|,\\quad\n V_n=-\\partial_tF/|\\nabla F|,\\\\\n \\partial_tF+\\boldsymbol u\\cdot\\nabla F=0&\\Rightarrow V_n=\\boldsymbol u\\cdot\\boldsymbol n,\\\\\n (\\boldsymbol T_g-\\boldsymbol T_l)\\boldsymbol n&=\\sigma\\kappa\\boldsymbol n,\\\\\n (\\boldsymbol T_g-\\boldsymbol T_l)\\boldsymbol n\n &=[(p_l-p_b)\\boldsymbol I-2\\mu\\boldsymbol D]\\boldsymbol n,\\\\\n \\boldsymbol n\\cdot\\boldsymbol n&=1,\\qquad\\boldsymbol s\\cdot\\boldsymbol n=0,\\\\\n p_l-p_b-2\\mu\\boldsymbol n\\cdot\\boldsymbol D\\boldsymbol n&=\\sigma\\kappa,\\\\\n -2\\mu\\boldsymbol s\\cdot\\boldsymbol D\\boldsymbol n&=0.\n \\end{aligned}\\\\[6pt]\\begin{aligned}\\boldsymbol D&=\\frac12\\left(\\nabla\\boldsymbol u+(\\nabla\\boldsymbol u)^\\mathsf T\\right),\\qquad \\kappa=\\nabla_s\\cdot\\boldsymbol n,\\\\V_n&=\\boldsymbol u\\cdot\\boldsymbol n,\\qquad p_l=p_b+\\sigma\\kappa+2\\mu\\boldsymbol n\\cdot\\boldsymbol D\\boldsymbol n,\\\\2\\mu\\boldsymbol s\\cdot\\boldsymbol D\\boldsymbol n&=0,\\qquad \\kappa_{\\mathrm{sphere}}=-\\frac2R.\\end{aligned}\\end{gathered}", "symbols_en": "$\\boldsymbol D$ is symmetric liquid strain-rate tensor (s⁻¹); $\\boldsymbol u$ is velocity (m s⁻¹); $\\nabla\\boldsymbol u$ is its spatial gradient, with superscript $\\mathsf T$ denoting transpose; $\\boldsymbol n$ is the unit normal from liquid into gas; $\\nabla_s\\cdot$ is surface divergence; $\\kappa$ is signed curvature (m⁻¹); $V_n$ is interface normal speed (m s⁻¹); $p_l,p_b$ are interface liquid and gas pressures (Pa); $\\sigma$ is surface tension (N m⁻¹); $\\mu$ is viscosity (Pa s); $\\boldsymbol s$ is any unit tangent; $R$ is spherical cavity radius (m). Dots denote tensor/vector contractions. $F$ is a smooth dimensionless level-set function increasing into gas, with nonzero gradient on the interface; $\\boldsymbol x$ is position (m), $t$ time (s); $\\boldsymbol T_g=-p_b\\boldsymbol I$ and $\\boldsymbol T_l=-p_l\\boldsymbol I+2\\mu\\boldsymbol D$ are gas and liquid stresses (Pa); $\\boldsymbol I$ is the identity tensor.", "symbols_zh": "$\\boldsymbol D$ 为液体对称应变率张量（s⁻¹）；$\\boldsymbol u$ 为速度（m s⁻¹）；$\\nabla\\boldsymbol u$ 为其空间梯度，上标 $\\mathsf T$ 表示转置；$\\boldsymbol n$ 为从液体指向气体的单位法向；$\\nabla_s\\cdot$ 为表面散度；$\\kappa$ 为带符号曲率（m⁻¹）；$V_n$ 为界面法向速度（m s⁻¹）；$p_l,p_b$ 为界面液体与气体压力（Pa）；$\\sigma$ 为表面张力（N m⁻¹）；$\\mu$ 为黏度（Pa s）；$\\boldsymbol s$ 为任意单位切向；$R$ 为球形空腔半径（m）。圆点表示张量或矢量缩并。 $F$ 为向气体侧增加的光滑无量纲水平集，界面上梯度非零；$\\boldsymbol x$ 为位置（m），$t$ 为时间（s）；$\\boldsymbol T_g=-p_b\\boldsymbol I$ 与 $\\boldsymbol T_l=-p_l\\boldsymbol I+2\\mu\\boldsymbol D$ 为气液应力（Pa）；$\\boldsymbol I$ 为单位张量。", "kind": "Kinematic condition and clean-interface traction", "kind_zh": "运动学条件与洁净界面牵引", "diagram": { "type": "jet", "labels": [ "Γ: liquid–gas interface", "n: liquid → gas", "s: tangent", "Vn = u·n", "u", "D", "pl", "pb", "σκ", "2μ n·D n", "κsphere = −2/R" ], "notes": [ "At a stationary no-slip solid wall use the wall velocity instead", "Phase transfer modifies the kinematic condition in Chapter 2" ] }, "caption_en": "Interface normals, signed curvature and material motion must use one consistent convention.", "caption_zh": "界面法向、带符号曲率与物质运动必须采用一致约定。" }, { "id": "C1-E28", "tex": "\\begin{gathered}\\begin{aligned}\n \\nabla\\cdot\\boldsymbol u=0,\\quad\\boldsymbol u=\\nabla\\phi\n &\\Rightarrow\\nabla\\cdot\\nabla\\phi=\\nabla^2\\phi=0,\\\\\n p_l&=p_b+\\sigma\\kappa,\\\\\n \\partial_t\\phi&=p_\\infty/\\rho-p_l/\\rho-|\\boldsymbol u|^2/2\\\\\n &=(p_\\infty-p_b-\\sigma\\kappa)/\\rho-|\\boldsymbol u|^2/2,\\\\\n \\frac{d\\phi(\\boldsymbol X(t),t)}{dt}\n &=\\partial_t\\phi+\\frac{d\\boldsymbol X}{dt}\\cdot\\nabla\\phi,\\\\\n \\frac{d\\boldsymbol X}{dt}=\\boldsymbol u&\\Rightarrow\n \\boldsymbol u\\cdot\\nabla\\phi=|\\boldsymbol u|^2,\\\\\n -|\\boldsymbol u|^2/2+|\\boldsymbol u|^2&=+|\\boldsymbol u|^2/2.\n \\end{aligned}\\\\[6pt]\\begin{aligned}\n \\boldsymbol u&=\\nabla\\phi,\\qquad\\nabla^2\\phi=0,\\\\\n \\frac{d\\boldsymbol X}{dt}&=\\boldsymbol u(\\boldsymbol X,t),\\\\\n \\partial_t\\phi&=(p_\\infty-p_b-\\sigma\\kappa)/\\rho-\\tfrac12|\\boldsymbol u|^2,\\\\\n \\frac{D\\phi}{Dt}&=\\partial_t\\phi+\\boldsymbol u\\cdot\\nabla\\phi\\\\\n &=(p_\\infty-p_b-\\sigma\\kappa)/\\rho+\\tfrac12|\\boldsymbol u|^2\n \\quad(\\boldsymbol X\\in\\Gamma).\n \\end{aligned}\\end{gathered}", "symbols_en": "$\\boldsymbol u$ is liquid velocity (m s⁻¹); $\\phi$ is velocity potential (m² s⁻¹); $\\nabla$ and $\\nabla^2$ are spatial gradient and Laplacian; $\\boldsymbol X$ is a material interface position (m); $t$ is time (s); $\\Gamma$ is the liquid–gas interface; $p_\\infty,p_b$ are far-field and bubble pressures (Pa); $\\sigma$ is surface tension (N m⁻¹); $\\kappa$ is curvature for the liquid-to-gas normal (m⁻¹); $\\rho$ is density (kg m⁻³); $D/Dt$ is the material derivative; $|\\boldsymbol u|$ is speed and the dot a vector contraction. The potential gauge is fixed at infinity. $p_l$ is liquid-side interface pressure (Pa). Interface markers $\\boldsymbol X(t)$ follow liquid particles; the material potential derivative has units m² s⁻². The construction assumes single-valued potential and a smooth pre-impact interface.", "symbols_zh": "$\\boldsymbol u$ 为液体速度（m s⁻¹）；$\\phi$ 为速度势（m² s⁻¹）；$\\nabla$ 与 $\\nabla^2$ 为空间梯度与拉普拉斯；$\\boldsymbol X$ 为物质界面点位置（m）；$t$ 为时间（s）；$\\Gamma$ 为液—气界面；$p_\\infty,p_b$ 为远场与气泡压力（Pa）；$\\sigma$ 为表面张力（N m⁻¹）；$\\kappa$ 为液体指向气体法向下的曲率（m⁻¹）；$\\rho$ 为密度（kg m⁻³）；$D/Dt$ 为物质导数；$|\\boldsymbol u|$ 为速率，圆点为矢量缩并。速度势参考值固定在无穷远。 $p_l$ 为界面液体侧压力（Pa）。界面标记点 $\\boldsymbol X(t)$ 随液体质点运动；速度势物质导数单位为 m² s⁻²。该构造假设速度势单值，且撞击前界面光滑。", "kind": "Inviscid pre-impact moving-interface closure", "kind_zh": "冲击前无黏运动界面的闭合关系", "diagram": { "type": "jet", "labels": [ "X(t) ∈ Γ", "u = ∇φ", "φ", "∇²φ = 0", "pb", "p∞", "σ", "κ", "ρ", "∂tφ vs Dφ/Dt" ], "notes": [ "Update shape and potential together", "Self-impact or jet piercing requires additional topology and compressibility treatment" ] }, "caption_en": "The interface shape evolves jointly with potential; changing geometry creates focusing.", "caption_zh": "界面形状与速度势共同演化，几何变化产生聚焦。" }, { "id": "C1-E29", "tex": "\\begin{gathered}\\begin{aligned}\n \\phi_\\Gamma&=-R^2\\dot R/R=-R\\dot R,\\\\\n d\\phi_\\Gamma/dt&=-(\\dot R\\dot R+R\\ddot R),\\\\\n -R\\ddot R-\\tfrac32\\dot R^2&=(p_\\infty-p_b+2\\sigma/R)/\\rho,\\\\\n R\\ddot R+\\tfrac32\\dot R^2&=(p_b-p_\\infty-2\\sigma/R)/\\rho.\n \\end{aligned}\\\\[6pt]\\begin{aligned}\\phi_\\Gamma&=-R\\dot R,\\qquad\\kappa=-\\frac2R,\\\\-\\dot R^2-R\\ddot R&=\\frac{p_\\infty-p_b+2\\sigma/R}{\\rho}+\\frac12\\dot R^2,\\\\\\rho\\left(R\\ddot R+\\frac32\\dot R^2\\right)&=p_b-p_\\infty-\\frac{2\\sigma}{R}.\\end{aligned}\\end{gathered}", "symbols_en": "$\\phi_\\Gamma$ is the velocity potential evaluated on the spherical interface (m² s⁻¹), with subscript $\\Gamma$ the interface label; $R>0$ is radius (m); $\\dot R,\\ddot R$ are speed (m s⁻¹) and acceleration (m s⁻²); $\\kappa$ is signed liquid-to-gas curvature (m⁻¹); $p_\\infty,p_b$ are pressures (Pa); $\\sigma$ is surface tension (N m⁻¹); $\\rho$ is carrier density (kg m⁻³). The potential is differentiated along the wall; viscosity is zero in this check. $t$ is time (s); $d/dt$ follows the spherical interface.", "symbols_zh": "$\\phi_\\Gamma$ 为球形界面处速度势（m² s⁻¹），下标 $\\Gamma$ 标识界面；$R>0$ 为半径（m）；$\\dot R,\\ddot R$ 为速度（m s⁻¹）与加速度（m s⁻²）；$\\kappa$ 为液体指向气体约定下的带符号曲率（m⁻¹）；$p_\\infty,p_b$ 为压力（Pa）；$\\sigma$ 为表面张力（N m⁻¹）；$\\rho$ 为载液密度（kg m⁻³）。速度势沿壁面求导；此检验中黏度为零。 $t$ 为时间（s）；$d/dt$ 跟随球形界面。", "kind": "Exact spherical recovery of the potential closure", "kind_zh": "势流闭合关系对球形模型的精确恢复", "diagram": { "type": "sphere", "labels": [ "φΓ = −R Ṙ", "κ = −2/R", "R", "Ṙ", "R̈", "pb", "p∞", "σ", "ρ" ], "notes": [ "Recover the same 3/2 coefficient from the material derivative" ] }, "caption_en": "The spatial potential formulation returns the spherical model under spherical geometry.", "caption_zh": "空间势流表述在球形几何下恢复球形模型。" }, { "id": "C1-E30", "tex": "\\begin{gathered}\\begin{aligned}\n \\partial_t\\rho'+\\rho_0\\nabla\\cdot\\boldsymbol u'&=0,\\quad\n \\rho_0\\partial_t\\boldsymbol u'=-\\nabla p',\\quad p'=c^2\\rho',\\\\\n \\partial_t^2\\rho'+\\rho_0\\nabla\\cdot\\partial_t\\boldsymbol u'&=0,\\quad\n \\rho_0\\nabla\\cdot\\partial_t\\boldsymbol u'=-\\nabla^2p',\\\\\n \\partial_t^2\\rho'-\\nabla^2p'&=0\\Rightarrow\\partial_t^2p'=c^2\\nabla^2p',\\\\\n L_e=ct_a&\\Rightarrow t_a=L_e/c,\\quad\\mathcal C_e=t_a/\\tau_e,\\\\\n t_a&=\\frac{100\\times10^{-6}}{1500}\\ \\mathrm s=66.6667\\ \\mathrm{ns},\\\\\n \\tau_e=10\\ \\mathrm{ns}&\\Rightarrow\\mathcal C_e=66.6667/10=6.66667,\\\\\n \\tau_e=1\\ \\mu\\mathrm s=1000\\ \\mathrm{ns}&\\Rightarrow\\mathcal C_e=66.6667/1000=0.0666667,\\\\\n |\\dot R|=15\\ \\mathrm{m\\,s^{-1}}&\\Rightarrow M_w=15/1500=0.01\\quad\\text{in both cases}.\n \\end{aligned}\\\\[6pt]\\begin{gathered}M_w=\\frac{|\\dot R|}{c}\\\\ \\mathcal C_e=\\frac{L_e}{c\\tau_e}\\end{gathered}\\end{gathered}", "symbols_en": "$M_w$ is wall Mach number and $\\mathcal C_e$ a pressure-communication ratio (both dimensionless); $\\dot R$ is wall speed (m s⁻¹); $c>0$ is carrier sound speed (m s⁻¹); $L_e>0$ is the distance across which loading must communicate (m); $\\tau_e>0$ is the event's relevant rise or change time (s); $|\\ |$ denotes magnitude; subscript $e$ labels the selected event scale. $\\rho_0>0$ is uniform background density (kg m⁻³); primes label small acoustic perturbations, with $\\rho'$ density (kg m⁻³), $p'$ pressure (Pa), $\\boldsymbol u'$ velocity (m s⁻¹); $\\nabla\\cdot$ and $\\nabla^2$ are spatial divergence and Laplacian; $\\partial_t$ is the fixed-position derivative; $t_a$ is one-way travel time (s); ns is 10⁻⁹ s. Linearization assumes inviscid barotropic liquid and constant $c$.", "symbols_zh": "$M_w$ 为壁面马赫数，$\\mathcal C_e$ 为压力通信比（两者均无量纲）；$\\dot R$ 为壁面速度（m s⁻¹）；$c>0$ 为载液声速（m s⁻¹）；$L_e>0$ 为载荷必须传播的距离（m）；$\\tau_e>0$ 为事件相关的上升或变化时间（s）；$|\\ |$ 表示大小；下标 $e$ 标识选定事件尺度。 $\\rho_0>0$ 为均匀背景密度（kg m⁻³）；撇号标记小幅声学扰动，$\\rho'$ 为密度（kg m⁻³），$p'$ 为压力（Pa），$\\boldsymbol u'$ 为速度（m s⁻¹）；$\\nabla\\cdot$ 和 $\\nabla^2$ 为空间散度与拉普拉斯；$\\partial_t$ 为固定位置导数；$t_a$ 为单程传播时间（s）；ns 为 10⁻⁹ s。线性化假设液体无黏、正压且声速恒定。", "kind": "Compressibility applicability diagnostics", "kind_zh": "可压缩性适用性诊断", "diagram": { "type": "timescale", "labels": [ "Mw = |Ṙ|/c", "Ce = Le/(cτe)", "Ṙ", "c", "Le", "τe", "Pressure propagation path" ], "notes": [ "Small wall Mach alone does not ensure fast pressure communication", "An impact rise time can be much shorter than the total collapse time" ] }, "caption_en": "A pressure pulse can require compressibility even while the bulk wall speed is modest.", "caption_zh": "即使总体壁面速率不高，压力脉冲仍可能要求考虑可压缩性。" }, { "id": "C1-E31", "tex": "\\begin{gathered}\\begin{aligned}\n p_l-p_\\infty&=p_b-p_\\infty-2\\sigma/R-4\\mu\\dot R/R,\\\\\n p_l-p_\\infty&=\\rho R\\ddot R+\\tfrac32\\rho\\dot R^2,\\\\\n \\rho R\\ddot R&=p_b-p_\\infty-2\\sigma/R-4\\mu\\dot R/R-\\tfrac32\\rho\\dot R^2,\\\\\n \\ddot R&=\\frac{p_b-p_\\infty-2\\sigma/R-4\\mu\\dot R/R-\\tfrac32\\rho\\dot R^2}{\\rho R}.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\rho\\left(R\\ddot R+\\frac32\\dot R^2\\right)=p_b-p_\\infty-\\frac{2\\sigma}{R}-\\frac{4\\mu\\dot R}{R}\\\\ p_b=p_{v,\\mathrm{PFC}}+p_{v,w}+p_g\\end{gathered}\\end{gathered}", "symbols_en": "$\\rho$ is carrier density (kg m⁻³); $R>0$ is cavity radius (m); $\\dot R,\\ddot R$ are wall speed (m s⁻¹) and acceleration (m s⁻²); $p_b,p_\\infty$ are total bubble and far-field pressures (Pa); $\\sigma$ is surface tension (N m⁻¹); $\\mu$ is viscosity (Pa s); $p_{v,\\mathrm{PFC}},p_{v,w},p_g$ are PFC-vapor, water-vapor and permanent-gas partial pressures (Pa). The equations are the spherical mechanical balance and the pressure-composition closure target. $p_l$ is liquid-side wall pressure (Pa).", "symbols_zh": "$\\rho$ 为载液密度（kg m⁻³）；$R>0$ 为空腔半径（m）；$\\dot R,\\ddot R$ 为壁面速度（m s⁻¹）与加速度（m s⁻²）；$p_b,p_\\infty$ 为气泡总压力与远场压力（Pa）；$\\sigma$ 为表面张力（N m⁻¹）；$\\mu$ 为黏度（Pa s）；$p_{v,\\mathrm{PFC}},p_{v,w},p_g$ 为 PFC 蒸气、水蒸气与非凝结气体分压（Pa）。两式分别为球形力学平衡与需要闭合的压力组成。 $p_l$ 为壁面液体侧压力（Pa）。", "kind": "Original formulas quoted for defense 1", "kind_zh": "答辩问题 1 引用的原始公式", "diagram": { "type": "phase", "labels": [ "PFC liquid → vapor", "pv,PFC", "pv,w", "pg", "pb", "p∞", "ρ carrier", "R", "Ṙ", "R̈", "σ", "μ" ], "notes": [ "The pressure history must connect to a specific liquid path" ] }, "caption_en": "The source pressure moves carrier liquid; geometry determines whether that motion is a useful jet.", "caption_zh": "源压力使载液运动；几何决定这种运动能否成为有用射流。" }, { "id": "C1-E32", "tex": "\\begin{gathered}\\begin{aligned}\n y&=\\dot R^2,\\quad\\frac{d(R^3y)}{dR}=-2\\Delta p_cR^2/\\rho,\\\\\n R^3y-0&=-\\frac{2\\Delta p_c}{3\\rho}(R^3-R_{\\max}^3),\\\\\n \\dot R&=-\\sqrt{y},\\quad\n t_c=R_{\\max}\\sqrt{\\frac{3\\rho}{2\\Delta p_c}}\\int_0^1\\frac{x^{3/2}}{\\sqrt{1-x^3}}\\,dx,\\\\\n K_l&=\\int_{V_{\\max}}^{V_b}-\\Delta p_c\\,dV=\\Delta p_c(V_{\\max}-V_b).\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\dot R=-\\sqrt{\\frac{2\\Delta p_c}{3\\rho}\\left[\\left(\\frac{R_{\\max}}R\\right)^3-1\\right]}\\\\ t_c=\\frac{B(5/6,1/2)}{\\sqrt6}R_{\\max}\\sqrt{\\frac\\rho{\\Delta p_c}}\\\\ K_l=\\frac{4\\pi}{3}\\Delta p_c(R_{\\max}^3-R^3)\\end{gathered}\\end{gathered}", "symbols_en": "$\\dot R\\le0$ is inward wall velocity (m s⁻¹); $R\\in(0,R_{\\max}]$ and $R_{\\max}$ are current and maximum radii (m); $\\Delta p_c>0$ is constant collapse pressure (Pa); $\\rho$ is carrier density (kg m⁻³); $t_c$ is ideal collapse time (s); $B$ is the dimensionless beta function defined in Eq. (C1-E16); $K_l$ is carrier kinetic energy (J); $\\pi$ is the circle constant. The initial cavity is at rest, with no gas, viscosity, capillarity or compressibility. $y$ is squared speed (m² s⁻²); $x=R/R_{\\max}$ is dimensionless radius; $V_b=4\\pi R^3/3$ and $V_{\\max}=4\\pi R_{\\max}^3/3$ are volumes (m³); $V$ is dummy volume (m³).", "symbols_zh": "$\\dot R\\le0$ 为向内壁面速度（m s⁻¹）；$R\\in(0,R_{\\max}]$ 与 $R_{\\max}$ 为当前与最大半径（m）；$\\Delta p_c>0$ 为常数塌缩压差（Pa）；$\\rho$ 为载液密度（kg m⁻³）；$t_c$ 为理想塌缩时间（s）；$B$ 为式（C1-E16）定义的无量纲贝塔函数；$K_l$ 为载液动能（J）；$\\pi$ 为圆周率。初始空腔静止，并忽略非凝结气体、黏性、毛细及可压缩性。 $y$ 为速度平方（m² s⁻²）；$x=R/R_{\\max}$ 为无量纲半径；$V_b=4\\pi R^3/3$ 与 $V_{\\max}=4\\pi R_{\\max}^3/3$ 为体积（m³）；$V$ 为积分哑体积（m³）。", "kind": "Original formulas quoted for defense 2", "kind_zh": "答辩问题 2 引用的原始公式", "diagram": { "type": "sphere", "labels": [ "Rmax", "R → 0", "Ṙ < 0", "Δpc", "ρ", "tc", "B(5/6,1/2)/√6", "Kl finite" ], "notes": [ "Finite time and finite energy coexist with an excluded wall-speed singularity" ] }, "caption_en": "The singular velocity belongs to an ideal radial trajectory, not a validated physical maximum.", "caption_zh": "奇异速度属于理想径向轨迹，而非经过验证的实际最大值。" }, { "id": "C1-E33", "tex": "\\begin{gathered}\\begin{aligned}\n \\Delta\\boldsymbol u&=-\\nabla\\Pi/\\rho\n -\\int_{t_0}^{t_1}(\\boldsymbol u\\cdot\\nabla)\\boldsymbol u\\,dt\n +\\nu\\int_{t_0}^{t_1}\\nabla^2\\boldsymbol u\\,dt,\\\\\n U\\tau/L\\ll1,\\quad\\nu\\tau/L^2\\ll1\n &\\Rightarrow\\Delta\\boldsymbol u\\simeq-\\nabla\\Pi/\\rho,\\\\\n \\nabla\\cdot\\Delta\\boldsymbol u=0&\\Rightarrow\\nabla^2\\Pi=0,\\\\\n \\Pi(0)=\\Pi_0,\\quad\\Pi(L)=0&\\Rightarrow\\Pi(z)=\\Pi_0(1-z/L),\\\\\n \\Delta u_z&=-\\frac1\\rho\\frac{d\\Pi}{dz}=\\frac{\\Pi_0}{\\rho L}.\n \\end{aligned}\\\\[6pt]\\begin{gathered}\\Pi(\\boldsymbol x)=\\int_{t_0}^{t_1}(p-p_{\\mathrm{ref}})dt\\\\\\Delta\\boldsymbol u\\simeq-\\frac{\\nabla\\Pi}{\\rho},\\quad\\nabla^2\\Pi=0\\\\ U_{\\mathrm{column}}=\\frac{\\Pi_0}{\\rho L},\\quad t_a=\\frac Lc\\end{gathered}\\end{gathered}", "symbols_en": "$\\Pi$ is local pressure impulse (Pa s); $\\boldsymbol x$ is fixed position (m); $t,t_0,t_1$ are time and interval endpoints (s); $p,p_{\\mathrm{ref}}$ are local and spatially uniform reference pressures (Pa); $\\Delta\\boldsymbol u$ is liquid speed change (m s⁻¹); $\\nabla,\\nabla^2$ denote spatial gradient and Laplacian; $\\rho$ is carrier density (kg m⁻³); $U_{\\mathrm{column}}$ is the ideal straight-column speed increment (m s⁻¹); $\\Pi_0$ is driven-end impulse (Pa s); $L$ is column length (m); $t_a$ is crossing time (s); $c$ is sound speed (m s⁻¹). The reduced momentum formula needs a frozen geometry and small omitted impulses. $\\boldsymbol u$ is velocity (m s⁻¹); $\\nu$ is kinematic viscosity (m² s⁻¹); $U$ is a characteristic speed (m s⁻¹); $\\tau=t_1-t_0$ is event duration (s); $z\\in[0,L]$ is column coordinate (m); $\\Delta u_z$ is axial speed increment (m s⁻¹); $\\ll$ is an asymptotic smallness condition.", "symbols_zh": "$\\Pi$ 为局部压力冲量（Pa s）；$\\boldsymbol x$ 为固定位置（m）；$t,t_0,t_1$ 为时间与区间端点（s）；$p,p_{\\mathrm{ref}}$ 为局部压力与空间均匀参考压力（Pa）；$\\Delta\\boldsymbol u$ 为液体速度变化（m s⁻¹）；$\\nabla,\\nabla^2$ 为空间梯度与拉普拉斯；$\\rho$ 为载液密度（kg m⁻³）；$U_{\\mathrm{column}}$ 为理想直液柱速度增量（m s⁻¹）；$\\Pi_0$ 为受驱动端冲量（Pa s）；$L$ 为液柱长度（m）；$t_a$ 为传播时间（s）；$c$ 为声速（m s⁻¹）。约化动量公式要求冻结几何，且被忽略的冲量项较小。 $\\boldsymbol u$ 为速度（m s⁻¹）；$\\nu$ 为运动黏度（m² s⁻¹）；$U$ 为特征速度（m s⁻¹）；$\\tau=t_1-t_0$ 为事件时长（s）；$z\\in[0,L]$ 为液柱坐标（m）；$\\Delta u_z$ 为轴向速度增量（m s⁻¹）；$\\ll$ 表示渐近很小条件。", "kind": "Original formulas quoted for defense 3", "kind_zh": "答辩问题 3 引用的原始公式", "diagram": { "type": "impulse", "labels": [ "Π(x)", "p − pref", "t₀ → t₁", "Δu ≈ −∇Π/ρ", "ρ", "Π₀", "L", "Ucolumn", "ta = L/c", "c" ], "notes": [ "Pa s is impulse per area; seconds is travel time", "A spatial gradient, not a scalar peak, sets velocity" ] }, "caption_en": "The impulse benchmark gives an early velocity field, followed by a moving-interface problem.", "caption_zh": "冲量基准给出早期速度场，之后还需解运动界面问题。" }, { "id": "C2-E01", "tex": "F(r)=F_0\\exp(-2r^2/w^2)", "symbols_en": "$F(r)$ is incident fluence at radial coordinate $r\\ge0$ (J m⁻²); $F_0>0$ is on-axis fluence (J m⁻²); $w>0$ is the transverse $e^{-2}$ beam radius (m); $\\exp$ is the natural exponential.", "symbols_zh": "$F(r)$ 为径向坐标 $r\\ge0$ 处的入射能量密度（J m⁻²）；$F_0>0$ 为轴上能量密度（J m⁻²）；$w>0$ 为横向 $e^{-2}$ 光束半径（m）；$\\exp$ 为自然指数函数。", "kind": "Prescribed optical profile", "kind_zh": "给定的光学分布", "diagram": { "type": "laser", "labels": [ "r: transverse radius", "w: e⁻² beam radius", "F(r), F₀: fluence" ], "notes": [ "Beam axis at r = 0", "Gaussian profile across the illuminated plane" ] }, "caption_en": "Fluence varies across the beam; the quoted radius fixes its definition.", "caption_zh": "能量密度沿光束横向变化；所给半径明确其定义。" }, { "id": "C2-E02", "tex": "\\begin{aligned}E_L&=2\\pi F_0\\int_0^\\infty r e^{-2r^2/w^2}\\,dr\\\\&=\\frac{\\pi w^2F_0}{2}\\int_0^\\infty e^{-x}\\,dx\\\\&=\\frac{\\pi w^2F_0}{2}[-e^{-x}]_0^\\infty=\\frac{\\pi w^2F_0}{2}.\\end{aligned}", "symbols_en": "$E_L$ is total incident pulse energy (J); $F_0$ is on-axis fluence (J m⁻²); $w>0$ and $r\\ge0$ are beam radius and transverse distance (m); $x=2r^2/w^2$ is a dimensionless substitution variable; $\\pi$ is the circular constant; $e^x$ is the exponential; $\\int$ and brackets denote integration and endpoint evaluation.", "symbols_zh": "$E_L$ 为总入射脉冲能量（J）；$F_0$ 为轴上能量密度（J m⁻²）；$w>0$ 和 $r\\ge0$ 分别为光束半径与横向距离（m）；$x=2r^2/w^2$ 为无量纲换元变量；$\\pi$ 为圆周率；$e^x$ 为指数函数；$\\int$ 与方括号表示积分及端点求值。", "kind": "Exact integral of the prescribed profile", "kind_zh": "给定分布的精确积分", "diagram": { "type": "laser", "labels": [ "E_L: sum over annuli", "F₀: central fluence", "r, w: transverse geometry", "x = 2r²/w²", "π: circular annuli" ], "notes": [ "Annulus area = 2πr dr", "Integral of e⁻ˣ from 0 to infinity = 1" ] }, "caption_en": "The Gaussian energy follows from summing annular contributions.", "caption_zh": "高斯光束能量来自环带贡献之和。" }, { "id": "C2-E03", "tex": "f_{\\mathrm{geo}}=\\frac{\\int_0^{b_a}F(r)2\\pi r\\,dr}{E_L}=1-\\exp(-2b_a^2/w^2),\\qquad 0\\le f_{\\mathrm{geo}}\\le1", "symbols_en": "$f_{\\mathrm{geo}}$ is intercepted energy fraction (dimensionless), with label geo meaning geometry; $b_a\\ge0$ is centered absorber-aperture radius (m); $F(r)$ is Gaussian fluence (J m⁻²); $r$ is transverse integration distance (m); $E_L>0$ is total pulse energy (J); $w>0$ is beam radius (m); $\\pi$ is the circular constant; $\\exp$ and $\\int$ denote exponential and integration.", "symbols_zh": "$f_{\\mathrm{geo}}$ 为截获能量比例（无量纲），geo 下标表示几何；$b_a\\ge0$ 为同轴吸收体窗口半径（m）；$F(r)$ 为高斯能量密度（J m⁻²）；$r$ 为横向积分距离（m）；$E_L>0$ 为总脉冲能量（J）；$w>0$ 为光束半径（m）；$\\pi$ 为圆周率；$\\exp$ 和 $\\int$ 表示指数函数与积分。", "kind": "Derived interception fraction", "kind_zh": "推导得到的截获比例", "diagram": { "type": "laser", "labels": [ "b_a: absorber aperture edge", "w: beam radius", "r, F(r): intercepted annuli", "E_L: full incident energy", "f_geo: intercepted fraction", "π: annulus geometry" ], "notes": [ "b_a → 0 gives no interception", "b_a ≫ w captures nearly all beam energy" ] }, "caption_en": "The aperture removes the outer annuli from the useful incident energy.", "caption_zh": "窗口使外侧环带不再计入有用入射能量。" }, { "id": "C2-E04", "tex": "\\int_{-\\infty}^{\\infty} f(t)\\,dt=1,\\qquad I(r,t)=F(r)f(t),\\qquad I_{\\mathrm{peak}}=F_0/\\tau_L\\quad\\text{(rectangular pulse)}", "symbols_en": "$f(t)\\ge0$ is normalized pulse shape (s⁻¹); $t$ is time (s); $I(r,t)$ is irradiance (W m⁻²); $F(r)$ and $F_0$ are local and central fluence (J m⁻²); $r$ is transverse distance (m); $I_{\\mathrm{peak}}$ is central peak irradiance (W m⁻²); $\\tau_L>0$ is rectangular laser-pulse duration (s), with L naming laser; $\\int$ is time integration.", "symbols_zh": "$f(t)\\ge0$ 为归一化脉冲形状（s⁻¹）；$t$ 为时间（s）；$I(r,t)$ 为辐照度（W m⁻²）；$F(r)$ 和 $F_0$ 为局部与中心能量密度（J m⁻²）；$r$ 为横向距离（m）；$I_{\\mathrm{peak}}$ 为中心峰值辐照度（W m⁻²）；$\\tau_L>0$ 为矩形激光脉冲时长（s），L 表示激光；$\\int$ 为时间积分。", "kind": "Temporal profile definition", "kind_zh": "时间分布的定义", "diagram": { "type": "laser", "labels": [ "t: pulse time", "τ_L: pulse width", "f(t): normalized profile", "I(r,t), I_peak: irradiance", "r: transverse position", "F(r), F₀: integrated fluence" ], "notes": [ "Area under f(t) is 1", "Peak irradiance and pulse energy are different observables" ] }, "caption_en": "A pulse waveform specifies when the energy is deposited.", "caption_zh": "脉冲波形规定能量何时沉积。" }, { "id": "C2-E05", "tex": "\\begin{aligned}\\partial_z I&=-\\mu_a I,& Q_{\\mathrm{abs}}&=\\mu_a I,\\\\ \\int_{I_{\\mathrm{in}}}^{I(z)}\\frac{dI^{\\prime}}{I^{\\prime}}&=-\\int_0^z\\mu_a\\,dz^{\\prime},&I(z)&=I_{\\mathrm{in}}e^{-\\mu_a z}.\\end{aligned}", "symbols_en": "$z\\ge0$ is absorber depth (m); $\\partial_z$ denotes differentiation in depth; $I(z)>0$ and $I_{\\mathrm{in}}>0$ are local and entry irradiance (W m⁻²), with in labeling entry; $\\mu_a>0$ is constant absorption coefficient (m⁻¹), with a labeling absorption; $Q_{\\mathrm{abs}}$ is volumetric optical heat deposition (W m⁻³); $I^{\\prime}$ and $z^{\\prime}$ are integration dummy variables with the same units as $I$ and $z$; $\\int$ denotes integration; $e^x$ denotes the exponential. Zero input gives identically zero deposition without dividing by irradiance.", "symbols_zh": "$z\\ge0$ 为吸收体深度（m）；$\\partial_z$ 表示对深度求导；$I(z)>0$ 和 $I_{\\mathrm{in}}>0$ 为局部与入口辐照度（W m⁻²），in 表示入口；$\\mu_a>0$ 为常数吸收系数（m⁻¹），a 表示吸收；$Q_{\\mathrm{abs}}$ 为光学体积生热率（W m⁻³）；$I^{\\prime}$ 与 $z^{\\prime}$ 为积分哑变量，单位分别与 $I$、$z$ 相同；$\\int$ 表示积分；$e^x$ 表示指数函数。零输入时沉积恒为零，无须除以辐照度。", "kind": "Beer–Lambert constitutive model and solution", "kind_zh": "Beer–Lambert 本构模型及其解", "diagram": { "type": "laser", "labels": [ "z: propagation depth", "I_in: entry face", "I(z): transmitted beam", "μ_a: absorption per depth", "Q_abs: heat inside slab", "I′, z′: integration variables" ], "notes": [ "Positive z points into absorber", "Photon power lost from beam becomes thermal source under this model" ] }, "caption_en": "The absorber deposits heat over a finite optical depth.", "caption_zh": "吸收体在有限光学深度内沉积热量。" }, { "id": "C2-E06", "tex": "\\begin{aligned}\\int_0^{h_a}Q_{\\mathrm{abs}}\\,dz&=I_{\\mathrm{in}}(1-e^{-\\mu_a h_a}),\\\\ A_\\lambda&=(1-\\mathcal R_\\lambda)(1-e^{-\\mu_a h_a}),\\\\ E_{\\mathrm{abs}}&=f_{\\mathrm{geo}}A_\\lambda E_L=\\int\\!\\int_{\\Omega_a}Q_{\\mathrm{abs}}\\,dV\\,dt.\\end{aligned}", "symbols_en": "$h_a>0$ is uniform absorber thickness (m); $z$ is depth (m); $Q_{\\mathrm{abs}}$ is volumetric heat source (W m⁻³); $I_{\\mathrm{in}}$ is irradiance entering the slab (W m⁻²); $\\mu_a$ is absorption coefficient (m⁻¹); $\\mathcal R_\\lambda\\in[0,1]$ is front-face reflected energy fraction at wavelength $\\lambda$ (m); $A_\\lambda$ is effective incident-light absorptance (dimensionless); $f_{\\mathrm{geo}}$ is intercepted fraction; $E_L,E_{\\mathrm{abs}}$ are incident and absorbed energies (J); $\\Omega_a$ is absorbing volume, $dV$ its volume element (m³); $t,dt$ denote time and its element (s); $\\int$ is integration; $e^x$ is the exponential.", "symbols_zh": "$h_a>0$ 为均匀吸收体厚度（m）；$z$ 为深度（m）；$Q_{\\mathrm{abs}}$ 为体积热源（W m⁻³）；$I_{\\mathrm{in}}$ 为进入薄层的辐照度（W m⁻²）；$\\mu_a$ 为吸收系数（m⁻¹）；$\\mathcal R_\\lambda\\in[0,1]$ 为波长 $\\lambda$（m）处前表面的反射能量比例；$A_\\lambda$ 为相对于入射光的有效吸收率（无量纲）；$f_{\\mathrm{geo}}$ 为截获比例；$E_L,E_{\\mathrm{abs}}$ 为入射与吸收能量（J）；$\\Omega_a$ 为吸收体体积，$dV$ 为其体积元（m³）；$t,dt$ 为时间及其微元（s）；$\\int$ 为积分；$e^x$ 为指数函数。", "kind": "Derived optical-energy balance", "kind_zh": "推导得到的光能平衡", "diagram": { "type": "laser", "labels": [ "h_a: slab thickness", "z: depth", "Ω_a, dV: absorbing volume", "I_in: light after reflection", "μ_a, Q_abs: optical heating", "λ, ℛ_λ, A_λ: wavelength and fractions", "f_geo, E_L, E_abs: energy accounting", "t: pulse integration" ], "notes": [ "Reflection occurs before entry", "Integrate the source once; do not add equivalent boundary heating again" ] }, "caption_en": "Interception, reflection, and internal absorption each act once.", "caption_zh": "截获、反射与内部吸收各作用一次。" }, { "id": "C2-E07", "tex": "\\boldsymbol q=-k\\nabla T,\\qquad \\rho c_p\\frac{DT}{Dt}=\\rho c_p(\\partial_tT+\\boldsymbol u\\cdot\\nabla T)=-\\nabla\\cdot\\boldsymbol q+Q_{\\mathrm{abs}}=\\nabla\\cdot(k\\nabla T)+Q_{\\mathrm{abs}}", "symbols_en": "$\\boldsymbol q$ is conductive heat flux (W m⁻²); $k>0$ is thermal conductivity (W m⁻¹ K⁻¹); $T$ is temperature (K); $\\rho>0$ is phase density (kg m⁻³); $c_p>0$ is constant specific heat capacity (J kg⁻¹ K⁻¹); $t$ is time (s); $\\boldsymbol u$ is material velocity (m s⁻¹); $Q_{\\mathrm{abs}}$ is optical heating (W m⁻³); $D/Dt$ is the material derivative; $\\partial_t$ is the fixed-position time derivative; $\\nabla$ and $\\nabla\\cdot$ are spatial gradient and divergence (m⁻¹); the centered dot here is a vector contraction, not punctuation.", "symbols_zh": "$\\boldsymbol q$ 为传导热通量（W m⁻²）；$k>0$ 为导热系数（W m⁻¹ K⁻¹）；$T$ 为温度（K）；$\\rho>0$ 为该相密度（kg m⁻³）；$c_p>0$ 为常数比热容（J kg⁻¹ K⁻¹）；$t$ 为时间（s）；$\\boldsymbol u$ 为物质速度（m s⁻¹）；$Q_{\\mathrm{abs}}$ 为光生热率（W m⁻³）；$D/Dt$ 为物质导数；$\\partial_t$ 为固定位置时间导数；$\\nabla$ 与 $\\nabla\\cdot$ 为空间梯度和散度（m⁻¹）；式中的中心点表示向量缩并，不是标点。", "kind": "Reduced heat equation and Fourier constitutive law", "kind_zh": "简化热传导方程与傅里叶本构定律", "diagram": { "type": "phase", "labels": [ "T: core temperature field", "ρ, c_p: stored sensible energy", "k, q: conduction toward colder material", "u: parcel advection", "Q_abs: absorber heating", "t, D/Dt, ∂_t: time changes", "∇, ∇·: spatial variation" ], "notes": [ "Absorber and PFC can occupy different locations", "This equation precedes phase change; latent transfer is added at the interface" ] }, "caption_en": "Heating the absorber and heating the PFC are separate parts of the thermal path.", "caption_zh": "加热吸收体与加热 PFC 是热路径中的不同环节。" }, { "id": "C2-E08", "tex": "q_n=\\boldsymbol q_A\\cdot\\boldsymbol n_{AB}=\\boldsymbol q_B\\cdot\\boldsymbol n_{AB},\\qquad T_A-T_B=\\mathcal R_T q_n", "symbols_en": "$q_n$ is signed normal heat flux (W m⁻²); $\\boldsymbol q_A,\\boldsymbol q_B$ are conductive fluxes in materials labeled A and B (W m⁻²); $\\boldsymbol n_{AB}$ is unit normal from A into B (dimensionless); $T_A,T_B$ are temperatures at the two contact sides (K); $\\mathcal R_T\\ge0$ is area-specific thermal contact resistance (m² K W⁻¹); $\\cdot$ denotes a vector dot product.", "symbols_zh": "$q_n$ 为带符号法向热通量（W m⁻²）；$\\boldsymbol q_A,\\boldsymbol q_B$ 为标记 A、B 的材料内的传导热通量（W m⁻²）；$\\boldsymbol n_{AB}$ 为从 A 指向 B 的单位法向（无量纲）；$T_A,T_B$ 为接触面两侧温度（K）；$\\mathcal R_T\\ge0$ 为单位面积热接触阻力（m² K W⁻¹）；$\\cdot$ 表示向量点积。", "kind": "Thermal-contact boundary conditions", "kind_zh": "热接触边界条件", "diagram": { "type": "phase", "labels": [ "A: absorber side, T_A, q_A", "B: PFC side, T_B, q_B", "n_AB: A → B", "q_n: continuous normal flux", "ℛ_T: contact resistance" ], "notes": [ "Hot A gives positive q_n when T_A > T_B", "No stored heat at the contact in this control" ] }, "caption_en": "Contact resistance delays heat transfer even after the light has been absorbed.", "caption_zh": "即使光已被吸收，接触热阻仍会延迟热传递。" }, { "id": "C2-E09", "tex": "\\alpha=\\frac{k}{\\rho c_p},\\qquad \\frac{\\Delta T}{t_{\\mathrm{th}}}\\sim\\alpha\\frac{\\Delta T}{L_h^2},\\qquad t_{\\mathrm{th}}\\sim\\frac{L_h^2}{\\alpha},\\qquad \\delta_T\\sim\\sqrt{\\alpha\\tau_h}", "symbols_en": "$\\alpha>0$ is thermal diffusivity (m² s⁻¹); $k,\\rho,c_p$ are phase conductivity (W m⁻¹ K⁻¹), density (kg m⁻³), and specific heat (J kg⁻¹ K⁻¹); $\\Delta T\\ne0$ is characteristic temperature change (K); $L_h>0$ is heating distance (m); $t_{\\mathrm{th}}$ is diffusion time estimate (s); $\\tau_h>0$ is available heating duration (s); $\\delta_T$ is penetration estimate (m); $\\sim$ means order-of-magnitude scaling, not equality.", "symbols_zh": "$\\alpha>0$ 为热扩散率（m² s⁻¹）；$k,\\rho,c_p$ 为该相导热系数（W m⁻¹ K⁻¹）、密度（kg m⁻³）及比热容（J kg⁻¹ K⁻¹）；$\\Delta T\\ne0$ 为特征温度变化（K）；$L_h>0$ 为加热距离（m）；$t_{\\mathrm{th}}$ 为扩散时间估计（s）；$\\tau_h>0$ 为可用加热时长（s）；$\\delta_T$ 为渗透深度估计（m）；$\\sim$ 表示量级尺度关系，不表示严格相等。", "kind": "Diffusive scaling derived from the heat equation", "kind_zh": "由热传导方程推导的扩散尺度关系", "diagram": { "type": "timescale", "labels": [ "L_h: absorber-to-core heating distance", "δ_T: penetration during τ_h", "α = k/(ρc_p)", "ΔT: thermal change", "t_th: diffusion time", "τ_h: available heating duration" ], "notes": [ "Long distance costs time quadratically", "Optical pulse duration is not automatically whole-core heating duration" ] }, "caption_en": "Compare heat penetration with the distance to the PFC, not only with beam width.", "caption_zh": "应比较热渗透尺度与到达 PFC 的距离，而不只是光束宽度。" }, { "id": "C2-E10", "tex": "\\begin{aligned}\\alpha_d&=\\frac{0.050}{1630(654)}=4.69034\\times10^{-8}\\ \\mathrm{m^2\\,s^{-1}},\\\\t_{\\mathrm{th}}&\\sim\\frac{(5\\times10^{-6})^2}{4.69034\\times10^{-8}}=5.33010\\times10^{-4}\\ \\mathrm{s},\\\\\\delta_T&\\sim\\sqrt{(4.69034\\times10^{-8})(10^{-8})}=2.16572\\times10^{-8}\\ \\mathrm{m}.\\end{aligned}", "symbols_en": "$\\alpha_d$ is declared PFC-liquid diffusivity (m² s⁻¹), with d labeling droplet liquid; $t_{\\mathrm{th}}$ is diffusion-time estimate (s); $\\delta_T$ is penetration estimate (m). The numerator 0.050 is conductivity in W m⁻¹ K⁻¹; 1630 is density in kg m⁻³; 654 is specific heat in J kg⁻¹ K⁻¹; $5\\times10^{-6}$ is distance in m; $10^{-8}$ is duration in s. Powers are numerical exponents, and $\\sim$ denotes a scale estimate.", "symbols_zh": "$\\alpha_d$ 为所声明 PFC 液体的热扩散率（m² s⁻¹），d 表示液滴液体；$t_{\\mathrm{th}}$ 为扩散时间估计（s）；$\\delta_T$ 为渗透深度估计（m）。分子 0.050 为导热系数，单位 W m⁻¹ K⁻¹；1630 为密度，单位 kg m⁻³；654 为比热容，单位 J kg⁻¹ K⁻¹；$5\\times10^{-6}$ 为距离，单位 m；$10^{-8}$ 为时长，单位 s。幂为数值指数，$\\sim$ 表示尺度估计。", "kind": "Declared-input thermal calculation", "kind_zh": "采用明确给定输入的热学计算", "diagram": { "type": "timescale", "labels": [ "α_d = 4.69034×10⁻⁸ m²/s", "L_h = 5 μm", "τ_h = 10 ns", "t_th ≈ 533 μs", "δ_T ≈ 21.66 nm", "k_d = 0.050, ρ_d = 1630, c_p,d = 654 (SI)" ], "notes": [ "Heated shell is much thinner than the core", "Distributed absorbers or later conduction must justify a uniform-core model" ] }, "caption_en": "A short pulse does not imply uniform temperature in a micron-scale core.", "caption_zh": "短脉冲不意味着微米级核心内温度均匀。" }, { "id": "C2-E11", "tex": "p_d\\simeq p_c+\\frac{2\\sigma_{pc}}{a}+\\Pi_{\\mathrm{shell}},\\qquad \\frac{2(0.020)}{5\\times10^{-6}}=8.00\\ \\mathrm{kPa},\\qquad\\frac{2(0.020)}{10^{-7}}=400\\ \\mathrm{kPa}", "symbols_en": "$p_d$ is initial PFC-liquid pressure and $p_c$ carrier pressure (Pa); $a>0$ is PFC-core radius (m); $\\sigma_{pc}\\ge0$ is PFC–carrier interfacial tension (N m⁻¹), with pc naming those phases; $\\Pi_{\\mathrm{shell}}$ is shell-supported excess pressure (Pa); $\\simeq$ is the static spherical approximation. The numerical interfacial tension 0.020 is a teaching input in N m⁻¹, the radii are in m, and 1 kPa = 1000 Pa.", "symbols_zh": "$p_d$ 为初始 PFC 液体压力，$p_c$ 为载液压力（Pa）；$a>0$ 为 PFC 核心半径（m）；$\\sigma_{pc}\\ge0$ 为 PFC–载液界面张力（N m⁻¹），pc 表示这两个相；$\\Pi_{\\mathrm{shell}}$ 为壳层支撑超压（Pa）；$\\simeq$ 表示静态球形近似。数值界面张力 0.020 是教学输入，单位 N m⁻¹；半径单位为 m；1 kPa = 1000 Pa。", "kind": "Static capillary and shell-pressure control", "kind_zh": "静态毛细压力与壳层压力对照模型", "diagram": { "type": "sphere", "labels": [ "a: PFC-core radius", "p_d: inside core", "p_c: aqueous carrier outside", "σ_pc: PFC–carrier interface", "Π_shell: shell pressure", "8 kPa at 5 μm; 400 kPa at 100 nm" ], "notes": [ "PFC-core radius is not the subsequent bubble radius", "A nanodroplet may need much more local pressure than a micron core" ] }, "caption_en": "The smaller inclusion has a larger capillary contribution at the same tension.", "caption_zh": "界面张力相同时，较小夹杂的毛细压力贡献更大。" }, { "id": "C2-E12", "tex": "\\log_{10}\\!\\left(\\frac{p_{\\mathrm{sat,PFP}}(T)}{10^5\\ \\mathrm{Pa}}\\right)=4.2063-\\frac{1103.454\\ \\mathrm{K}}{T-39.77\\ \\mathrm{K}},\\qquad282.82\\ \\mathrm{K}\\le T\\le337.94\\ \\mathrm{K}", "symbols_en": "$p_{\\mathrm{sat,PFP}}(T)$ is bulk equilibrium PFP vapor pressure (Pa), sat labeling saturation; $T$ is absolute temperature in the stated range (K); $\\log_{10}$ is the base-ten logarithm. One bar equals $10^5$ Pa. Empirical coefficient 4.2063 is dimensionless; 1103.454 K and 39.77 K have temperature units. The denominator is positive throughout the stated domain.", "symbols_zh": "$p_{\\mathrm{sat,PFP}}(T)$ 为体相平衡 PFP 蒸汽压（Pa），sat 表示饱和；$T$ 为所给范围内的绝对温度（K）；$\\log_{10}$ 为常用对数。一巴等于 $10^5$ Pa。经验系数 4.2063 无量纲；1103.454 K 和 39.77 K 具有温度单位。所给范围内分母为正。", "kind": "Verified empirical PFP property correlation", "kind_zh": "已核验的 PFP 物性经验关联式", "diagram": { "type": "phase", "labels": [ "T: interface temperature", "p_sat,PFP: bulk equilibrium vapor pressure", "282.82–337.94 K: fit domain", "1 bar = 10⁵ Pa", "4.2063; 1103.454 K; 39.77 K: coefficients" ], "notes": [ "Liquid and vapor coexist only when inventory and exchange permit", "This curve is not a laser-fluence threshold" ] }, "caption_en": "The saturation curve concerns an equilibrium state, not automatic activation.", "caption_zh": "饱和曲线描述平衡状态，而不是自动激活。" }, { "id": "C2-E13", "tex": "\\log_{10}\\!\\left(\\frac{p_{\\mathrm{sat,w}}(T)}{10^5\\ \\mathrm{Pa}}\\right)=\\begin{cases}5.40221-1838.675\\ \\mathrm{K}/(T-31.737\\ \\mathrm{K}),&273\\le T/\\mathrm{K}\\le303,\\\\5.20389-1733.926\\ \\mathrm{K}/(T-39.485\\ \\mathrm{K}),&304\\le T/\\mathrm{K}\\le333.\\end{cases}", "symbols_en": "$p_{\\mathrm{sat,w}}$ is bulk equilibrium water vapor pressure (Pa), w labeling water; $T$ is temperature (K), restricted to either listed interval; $\\log_{10}$ is the base-ten logarithm of pressure relative to one bar ($10^5$ Pa). The leading coefficients are dimensionless; all coefficients multiplied by K have temperature units. $T/\\mathrm K$ is the numerical temperature in kelvin. No value is assigned to the 303–304 K gap by these two branches.", "symbols_zh": "$p_{\\mathrm{sat,w}}$ 为体相平衡水蒸汽压（Pa），w 表示水；$T$ 为温度（K），限定在两个所列区间之一；$\\log_{10}$ 为相对于一巴（$10^5$ Pa）的压力比值的常用对数。首项系数无量纲；乘以 K 的系数具有温度单位。$T/\\mathrm K$ 为以开尔文计的温度数值。这两个分支未给出 303–304 K 间隙的值。", "kind": "Verified empirical water property correlations", "kind_zh": "已核验的水物性经验关联式", "diagram": { "type": "phase", "labels": [ "T: water temperature", "p_sat,w: water equilibrium pressure", "273–303 K; 304–333 K: separate branches", "10⁵ Pa: pressure reference", "5.40221, 1838.675 K, 31.737 K", "5.20389, 1733.926 K, 39.485 K" ], "notes": [ "Use the coefficient branch belonging to the temperature", "The hot PFC and carrier may have different local temperatures" ] }, "caption_en": "Water provides a matched-temperature reference, not a guaranteed vapor reservoir.", "caption_zh": "水提供同温参照，却不是有保证的蒸汽库。" }, { "id": "C2-E14", "tex": "\\Delta p_n=p_{\\mathrm{sat,PFC}}(T_i)-p_d,\\qquad W(r_n)=4\\pi\\sigma_{vp}r_n^2-\\frac{4\\pi}{3}\\Delta p_n r_n^3", "symbols_en": "$\\Delta p_n$ is vapor-nucleation driving pressure (Pa), n labeling nucleation; $p_{\\mathrm{sat,PFC}}(T_i)$ is bulk PFC saturation pressure at local interface temperature $T_i$ (Pa and K); $p_d$ is PFC-liquid pressure (Pa); $W(r_n)$ is nucleus formation free energy (J); $r_n\\ge0$ is vapor-nucleus radius (m); $\\sigma_{vp}>0$ is vapor–PFC tension (N m⁻¹), vp naming the interface; $\\pi$ is the circular constant. The model assumes fixed $T_i,p_d$, and $r_n$ small compared with the PFC core.", "symbols_zh": "$\\Delta p_n$ 为汽化成核驱动压差（Pa），n 表示成核；$p_{\\mathrm{sat,PFC}}(T_i)$ 为局部界面温度 $T_i$ 下的体相 PFC 饱和蒸汽压（Pa 和 K）；$p_d$ 为 PFC 液体压力（Pa）；$W(r_n)$ 为汽核形成自由能（J）；$r_n\\ge0$ 为汽核半径（m）；$\\sigma_{vp}>0$ 为蒸汽–PFC 张力（N m⁻¹），vp 表示该界面；$\\pi$ 为圆周率。模型假设 $T_i,p_d$ 固定，且 $r_n$ 远小于 PFC 核心。", "kind": "Classical homogeneous capillarity approximation", "kind_zh": "经典均相成核的毛细近似", "diagram": { "type": "nucleation", "labels": [ "r_n: nucleus radius inside PFC", "σ_vp: newly created vapor–PFC interface", "T_i: local nucleus temperature", "p_d: surrounding PFC liquid", "p_sat,PFC: equilibrium vapor reference", "Δp_n: volume driving", "W: surface cost minus volume gain", "π: spherical geometry" ], "notes": [ "The outer PFC–carrier tension is a different interface", "The whole core cannot be treated as an infinite liquid reservoir" ] }, "caption_en": "The barrier belongs to creating a new internal vapor interface.", "caption_zh": "势垒来自生成新的内部蒸汽界面。" }, { "id": "C2-E15", "tex": "\\begin{aligned}\\frac{dW}{dr_n}&=8\\pi\\sigma_{vp}r_n-4\\pi\\Delta p_n r_n^2=4\\pi r_n(2\\sigma_{vp}-\\Delta p_n r_n),\\\\r_*&=2\\sigma_{vp}/\\Delta p_n\\quad(\\Delta p_n>0),\\\\\\left.\\frac{d^2W}{dr_n^2}\\right|_{r_*}&=8\\pi\\sigma_{vp}-8\\pi\\Delta p_n r_*=-8\\pi\\sigma_{vp}<0.\\end{aligned}", "symbols_en": "$W$ is nucleus free energy (J); $r_n>0$ is nucleus radius (m); $\\sigma_{vp}>0$ is vapor–PFC interfacial tension (N m⁻¹); $\\Delta p_n$ is fixed driving pressure (Pa); $r_*$ is the positive critical radius (m), with star labeling the stationary point and requiring $\\Delta p_n>0$; $d/dr_n$ is radius differentiation, $|_{r_*}$ denotes evaluation there, and $\\pi$ is the circular constant. The first derivative has units J m⁻¹ and the second J m⁻².", "symbols_zh": "$W$ 为汽核自由能（J）；$r_n>0$ 为汽核半径（m）；$\\sigma_{vp}>0$ 为蒸汽–PFC 界面张力（N m⁻¹）；$\\Delta p_n$ 为固定驱动压差（Pa）；$r_*$ 为正临界半径（m），星号标记驻点，且要求 $\\Delta p_n>0$；$d/dr_n$ 为对半径求导，$|_{r_*}$ 表示在该点求值，$\\pi$ 为圆周率。一阶导数单位为 J m⁻¹，二阶导数为 J m⁻²。", "kind": "Derived critical radius and branch check", "kind_zh": "推导得到的临界半径与分支检验", "diagram": { "type": "nucleation", "labels": [ "r_n: horizontal radius coordinate", "W: free-energy curve", "r_*: maximum for Δp_n > 0", "σ_vp, Δp_n: barrier controls", "dW/dr_n = 0 at r_*", "d²W/dr_n² < 0", "π: spherical factors" ], "notes": [ "Below r_* the surface term drives shrinkage in the static control", "No positive critical radius when Δp_n ≤ 0" ] }, "caption_en": "The stationary point is a maximum, not a stable equilibrium nucleus.", "caption_zh": "该驻点是最大值，不是稳定平衡汽核。" }, { "id": "C2-E16", "tex": "\\begin{aligned}W_*&=4\\pi\\sigma_{vp}\\frac{4\\sigma_{vp}^2}{\\Delta p_n^2}-\\frac{4\\pi}{3}\\Delta p_n\\frac{8\\sigma_{vp}^3}{\\Delta p_n^3}\\\\&=\\left(16-\\frac{32}{3}\\right)\\frac{\\pi\\sigma_{vp}^3}{\\Delta p_n^2}=\\frac{16\\pi\\sigma_{vp}^3}{3\\Delta p_n^2}.\\end{aligned}", "symbols_en": "$W_*=W(r_*)$ is critical-nucleus free-energy barrier (J), with star labeling the critical value; $\\sigma_{vp}>0$ is vapor–PFC interfacial tension (N m⁻¹); $\\Delta p_n>0$ is driving pressure (Pa); $\\pi$ is the circular constant; superscripts 2 and 3 are powers. This result inherits the homogeneous, locally isothermal, small-nucleus assumptions.", "symbols_zh": "$W_*=W(r_*)$ 为临界汽核自由能势垒（J），星号标记临界值；$\\sigma_{vp}>0$ 为蒸汽–PFC 界面张力（N m⁻¹）；$\\Delta p_n>0$ 为驱动压差（Pa）；$\\pi$ 为圆周率；上标 2、3 表示幂。该结果继承均匀成核、局部等温及小汽核假设。", "kind": "Derived homogeneous barrier height", "kind_zh": "推导得到的均相成核势垒高度", "diagram": { "type": "nucleation", "labels": [ "W_*: free-energy maximum", "r_*: critical radius", "σ_vp³: strong tension dependence", "Δp_n²: driving-pressure dependence", "π: spherical geometry" ], "notes": [ "Surface energy and displaced-volume contribution evaluated at r_*", "A positive driving pressure still leaves a finite barrier" ] }, "caption_en": "Positive superheat driving does not make the nucleation barrier vanish.", "caption_zh": "正的过热驱动力并不会使成核势垒消失。" }, { "id": "C2-E17", "tex": "\\begin{aligned}\\Lambda(t)&=\\int_{V_d(t)}J(T, p)\\,dV,\\qquad\\frac{dS}{dt}=-\\Lambda(t)S,\\qquad S(0)=1,\\\\\\ln S(t)&=-\\int_0^t\\Lambda(t^{\\prime})\\,dt^{\\prime},\\qquad P_{\\mathrm{act}}(t)=1-S(t)=1-e^{-\\int_0^t\\int_{V_d(t^{\\prime})}J(T,p)\\,dV\\,dt^{\\prime}}.\\end{aligned}", "symbols_en": "$J(T,p)\\ge0$ is prescribed nucleation rate (m⁻³ s⁻¹) at local temperature $T$ (K) and pressure $p$ (Pa); $V_d(t)$ denotes the remaining liquid-core region and its volume (m³), d labeling droplet; $\\Lambda(t)$ is total event rate (s⁻¹); $S(t)$ is no-event survival probability and $P_{\\mathrm{act}}$ activation probability (dimensionless); $t\\ge0,t^{\\prime}$ are time and dummy time (s); $dV$ is volume element (m³); $d/dt$, $\\int$, $\\ln$, and $e^x$ denote differentiation, integration, natural logarithm, and exponential. The derivation assumes finite integrated nonnegative rate and independent Poisson events.", "symbols_zh": "$J(T,p)\\ge0$ 为给定的成核速率（m⁻³ s⁻¹），取决于局部温度 $T$（K）与压力 $p$（Pa）；$V_d(t)$ 表示剩余液核区域及其体积（m³），d 表示液滴；$\\Lambda(t)$ 为总事件速率（s⁻¹）；$S(t)$ 为未发生事件的概率，$P_{\\mathrm{act}}$ 为激活概率（无量纲）；$t\\ge0,t^{\\prime}$ 为时间及哑时间（s）；$dV$ 为体积元（m³）；$d/dt$、$\\int$、$\\ln$、$e^x$ 表示求导、积分、自然对数和指数函数。推导要求积分非负速率有限，且事件满足独立 Poisson 模型。", "kind": "Derived probability under a prescribed Poisson rate", "kind_zh": "给定泊松事件率下推导的概率", "diagram": { "type": "nucleation", "labels": [ "V_d(t): remaining liquid region", "T, p: local state", "J: events per volume per time", "Λ(t): integrated event rate", "S(0) = 1; S(t): survival", "P_act = 1 − S", "t, t′, dV: integration domain" ], "notes": [ "Integrate the hot, pressurized region rather than only its average temperature", "An absent event can mean low probability, not impossibility" ] }, "caption_en": "Activation probability requires the rate and the time spent in the activating state.", "caption_zh": "激活概率需要速率及处于可激活状态的时长。" }, { "id": "C2-E18", "tex": "m_{\\mathrm{PFC},0}=\\frac{4\\pi}{3}\\rho_da_0^3,\\qquad m_l+m_v+m_{\\mathrm{diss}}+m_{\\mathrm{esc}}=m_{\\mathrm{PFC},0},\\qquad 0\\le m_v\\le m_{\\mathrm{PFC},0}", "symbols_en": "$m_{\\mathrm{PFC},0}$ is initial PFC mass (kg), 0 labeling the initial state; $\\rho_d>0$ is initial PFC-liquid density (kg m⁻³), d labeling droplet; $a_0>0$ is initial core radius (m); $m_l,m_v,m_{\\mathrm{diss}},m_{\\mathrm{esc}}\\ge0$ are remaining liquid PFC, in-domain vapor PFC, dissolved PFC, and cumulative escaped PFC (kg); the l, v, diss, and esc labels name those compartments; $\\pi$ is the circular constant. The bound assumes no subsequent PFC supply.", "symbols_zh": "$m_{\\mathrm{PFC},0}$ 为初始 PFC 质量（kg），0 表示初始状态；$\\rho_d>0$ 为初始 PFC 液体密度（kg m⁻³），d 表示液滴；$a_0>0$ 为初始核心半径（m）；$m_l,m_v,m_{\\mathrm{diss}},m_{\\mathrm{esc}}\\ge0$ 为剩余液态 PFC、域内 PFC 蒸汽、溶解 PFC 及累计逸出 PFC（kg）；l、v、diss、esc 分别标记这些部分；$\\pi$ 为圆周率。质量上界假设后续没有 PFC 供应。", "kind": "Exact compound-mass ledger under the stated closed supply", "kind_zh": "所述封闭供给条件下化合物质量的精确收支", "diagram": { "type": "phase", "labels": [ "a₀, ρ_d: original liquid core", "m_PFC,0: fixed initial inventory", "m_l: liquid remainder", "m_v: bubble vapor", "m_diss: carrier dissolution", "m_esc: mass crossing outer boundary", "π: initial sphere geometry" ], "notes": [ "Mass can change compartment without being created", "Once venting begins, in-domain mass can decrease" ] }, "caption_en": "The compound ledger survives expansion, condensation, dissolution, and venting.", "caption_zh": "化合物账目在膨胀、凝结、溶解及排气过程中均成立。" }, { "id": "C2-E19", "tex": "\\left.\\dot m_{v,s}\\right|_{\\mathrm{phase}}=\\int_{\\Gamma_s}j_s\\,dA,\\qquad \\left.\\dot m_{v,s}\\right|_{\\mathrm{phase}}=4\\pi R^2j_s\\quad\\text{only for uniform complete spherical contact}", "symbols_en": "$m_{v,s}$ is vapor mass of species $s$ (kg), where $s$ labels a specified species such as PFC or water; a dot is a time derivative (kg s⁻¹); the phase label isolates the phase-change contribution, excluding escape and dissolution; $j_s$ is signed liquid-to-vapor mass flux (kg m⁻² s⁻¹); $\\Gamma_s$ is the species-contacting interface; $dA$ is its area element (m²); $R>0$ is spherical interface radius (m) only in the special second formula; $\\pi$ is the circular constant; $\\int$ is surface integration.", "symbols_zh": "$m_{v,s}$ 为组分 $s$ 的蒸汽质量（kg），$s$ 标记 PFC 或水等明确组分；上点表示时间导数（kg s⁻¹）；phase 标记单独的相变贡献，不含逸出与溶解；$j_s$ 为带符号液到汽质量通量（kg m⁻² s⁻¹）；$\\Gamma_s$ 为该组分接触的界面；$dA$ 为面积元（m²）；$R>0$ 仅在第二个特殊式中表示球形界面半径（m）；$\\pi$ 为圆周率；$\\int$ 为面积积分。", "kind": "Species phase-mass balance", "kind_zh": "组分的分相质量平衡", "diagram": { "type": "phase", "labels": [ "Γ_s: actual species interface", "j_s: liquid → vapor", "m_v,s, ṁ_v,s: vapor inventory and rate", "R: complete bubble radius only when applicable", "dA: contacted surface patch", "π: full-sphere special case" ], "notes": [ "PFC may occupy only a residual inclusion", "Do not give every species the whole bubble area automatically" ] }, "caption_en": "A bubble surface and a PFC evaporation surface need not be identical.", "caption_zh": "气泡表面与 PFC 蒸发表面未必相同。" }, { "id": "C2-E20", "tex": "\\begin{aligned}j_s&=\\rho_l(\\boldsymbol u_l-\\boldsymbol v_\\Gamma)\\cdot\\boldsymbol n=\\rho_v(\\boldsymbol u_v-\\boldsymbol v_\\Gamma)\\cdot\\boldsymbol n,\\\\\\boldsymbol n&=-\\boldsymbol e_r,\\quad\\boldsymbol v_\\Gamma=\\dot R\\boldsymbol e_r,\\quad j_s=\\rho_l[\\dot R-u_l(R)],\\\\u_l(R)&=\\dot R-j_s/\\rho_l.\\end{aligned}", "symbols_en": "$j_s$ is single-component phase mass flux (kg m⁻² s⁻¹), s labeling that component; $\\rho_l,\\rho_v>0$ are adjacent liquid and vapor densities (kg m⁻³); $\\boldsymbol u_l,\\boldsymbol u_v$ are their velocities (m s⁻¹); $\\boldsymbol v_\\Gamma$ is interface velocity (m s⁻¹); $\\boldsymbol n$ is unit normal liquid→vapor and $\\boldsymbol e_r$ outward radial unit vector (dimensionless); $R>0$ is bubble radius (m), $\\dot R$ its time derivative (m s⁻¹); $u_l(R)$ is outward radial liquid velocity at the interface (m s⁻¹); $\\cdot$ is a vector dot product.", "symbols_zh": "$j_s$ 为单组分相变质量通量（kg m⁻² s⁻¹），s 标记该组分；$\\rho_l,\\rho_v>0$ 为邻接液体与蒸汽密度（kg m⁻³）；$\\boldsymbol u_l,\\boldsymbol u_v$ 为两侧速度（m s⁻¹）；$\\boldsymbol v_\\Gamma$ 为界面速度（m s⁻¹）；$\\boldsymbol n$ 为液→汽单位法向，$\\boldsymbol e_r$ 为向外径向单位向量（无量纲）；$R>0$ 为气泡半径（m），$\\dot R$ 为其时间导数（m s⁻¹）；$u_l(R)$ 为界面处向外径向液速（m s⁻¹）；$\\cdot$ 为向量点积。", "kind": "Exact single-component interface mass jump", "kind_zh": "单组分界面的精确质量跃迁条件", "diagram": { "type": "phase", "labels": [ "R, Ṙ: moving bubble surface", "e_r: outward radial direction", "n = −e_r: liquid → vapor", "u_l(R), u_l: outside-liquid motion", "u_v: vapor-side motion", "v_Γ: interface motion", "ρ_l, ρ_v: adjoining densities", "j_s: evaporation through interface" ], "notes": [ "Positive j_s makes liquid speed smaller than outward wall speed", "For mixtures, species diffusion augments the component fluxes" ] }, "caption_en": "The interface need not move at the same speed as the adjacent liquid.", "caption_zh": "界面不必以邻接液体的速度运动。" }, { "id": "C2-E21", "tex": "L_{v,s}=h_{v,s}-h_{l,s}>0,\\qquad j_sL_{v,s}=(\\boldsymbol q_l-\\boldsymbol q_v)\\cdot\\boldsymbol n,\\qquad\\boldsymbol q_\\eta=-k_\\eta\\nabla T_\\eta\\quad(\\eta=l,v)", "symbols_en": "$L_{v,s}>0$ is latent enthalpy per species mass (J kg⁻¹); $h_{v,s},h_{l,s}$ are vapor and liquid specific enthalpies of species s at the interface (J kg⁻¹); $j_s$ is signed evaporation flux (kg m⁻² s⁻¹); $\\boldsymbol q_l,\\boldsymbol q_v$ are conductive heat fluxes on liquid and vapor sides (W m⁻²); $\\boldsymbol n$ points liquid→vapor; $\\eta\\in\\{l,v\\}$ labels phase; $k_\\eta$ and $T_\\eta$ are conductivity (W m⁻¹ K⁻¹) and temperature (K); $\\nabla$ is spatial gradient (m⁻¹); $\\cdot$ is a vector dot product. All terms in the heat jump have units W m⁻².", "symbols_zh": "$L_{v,s}>0$ 为单位组分质量的潜焓（J kg⁻¹）；$h_{v,s},h_{l,s}$ 为界面处组分 s 的蒸汽与液体比焓（J kg⁻¹）；$j_s$ 为带符号蒸发通量（kg m⁻² s⁻¹）；$\\boldsymbol q_l,\\boldsymbol q_v$ 为液、汽两侧传导热通量（W m⁻²）；$\\boldsymbol n$ 从液指向汽；$\\eta\\in\\{l,v\\}$ 表示相；$k_\\eta,T_\\eta$ 为导热系数（W m⁻¹ K⁻¹）与温度（K）；$\\nabla$ 为空间梯度（m⁻¹）；$\\cdot$ 为向量点积。热跳跃式各项单位均为 W m⁻²。", "kind": "Reduced interfacial energy jump and Fourier law", "kind_zh": "简化界面能量跃迁条件与傅里叶定律", "diagram": { "type": "phase", "labels": [ "n: liquid → vapor", "q_l: heat arriving from liquid", "q_v: heat leaving into vapor", "j_s: phase mass flux", "h_l,s → h_v,s: enthalpy change", "L_v,s: latent requirement", "k_l, k_v; T_l, T_v: thermal fields", "η = l or v; ∇: local gradient" ], "notes": [ "Evaporation j_s > 0 consumes latent enthalpy", "Condensation j_s < 0 supplies heat to surrounding material" ] }, "caption_en": "The liquid-to-vapor normal fixes the signs of heat and phase flux.", "caption_zh": "液到汽法向确定热通量及相变通量符号。" }, { "id": "C2-E22", "tex": "\\dot U_b=\\dot Q_b-p_b\\dot V_b+\\sum_{s=1}^{N_s}h_{v,s}\\dot m_{v,s},\\qquad V_b=\\frac{4\\pi}{3}R^3,\\qquad\\dot V_b=4\\pi R^2\\dot R", "symbols_en": "$U_b$ is bubble internal energy (J); $\\dot Q_b$ is net conductive heat into the bubble (W), excluding transported mass enthalpy; $p_b$ is uniform absolute bubble pressure (Pa); $V_b$ is bubble volume (m³); $R>0$ is radius in the spherical control (m); $h_{v,s}$ is boundary specific vapor enthalpy of species s (J kg⁻¹); $\\dot m_{v,s}$ is its signed net mass-entry rate (kg s⁻¹); $s=1,\\ldots,N_s$ indexes the $N_s$ species, a dimensionless count; $\\sum$ is summation; a dot is a time derivative, and $\\pi$ is the circular constant. Distinct entry/exit states require distinct summed boundary terms.", "symbols_zh": "$U_b$ 为气泡内能（J）；$\\dot Q_b$ 为进入气泡的净传导热（W），不含随质量输运的焓；$p_b$ 为均匀绝对气泡压力（Pa）；$V_b$ 为气泡体积（m³）；$R>0$ 为球形参照的半径（m）；$h_{v,s}$ 为组分 s 的边界蒸汽比焓（J kg⁻¹）；$\\dot m_{v,s}$ 为其带符号净流入质量速率（kg s⁻¹）；$s=1,\\ldots,N_s$ 遍历 $N_s$ 个组分，$N_s$ 为无量纲数量；$\\sum$ 为求和；上点为时间导数，$\\pi$ 为圆周率。不同流入／流出状态要求分别求和其边界项。", "kind": "Uniform-state open-bubble first-law model", "kind_zh": "均匀状态开放气泡的热力学第一定律模型", "diagram": { "type": "energy", "labels": [ "U_b: vapor energy storage", "Q̇_b: conductive heat only", "p_b, V_b, V̇_b: pressure work", "R, Ṙ: bubble geometry and speed", "h_v,s, ṁ_v,s: mass enthalpy entry", "s = 1…N_s: species ledger", "π: spherical volume" ], "notes": [ "Expansion removes energy through p_b V̇_b", "Do not add a second latent-heat source to transported vapor enthalpy" ] }, "caption_en": "Mass enthalpy and conductive heat are different terms in the bubble ledger.", "caption_zh": "质量焓与传导热是气泡账目中的不同项。" }, { "id": "C2-E23", "tex": "\\begin{aligned}U_b&=\\sum_{s=1}^{N_s}m_{v,s}e_s(T_b),\\qquad c_{v,s}=de_s/dT_b,\\\\\\dot U_b&=\\left(\\sum_s m_{v,s}c_{v,s}\\right)\\dot T_b+\\sum_s e_s\\dot m_{v,s},\\\\\\left(\\sum_s m_{v,s}c_{v,s}\\right)\\dot T_b&=\\dot Q_b-p_b\\dot V_b+\\sum_s(h_{v,s}-e_s)\\dot m_{v,s}.\\end{aligned}", "symbols_en": "$U_b$ is internal energy (J); $m_{v,s}$ is species vapor mass (kg); $e_s(T_b)$ is specific internal energy using a consistent reference (J kg⁻¹); $T_b$ is common bubble temperature (K); $c_{v,s}=de_s/dT_b$ is constant-volume specific heat (J kg⁻¹ K⁻¹); $h_{v,s}$ is actual boundary specific enthalpy (J kg⁻¹); $\\dot Q_b$ is conductive heat input (W); $p_b,V_b$ are pressure (Pa) and volume (m³); $s=1,\\ldots,N_s$ is species index and $\\sum_s$ sums that range; a dot denotes a time derivative; $d/dT_b$ is temperature differentiation. Boundary temperature may differ from $T_b$ and must then be used in $h_{v,s}$.", "symbols_zh": "$U_b$ 为内能（J）；$m_{v,s}$ 为组分蒸汽质量（kg）；$e_s(T_b)$ 为采用一致参考的比内能（J kg⁻¹）；$T_b$ 为共同气泡温度（K）；$c_{v,s}=de_s/dT_b$ 为定容比热（J kg⁻¹ K⁻¹）；$h_{v,s}$ 为实际边界比焓（J kg⁻¹）；$\\dot Q_b$ 为传导热输入（W）；$p_b,V_b$ 为压力（Pa）与体积（m³）；$s=1,\\ldots,N_s$ 为组分指标，$\\sum_s$ 对该范围求和；上点表示时间导数；$d/dT_b$ 为温度求导。边界温度可不同于 $T_b$，此时 $h_{v,s}$ 必须使用边界温度。", "kind": "Product-rule derivation of thermal closure", "kind_zh": "通过乘积求导法则推导热学闭合关系", "diagram": { "type": "energy", "labels": [ "T_b, Ṫ_b: common vapor temperature", "m_v,s, ṁ_v,s: changing species mass", "e_s, c_v,s: internal-energy storage", "h_v,s: boundary enthalpy", "U_b: total stored energy", "Q̇_b: conduction", "p_b V̇_b: work", "s = 1…N_s: summation" ], "notes": [ "The product rule includes both temperature and mass changes", "Consistent energy references preserve the balance" ] }, "caption_en": "A variable vapor mass makes a fixed-mass polytropic shortcut conditional.", "caption_zh": "蒸汽质量变化，使固定质量多方捷径只能有条件成立。" }, { "id": "C2-E24", "tex": "p_{v,s}V_b=\\frac{m_{v,s}}{M_s}R_uT_b,\\qquad p_b=\\sum_{s=1}^{N_s}p_{v,s},\\qquad p_b=\\mathcal P(\\rho_b,e_b,\\boldsymbol Y)\\quad\\text{for a stated real-mixture EOS}", "symbols_en": "$p_{v,s}$ is ideal partial pressure of species s (Pa), v naming the vapor-domain component; $V_b>0$ is bubble volume (m³); $m_{v,s}\\ge0$ is its mass (kg); $M_s>0$ is molar mass (kg mol⁻¹); $R_u=8.314462618$ J mol⁻¹ K⁻¹ is the universal gas constant; $T_b>0$ is bulk temperature (K); $p_b$ is total pressure (Pa); $s=1,\\ldots,N_s$ indexes all components including permanent gas; $\\sum$ is summation. $\\mathcal P$ is a specified equation-of-state function, $\\rho_b$ mixture density (kg m⁻³), $e_b$ specific internal energy (J kg⁻¹), and $\\boldsymbol Y$ the species mass-fraction vector (dimensionless, entries summing to one); EOS means equation of state.", "symbols_zh": "$p_{v,s}$ 为组分 s 的理想分压（Pa），v 标记蒸汽域组分；$V_b>0$ 为气泡体积（m³）；$m_{v,s}\\ge0$ 为其质量（kg）；$M_s>0$ 为摩尔质量（kg mol⁻¹）；$R_u=8.314462618$ J mol⁻¹ K⁻¹ 为通用气体常数；$T_b>0$ 为体相温度（K）；$p_b$ 为总压力（Pa）；$s=1,\\ldots,N_s$ 遍历包含永久气体的所有组分；$\\sum$ 为求和。$\\mathcal P$ 为明确的状态方程函数，$\\rho_b$ 为混合物密度（kg m⁻³），$e_b$ 为比内能（J kg⁻¹），$\\boldsymbol Y$ 为组分质量分数向量（无量纲，各分量之和为一）；EOS 指状态方程。", "kind": "Ideal-mixture constitutive control and required refinement", "kind_zh": "理想混合物本构对照模型与所需改进", "diagram": { "type": "phase", "labels": [ "m_v,s, M_s: species mole inventory", "T_b: common temperature", "V_b: finite bubble volume", "p_v,s: component pressure", "p_b: total mixture pressure", "R_u: gas constant", "ρ_b, e_b, Y: real-mixture state", "s = 1…N_s; P: EOS function" ], "notes": [ "Permanent gas remains after condensable vapor is removed", "Dense compression requires a calibrated EOS" ] }, "caption_en": "The pressure responds to species masses, temperature, and changing volume.", "caption_zh": "压力响应组分质量、温度及体积变化。" }, { "id": "C2-E25", "tex": "\\frac{\\dot p_{v,s}}{p_{v,s}}=\\frac{\\dot m_{v,s}}{m_{v,s}}+\\frac{\\dot T_b}{T_b}-\\frac{\\dot V_b}{V_b}=\\frac{\\dot m_{v,s}}{m_{v,s}}+\\frac{\\dot T_b}{T_b}-3\\frac{\\dot R}{R}\\quad\\text{(sphere)}", "symbols_en": "$p_{v,s}>0$ is species partial pressure (Pa); $m_{v,s}>0$ is its vapor-domain mass (kg), s labeling species; $T_b>0$ is bubble temperature (K); $V_b>0$ is volume (m³); $R>0$ is radius for a spherical bubble (m); a dot is a time derivative. Every ratio has units s⁻¹; the final equality uses $V_b=4\\pi R^3/3$ and is restricted to spherical geometry.", "symbols_zh": "$p_{v,s}>0$ 为组分分压（Pa）；$m_{v,s}>0$ 为其蒸汽域质量（kg），s 表示组分；$T_b>0$ 为气泡温度（K）；$V_b>0$ 为体积（m³）；$R>0$ 为球形气泡半径（m）；上点表示时间导数。各比值单位均为 s⁻¹；最后一个等式使用 $V_b=4\\pi R^3/3$，仅限球形几何。", "kind": "Derived ideal-pressure evolution identity", "kind_zh": "推导得到的理想气体压力演化恒等式", "diagram": { "type": "phase", "labels": [ "p_v,s, ṗ_v,s: partial pressure change", "m_v,s, ṁ_v,s: evaporation or loss", "T_b, Ṫ_b: heating or cooling", "V_b, V̇_b: geometric expansion", "R, Ṙ: spherical volume change", "s: species index" ], "notes": [ "Expansion lowers pressure at fixed mass and temperature", "Evaporation or heating can oppose the geometric pressure decrease" ] }, "caption_en": "A radius-only gas law hides mass and thermal changes.", "caption_zh": "仅由半径决定的气体定律会隐藏质量与热变化。" }, { "id": "C2-E26", "tex": "m_{v,\\mathrm{eq}}=\\min\\!\\left[m_{\\mathrm{PFC},0},\\frac{M_{\\mathrm{PFC}}p_{\\mathrm{sat,PFC}}(T_b)V_b}{R_uT_b}\\right],\\qquad p_{v,\\mathrm{PFC,eq}}=\\min\\!\\left[p_{\\mathrm{sat,PFC}}(T_b),\\frac{m_{\\mathrm{PFC},0}R_uT_b}{M_{\\mathrm{PFC}}V_b}\\right]", "symbols_en": "$m_{v,\\mathrm{eq}}$ is equilibrium PFC vapor mass (kg), eq labeling the restricted equilibrium control; $m_{\\mathrm{PFC},0}$ is finite initial supply (kg); $M_{\\mathrm{PFC}}$ is PFC molar mass (kg mol⁻¹); $p_{\\mathrm{sat,PFC}}(T_b)$ is bulk saturation pressure (Pa); $p_{v,\\mathrm{PFC,eq}}$ is ideal equilibrium PFC partial pressure (Pa); $T_b>0$ is prescribed temperature (K); $V_b>0$ is available vapor volume (m³); $R_u$ is universal gas constant (J mol⁻¹ K⁻¹); $\\min$ selects the lesser nonnegative argument. The supply is closed and nondissolving, and mass exchange is assumed fast.", "symbols_zh": "$m_{v,\\mathrm{eq}}$ 为平衡 PFC 蒸汽质量（kg），eq 标记受限平衡参照；$m_{\\mathrm{PFC},0}$ 为有限初始供应（kg）；$M_{\\mathrm{PFC}}$ 为 PFC 摩尔质量（kg mol⁻¹）；$p_{\\mathrm{sat,PFC}}(T_b)$ 为体相饱和蒸汽压（Pa）；$p_{v,\\mathrm{PFC,eq}}$ 为理想平衡 PFC 分压（Pa）；$T_b>0$ 为给定温度（K）；$V_b>0$ 为可用蒸汽体积（m³）；$R_u$ 为通用气体常数（J mol⁻¹ K⁻¹）；$\\min$ 选择较小非负值。供应闭合且不溶解，并假设质量交换迅速。", "kind": "Finite-inventory equilibrium benchmark", "kind_zh": "有限存量的平衡基准模型", "diagram": { "type": "phase", "labels": [ "m_PFC,0: total available compound", "m_v,eq: actual equilibrium vapor mass", "M_PFC, R_u: mole-to-mass conversion", "T_b: prescribed thermal state", "V_b: available vapor volume", "p_sat,PFC: coexistence ceiling", "p_v,PFC,eq: inventory-limited pressure", "min: choose supply or saturation branch" ], "notes": [ "Small vapor volume can coexist with remaining liquid", "A fully vaporized large volume cannot maintain saturation without more PFC" ] }, "caption_en": "The saturation branch ends when the finite liquid inventory is exhausted.", "caption_zh": "有限液体库存耗尽时，饱和分支终止。" }, { "id": "C2-E27", "tex": "m_{\\mathrm{PFC},0}=\\frac{4\\pi}{3}(1630)(5\\times10^{-6})^3=8.534660\\times10^{-13}\\ \\mathrm{kg}=0.853466\\ \\mathrm{ng}", "symbols_en": "$m_{\\mathrm{PFC},0}$ is initial PFC mass (kg), 0 labeling initial state; 1630 is density in kg m⁻³; $5\\times10^{-6}$ is initial radius in m; $\\pi$ is the circular constant; the cube is a power; ng denotes nanogram, $1$ ng = $10^{-12}$ kg.", "symbols_zh": "$m_{\\mathrm{PFC},0}$ 为初始 PFC 质量（kg），0 表示初态；1630 为以 kg m⁻³ 计的密度；$5\\times10^{-6}$ 为以 m 计的初始半径；$\\pi$ 为圆周率；立方表示幂；ng 为纳克，$1$ ng = $10^{-12}$ kg。", "kind": "Finite-mass worked calculation", "kind_zh": "有限质量的完整示例计算", "diagram": { "type": "sphere", "labels": [ "a₀ = 5 μm", "ρ_d = 1630 kg/m³", "m_PFC,0 = 0.853466 ng", "π: spherical volume" ], "notes": [ "This is a micron-scale teaching core", "All later PFC phases share this mass inventory" ] }, "caption_en": "A mass ledger begins with the actual core volume.", "caption_zh": "质量账目从实际核心体积开始。" }, { "id": "C2-E28", "tex": "\\begin{aligned}Q_{\\mathrm{sens}}&\\approx m_{\\mathrm{PFC},0}c_{p,d}(T_*-T_0)=(8.534660\\times10^{-13})(654)(323-293)=16.7450\\ \\mathrm{nJ},\\\\Q_{\\mathrm{lat}}&\\approx m_{\\mathrm{PFC},0}L_v=(8.534660\\times10^{-13})(95000)=81.0793\\ \\mathrm{nJ},\\\\Q_{\\mathrm{prep}}&\\approx Q_{\\mathrm{sens}}+Q_{\\mathrm{lat}}=97.8243\\ \\mathrm{nJ}.\\end{aligned}", "symbols_en": "$Q_{\\mathrm{sens}},Q_{\\mathrm{lat}},Q_{\\mathrm{prep}}$ are estimated sensible, latent, and total preparation energies (J), named by their subscripts; $m_{\\mathrm{PFC},0}$ is mass (kg); $c_{p,d}=654$ J kg⁻¹ K⁻¹ is constant liquid heat capacity; $T_*=323$ K is final reference temperature, star labeling the selected state rather than a critical nucleus; $T_0=293$ K is initial temperature; $L_v=95000$ J kg⁻¹ is declared latent enthalpy; nJ means $10^{-9}$ J; $\\approx$ denotes the constant-property preparation estimate.", "symbols_zh": "$Q_{\\mathrm{sens}},Q_{\\mathrm{lat}},Q_{\\mathrm{prep}}$ 为显热、潜热及总制备能量估计（J），下标表示其含义；$m_{\\mathrm{PFC},0}$ 为质量（kg）；$c_{p,d}=654$ J kg⁻¹ K⁻¹ 为常液相比热容；$T_*=323$ K 为最终参照温度，此处星号标记所选状态，不是临界汽核；$T_0=293$ K 为初始温度；$L_v=95000$ J kg⁻¹ 为声明的潜焓；nJ 表示 $10^{-9}$ J；$\\approx$ 表示常性质制备估计。", "kind": "Declared preparation-path enthalpy estimate", "kind_zh": "给定制备路径的焓估算", "diagram": { "type": "energy", "labels": [ "m_PFC,0: finite core mass", "T₀ = 293 K → T_* = 323 K", "c_p,d = 654 J/(kg K)", "L_v = 95 kJ/kg", "Q_sens = 16.7450 nJ", "Q_lat = 81.0793 nJ", "Q_prep = 97.8243 nJ" ], "notes": [ "Sensible plus latent preparation energy", "Carrier losses and mechanical work require a broader energy ledger" ] }, "caption_en": "Most of this declared preparation estimate is latent enthalpy.", "caption_zh": "该制备估计中的大部分能量是潜焓。" }, { "id": "C2-E29", "tex": "Q_{\\mathrm{iso}}=m_{\\mathrm{PFC},0}\\!\\left[\\int_{T_0}^{T_{\\mathrm{sat}}}c_{p,l}(T,p)\\,dT+L_v(T_{\\mathrm{sat}},p)+\\int_{T_{\\mathrm{sat}}}^{T_*}c_{p,v}(T,p)\\,dT\\right],\\qquad T_00$ is ideal inventory-state bubble radius (m), b naming bubble and inv naming inventory; $a_0=5$ μm is initial core radius; $m_{\\mathrm{PFC},0}$ is initial mass (kg); $\\rho_d=1630$ kg m⁻³ is original liquid density; $R_u=8.314462618$ J mol⁻¹ K⁻¹ is gas constant; $T_*=323$ K is chosen final reference; $M_{\\mathrm{PFC}}=0.2880343$ kg mol⁻¹ is PFP molar mass; $p_{v,\\mathrm{PFC}}=10^5$ Pa is chosen partial pressure; $\\pi$ is the circular constant; powers 3 and 1/3 denote cube and positive cube root. The ratio is dimensionless.", "symbols_zh": "$R_{b,\\mathrm{inv}}>0$ 为理想库存状态气泡半径（m），b 表示气泡，inv 表示库存；$a_0=5$ μm 为初始核心半径；$m_{\\mathrm{PFC},0}$ 为初始质量（kg）；$\\rho_d=1630$ kg m⁻³ 为初始液体密度；$R_u=8.314462618$ J mol⁻¹ K⁻¹ 为气体常数；$T_*=323$ K 为选定最终参照；$M_{\\mathrm{PFC}}=0.2880343$ kg mol⁻¹ 为 PFP 摩尔质量；$p_{v,\\mathrm{PFC}}=10^5$ Pa 为所选分压；$\\pi$ 为圆周率；幂 3 和 1/3 表示立方及正立方根。该比值无量纲。", "kind": "Derived ideal-vapor inventory radius", "kind_zh": "由理想蒸气存量推导的半径", "diagram": { "type": "sphere", "labels": [ "a₀ = 5 μm: original liquid", "R_b,inv = 26.6827 μm: assumed vapor state", "m_PFC,0, ρ_d: original inventory", "T_* = 323 K", "p_v,PFC = 100 kPa", "M_PFC, R_u: ideal mole relation", "151.9778: volume expansion", "π: sphere geometry" ], "notes": [ "Volume ratio is about 152; radius ratio about 5.34", "Inventory radius is not dynamical R_max" ] }, "caption_en": "The expansion ratio follows from mass conservation at a stated pressure and temperature.", "caption_zh": "膨胀比来自指定压力与温度下的质量守恒。" }, { "id": "C2-E31", "tex": "\\begin{aligned}f_{\\mathrm{geo}}&=1-e^{-2(10/20)^2}=0.393469,\\\\A_\\lambda&=(1-0.10)[1-e^{-(2\\times10^5)(5\\times10^{-6})}]=0.568909,\\\\E_{\\mathrm{abs}}&=(0.393469)(0.568909)(1\\ \\mu\\mathrm J)=223.848\\ \\mathrm{nJ},\\\\\\eta_{\\mathrm{th,min}}&=Q_{\\mathrm{prep}}/E_{\\mathrm{abs}}=97.8243/223.848=0.437012.\\end{aligned}", "symbols_en": "$f_{\\mathrm{geo}}$ is interception fraction; $A_\\lambda$ is effective absorptance at the chosen wavelength $\\lambda$ (m); both are dimensionless. The 10 and 20 are aperture and beam radii in the same μm unit; 0.10 is reflection fraction; $2\\times10^5$ is absorption coefficient in m⁻¹; $5\\times10^{-6}$ is thickness in m. $E_{\\mathrm{abs}}$ is absorbed energy (J); μJ and nJ mean $10^{-6}$ J and $10^{-9}$ J. $Q_{\\mathrm{prep}}$ is declared preparation-energy estimate (J); $\\eta_{\\mathrm{th,min}}$ is the minimum heat-delivery fraction in this loss-excluding estimate, th naming thermal and min minimum; $e^x$ is the exponential.", "symbols_zh": "$f_{\\mathrm{geo}}$ 为截获比例；$A_\\lambda$ 为所选波长 $\\lambda$（m）处的有效吸收率；两者无量纲。10 和 20 是使用相同 μm 单位的窗口与光束半径；0.10 是反射比例；$2\\times10^5$ 是以 m⁻¹ 计的吸收系数；$5\\times10^{-6}$ 是以 m 计的厚度。$E_{\\mathrm{abs}}$ 为吸收能量（J）；μJ 与 nJ 分别为 $10^{-6}$ J 和 $10^{-9}$ J。$Q_{\\mathrm{prep}}$ 为声明的制备能量估计（J）；$\\eta_{\\mathrm{th,min}}$ 为该不含损失估计中的最低热输送比例，th 表示热学，min 表示最低；$e^x$ 为指数函数。", "kind": "Declared optical-to-heat energy screen", "kind_zh": "给定光能转热能过程的能量筛查", "diagram": { "type": "energy", "labels": [ "E_L = 1 μJ: incident", "f_geo = 0.393469: interception", "A_λ = 0.568909: absorption", "E_abs = 223.848 nJ: absorber heat", "Q_prep = 97.8243 nJ: specified PFC preparation", "η_th,min = 0.437012", "b_a = 10 μm; w = 20 μm", "μ_a = 2×10⁵ m⁻¹; h_a = 5 μm; ℛ_λ = 0.10" ], "notes": [ "Not all absorbed energy reaches the PFC", "A surplus absorber budget is not evidence of nucleation or jetting" ] }, "caption_en": "The preparation estimate demands at least 43.7% of the declared absorbed energy.", "caption_zh": "该制备估计要求至少 43.7% 的所声明吸收能量。" }, { "id": "C2-E32", "tex": "E_L=\\frac{\\pi w^2F_0}{2},\\qquad E_{\\mathrm{abs}}=f_{\\mathrm{geo}}A_\\lambda E_L,\\qquad t_{\\mathrm{th}}\\sim L_h^2/\\alpha,\\qquad\\delta_T\\sim\\sqrt{\\alpha\\tau_h}", "symbols_en": "$E_L,E_{\\mathrm{abs}}$ are incident and absorbed energy (J); $w>0$ is Gaussian beam radius (m); $F_0$ is central fluence (J m⁻²); $\\pi$ is the circular constant; $f_{\\mathrm{geo}},A_\\lambda$ are interception and effective absorptance (dimensionless), $\\lambda$ labeling wavelength; $t_{\\mathrm{th}}$ is diffusion-time estimate (s); $L_h>0$ is heating distance (m); $\\alpha>0$ is diffusivity (m² s⁻¹); $\\tau_h$ is heating duration (s); $\\delta_T$ is penetration scale (m); $\\sim$ denotes scaling.", "symbols_zh": "$E_L,E_{\\mathrm{abs}}$ 为入射与吸收能量（J）；$w>0$ 为高斯光束半径（m）；$F_0$ 为中心能量密度（J m⁻²）；$\\pi$ 为圆周率；$f_{\\mathrm{geo}},A_\\lambda$ 为截获比例与有效吸收率（无量纲），$\\lambda$ 标记波长；$t_{\\mathrm{th}}$ 为扩散时间估计（s）；$L_h>0$ 为加热距离（m）；$\\alpha>0$ 为扩散率（m² s⁻¹）；$\\tau_h$ 为加热时长（s）；$\\delta_T$ 为渗透尺度（m）；$\\sim$ 表示尺度关系。", "kind": "Original formulas for Defense 1", "kind_zh": "答辩问题 1 的原始公式", "diagram": { "type": "laser", "labels": [ "E_L, w, F₀: incident beam", "f_geo, A_λ: interception and absorption", "E_abs: absorber budget", "L_h: distance to PFC", "α, τ_h: heat propagation", "t_th, δ_T: thermal tests", "π: Gaussian area factor" ], "notes": [ "Light enters the absorber first", "Heat then travels to the phase-changing material" ] }, "caption_en": "Original optical and thermal formulas used in the reference answer.", "caption_zh": "参考答案使用的原始光学与热学公式。" }, { "id": "C2-E33", "tex": "p_d\\simeq p_c+2\\sigma_{pc}/a+\\Pi_{\\mathrm{shell}},\\quad \\Delta p_n=p_{\\mathrm{sat,PFC}}(T_i)-p_d,\\quad r_*=2\\sigma_{vp}/\\Delta p_n,\\quad W_*=16\\pi\\sigma_{vp}^3/(3\\Delta p_n^2)\\quad(\\Delta p_n>0)", "symbols_en": "$p_d,p_c,\\Pi_{\\mathrm{shell}}$ are core, carrier, and shell excess pressures (Pa); $a>0$ is core radius (m); $\\sigma_{pc}$ is PFC–carrier tension (N m⁻¹); $p_{\\mathrm{sat,PFC}}(T_i)$ is equilibrium PFC vapor pressure (Pa) at local interface temperature $T_i$ (K); $\\Delta p_n>0$ is nucleation driving pressure (Pa); $\\sigma_{vp}>0$ is vapor–PFC tension (N m⁻¹); $r_*$ is critical nucleus radius (m); $W_*$ is barrier (J); star labels critical values; $\\pi$ is the circular constant; $\\simeq$ denotes the static core-pressure approximation. Critical formulas assume homogeneous small-nucleus capillarity.", "symbols_zh": "$p_d,p_c,\\Pi_{\\mathrm{shell}}$ 为核心、载液及壳层超压（Pa）；$a>0$ 为核心半径（m）；$\\sigma_{pc}$ 为 PFC–载液张力（N m⁻¹）；$p_{\\mathrm{sat,PFC}}(T_i)$ 为局部界面温度 $T_i$（K）处的平衡 PFC 蒸汽压（Pa）；$\\Delta p_n>0$ 为成核驱动压差（Pa）；$\\sigma_{vp}>0$ 为蒸汽–PFC 张力（N m⁻¹）；$r_*$ 为临界汽核半径（m）；$W_*$ 为势垒（J）；星号标记临界值；$\\pi$ 为圆周率；$\\simeq$ 表示静态核压力近似。临界公式假设均匀小汽核毛细模型。", "kind": "Original formulas for Defense 2", "kind_zh": "答辩问题 2 的原始公式", "diagram": { "type": "nucleation", "labels": [ "a: core outer radius", "σ_pc: outer interface", "Π_shell, p_c: confinement", "p_d: core liquid state", "T_i, p_sat,PFC: thermal vapor driving", "Δp_n: net driving", "r_*, W_*: internal barrier", "σ_vp: internal nucleus interface", "π: spherical factors" ], "notes": [ "Outer confinement and internal barrier are separate", "Positive vapor driving is necessary in this control but not a guarantee" ] }, "caption_en": "Original confinement and nucleation formulas used in the reference answer.", "caption_zh": "参考答案使用的原始约束与成核公式。" }, { "id": "C2-E34", "tex": "P_{\\mathrm{act}}=1-\\exp\\!\\left[-\\int_0^t\\int_{V_d(t^{\\prime})}J(T,p)\\,dV\\,dt^{\\prime}\\right]", "symbols_en": "$P_{\\mathrm{act}}$ is activation probability (dimensionless); $J(T,p)\\ge0$ is supplied Poisson event rate (m⁻³ s⁻¹), evaluated at local temperature $T$ (K) and pressure $p$ (Pa); $V_d(t^{\\prime})$ is remaining PFC-liquid region, d labeling droplet; $t\\ge0,t^{\\prime}$ are elapsed and dummy time (s); $dV$ is volume element (m³); $\\int$ is integration; $\\exp$ is the natural exponential. Independent events and finite integrated rate are assumed.", "symbols_zh": "$P_{\\mathrm{act}}$ 为激活概率（无量纲）；$J(T,p)\\ge0$ 为给定 Poisson 事件速率（m⁻³ s⁻¹），取局部温度 $T$（K）与压力 $p$（Pa）；$V_d(t^{\\prime})$ 为剩余 PFC 液体区域，d 表示液滴；$t\\ge0,t^{\\prime}$ 为经过时间与哑时间（s）；$dV$ 为体积元（m³）；$\\int$ 为积分；$\\exp$ 为自然指数函数。假设事件独立且积分速率有限。", "kind": "Original activation-probability formula", "kind_zh": "原始激活概率公式", "diagram": { "type": "nucleation", "labels": [ "T, p: local hot state", "J: supplied event rate", "V_d(t′): available liquid", "t, t′, dV: event integration", "P_act: activation probability" ], "notes": [ "A rate law or calibrated activation law is still required", "One absent event does not prove impossibility" ] }, "caption_en": "The probability formula does not replace the missing material-specific rate.", "caption_zh": "概率公式不能替代缺失的材料特定速率。" }, { "id": "C2-E35", "tex": "m_{\\mathrm{PFC},0}=\\frac{4\\pi}{3}\\rho_da_0^3,\\qquad m_l+m_v+m_{\\mathrm{diss}}+m_{\\mathrm{esc}}=m_{\\mathrm{PFC},0},\\qquad p_{v,\\mathrm{PFC}}V_b=\\frac{m_v}{M_{\\mathrm{PFC}}}R_uT_b", "symbols_en": "$m_{\\mathrm{PFC},0}$ is initial PFC mass (kg); $\\rho_d$ is initial liquid density (kg m⁻³); $a_0$ is initial core radius (m); $m_l,m_v,m_{\\mathrm{diss}},m_{\\mathrm{esc}}$ are remaining liquid, in-domain vapor, dissolved, and cumulative escaped PFC masses (kg); 0 denotes initial state and other subscripts name compartments. $p_{v,\\mathrm{PFC}}$ is ideal PFC partial pressure (Pa); $V_b>0$ is bubble volume (m³); $M_{\\mathrm{PFC}}$ is molar mass (kg mol⁻¹); $R_u$ is universal gas constant (J mol⁻¹ K⁻¹); $T_b>0$ is bulk temperature (K); $\\pi$ is the circular constant. No external compound supply is allowed.", "symbols_zh": "$m_{\\mathrm{PFC},0}$ 为初始 PFC 质量（kg）；$\\rho_d$ 为初始液体密度（kg m⁻³）；$a_0$ 为初始核心半径（m）；$m_l,m_v,m_{\\mathrm{diss}},m_{\\mathrm{esc}}$ 为剩余液体、域内蒸汽、溶解及累计逸出 PFC 质量（kg）；0 表示初态，其余下标标记分区。$p_{v,\\mathrm{PFC}}$ 为理想 PFC 分压（Pa）；$V_b>0$ 为气泡体积（m³）；$M_{\\mathrm{PFC}}$ 为摩尔质量（kg mol⁻¹）；$R_u$ 为通用气体常数（J mol⁻¹ K⁻¹）；$T_b>0$ 为体相温度（K）；$\\pi$ 为圆周率。不允许外部化合物供应。", "kind": "Original inventory and pressure formulas for Defense 3", "kind_zh": "答辩问题 3 的原始存量与压力公式", "diagram": { "type": "phase", "labels": [ "a₀, ρ_d: initial core", "m_PFC,0: total supply", "m_l, m_v, m_diss, m_esc: compartments", "p_v,PFC: partial pressure", "V_b, T_b: bubble state", "M_PFC, R_u: ideal conversion", "π: initial sphere" ], "notes": [ "All compound compartments sum to the original mass", "Water cannot be a source of unlimited PFC vapor" ] }, "caption_en": "Original finite-inventory formulas used in the reference answer.", "caption_zh": "参考答案使用的原始有限库存公式。" }, { "id": "C2-E36", "tex": "j_sL_{v,s}=(\\boldsymbol q_l-\\boldsymbol q_v)\\cdot\\boldsymbol n,\\qquad\\dot U_b=\\dot Q_b-p_b\\dot V_b+\\sum_s h_{v,s}\\dot m_{v,s},\\qquad u_l(R)=\\dot R-j_s/\\rho_l", "symbols_en": "$j_s$ is signed phase flux (kg m⁻² s⁻¹); $L_{v,s}$ is latent enthalpy (J kg⁻¹); $\\boldsymbol q_l,\\boldsymbol q_v$ are liquid/vapor conductive fluxes (W m⁻²); $\\boldsymbol n$ is unit liquid→vapor normal; $U_b$ is bubble internal energy (J); $\\dot Q_b$ is conductive heat input excluding mass enthalpy (W); $p_b$ is pressure (Pa); $V_b$ is volume (m³); $h_{v,s}$ is boundary specific vapor enthalpy (J kg⁻¹); $m_{v,s}$ is species mass (kg); $\\sum_s$ sums all declared species; $R$ is spherical bubble radius (m), $\\dot R$ its wall speed (m s⁻¹), $u_l(R)$ adjacent radial liquid speed (m s⁻¹), and $\\rho_l$ liquid density (kg m⁻³). A dot is a time derivative and $\\cdot$ is a vector dot product. Interface heat and slip formulas assume the stated single-component reduced jump.", "symbols_zh": "$j_s$ 为带符号相变通量（kg m⁻² s⁻¹）；$L_{v,s}$ 为潜焓（J kg⁻¹）；$\\boldsymbol q_l,\\boldsymbol q_v$ 为液／汽传导热通量（W m⁻²）；$\\boldsymbol n$ 为液→汽单位法向；$U_b$ 为气泡内能（J）；$\\dot Q_b$ 为不含质量焓的传导热输入（W）；$p_b$ 为压力（Pa）；$V_b$ 为体积（m³）；$h_{v,s}$ 为边界蒸汽比焓（J kg⁻¹）；$m_{v,s}$ 为组分质量（kg）；$\\sum_s$ 对所有声明组分求和；$R$ 为球形气泡半径（m），$\\dot R$ 为壁速（m s⁻¹），$u_l(R)$ 为邻接径向液速（m s⁻¹），$\\rho_l$ 为液体密度（kg m⁻³）。上点表示时间导数，$\\cdot$ 表示向量点积。界面热及滑移公式采用所声明单组分简化跳跃假设。", "kind": "Original phase-energy and velocity-slip formulas", "kind_zh": "原始相变能量与速度滑移公式", "diagram": { "type": "energy", "labels": [ "j_s, L_v,s: phase conversion", "q_l, q_v, n: heat jump signs", "U_b, Q̇_b: bubble thermal ledger", "p_b V̇_b: pressure work", "h_v,s, ṁ_v,s: transported enthalpy", "R, Ṙ, u_l(R): moving interface", "ρ_l: adjacent liquid density", "s: species index" ], "notes": [ "Latent conversion is already included consistently in phase and enthalpy balances", "Evaporation and condensation have opposite signs" ] }, "caption_en": "Original conservation formulas prevent unlimited vapor and duplicated latent energy.", "caption_zh": "原始守恒公式防止无限蒸汽及重复潜热计入。" }, { "id": "C3-E01", "tex": "\\begin{aligned}\\mathbb E[N_b]&=N_dp_a,\\quad \\operatorname{Var}(N_b)=N_dp_a(1-p_a),\\\\ \\mathrm{CV}(N_b)&=\\frac{\\sqrt{N_dp_a(1-p_a)}}{N_dp_a}=\\sqrt{\\frac{1-p_a}{N_dp_a}},\\quad \\Pr(N_b=N_d)=p_a^{N_d}.\\end{aligned}", "symbols_en": "$N_d$ is positive integer site count; $N_b$ is the random activated count; $p_a\\in(0,1]$ is identical independent activation probability. $\\mathbb E$, $\\operatorname{Var}$ and $\\Pr$ denote expectation, variance and probability; $\\mathrm{CV}$ is standard deviation divided by nonzero mean. All are dimensionless.", "symbols_zh": "$N_d$ 是正整数位点数；$N_b$ 为随机已激活数量；$p_a\\in(0,1]$ 为相同且相互独立的激活概率。$\\mathbb E$、$\\operatorname{Var}$、$\\Pr$ 分别表示期望、方差、概率；$\\mathrm{CV}$ 是标准差除以非零均值。所有量均无量纲。", "kind": "Independent-event statistical model", "kind_zh": "独立事件的统计模型", "diagram": { "type": "array", "labels": [ "N_d fabricated sites", "N_b activated sites", "p_a activation probability", "E[N_b], Var, CV, Pr: count statistics" ], "notes": [ "Missing sites change symmetry and timing", "Independent activation is an assumption" ] }, "caption_en": "Count reproducibility and complete activation are different tests.", "caption_zh": "数量重复性与全部激活是不同检验。" }, { "id": "C3-E02", "tex": "\\frac{F(R_A)}{F_0}=\\exp(-2R_A^2/w^2)\\ge1-\\epsilon\\ \\Longrightarrow\\ w^2\\ge\\frac{2R_A^2}{-\\ln(1-\\epsilon)}\\ \\Longrightarrow\\ w\\ge R_A\\sqrt{\\frac{2}{-\\ln(1-\\epsilon)}}.", "symbols_en": "$F(r)$ is Gaussian pulse fluence at radial distance $r$ (J m⁻²); $F_0>0$ is central fluence. $R_A>0$ encloses the array and $w>0$ is the Gaussian 1/e² radius (m). $\\epsilon\\in(0,1)$ is allowed fractional edge decrease. $\\exp$ and $\\ln$ are exponential and natural logarithm of dimensionless arguments.", "symbols_zh": "$F(r)$ 是径向距离 $r$ 处的 Gaussian 脉冲通量（J m⁻²）；$F_0>0$ 为中心通量。$R_A>0$ 包围阵列，$w>0$ 为 Gaussian 的 1/e² 半径（m）。$\\epsilon\\in(0,1)$ 是允许的边缘相对降低量。$\\exp$、$\\ln$ 分别为无量纲自变量的指数及自然对数。", "kind": "Derived optical-uniformity condition", "kind_zh": "推导得到的光学均匀性条件", "diagram": { "type": "laser", "labels": [ "F₀ central fluence", "F(R_A) edge fluence", "R_A array radius", "w beam radius", "ε allowed decrease; exp, ln" ], "notes": [ "Beam axis passes through array center", "Uniform fluence does not prove uniform activation" ] }, "caption_en": "An optical condition can be checked before solving cavity mechanics.", "caption_zh": "求解腔体力学前即可检验光学条件。" }, { "id": "C3-E03", "tex": "\\begin{aligned}4\\pi r_j^2u_j(r_j,t)&=4\\pi R_j^2\\dot R_j,\\\\ \\phi_j(r_j,t)&=\\int_{\\infty}^{r_j}\\frac{R_j^2\\dot R_j}{\\xi^2}\\,d\\xi=-\\frac{R_j^2\\dot R_j}{r_j},\\\\ \\partial_t\\phi_j\\big|_{r_j=d_{ij}}&=-\\frac{2R_j\\dot R_j^2+R_j^2\\ddot R_j}{d_{ij}}.\\end{aligned}", "symbols_en": "$j$ labels the emitting bubble and $i\\ne j$ a receiving center. $R_j(t)>0$, $r_j\\ge R_j$, dummy radius $\\xi$ and fixed $d_{ij}>0$ are lengths (m); $t$ is time (s). $\\dot R_j,\\ddot R_j$ are radius derivatives (m s⁻¹, m s⁻²). $u_j$ is radial velocity (m s⁻¹); $\\phi_j$ is velocity potential (m² s⁻¹). $\\partial_t$ holds position fixed; $\\int$ is radial integration, $\\infty$ the zero-potential far field; $\\pi$ is dimensionless.", "symbols_zh": "$j$ 标记发射气泡，$i\\ne j$ 标记接收中心。$R_j(t)>0$、$r_j\\ge R_j$、积分半径 $\\xi$ 及固定 $d_{ij}>0$ 均为长度（m）；$t$ 为时间（s）。$\\dot R_j$、$\\ddot R_j$ 为半径时间导数（m s⁻¹、m s⁻²）。$u_j$ 为径向速度（m s⁻¹）；$\\phi_j$ 为速度势（m² s⁻¹）。$\\partial_t$ 保持位置固定；$\\int$ 为径向积分，$\\infty$ 为速度势为零的远场；$\\pi$ 无量纲。", "kind": "Continuity and potential integration", "kind_zh": "连续性关系与势函数积分", "diagram": { "type": "array", "labels": [ "R_j, Ṙ_j, R̈_j: source bubble", "r_j, ξ: radial locations", "d_ij: center separation", "u_j: radial liquid velocity", "φ_j, ∂ₜφ_j: source potential", "i receiving center; t time; π" ], "notes": [ "Centers fixed; far separation", "The derivative is at a fixed receiving position" ] }, "caption_en": "A moving spherical volume produces a monopole potential.", "caption_zh": "运动球体体积产生单极子速度势。" }, { "id": "C3-E04", "tex": "p'_{j\\to i}\\simeq-\\rho\\partial_t\\phi_j(d_{ij},t)=\\frac{\\rho}{d_{ij}}(R_j^2\\ddot R_j+2R_j\\dot R_j^2),\\qquad p_{\\mathrm{ext},i}\\simeq p_\\infty+\\sum_{j\\ne i}p'_{j\\to i}.", "symbols_en": "$i,j=1,\\ldots,N_b$ are distinct bubble indices; $p'_{j\\to i}$ is source $j$'s pressure disturbance at $i$, $p_{\\mathrm{ext},i}$ external pressure and $p_\\infty$ remote pressure (Pa). $\\rho$ is constant carrier density (kg m⁻³); $\\phi_j$ is potential (m² s⁻¹); $t$ is time (s), $\\partial_t$ its fixed-position derivative. $R_j,d_{ij}$ are positive lengths (m), dots time derivatives. $\\sum_{j\\ne i}$ adds all other sources; $\\simeq$ marks the leading far-separated approximation.", "symbols_zh": "$i,j=1,\\ldots,N_b$ 为不同气泡编号；$p'_{j\\to i}$ 是源 $j$ 在 $i$ 处的压力扰动，$p_{\\mathrm{ext},i}$ 为外压，$p_\\infty$ 为远场压力（Pa）。$\\rho$ 为恒定载液密度（kg m⁻³）；$\\phi_j$ 为速度势（m² s⁻¹）；$t$ 为时间（s），$\\partial_t$ 为固定位置导数。$R_j$、$d_{ij}$ 为正长度（m），点表示时间导数。$\\sum_{j\\ne i}$ 对其他源求和；$\\simeq$ 表示充分分离时的首阶近似。", "kind": "Leading unsteady-Bernoulli approximation", "kind_zh": "非定常伯努利方程的首阶近似", "diagram": { "type": "array", "labels": [ "j source → i receiver", "R_j, Ṙ_j, R̈_j", "d_ij center separation", "p′_j→i near receiver", "p_ext,i = p∞ + Σ p′", "ρ, φ_j, ∂ₜ, t, N_b: field legend" ], "notes": [ "Positive source-volume acceleration raises surrounding pressure", "Convective cross terms and shape gradients omitted" ] }, "caption_en": "The correction changes surrounding pressure; it is not a new energy reservoir.", "caption_zh": "该修正改变周围压力，并不增加新的能量库。" }, { "id": "C3-E05", "tex": "\\begin{aligned}R_i\\ddot R_i+\\frac32\\dot R_i^2+\\sum_{j\\ne i}\\frac{R_j^2\\ddot R_j+2R_j\\dot R_j^2}{d_{ij}}&=B_i,\\\\ B_i&=\\frac{p_{b,i}-p_\\infty-2\\sigma_b/R_i-4\\mu\\dot R_i/R_i}{\\rho}.\\end{aligned}", "symbols_en": "$i,j$ index activated bubbles; $R_i,R_j,d_{ij}>0$ are lengths (m), dots are time derivatives. $p_{b,i}$ is bubble $i$'s internal pressure and $p_\\infty$ remote pressure (Pa). $\\sigma_b$ is bubble–carrier tension (N m⁻¹), $\\mu$ carrier dynamic viscosity (Pa s), $\\rho$ carrier density (kg m⁻³). $B_i$ is the isolated radial driving expression (m² s⁻²); $\\sum_{j\\ne i}$ sums other sources. Every term has unit m² s⁻².", "symbols_zh": "$i,j$ 标记已激活气泡；$R_i,R_j,d_{ij}>0$ 为长度（m），点为时间导数。$p_{b,i}$ 为第 $i$ 个气泡内压，$p_\\infty$ 为远场压力（Pa）。$\\sigma_b$ 为泡—载液张力（N m⁻¹），$\\mu$ 为载液动力黏度（Pa s），$\\rho$ 为载液密度（kg m⁻³）。$B_i$ 为孤立泡径向驱动表达式（m² s⁻²）；$\\sum_{j\\ne i}$ 对其他源求和。各项单位均为 m² s⁻²。", "kind": "Coupled spherical approximation", "kind_zh": "耦合球形近似", "diagram": { "type": "array", "labels": [ "R_i, Ṙ_i, R̈_i at receiver i", "R_j, Ṙ_j, R̈_j at source j", "d_ij between centers", "p_b,i inside bubble", "p∞ remote liquid", "B_i, σ_b, μ, ρ: radial balance legend" ], "notes": [ "All accelerations solved together", "No jet shape or film boundary is yet resolved" ] }, "caption_en": "Substitute the corrected external pressure into each radial balance.", "caption_zh": "将修正外压代入各径向平衡。" }, { "id": "C3-E06", "tex": "\\begin{aligned}d_k&=s\\frac{\\sin(k\\pi/N)}{\\sin(\\pi/N)},\\qquad S_N=\\sum_{k=1}^{N-1}\\frac1{d_k},\\\\ S_3&=\\frac2s,\\qquad S_4=\\frac{2+1/\\sqrt2}{s},\\qquad S_5=\\frac{2+2/\\varphi_g}{s},\\quad\\varphi_g=\\frac{1+\\sqrt5}{2}.\\end{aligned}", "symbols_en": "$N\\in\\{3,4,5\\}$ is regular-polygon bubble count; $k=1,\\ldots,N-1$ counts steps from one vertex. $s>0$ and $d_k>0$ are neighboring-vertex and $k$-step center distances (m). $S_N$ is the reciprocal-distance sum (m⁻¹). $\\varphi_g$ is the dimensionless golden ratio; $\\sin$ takes angles in radians, $\\pi$ is dimensionless, and $\\sum$ sums the stated steps.", "symbols_zh": "$N\\in\\{3,4,5\\}$ 为正多边形气泡数；$k=1,\\ldots,N-1$ 为从一个顶点出发的步数。$s>0$、$d_k>0$ 为相邻及相隔 $k$ 步的中心距离（m）。$S_N$ 为距离倒数和（m⁻¹）。$\\varphi_g$ 为无量纲黄金比；$\\sin$ 的角度采用弧度，$\\pi$ 无量纲；$\\sum$ 对规定步数求和。", "kind": "Exact polygon geometry", "kind_zh": "精确的多边形几何关系", "diagram": { "type": "array", "labels": [ "N = 3, 4, 5 vertices", "s neighbor pitch", "d_k chord distances", "S_N = Σ 1/d_k", "k, π, sin, φ_g: geometry legend" ], "notes": [ "Triangle: two neighbors at s", "Square: two at s, one at √2 s", "Pentagon: two at s, two at φ_g s" ] }, "caption_en": "Equal geometric sums justify a shared radial trajectory in this constrained control.", "caption_zh": "相同几何和使这一约束对照可采用共同径向轨迹。" }, { "id": "C3-E07", "tex": "\\begin{aligned}\\Delta p_c&=p_\\infty-p_b>0,\\qquad S=S_N,\\\\ (1+SR)R\\ddot R+\\left(\\frac32+2SR\\right)\\dot R^2&=-\\frac{\\Delta p_c}{\\rho},\\qquad R(0)=R_{\\max},\\quad\\dot R(0)=0.\\end{aligned}", "symbols_en": "$p_\\infty,p_b$ are constant ambient and internal pressure (Pa), $\\Delta p_c>0$ their collapse-driving difference. $S=S_N\\ge0$ is the fixed reciprocal-distance sum (m⁻¹); $R(t)>0$ and $R_{\\max}>0$ are shared and initial maximum radius (m). Dots are time derivatives, $t=0$ the initial time (s); $\\rho>0$ is carrier density (kg m⁻³). Surface tension and viscosity are omitted in this ideal control; $SR$ is dimensionless.", "symbols_zh": "$p_\\infty$、$p_b$ 为恒定环境及泡内压力（Pa），$\\Delta p_c>0$ 为其塌缩驱动压差。$S=S_N\\ge0$ 为固定距离倒数和（m⁻¹）；$R(t)>0$、$R_{\\max}>0$ 为共同及初始最大半径（m）。点为时间导数，$t=0$ 为初始时刻（s）；$\\rho>0$ 为载液密度（kg m⁻³）。本理想对照忽略表面张力及黏度；$SR$ 无量纲。", "kind": "Symmetric constant-pressure collapse control", "kind_zh": "对称恒压塌缩对照模型", "diagram": { "type": "array", "labels": [ "R = R_max at t = 0", "Ṙ = 0 initially; R̈ inward", "S, S_N neighbor sum", "p_b inside; p∞ outside", "Δp_c = p∞ − p_b", "ρ shared liquid; SR dimensionless" ], "notes": [ "The shared-liquid inertia factor is 1 + SR", "This is a constrained radial control, not a jet simulation" ] }, "caption_en": "Substitution of the common radius collects the two interaction contributions.", "caption_zh": "代入共同半径后合并两种相互作用贡献。" }, { "id": "C3-E08", "tex": "\\begin{aligned}y(R)&=\\dot R^2,\\quad \\frac{dy}{dR}=\\frac{2\\dot R\\ddot R}{\\dot R}=2\\ddot R\\quad(\\dot R\\ne0),\\\\ y'+\\frac{3+4SR}{R(1+SR)}y&=-\\frac{2\\Delta p_c}{\\rho R(1+SR)},\\\\ \\frac{3+4SR}{R(1+SR)}&=\\frac3R+\\frac S{1+SR}=\\frac{d}{dR}\\ln[R^3(1+SR)/\\ell_*^3],\\\\ \\frac{d}{dR}\\left[R^3(1+SR)y\\right]&=-\\frac{2\\Delta p_c}{\\rho}R^2.\\end{aligned}", "symbols_en": "$y(R)\\ge0$ is squared wall speed (m² s⁻²), $y'=dy/dR$ (m s⁻²); $R>0$ is radius (m) and dots its time derivatives. $S\\ge0$ is neighbor sum (m⁻¹), $\\rho>0$ density (kg m⁻³), $\\Delta p_c>0$ pressure difference (Pa). $\\ell_*>0$ is an arbitrary constant reference length (m), inserted solely to make the logarithm dimensionless; it cancels from the integrating factor. $\\ln$ is natural logarithm; $d/dR$ differentiates in radius. Division by $\\dot R$ is used only on the moving branch.", "symbols_zh": "$y(R)\\ge0$ 为壁速平方（m² s⁻²），$y'=dy/dR$（m s⁻²）；$R>0$ 为半径（m），点表示其时间导数。$S\\ge0$ 为邻距和（m⁻¹），$\\rho>0$ 为密度（kg m⁻³），$\\Delta p_c>0$ 为压差（Pa）。$\\ell_*>0$ 为任意恒定参考长度（m），仅用来使对数无量纲，并在积分因子中抵消。$\\ln$ 为自然对数；$d/dR$ 对半径求导。仅在运动分支除以 $\\dot R$。", "kind": "Chain rule and integrating-factor derivation", "kind_zh": "链式法则与积分因子推导", "diagram": { "type": "array", "labels": [ "R, Ṙ, R̈: contracting bubble", "y = Ṙ²; y′ = dy/dR", "S, ρ, Δp_c", "R³(1 + SR) liquid-inertia factor", "ℓ_*: dimensional reference; ln, d/dR" ], "notes": [ "Restrict cancellation to Ṙ ≠ 0", "Continuous initial limit restores Ṙ = 0" ] }, "caption_en": "Every term of the integrating factor can be reconstructed from the radial equation.", "caption_zh": "积分因子的每一项均可从径向方程重构。" }, { "id": "C3-E09", "tex": "\\begin{aligned}[R^3(1+SR)y]_{R_{\\max}}^R&=-\\frac{2\\Delta p_c}{\\rho}\\int_{R_{\\max}}^R\\xi^2d\\xi=\\frac{2\\Delta p_c}{3\\rho}(R_{\\max}^3-R^3),\\\\ \\dot R^2&=\\frac{2\\Delta p_c}{3\\rho}\\frac{R_{\\max}^3-R^3}{R^3(1+SR)},\\qquad \\dot R=-\\sqrt{\\frac{2\\Delta p_c}{3\\rho}\\frac{R_{\\max}^3-R^3}{R^3(1+SR)}}.\\end{aligned}", "symbols_en": "$R\\in(0,R_{\\max}]$ and $R_{\\max}$ are current/maximum radius (m); $\\xi$ is dummy radius (m), dots radius time derivatives. $y=\\dot R^2$ (m² s⁻²), $S\\ge0$ neighbor sum (m⁻¹), $\\rho>0$ density (kg m⁻³), $\\Delta p_c>0$ pressure difference (Pa). Brackets mean upper minus lower endpoint, $\\int$ definite integration and $\\sqrt{\\ }$ the nonnegative root; the explicit minus sign selects collapse.", "symbols_zh": "$R\\in(0,R_{\\max}]$、$R_{\\max}$ 为当前／最大半径（m）；$\\xi$ 为积分半径（m），点为半径时间导数。$y=\\dot R^2$（m² s⁻²），$S\\ge0$ 为邻距和（m⁻¹），$\\rho>0$ 为密度（kg m⁻³），$\\Delta p_c>0$ 为压差（Pa）。方括号表示上端减下端，$\\int$ 为定积分，$\\sqrt{\\ }$ 取非负根；显式负号选择塌缩。", "kind": "Integrated solution on the inward branch", "kind_zh": "向内运动分支上的积分解", "diagram": { "type": "array", "labels": [ "R_max initial radius", "R contracting radius; ξ integration radius", "Ṙ < 0; y = Ṙ²", "S, ρ, Δp_c", "Endpoint condition y(R_max) = 0" ], "notes": [ "At identical relative radius, interaction lowers speed", "The zero-radius endpoint remains formal" ] }, "caption_en": "The initial condition removes the integration constant and the physical branch fixes the sign.", "caption_zh": "初值消去积分常数，物理分支固定符号。" }, { "id": "C3-E10", "tex": "\\begin{aligned}t_c&=\\int_0^{R_{\\max}}\\frac{dR}{|\\dot R|}=R_{\\max}\\sqrt{\\frac{\\rho}{\\Delta p_c}}\\,C(\\chi),\\quad x=R/R_{\\max},\\quad\\chi=SR_{\\max},\\\\ C(\\chi)&=\\sqrt{\\frac32}\\int_0^1\\sqrt{\\frac{x^3(1+\\chi x)}{1-x^3}}\\,dx,\\\\ C'(\\chi)&=\\frac12\\sqrt{\\frac32}\\int_0^1\\frac{x^{5/2}}{\\sqrt{(1-x^3)(1+\\chi x)}}\\,dx>0\\quad(\\chi\\ge0).\\end{aligned}", "symbols_en": "$t_c$ is formal collapse time (s); $R,R_{\\max}>0$ are radii (m), $\\dot R$ wall velocity (m s⁻¹), $|\\ |$ absolute value. $\\rho$ is density (kg m⁻³), $\\Delta p_c>0$ pressure difference (Pa), $S\\ge0$ neighbor sum (m⁻¹). $x\\in[0,1]$ is dimensionless integration radius and $\\chi\\ge0$ interaction strength. $C$ and $C'=dC/d\\chi$ are dimensionless; integrals are convergent improper integrals at $x=1$.", "symbols_zh": "$t_c$ 为形式塌缩时间（s）；$R,R_{\\max}>0$ 为半径（m），$\\dot R$ 为壁速（m s⁻¹），$|\\ |$ 为绝对值。$\\rho$ 为密度（kg m⁻³），$\\Delta p_c>0$ 为压差（Pa），$S\\ge0$ 为邻距和（m⁻¹）。$x\\in[0,1]$ 为无量纲积分半径，$\\chi\\ge0$ 为相互作用强度。$C$ 及 $C'=dC/d\\chi$ 无量纲；积分在 $x=1$ 处为收敛广义积分。", "kind": "Collapse-time quadrature and sign check", "kind_zh": "塌缩时间的求积表达式与符号检验", "diagram": { "type": "timescale", "labels": [ "t_c radial event time", "R_max, R; x = R/R_max", "χ = S R_max", "C(χ), C′(χ) > 0", "ρ / Δp_c sets time scale", "Ṙ inward speed; |Ṙ| positive" ], "notes": [ "x = 1 endpoint behaves as (1 − x)^−1/2", "χ = 0 recovers the Rayleigh coefficient" ] }, "caption_en": "Shared-liquid interaction increases this constrained collapse time.", "caption_zh": "共享液体相互作用使这一约束塌缩时间增加。" }, { "id": "C3-E11", "tex": "\\begin{aligned}b_i&=R_i^2\\dot R_i,\\quad \\phi_i=-b_i/r_i,\\quad K_N\\simeq\\frac{\\rho}{2}\\sum_{i,j=1}^{N}\\int_\\Omega\\nabla\\phi_i\\cdot\\nabla\\phi_j\\,dV,\\\\ \\int_\\Omega|\\nabla\\phi_i|^2\\,dV&\\simeq4\\pi b_i^2\\int_{R_i}^\\infty r_i^{-2}\\,dr_i=4\\pi R_i^3\\dot R_i^2,\\\\ \\int_\\Omega\\nabla\\phi_i\\cdot\\nabla\\phi_j\\,dV&\\simeq\\int_{\\Gamma_j}\\phi_i\\partial_n\\phi_j\\,dA\\simeq(-b_i/d_{ij})(-\\dot R_j)4\\pi R_j^2=\\frac{4\\pi b_ib_j}{d_{ij}}\\quad(i\\ne j),\\\\ K_N&\\simeq2\\pi\\rho\\sum_i R_i^3\\dot R_i^2+4\\pi\\rho\\sum_{i0$ representative-region volume (m³), $N_b$ bubble count, $R_i,R,s$ radius/common cubic-lattice radius/pitch (m), $\\pi$ dimensionless. $\\phi(\\boldsymbol x,t)$ is weak far-field potential (m² s⁻¹), $p'$ pressure disturbance (Pa), $\\boldsymbol x$ observation position (m), $t$ time (s), $r_i>0$ distance to source $i$ (m), $c$ sound speed (m s⁻¹), $\\rho$ density (kg m⁻³). Dots on $V_i$ are volume time derivatives (m³ s⁻¹, m³ s⁻²); $t-r_i/c$ is retarded time. $\\sum$ sums sources; $\\simeq$ is a distant, compact, weak-source approximation.", "symbols_zh": "$\\phi_b$ 为无量纲瞬时气泡体积分数；$V_i$ 为泡体积、$V_\\Omega>0$ 为代表区体积（m³），$N_b$ 为泡数，$R_i,R,s$ 为半径／立方晶格共同半径／间距（m），$\\pi$ 无量纲。$\\phi(\\boldsymbol x,t)$ 为弱扰动远场势（m² s⁻¹），$p'$ 为压力扰动（Pa），$\\boldsymbol x$ 为观察位置（m），$t$ 为时间（s），$r_i>0$ 为至源 $i$ 的距离（m），$c$ 为声速（m s⁻¹），$\\rho$ 为密度（kg m⁻³）。$V_i$ 上的点为体积时间导数（m³ s⁻¹、m³ s⁻²）；$t-r_i/c$ 为延迟时间。$\\sum$ 对源求和；$\\simeq$ 为远距、紧致、弱源近似。", "kind": "Volume-fraction definition and retarded monopole approximation", "kind_zh": "体积分数定义与延迟单极源近似", "diagram": { "type": "array", "labels": [ "V_i, R_i: bubble sites", "V_Ω representative region", "φ_b = ΣV_i / V_Ω", "s, R: cubic-cell geometry", "r_i → observation x", "p′, φ at time t", "ρ, c; V̇_i, V̈_i; t − r_i/c" ], "notes": [ "Planar arrays require a declared layer thickness", "Activation and propagation delays affect overlap", "Weak far-field sources do not predict impact pressure" ] }, "caption_en": "A spatial sum needs source histories and travel times, not only site count.", "caption_zh": "空间求和需要源历程及传播时间，不能只给位点数。" }, { "id": "C3-E13", "tex": "\\boldsymbol u=\\nabla\\phi,\\qquad \\nabla\\cdot\\boldsymbol u=0\\ \\Longrightarrow\\ \\nabla^2\\phi=0\\quad\\text{in }\\Omega(t).", "symbols_en": "$\\boldsymbol u(\\boldsymbol x,t)$ is velocity (m s⁻¹), $\\phi(\\boldsymbol x,t)$ potential (m² s⁻¹), $\\boldsymbol x$ spatial position (m), $t$ time (s), and $\\Omega(t)$ the moving liquid domain. $\\nabla$, $\\nabla\\cdot$ and $\\nabla^2$ are spatial gradient, divergence and Laplacian; the central dot is an actual contraction, not an equation separator.", "symbols_zh": "$\\boldsymbol u(\\boldsymbol x,t)$ 为速度（m s⁻¹），$\\phi(\\boldsymbol x,t)$ 为势（m² s⁻¹），$\\boldsymbol x$ 为空间位置（m），$t$ 为时间（s），$\\Omega(t)$ 为运动液体域。$\\nabla$、$\\nabla\\cdot$、$\\nabla^2$ 为空间梯度、散度、Laplace 算子；中心点是真正缩并，并非方程分隔符。", "kind": "Potential-flow reduction", "kind_zh": "势流简化模型", "diagram": { "type": "jet", "labels": [ "Ω(t) liquid domain", "φ(x,t) velocity potential", "u = ∇φ liquid velocity", "∇·u = 0; ∇²φ = 0", "x spatial point; t event time" ], "notes": [ "Actual meniscus and bubble surfaces are separate boundaries", "Zero vorticity and negligible bulk viscosity assumed" ] }, "caption_en": "Geometry enters through the domain and its boundary conditions.", "caption_zh": "几何通过区域及边界条件进入。" }, { "id": "C3-E14", "tex": "\\begin{aligned}\\frac{d\\boldsymbol X}{dt}&=\\boldsymbol u(\\boldsymbol X,t),\\quad V_n=\\partial_n\\phi,\\quad \\partial_n\\phi=\\boldsymbol V_w\\cdot\\boldsymbol n\\quad\\text{on walls},\\\\ (\\Gamma,\\phi_\\Gamma)_{t=0}&=(\\Gamma_0,\\phi_0),\\qquad \\phi\\to0\\quad\\text{in a quiescent far field}.\\end{aligned}", "symbols_en": "$\\boldsymbol X$ is a material interface point (m); $t$ is time (s), $d/dt$ its trajectory derivative. $\\boldsymbol u$ is liquid velocity and $\\boldsymbol V_w$ prescribed wall velocity (m s⁻¹). $\\boldsymbol n$ is outward from liquid toward gas or solid, unitless; $V_n$ normal interface speed (m s⁻¹), $\\partial_n\\phi=\\nabla\\phi\\cdot\\boldsymbol n$, and $\\phi$ potential (m² s⁻¹). $\\Gamma$ is the complete moving interface; $\\phi_\\Gamma$ its potential; $\\Gamma_0,\\phi_0$ are initial data. Arrows indicate the stated far-field limit.", "symbols_zh": "$\\boldsymbol X$ 为材料界面点（m）；$t$ 为时间（s），$d/dt$ 为轨迹导数。$\\boldsymbol u$ 为液速，$\\boldsymbol V_w$ 为规定壁速（m s⁻¹）。$\\boldsymbol n$ 从液体指向气体或固体，是无量纲单位法向；$V_n$ 为界面法向速度（m s⁻¹），$\\partial_n\\phi=\\nabla\\phi\\cdot\\boldsymbol n$，$\\phi$ 为势（m² s⁻¹）。$\\Gamma$ 为完整运动界面；$\\phi_\\Gamma$ 为其势；$\\Gamma_0,\\phi_0$ 为初值。箭头表示规定远场极限。", "kind": "Kinematic, initial and wall conditions", "kind_zh": "运动学条件、初始条件与壁面条件", "diagram": { "type": "jet", "labels": [ "Γ, Γ₀: meniscus and cavity shapes", "X moves with u", "n liquid → gas or solid", "V_n = ∂ₙφ", "V_w wall velocity", "φ_Γ, φ₀; t = 0", "φ → 0 far away" ], "notes": [ "Outlet geometry and wall compliance must be supplied", "Phase flux changes the material-interface condition" ] }, "caption_en": "A potential equation without initial shape and wall data cannot predict a jet.", "caption_zh": "无初始形状及壁面数据的势方程无法预测射流。" }, { "id": "C3-E15", "tex": "\\begin{aligned}p_l&=p_g+\\sigma\\kappa,\\quad\\kappa=\\nabla_s\\cdot\\boldsymbol n,\\quad\\kappa_{\\mathrm{cavity}}=-2/R,\\quad\\kappa_{\\mathrm{jet}}=1/a_j,\\\\ \\frac{D\\phi_\\Gamma}{Dt}&=\\frac12|\\nabla\\phi|^2+\\frac{p_\\infty-p_g-\\sigma\\kappa}{\\rho},\\qquad \\frac D{Dt}=\\partial_t+\\boldsymbol u\\cdot\\nabla.\\end{aligned}", "symbols_en": "$p_l,p_g,p_\\infty$ are interface-liquid, gas and remote pressures (Pa); $\\sigma$ is the tension appropriate to that interface (N m⁻¹). $\\kappa$ is signed total curvature (m⁻¹), $\\nabla_s\\cdot$ surface divergence, $\\boldsymbol n$ the liquid-to-gas unit normal, $R$ inner-cavity radius and $a_j$ exterior-cylinder radius (m). $\\phi_\\Gamma$ and $\\phi$ are surface/bulk potential (m² s⁻¹); $\\boldsymbol u$ velocity (m s⁻¹), $\\rho$ density (kg m⁻³), $t$ time (s). $D/Dt$ is the material derivative, $\\partial_t$ fixed-position derivative, $\\nabla$ spatial gradient, $|\\ |$ Euclidean norm; labels cavity and jet select geometries.", "symbols_zh": "$p_l,p_g,p_\\infty$ 为界面液压、气压、远压（Pa）；$\\sigma$ 为对应界面张力（N m⁻¹）。$\\kappa$ 为带符号总曲率（m⁻¹），$\\nabla_s\\cdot$ 为表面散度，$\\boldsymbol n$ 为液体指向气体的单位法向，$R$ 为内部腔体半径、$a_j$ 为外部圆柱半径（m）。$\\phi_\\Gamma$、$\\phi$ 为表面／体内势（m² s⁻¹）；$\\boldsymbol u$ 为速度（m s⁻¹），$\\rho$ 为密度（kg m⁻³），$t$ 为时间（s）。$D/Dt$ 为材料导数，$\\partial_t$ 为固定位置导数，$\\nabla$ 为空间梯度，$|\\ |$ 为 Euclidean 范数；cavity、jet 标签选择几何。", "kind": "Normal stress and material Bernoulli condition", "kind_zh": "法向应力与随界面运动的伯努利条件", "diagram": { "type": "jet", "labels": [ "p_g gas; p_l adjacent liquid", "n liquid → gas; κ = ∇ₛ·n", "R cavity: κ = −2/R", "a_j exterior jet: κ = 1/a_j", "φ_Γ, ∇φ = u", "p∞, σ, ρ; D/Dt, ∂ₜ, t" ], "notes": [ "Surface shape and potential evolve together", "Different interfaces can have different surface tensions" ] }, "caption_en": "Signed curvature and the material derivative keep the dynamic condition consistent.", "caption_zh": "带符号曲率及材料导数使动力学条件自洽。" }, { "id": "C3-E16", "tex": "\\begin{aligned}\\Pi(\\boldsymbol x)&=\\int_{t_0}^{t_0+\\tau_p}(p-p_{\\mathrm{ref}})\\,dt,\\quad \\Delta\\boldsymbol u\\simeq-\\nabla\\Pi/\\rho,\\quad\\nabla^2\\Pi=0,\\\\ \\Pi(z)&=\\Pi_0(1-z/L),\\quad U=\\frac{\\Pi_0}{\\rho L},\\quad m=\\rho AL,\\quad\\mathcal J=\\Pi_0A=mU,\\\\ U\\big|_{\\mathcal J\\ \\mathrm{fixed}}&=\\frac{\\mathcal J}{m},\\quad E=\\frac{\\mathcal J^2}{2m},\\qquad U\\big|_{E\\ \\mathrm{fixed}}=\\sqrt{\\frac{2E}{m}}.\\end{aligned}", "symbols_en": "$\\Pi(\\boldsymbol x),\\Pi_0$ are pressure impulse per area (Pa s), $p,p_{\\mathrm{ref}}$ total and reference pressure (Pa), $\\boldsymbol x$ position (m), $t,t_0,\\tau_p>0$ time/start/pulse duration (s). $\\Delta\\boldsymbol u$ velocity increment and $U$ column speed (m s⁻¹), $\\rho>0$ density (kg m⁻³), $z\\in[0,L]$ axial coordinate, $L>0$ length (m), $A>0$ area (m²), $m>0$ liquid mass (kg). $\\mathcal J$ is total directional force impulse (N s), $E$ kinetic energy (J). $\\nabla,\\nabla^2$ are gradient/Laplacian; $\\int$ time integration and vertical bars denote a stated fixed-budget branch. The pulse approximation freezes geometry and neglects integrated convection and viscosity.", "symbols_zh": "$\\Pi(\\boldsymbol x),\\Pi_0$ 为单位面积压力冲量（Pa s），$p,p_{\\mathrm{ref}}$ 为总／参考压力（Pa），$\\boldsymbol x$ 为位置（m），$t,t_0,\\tau_p>0$ 为时间／开始时刻／脉冲时长（s）。$\\Delta\\boldsymbol u$ 为速度增量，$U$ 为液柱速度（m s⁻¹），$\\rho>0$ 为密度（kg m⁻³），$z\\in[0,L]$ 为轴坐标，$L>0$ 为长度（m），$A>0$ 为面积（m²），$m>0$ 为液体质量（kg）。$\\mathcal J$ 为总方向力冲量（N s），$E$ 为动能（J）。$\\nabla,\\nabla^2$ 为梯度／Laplace 算子；$\\int$ 为时间积分，竖线表示指定固定预算分支。脉冲近似冻结几何并忽略积分后的对流及黏性。", "kind": "Pressure-impulse control and distinct focusing budgets", "kind_zh": "压力冲量对照模型与不同聚焦过程的收支", "diagram": { "type": "impulse", "labels": [ "Π₀ at z = 0 source", "Π = 0 at z = L free end", "A column area; L length", "Π(z), ∇Π → Δu", "U = Π₀/(ρL)", "m = ρAL; J = Π₀A", "E, fixed J vs fixed E", "p, p_ref, x, t₀, τ_p, t" ], "notes": [ "Fixed impulse per area is not fixed total impulse", "At fixed total impulse, lower mass needs more energy" ] }, "caption_en": "The pressure-impulse field and the total energy constraint must be compatible.", "caption_zh": "压力冲量场必须与总能量约束相容。" }, { "id": "C3-E17", "tex": "\\begin{aligned}Q&=\\pi a_{\\mathrm{in}}^2U_{\\mathrm{in}}=\\pi a_j^2U_j,\\qquad \\alpha=(a_j/a_{\\mathrm{in}})^2,\\quad U_{\\mathrm{in}}=\\alpha U_j,\\\\ p_{\\mathrm{in}}-p_g&=\\frac12\\rho(U_j^2-U_{\\mathrm{in}}^2)+\\frac{\\sigma_j}{a_j}+\\Delta p_{\\mathrm{loss}},\\\\ U_j&=\\sqrt{\\frac{2[p_{\\mathrm{in}}-p_g-\\sigma_j/a_j-\\Delta p_{\\mathrm{loss}}]}{\\rho(1-\\alpha^2)}}\\quad(0<\\alpha<1).\\end{aligned}", "symbols_en": "$Q$ is steady volume flow (m³ s⁻¹), $a_{\\mathrm{in}}>a_j>0$ solid-tube inlet and free cylindrical-jet radii (m); $U_{\\mathrm{in}},U_j$ are uniform speeds (m s⁻¹), $\\alpha$ dimensionless area ratio. $p_{\\mathrm{in}},p_g$ inlet and gas pressures, $\\Delta p_{\\mathrm{loss}}\\ge0$ prescribed loss (Pa); $\\rho>0$ density (kg m⁻³), $\\sigma_j$ jet–gas tension (N m⁻¹), $\\pi$ dimensionless. The square root requires nonnegative numerator. This is a steady inviscid-core nozzle control with optional loss, not a transient cavity solution.", "symbols_zh": "$Q$ 为稳态体积流量（m³ s⁻¹），$a_{\\mathrm{in}}>a_j>0$ 为固体管入口及自由圆柱射流半径（m）；$U_{\\mathrm{in}},U_j$ 为均匀速度（m s⁻¹），$\\alpha$ 为无量纲面积比。$p_{\\mathrm{in}},p_g$ 为入口及气压，$\\Delta p_{\\mathrm{loss}}\\ge0$ 为给定损失（Pa）；$\\rho>0$ 为密度（kg m⁻³），$\\sigma_j$ 为射流—气体张力（N m⁻¹），$\\pi$ 无量纲。平方根要求分子非负。它是允许损失的稳态无黏核心喷嘴对照，并非瞬态腔体解。", "kind": "Continuity plus Bernoulli focusing benchmark", "kind_zh": "连续性关系与伯努利方程的聚焦基准模型", "diagram": { "type": "jet", "labels": [ "a_in solid inlet; U_in, p_in", "a_j free outlet; U_j", "Q = inlet flux = outlet flux", "p_g outside jet", "σ_j/a_j outlet capillary pressure", "ρ, α, Δp_loss: budget legend" ], "notes": [ "A solid inlet has no imposed σ_j/a_in free-surface jump", "Required pressure rises with speed gain" ] }, "caption_en": "Specify either source pressure or flow and solve the compatible remaining quantity.", "caption_zh": "指定源压力或流量，再求其余相容量。" }, { "id": "C3-E18", "tex": "\\begin{aligned}A_j&=\\pi d_j^2/4,\\qquad m_j=\\rho A_jL_j,\\qquad E_j=\\frac12\\int_{\\mathcal M_j}|\\boldsymbol u|^2dm=\\frac12m_jU_{\\mathrm{rms}}^2,\\\\ P_j&=\\int_{\\mathcal M_j}(\\boldsymbol u\\cdot\\boldsymbol e_z)\\,dm,\\\\ |P_j|^2&\\le\\left(\\int_{\\mathcal M_j}1\\,dm\\right)\\left(\\int_{\\mathcal M_j}|\\boldsymbol u\\cdot\\boldsymbol e_z|^2dm\\right)\\le2m_jE_j,\\qquad U_{\\mathrm{rms}}\\le\\sqrt{2E_{\\mathrm{avail}}/m_j}.\\end{aligned}", "symbols_en": "$A_j$ is area (m²), $d_j,L_j>0$ diameter/emitted length (m), $\\rho$ carrier density (kg m⁻³), $m_j>0$ emitted mass (kg); the cylinder expression assumes uniform area/density. $\\mathcal M_j$ is the emitted material collection, $dm$ mass element (kg), $\\boldsymbol u$ velocity field, $U_{\\mathrm{rms}}$ mass-weighted root-mean-square speed (m s⁻¹). $\\boldsymbol e_z$ is a unit vector along intended transfer; $P_j$ its signed momentum (N s), $E_j$ kinetic energy and $E_{\\mathrm{avail}}\\ge E_j$ available mechanical energy (J). $\\int$ integrates over emitted mass; $|\\ |$ denotes vector norm or scalar magnitude; $\\pi$ is dimensionless.", "symbols_zh": "$A_j$ 为面积（m²），$d_j,L_j>0$ 为直径／喷出长度（m），$\\rho$ 为载液密度（kg m⁻³），$m_j>0$ 为喷出质量（kg）；圆柱式假设面积及密度均匀。$\\mathcal M_j$ 为喷出材料集合，$dm$ 为质量微元（kg），$\\boldsymbol u$ 为速度场，$U_{\\mathrm{rms}}$ 为质量加权均方根速度（m s⁻¹）。$\\boldsymbol e_z$ 为期望转印方向的单位向量；$P_j$ 为该方向带符号动量（N s），$E_j$ 为动能、$E_{\\mathrm{avail}}\\ge E_j$ 为可用机械能（J）。$\\int$ 对喷出质量积分；$|\\ |$ 为向量范数或标量大小；$\\pi$ 无量纲。", "kind": "Finite-mass definitions and Cauchy–Schwarz bound", "kind_zh": "有限质量定义与柯西–施瓦茨界限", "diagram": { "type": "jet", "labels": [ "d_j, A_j cross section", "L_j emitted length", "M_j emitted liquid; m_j = ∫dm", "u field; e_z transfer direction", "P_j directional momentum", "E_j, E_avail energy budget", "U_rms finite-mass speed; ρ, π" ], "notes": [ "Uniform aligned velocity is required for momentum equality", "A mathematical tip can have negligible mass" ] }, "caption_en": "Finite jet mass converts an energy budget into a meaningful speed and momentum bound.", "caption_zh": "有限射流质量将能量预算转化为有意义的速度及动量界。" }, { "id": "C3-E19", "tex": "\\begin{aligned}A(z,t)&=\\pi a(z,t)^2,\\\\ \\partial_t\\int_z^{z+\\Delta z}A(\\xi,t)\\,d\\xi&=[Av](z,t)-[Av](z+\\Delta z,t),\\\\ \\partial_t\\frac{1}{\\Delta z}\\int_z^{z+\\Delta z}A\\,d\\xi+\\frac{[Av](z+\\Delta z,t)-[Av](z,t)}{\\Delta z}&=0,\\\\ \\Delta z\\to0:\\quad \\partial_tA+\\partial_z(Av)&=0\\ \\Longrightarrow\\ \\partial_ta+v\\partial_za=-\\frac a2\\partial_zv.\\end{aligned}", "symbols_en": "$z$ is axial position (m), $t$ time (s), $\\Delta z>0$ fixed finite slice length (m), and $\\xi$ its dummy axial coordinate with element $d\\xi$ (m). $a(z,t)>0$ is local jet radius (m), $A(z,t)$ area (m²), $v(z,t)$ cross-sectional average axial velocity (m s⁻¹), and $\\pi$ dimensionless. $\\partial_t,\\partial_z$ are fixed-coordinate derivatives; $[Av]$ denotes the volume flux (m³ s⁻¹) evaluated at the stated end. $\\int$ integrates over the slice; the differential limit requires differentiable fields. The last step divides by positive $2\\pi a$.", "symbols_zh": "$z$ 为轴向位置（m），$t$ 为时间（s），$\\Delta z>0$ 为固定有限切片长度（m），$\\xi$ 为轴向积分坐标，$d\\xi$ 为其微元（m）。$a(z,t)>0$ 为局部射流半径（m），$A(z,t)$ 为面积（m²），$v(z,t)$ 为截面平均轴速（m s⁻¹），$\\pi$ 无量纲。$\\partial_t,\\partial_z$ 为固定坐标导数；$[Av]$ 表示在指定端点评价的体积流率（m³ s⁻¹）。$\\int$ 为切片积分；微分极限要求场可微。最后一步除以正的 $2\\pi a$。", "kind": "Slender-jet volume conservation", "kind_zh": "细长射流的体积守恒", "diagram": { "type": "jet", "labels": [ "z → axial flight direction", "Δz fixed finite slice; ξ inside", "a(z,t), A = πa²", "Av incoming and outgoing flux", "v average axial velocity", "∂ₜ, ∂z, t: evolution legend" ], "notes": [ "Positive axial strain thins the jet", "Initial/end data are required" ] }, "caption_en": "Axial stretching changes radius even before capillary necking grows.", "caption_zh": "在毛细颈缩增长前，轴向拉伸即会改变半径。" }, { "id": "C3-E20", "tex": "\\begin{aligned}\\partial_r u_r+u_r/r+\\partial_zv&=0\\ \\Longrightarrow\\ u_r=-\\frac r2\\partial_zv,\\\\ \\tau_{zz}&=2\\mu\\partial_zv,\\quad\\tau_{rr}=2\\mu\\partial_ru_r=-\\mu\\partial_zv,\\quad\\tau_{zz}-\\tau_{rr}=3\\mu\\partial_zv,\\\\ \\partial_tv+v\\partial_zv&=-\\frac{\\sigma_j}{\\rho}\\partial_z\\kappa+\\frac{3\\mu}{\\rho A}\\partial_z(A\\partial_zv),\\\\ \\kappa&=\\frac1{a\\sqrt{1+(\\partial_za)^2}}-\\frac{\\partial_{zz}a}{[1+(\\partial_za)^2]^{3/2}}.\\end{aligned}", "symbols_en": "$r\\in[0,a]$ and $z$ are radial/axial coordinates (m), $t$ time (s); $u_r$ is radial and $v$ axial velocity (m s⁻¹), $a>0$ radius (m), $A=\\pi a^2$ area (m²). $\\rho$ is density (kg m⁻³), $\\mu$ Newtonian viscosity (Pa s), $\\sigma_j$ jet–gas tension (N m⁻¹); $\\tau_{zz},\\tau_{rr}$ are viscous normal stresses (Pa), subscripts coordinate components. $\\kappa$ is outward liquid-to-gas curvature (m⁻¹). $\\partial_r,\\partial_z,\\partial_t$ are partial derivatives; $\\partial_{zz}$ is a second axial derivative. The axial momentum law is a slender approximation although the geometric curvature is retained in full.", "symbols_zh": "$r\\in[0,a]$、$z$ 为径向／轴向坐标（m），$t$ 为时间（s）；$u_r$ 为径向、$v$ 为轴向速度（m s⁻¹），$a>0$ 为半径（m），$A=\\pi a^2$ 为面积（m²）。$\\rho$ 为密度（kg m⁻³），$\\mu$ 为 Newton 黏度（Pa s），$\\sigma_j$ 为射流—气体张力（N m⁻¹）；$\\tau_{zz},\\tau_{rr}$ 为黏性法向应力（Pa），下标为坐标分量。$\\kappa$ 为液体朝气体外法向曲率（m⁻¹）。$\\partial_r,\\partial_z,\\partial_t$ 为偏导数；$\\partial_{zz}$ 为轴向二阶导数。虽然保留完整几何曲率，轴向动量规律仍为细长近似。", "kind": "Newtonian stress and slender axial momentum", "kind_zh": "牛顿流体应力与细长射流的轴向动量", "diagram": { "type": "jet", "labels": [ "a(z,t), A jet section", "r radial; z axial", "u_r radial; v axial", "τ_zz − τ_rr = 3μ ∂zv", "κ surface curvature", "σ_j, ρ, μ", "∂r, ∂z, ∂zz, ∂ₜ, t" ], "notes": [ "Regular radial velocity vanishes at r = 0", "Curvature gradients drive axial redistribution", "Surrounding dense liquid requires a two-fluid treatment" ] }, "caption_en": "The factor three follows from extensional stress, rather than ordinary shear drag.", "caption_zh": "因子三来自拉伸应力，而不是普通剪切阻力。" }, { "id": "C3-E21", "tex": "\\begin{aligned}a&=a_0+\\delta(t)\\cos[k(z-U_jt)],\\quad \\kappa\\simeq a_0^{-1}+(k^2-a_0^{-2})\\delta\\cos[k(z-U_jt)],\\\\ \\psi&=B I_0(kr)e^{gt}\\cos[k(z-U_jt)],\\quad\\delta=\\delta_0e^{gt},\\\\ g\\delta_0&=BkI_1(ka_0),\\quad \\rho gB I_0(ka_0)=\\sigma_j(a_0^{-2}-k^2)\\delta_0,\\\\ g^2&=\\frac{\\sigma_j}{\\rho a_0^3}\\,q(1-q^2)\\frac{I_1(q)}{I_0(q)},\\quad q=ka_0\\in(0,1).\\end{aligned}", "symbols_en": "$a$ is perturbed radius and $a_0>0$ base-cylinder radius (m); $0<\\delta_0\\le\\delta(t)\\ll a_0$ are initial/small disturbance amplitudes (m), $z,r$ axial/radial positions (m), $t$ time (s), $U_j$ base axial speed (m s⁻¹), $k>0$ wavenumber (m⁻¹). $\\kappa$ is curvature (m⁻¹); $\\psi$ perturbation potential and $B\\ne0$ its amplitude (m² s⁻¹), $g>0$ temporal growth rate (s⁻¹); $g$ does not denote gravity in this chapter. $I_0,I_1$ are dimensionless modified Bessel functions, orders zero/one; $q$ dimensionless wavenumber. $\\rho$ density (kg m⁻³), $\\sigma_j$ tension (N m⁻¹); cosine and exponential have dimensionless arguments. This inviscid infinite-cylinder linear control neglects surrounding-gas dynamics.", "symbols_zh": "$a$ 为扰动半径，$a_0>0$ 为基态圆柱半径（m）；$0<\\delta_0\\le\\delta(t)\\ll a_0$ 为初始／小扰动振幅（m），$z,r$ 为轴向／径向位置（m），$t$ 为时间（s），$U_j$ 为基态轴速（m s⁻¹），$k>0$ 为波数（m⁻¹）。$\\kappa$ 为曲率（m⁻¹）；$\\psi$ 为扰动势，$B\\ne0$ 为其振幅（m² s⁻¹），$g>0$ 为时间增长率（s⁻¹）；本章 $g$ 不表示重力。$I_0,I_1$ 为零／一阶无量纲修正 Bessel 函数；$q$ 为无量纲波数。$\\rho$ 为密度（kg m⁻³），$\\sigma_j$ 为张力（N m⁻¹）；余弦及指数自变量无量纲。该无黏无限圆柱线性对照忽略周围气体动力学。", "kind": "Inviscid-cylinder instability derivation", "kind_zh": "无黏液柱失稳的推导", "diagram": { "type": "jet", "labels": [ "a₀ base radius; a perturbed radius", "δ, δ₀ neck amplitude", "z − U_j t moving axial position", "r, k; q = k a₀", "ψ, B; I₀, I₁ radial solution", "g growth rate; exp(gt), cos", "κ, σ_j, ρ surface balance" ], "notes": [ "Unstable wavelengths exceed cylinder circumference", "This is capillary breakup of an emitted jet, not acoustic bubble forcing" ] }, "caption_en": "Kinematic and pressure conditions eliminate the potential amplitude and determine growth.", "caption_zh": "运动学及压力条件消去势振幅，从而确定增长。" }, { "id": "C3-E22", "tex": "\\begin{aligned}t_\\sigma&=\\sqrt{\\rho a_0^3/\\sigma_j},\\quad g_{\\max}=0.343339/t_\\sigma,\\quad\\lambda_{\\max}=2\\pi a_0/0.697019,\\\\ t_{\\mathrm{lin}}&=\\frac1{g_{\\max}}\\ln\\left(\\frac{\\delta_{\\mathrm{crit}}}{\\delta_0}\\right),\\qquad\\frac{\\delta_{\\mathrm{arr}}}{a_0}=\\frac{\\delta_0}{a_0}\\exp\\left(g_{\\max}\\frac H{U_j}\\right),\\quad t_{\\mathrm{flight}}=H/U_j,\\\\ \\mathrm{Re}_j&=\\frac{\\rho U_jd_j}{\\mu},\\quad\\mathrm{We}_j=\\frac{\\rho U_j^2d_j}{\\sigma_j},\\quad\\mathrm{Oh}_j=\\frac{\\mu}{\\sqrt{\\rho\\sigma_jd_j}},\\qquad d_j=2a_0.\\end{aligned}", "symbols_en": "$t_\\sigma,t_{\\mathrm{lin}},t_{\\mathrm{flight}}$ are capillary, linear-threshold and flight times (s); $\\rho$ density (kg m⁻³), $a_0,d_j,H$ base radius/diameter/gap (m), $\\sigma_j$ tension (N m⁻¹), $\\mu$ viscosity (Pa s), $U_j>0$ uniform speed (m s⁻¹). $g_{\\max}>0$ fastest inviscid growth rate (s⁻¹), $\\lambda_{\\max}$ its wavelength (m), $\\delta_0<\\delta_{\\mathrm{crit}}\\ll a_0$ initial/chosen small threshold and $\\delta_{\\mathrm{arr}}$ arrival amplitude (m). $\\ln,\\exp$ are natural logarithm/exponential, $\\pi$ dimensionless; $\\mathrm{Re}_j,\\mathrm{We}_j,\\mathrm{Oh}_j$ are dimensionless diameter-based Reynolds, Weber and Ohnesorge numbers. Constant base state and a sufficiently long cylinder are assumed for growth.", "symbols_zh": "$t_\\sigma,t_{\\mathrm{lin}},t_{\\mathrm{flight}}$ 为毛细／线性阈值／飞行时间（s）；$\\rho$ 为密度（kg m⁻³），$a_0,d_j,H$ 为基态半径／直径／间隙（m），$\\sigma_j$ 为张力（N m⁻¹），$\\mu$ 为黏度（Pa s），$U_j>0$ 为均匀速度（m s⁻¹）。$g_{\\max}>0$ 为最快无黏增长率（s⁻¹），$\\lambda_{\\max}$ 为其波长（m），$\\delta_0<\\delta_{\\mathrm{crit}}\\ll a_0$ 为初始／选定小阈值振幅，$\\delta_{\\mathrm{arr}}$ 为到达振幅（m）。$\\ln,\\exp$ 为自然对数／指数，$\\pi$ 无量纲；$\\mathrm{Re}_j,\\mathrm{We}_j,\\mathrm{Oh}_j$ 为以直径定义的无量纲 Reynolds、Weber、Ohnesorge 数。增长计算假设基态恒定且圆柱足够长。", "kind": "Finite-flight growth and regime diagnostics", "kind_zh": "有限飞行时间内的扰动增长与适用区间诊断", "diagram": { "type": "timescale", "labels": [ "H flight gap; U_j travel speed", "a₀, d_j = 2a₀", "δ₀ → δ_arr → δ_crit", "t_flight, t_lin, t_σ", "g_max growth; λ_max wavelength", "ρ, σ_j, μ; Re_j, We_j, Oh_j", "ln, exp: growth legend" ], "notes": [ "Radius defines capillary time; diameter defines the three groups", "A radius-sized disturbance is outside linear validity" ] }, "caption_en": "Compare growth during actual flight against a declared small-amplitude threshold.", "caption_zh": "比较实际飞行期间增长与明确的小振幅阈值。" }, { "id": "C3-E23", "tex": "q_j=\\frac12\\rho U_j^2,\\qquad\\dot m=\\rho A_jU_j,\\qquad F_{\\mathrm{steady}}=\\dot mU_j=\\rho A_jU_j^2=2q_jA_j.", "symbols_en": "$q_j$ is jet dynamic-pressure scale (Pa), $\\rho$ density (kg m⁻³), $U_j>0$ uniform speed (m s⁻¹), $A_j$ incident area (m²), $\\dot m$ positive mass-flow rate (kg s⁻¹; dot on $m$ here denotes throughflow, not changing the finite emitted mass), and $F_{\\mathrm{steady}}$ receiver-normal steady force (N). Complete lateral redirection, fixed receiver, negligible other axial forces and uniform inlet are assumed.", "symbols_zh": "$q_j$ 为射流动压尺度（Pa），$\\rho$ 为密度（kg m⁻³），$U_j>0$ 为均匀速度（m s⁻¹），$A_j$ 为入射面积（m²），$\\dot m$ 为正的质量流率（kg s⁻¹；此处 $m$ 上的点表示通过流率，并非有限喷出质量变化），$F_{\\mathrm{steady}}$ 为接收体法向稳态力（N）。假设完全侧向转流、接收体固定、其他轴向力可忽略且入口均匀。", "kind": "Dynamic-pressure definition and steady momentum flux", "kind_zh": "动压定义与稳态动量通量", "diagram": { "type": "impact", "labels": [ "U_j incident jet speed", "A_j incident area", "ρ liquid; ṁ = ρA_jU_j", "q_j dynamic-pressure scale", "F_steady receiver force", "Lateral outlet carries zero axial momentum" ], "notes": [ "q_j is a scale, not a universal mean or local traction", "A moving receiver changes relative speed and force" ] }, "caption_en": "Steady force follows from momentum flux, rather than assigning dynamic pressure everywhere.", "caption_zh": "稳态力来自动量流率，而非将动压指定至全部表面。" }, { "id": "C3-E24", "tex": "\\begin{aligned}\\partial_tu&=-\\rho^{-1}\\partial_zp',\\qquad\\partial_tp'=-\\rho c^2\\partial_zu,\\quad Z_l=\\rho c,\\\\ (\\partial_t+c\\partial_z)(u+p'/Z_l)&=0,\\quad(\\partial_t-c\\partial_z)(u-p'/Z_l)=0,\\\\ p_{\\mathrm{early}}&=Z_l(U_j-v_i)=Z_rv_i\\ \\Longrightarrow\\ v_i=\\frac{Z_l}{Z_l+Z_r}U_j,\\quad p_{\\mathrm{early}}=\\frac{Z_lZ_r}{Z_l+Z_r}U_j.\\end{aligned}", "symbols_en": "$u$ is liquid normal velocity, $U_j$ its initial incident speed and $v_i$ common contact velocity (m s⁻¹); $p'$ is linear compressive pressure disturbance and $p_{\\mathrm{early}}$ contact value (Pa). $\\rho$ is liquid density (kg m⁻³), $c$ liquid longitudinal sound speed (m s⁻¹), $Z_l=\\rho c>0$ liquid and $Z_r>0$ receiver longitudinal impedance (Pa s m⁻¹). $z$ is normal coordinate (m), $t$ time (s), and $\\partial_t,\\partial_z$ are partial derivatives. The two differential operators follow right/left characteristics. Both media are locally semi-infinite and initially unpressurized relative to the same contact reference; the receiver is initially at rest.", "symbols_zh": "$u$ 为液体法向速度，$U_j$ 为初始入射速度，$v_i$ 为共同接触速度（m s⁻¹）；$p'$ 为线性压缩压力扰动，$p_{\\mathrm{early}}$ 为接触值（Pa）。$\\rho$ 为液体密度（kg m⁻³），$c$ 为液体纵向声速（m s⁻¹），$Z_l=\\rho c>0$ 为液体阻抗，$Z_r>0$ 为接收体纵向阻抗（Pa s m⁻¹）。$z$ 为法向坐标（m），$t$ 为时间（s），$\\partial_t,\\partial_z$ 为偏导数。两个微分算子沿右／左行特征。两介质局部半无限，相对同一接触参考值初始无压力扰动；接收体初始静止。", "kind": "Linear transient wave equations and impact matching", "kind_zh": "线性瞬态波动方程与冲击匹配条件", "diagram": { "type": "impact", "labels": [ "U_j incoming liquid", "v_i common interface velocity", "p_early compressive contact pressure", "Z_l = ρc liquid side", "Z_r receiver side", "u, p′, z, t characteristic fields", "∂ₜ ± c∂z; c wave speed" ], "notes": [ "Liquid deceleration wave travels upstream", "Receiver wave travels into receiver", "No pre-existing incident pressure wave is separately added" ] }, "caption_en": "Pressure continuity and equal normal velocity determine the early contact state.", "caption_zh": "压力连续及相同法向速度确定早期接触状态。" }, { "id": "C3-E25", "tex": "\\begin{aligned}\\mathcal J_{\\mathrm{rec}}&=P_{\\mathrm{in}}-P_{\\mathrm{out}},\\quad P_{\\mathrm{in}}=m_jU_j=\\rho A_jL_jU_j,\\\\ (\\rho cU_j)A_j\\tau_{\\mathrm{eq}}&=m_jU_j\\ \\Longrightarrow\\ \\tau_{\\mathrm{eq}}=L_j/c,\\\\ t_{\\mathrm{side}}&=a_j/c,\\quad t_{\\mathrm{axial}}=L_j/c,\\quad t_{\\mathrm{early}}\\ll\\min(t_{\\mathrm{side}},t_{\\mathrm{axial}},t_{\\mathrm{return}}).\\end{aligned}", "symbols_en": "$\\mathcal J_{\\mathrm{rec}}$ is receiver normal contact impulse, $P_{\\mathrm{in}},P_{\\mathrm{out}}$ incoming/outgoing axial liquid momentum (N s). $m_j$ emitted mass (kg), $\\rho$ liquid density (kg m⁻³), $A_j$ area (m²), $a_j,L_j$ radius/length (m), $U_j$ incident speed and $c$ sound speed (m s⁻¹). $\\tau_{\\mathrm{eq}}$ is an impulse-equivalent duration (s) for a fictitious uniform rigid-contact rectangle with zero outgoing axial momentum. $t_{\\mathrm{side}},t_{\\mathrm{axial}},t_{\\mathrm{early}},t_{\\mathrm{return}}$ are side-release, axial-release, early observation and first receiver-return times (s); $\\min$ selects the earliest. The liquid momentum balance excludes additional direct source forces or a separate collapse shock; receiver support reactions affect receiver motion but are not counted as a second direct liquid impulse.", "symbols_zh": "$\\mathcal J_{\\mathrm{rec}}$ 为接收体法向接触冲量，$P_{\\mathrm{in}},P_{\\mathrm{out}}$ 为入／出射液体轴向动量（N s）。$m_j$ 为喷出质量（kg），$\\rho$ 为密度（kg m⁻³），$A_j$ 为面积（m²），$a_j,L_j$ 为半径／长度（m），$U_j$ 为入射速度，$c$ 为声速（m s⁻¹）。$\\tau_{\\mathrm{eq}}$ 为出射轴向动量为零时，假想均匀刚性接触矩形载荷的冲量等效时长（s）。$t_{\\mathrm{side}},t_{\\mathrm{axial}},t_{\\mathrm{early}},t_{\\mathrm{return}}$ 为侧向释放／轴向释放／早期观察／首个接收体回波时间（s）；$\\min$ 取最早值。液体动量平衡排除额外直接源力或独立塌缩激波；接收体支撑反力影响其运动，但不另算为第二个直接液体冲量。", "kind": "Finite momentum and early-impact duration conditions", "kind_zh": "有限动量与冲击早期持续时间的条件", "diagram": { "type": "impact", "labels": [ "m_j, U_j → P_in", "P_out after redirection or rebound", "J_rec delivered normal impulse", "A_j, a_j, L_j finite plug", "ρcU_j fictitious rectangular pressure", "τ_eq = L_j/c", "t_side, t_axial, t_early, t_return" ], "notes": [ "Equivalent duration is not a predicted pulse shape", "A free side can release the central pressure earlier" ] }, "caption_en": "A very large initial pressure has a finite momentum budget and restricted duration.", "caption_zh": "很大的初始压力仍受有限动量预算及持续时间限制。" }, { "id": "C3-E26", "tex": "\\begin{aligned}p_{\\mathrm{obs}}(t)&=\\frac1{A_o\\tau_o}\\int_{t-\\tau_o}^t\\int_{A_o}[p(\\boldsymbol x,\\xi)-p_{\\mathrm{ref}}]\\,dA\\,d\\xi,\\\\ 0\\le p_{\\mathrm{obs}}(t)&\\le\\frac{\\mathcal J_{\\mathrm{rec}}}{A_o\\tau_o},\\\\ &\\text{for nonnegative excess pressure, negligible normal viscous stress,}\\\\ &\\text{and full-event footprint impulse }\\mathcal J_{\\mathrm{rec}}.\\end{aligned}", "symbols_en": "$p_{\\mathrm{obs}}$ is signed area–time mean excess pressure (Pa), $p$ local pressure and $p_{\\mathrm{ref}}$ fixed reference (Pa); $\\boldsymbol x$ position (m), $t$ current time and $\\xi$ dummy time (s). $A_o>0$ is the fixed observer area (m²), $\\tau_o>0$ averaging duration (s), $dA$ area element (m²), $\\mathcal J_{\\mathrm{rec}}\\ge0$ total compressive normal impulse over the observed footprint (N s). Integrals sum the area and preceding time window. The bound assumes negligible normal viscous stress, so excess pressure equals compressive normal traction; this excess must be nonnegative throughout the full-event footprint budget, with no omitted forces. An arbitrary signed waveform, finite normal viscous stress or a different interface does not satisfy these premises.", "symbols_zh": "$p_{\\mathrm{obs}}$ 为带符号面积—时间平均超压（Pa），$p$ 为局部压力、$p_{\\mathrm{ref}}$ 为固定参考值（Pa）；$\\boldsymbol x$ 为位置（m），$t$ 为当前时间、$\\xi$ 为积分时间（s）。$A_o>0$ 为固定观察面积（m²），$\\tau_o>0$ 为平均时长（s），$dA$ 为面积微元（m²），$\\mathcal J_{\\mathrm{rec}}\\ge0$ 为观察载荷区域上的总法向压缩冲量（N s）。积分对面积及前一时间窗口求和。界限假设法向黏性应力可忽略，因而超压等于法向压缩牵引；在采用完整事件冲量预算的整个区域内，该超压须始终非负，且不存在漏项力。任意带符号波形、有限法向黏性应力或另一界面不满足这些前提。", "kind": "Observer definition and conditional impulse bound", "kind_zh": "观测量定义与有条件的冲量界限", "diagram": { "type": "impact", "labels": [ "A_o fixed sensor footprint", "τ_o averaging window", "p(x,ξ) local surface pressure", "p_ref reference pressure", "p_obs(t) observed mean", "J_rec total footprint impulse", "dA area element; x, ξ, t" ], "notes": [ "Fluid, exposed surface and release interface are different locations", "Bandwidth and sign convention belong to the observation" ] }, "caption_en": "A reported maximum is meaningful only for an unchanged observer.", "caption_zh": "仅在观察定义不变时，报告的最大值才具有可比较含义。" }, { "id": "C3-E27", "tex": "\\begin{aligned}E_w&=\\int_0^{\\tau_o}\\int_{A_o}\\frac{p'^2}{Z_l}\\,dA\\,dt,\\qquad \\overline p=\\frac1{A_o\\tau_o}\\int_0^{\\tau_o}\\int_{A_o}p'\\,dA\\,dt,\\\\ |\\overline p|^2&\\le\\frac1{A_o\\tau_o}\\int_0^{\\tau_o}\\int_{A_o}p'^2\\,dA\\,dt=\\frac{Z_lE_w}{A_o\\tau_o}.\\end{aligned}", "symbols_en": "$E_w$ is outgoing linear plane-wave energy crossing the stated footprint/window (J), $p'$ its pressure disturbance and $\\overline p$ signed area–time mean (Pa), $Z_l>0$ constant wave impedance (Pa s m⁻¹), $A_o>0$ area (m²), $\\tau_o>0$ duration and $t$ time (s), $dA$ area element (m²). $\\int$ is area/time integration and $|\\ |$ magnitude. The premise is one-way linear pressure–velocity relation $u'=p'/Z_l$, so intensity is $p'^2/Z_l$. This is not an energy bound on arbitrary standing-wave or solid contact traction.", "symbols_zh": "$E_w$ 为通过规定面积／时间窗口的单向线性平面波能量（J），$p'$ 为其压力扰动、$\\overline p$ 为带符号面积—时间均值（Pa），$Z_l>0$ 为恒定波阻抗（Pa s m⁻¹），$A_o>0$ 为面积（m²），$\\tau_o>0$ 为时长、$t$ 为时间（s），$dA$ 为面积微元（m²）。$\\int$ 为面积／时间积分，$|\\ |$ 为大小。前提为单向线性压力—速度关系 $u'=p'/Z_l$，故强度为 $p'^2/Z_l$。它不是任意驻波或固体接触牵引的能量界。", "kind": "One-way-wave energy and finite-observer Cauchy–Schwarz bound", "kind_zh": "单向行波能量与有限观测条件下的柯西–施瓦茨界限", "diagram": { "type": "energy", "labels": [ "p′ outgoing wave through A_o", "Z_l wave impedance", "τ_o, t observation window", "E_w = ∫p′²/Z_l", "p̄ signed observed mean", "dA spatial averaging" ], "notes": [ "Wave energy must be allocated from the same source budget", "This is a conditional wave bound, not universal film traction" ] }, "caption_en": "Finite area and time prevent an unmeasured point spike from standing in for delivered loading.", "caption_zh": "有限面积及时间避免用未测量点尖峰代替实际传递载荷。" }, { "id": "C3-E28", "tex": "\\begin{aligned}\\partial_t\\rho+\\nabla\\cdot(\\rho\\boldsymbol u)&=0,\\\\ \\partial_t(\\rho\\boldsymbol u)+\\nabla\\cdot(\\rho\\boldsymbol u\\otimes\\boldsymbol u+p\\boldsymbol I-\\boldsymbol\\tau)&=0,\\\\ \\partial_t(\\rho e_t)+\\nabla\\cdot[(\\rho e_t+p)\\boldsymbol u-\\boldsymbol\\tau\\cdot\\boldsymbol u+\\boldsymbol q]&=Q_{\\mathrm{abs}},\\quad e_t=e+|\\boldsymbol u|^2/2,\\quad p=\\mathcal P(\\rho,e,\\{Y_k\\}).\\end{aligned}", "symbols_en": "$\\rho(\\boldsymbol x,t)$ is compressible density (kg m⁻³), $\\boldsymbol u$ velocity (m s⁻¹), $p$ absolute thermodynamic pressure (Pa), $e$ specific internal energy and $e_t$ total specific energy (J kg⁻¹). $\\boldsymbol I$ is identity tensor; $\\boldsymbol\\tau$ viscous stress (Pa), $\\boldsymbol q$ conductive heat flux (W m⁻²), $Q_{\\mathrm{abs}}$ optical volumetric heating (W m⁻³; zero after heating if no continued absorption). $\\mathcal P$ is the adopted equation-of-state function; $Y_k$ species mass fractions, $k$ species index (dimensionless). $\\boldsymbol x$ position (m), $t$ time (s), $\\partial_t$ fixed-position derivative; $\\nabla\\cdot$ tensor/vector divergence, $\\otimes$ tensor product and $\\cdot$ contraction; $|\\ |$ velocity norm.", "symbols_zh": "$\\rho(\\boldsymbol x,t)$ 为可压缩密度（kg m⁻³），$\\boldsymbol u$ 为速度（m s⁻¹），$p$ 为绝对热力学压力（Pa），$e$ 为比内能、$e_t$ 为总比能（J kg⁻¹）。$\\boldsymbol I$ 为单位张量；$\\boldsymbol\\tau$ 为黏性应力（Pa），$\\boldsymbol q$ 为导热通量（W m⁻²），$Q_{\\mathrm{abs}}$ 为光学体积加热（W m⁻³；加热后若无继续吸收则为零）。$\\mathcal P$ 为采用的状态方程函数；$Y_k$ 为组分质量分数，$k$ 为组分编号（无量纲）。$\\boldsymbol x$ 为位置（m），$t$ 为时间（s），$\\partial_t$ 为固定位置导数；$\\nabla\\cdot$ 为张量／向量散度，$\\otimes$ 为张量积、$\\cdot$ 为缩并；$|\\ |$ 为速度范数。", "kind": "Compressible bulk conservation and required constitutive closure", "kind_zh": "可压缩体相守恒律与所需本构闭合关系", "diagram": { "type": "impact", "labels": [ "ρ, u, p compressible fields", "e internal; e_t total energy", "τ viscous stress; q heat flux", "Q_abs optical source counted once", "P(ρ,e,{Y_k}) equation of state", "I identity; ⊗, ·, ∇· operators", "x, t coordinates; k species index" ], "notes": [ "Initial jet state is inherited, not recreated with extra pressure", "Species and interfacial jump conditions must also be supplied" ] }, "caption_en": "Strong compression requires conservative evolution with an appropriate equation of state.", "caption_zh": "强压缩需要采用适当状态方程的守恒演化。" }, { "id": "C3-E29", "tex": "\\begin{aligned}E_{\\mathrm{abs,tot}}&=f_{\\mathrm{geo}}A_\\lambda E_L=(0.8)(0.5)(25\\,\\mu\\mathrm J)=10\\,\\mu\\mathrm J,\\\\ E_{\\mathrm{abs,site}}&=E_{\\mathrm{abs,tot}}/N_d=400\\,\\mathrm{nJ},\\quad E_{\\mathrm{PFC,deadline}}=f_T E_{\\mathrm{abs,site}}=160\\,\\mathrm{nJ},\\\\ E_j&=\\eta_j E_{\\mathrm{abs,site}}=2.00\\,\\mathrm{nJ},\\quad E_{j,\\mathrm{tot}}=N_dE_j=50.0\\,\\mathrm{nJ}.\\end{aligned}", "symbols_en": "$E_L$ is incident laser energy, $E_{\\mathrm{abs,tot}}$ total absorbed energy, $E_{\\mathrm{abs,site}}$ per-site absorbed energy, $E_{\\mathrm{PFC,deadline}}$ energy delivered to that PFC inventory by the activation deadline, $E_j$ final per-jet kinetic energy and $E_{j,\\mathrm{tot}}$ total jet energy (J; μJ = 10⁻⁶ J, nJ = 10⁻⁹ J). $N_d=25$ is site count; $f_{\\mathrm{geo}}=0.8$ intercepted fraction, $A_\\lambda=0.5$ effective absorptance at laser wavelength $\\lambda$ (m), $f_T=0.40$ stipulated thermal fraction and $\\eta_j=0.005$ stipulated absorbed-to-jet kinetic efficiency (dimensionless). Subscripts label reservoirs/stages.", "symbols_zh": "$E_L$ 为入射激光能，$E_{\\mathrm{abs,tot}}$ 为总吸收能，$E_{\\mathrm{abs,site}}$ 为单个位点吸收能，$E_{\\mathrm{PFC,deadline}}$ 为激活截止前到达该 PFC 储量的能量，$E_j$ 为最终单射流动能、$E_{j,\\mathrm{tot}}$ 为总射流动能（J；μJ = 10⁻⁶ J，nJ = 10⁻⁹ J）。$N_d=25$ 为位点数；$f_{\\mathrm{geo}}=0.8$ 为截获比例，$A_\\lambda=0.5$ 为激光波长 $\\lambda$（m）处有效吸收率，$f_T=0.40$ 为假设热比例，$\\eta_j=0.005$ 为假设吸收能至射流动能效率（无量纲）。下标区分能量库／阶段。", "kind": "Declared optical and mechanical allocation", "kind_zh": "明确给定的光能与机械能分配", "diagram": { "type": "laser", "labels": [ "E_L = 25 μJ incoming", "f_geo = 0.8 intercepted", "A_λ = 0.5 absorbed; λ wavelength", "N_d = 25 sites", "E_abs,tot = 10 μJ", "E_abs,site = 400 nJ", "f_T = 0.40 → E_PFC,deadline", "η_j = 0.005 → E_j, E_j,tot" ], "notes": [ "Thermal preparation is checked on its own timescale", "η_j is an assumption to be measured or solved" ] }, "caption_en": "Energy allocations follow the project chain without double counting.", "caption_zh": "沿项目因果链分配能量，并避免重复计数。" }, { "id": "C3-E30", "tex": "\\begin{aligned}m_j&=(1000\\,\\mathrm{kg\\,m^{-3}})\\frac{\\pi(10\\times10^{-6}\\,\\mathrm m)^2}{4}(50\\times10^{-6}\\,\\mathrm m)\\simeq3.926991\\times10^{-12}\\,\\mathrm{kg},\\\\ U_j&=\\sqrt{\\frac{2(2.00\\times10^{-9}\\,\\mathrm J)}{3.926991\\times10^{-12}\\,\\mathrm{kg}}}\\simeq31.91538\\,\\mathrm{m\\,s^{-1}},\\\\ q_j&\\simeq0.5092958\\,\\mathrm{MPa},\\quad p_{\\mathrm{rigid,early}}=\\rho cU_j\\simeq47.23477\\,\\mathrm{MPa},\\\\ P_j&=m_jU_j\\simeq1.253314\\times10^{-10}\\,\\mathrm{N\\,s},\\quad P_{\\mathrm{tot}}=25P_j\\simeq3.133285\\times10^{-9}\\,\\mathrm{N\\,s}.\\end{aligned}", "symbols_en": "$m_j$ is per-jet carrier mass (kg), $\\rho=1000$ kg m⁻³ carrier density; $d_j=10\\times10^{-6}$ m diameter and $L_j=50\\times10^{-6}$ m emitted length supply the first row. $E_j=2.00\\times10^{-9}$ J is per-jet kinetic energy; $U_j$ uniform speed (m s⁻¹), $q_j=\\rho U_j^2/2$ dynamic pressure and $p_{\\mathrm{rigid,early}}$ early rigid-contact pressure (Pa; MPa = 10⁶ Pa). $c=1480$ m s⁻¹ is stipulated sound speed; $P_j,P_{\\mathrm{tot}}$ per-jet and total aligned incoming momentum (N s); $\\pi$ dimensionless. Uniform direction, full activation and independent cells are assumed.", "symbols_zh": "$m_j$ 为单射流载液质量（kg），$\\rho=1000$ kg m⁻³ 为载液密度；$d_j=10\\times10^{-6}$ m 直径及 $L_j=50\\times10^{-6}$ m 喷出长度用于第一行。$E_j=2.00\\times10^{-9}$ J 为单射流动能；$U_j$ 为均匀速度（m s⁻¹），$q_j=\\rho U_j^2/2$ 为动压，$p_{\\mathrm{rigid,early}}$ 为早期刚性接触压力（Pa；MPa = 10⁶ Pa）。$c=1480$ m s⁻¹ 为给定声速；$P_j,P_{\\mathrm{tot}}$ 为单射流及总对齐入射动量（N s）；$\\pi$ 无量纲。假设方向相同、全部激活且单元独立。", "kind": "Worked finite jet mass, speed and pressure scales", "kind_zh": "有限射流质量、速度与压力尺度的示例计算", "diagram": { "type": "impact", "labels": [ "d_j = 10 μm; L_j = 50 μm", "m_j = 3.926991 ng", "E_j = 2 nJ; U_j = 31.91538 m/s", "q_j = 0.509296 MPa", "p_rigid,early = 47.23477 MPa", "P_j, P_tot aligned momentum", "ρ = 1000 kg/m³; c = 1480 m/s; π" ], "notes": [ "Carrier mass, not PFC vapor mass, is emitted", "High early pressure does not create additional energy" ] }, "caption_en": "The finite state carried to Chapter 4 includes mass and energy, not only a pressure peak.", "caption_zh": "传入第四章的有限状态包含质量及能量，而非只有压力峰值。" }, { "id": "C3-E31", "tex": "\\begin{aligned}R_i\\ddot R_i+\\frac32\\dot R_i^2&=\\frac{p_{b,i}-p_\\infty-2\\sigma_b/R_i-4\\mu\\dot R_i/R_i}{\\rho}-\\sum_{j\\ne i}\\frac{R_j^2\\ddot R_j+2R_j\\dot R_j^2}{d_{ij}},\\\\ E_{B,\\mathrm{tot}}&=N\\frac{4\\pi}{3}\\Delta p_cR_{\\max,N}^3,\\qquad R_{\\max,N}=R_*N^{-1/3}.\\end{aligned}", "symbols_en": "Original formulas: $i,j$ index bubbles; $R_i,R_j,d_{ij}$ are radius/separation (m), dots time derivatives; $p_{b,i},p_\\infty,\\Delta p_c>0$ are bubble pressure, ambient pressure and constant ideal collapse difference (Pa). $\\sigma_b$ tension (N m⁻¹), $\\mu$ viscosity (Pa s), $\\rho$ carrier density (kg m⁻³), $\\sum$ sums neighbors. $N$ is count, $E_{B,\\mathrm{tot}}$ fixed total initial pressure work (J), $R_{\\max,N}$ per-bubble maximum radius and $R_*$ isolated reference radius (m); $\\pi$ dimensionless, $*$ reference label. The coupled row requires far separation; the fixed-work row assumes identical constant-pressure ideal cavities.", "symbols_zh": "原始公式：$i,j$ 为气泡编号；$R_i,R_j,d_{ij}$ 为半径／间距（m），点为时间导数；$p_{b,i},p_\\infty,\\Delta p_c>0$ 为泡内压力、环境压力、恒定理想塌缩压差（Pa）。$\\sigma_b$ 为张力（N m⁻¹），$\\mu$ 为黏度（Pa s），$\\rho$ 为载液密度（kg m⁻³），$\\sum$ 对邻泡求和。$N$ 为数量，$E_{B,\\mathrm{tot}}$ 为固定初始总压力做功（J），$R_{\\max,N}$ 为单泡最大半径、$R_*$ 为孤立参考半径（m）；$\\pi$ 无量纲，$*$ 为参考标签。耦合行要求充分分离；固定做功行假设相同恒定压力理想腔体。", "kind": "Reference answer: original interaction and allocation formulas", "kind_zh": "参考答案：原始相互作用与分配公式", "diagram": { "type": "array", "labels": [ "R_i, R_j bubble radii", "d_ij separations; N sites", "p_b,i inside; p∞ outside", "Ṙ_i, R̈_i, Ṙ_j, R̈_j", "σ_b, μ, ρ balance parameters", "E_B,tot fixed work; Δp_c", "R_max,N = R_* N^−1/3; π" ], "notes": [ "Hydraulic connectivity decides whether sources are independent", "Fixed total work changes each cavity scale" ] }, "caption_en": "Use connectivity and source allocation before any sum of outputs.", "caption_zh": "求和输出前先确定液体连通性及源分配。" }, { "id": "C3-E32", "tex": "U_{\\mathrm{rms}}\\le\\sqrt{\\frac{2E_{\\mathrm{avail}}}{m_j}},\\qquad P_j^2\\le2m_jE_j,\\qquad \\frac{\\delta_{\\mathrm{arr}}}{a_0}=\\frac{\\delta_0}{a_0}\\exp(g_{\\max}H/U_j).", "symbols_en": "Original formulas: $U_{\\mathrm{rms}}$ is mass-weighted speed, $U_j$ uniform benchmark speed (m s⁻¹), $m_j>0$ emitted carrier mass (kg), $E_{\\mathrm{avail}},E_j$ available and kinetic energy (J), $P_j$ directional momentum (N s). $a_0,H,\\delta_0,\\delta_{\\mathrm{arr}}$ are base radius, gap, initial and arriving disturbance amplitude (m); $g_{\\max}$ fastest cylinder growth rate (s⁻¹), $\\exp$ exponential of dimensionless argument. Growth assumes constant inviscid base cylinder and a small perturbation in negligible surrounding gas.", "symbols_zh": "原始公式：$U_{\\mathrm{rms}}$ 为质量加权速度，$U_j$ 为均匀对照速度（m s⁻¹），$m_j>0$ 为喷出载液质量（kg），$E_{\\mathrm{avail}},E_j$ 为可用能及动能（J），$P_j$ 为方向动量（N s）。$a_0,H,\\delta_0,\\delta_{\\mathrm{arr}}$ 为基态半径、间隙、初始及到达扰动振幅（m）；$g_{\\max}$ 为最快圆柱增长率（s⁻¹），$\\exp$ 为无量纲自变量的指数。增长假设恒定无黏基态圆柱、微小扰动及周围气体动力学可忽略。", "kind": "Reference answer: original finite-mass and growth formulas", "kind_zh": "参考答案：原始有限质量与扰动增长公式", "diagram": { "type": "jet", "labels": [ "m_j finite liquid mass", "E_avail, E_j energy budget", "U_rms, U_j velocities", "P_j useful axial momentum", "a₀, δ₀ → δ_arr", "H flight gap; g_max growth", "exp(g_max H/U_j) flight amplification" ], "notes": [ "Fastest tip speed differs from a finite-mass velocity", "Coherent arrival depends on initial disturbance and finite geometry" ] }, "caption_en": "Speed, focusing and survival must refer to the same emitted liquid state.", "caption_zh": "速度、聚焦及存活须指向同一喷出液体状态。" }, { "id": "C3-E33", "tex": "p_{\\mathrm{early}}\\simeq\\frac{Z_lZ_r}{Z_l+Z_r}U_j,\\quad Z_l=\\rho c,\\qquad \\tau_{\\mathrm{eq}}=\\frac{L_j}{c},\\qquad p_{\\mathrm{obs}}(t)=\\frac1{A_o\\tau_o}\\int_{t-\\tau_o}^t\\int_{A_o}(p-p_{\\mathrm{ref}})\\,dA\\,d\\xi.", "symbols_en": "Original formulas: $p_{\\mathrm{early}}$ is early compressive impact scale and $p_{\\mathrm{obs}}$ signed finite-observer mean (Pa), $Z_l,Z_r>0$ liquid/receiver impedances (Pa s m⁻¹), $\\rho$ carrier density (kg m⁻³), $c$ sound speed and $U_j$ incident speed (m s⁻¹), $L_j$ finite jet length (m), $\\tau_{\\mathrm{eq}}$ rigid stopping-impulse equivalent duration (s). $A_o$ observation area and $dA$ element (m²), $\\tau_o$ averaging duration, $t$ current time and $\\xi$ dummy time (s), $p,p_{\\mathrm{ref}}$ local/reference pressure (Pa). $\\int$ integrates local pressure $p(\\boldsymbol x,\\xi)$ over the footprint/window, with position $\\boldsymbol x$ (m) implicit. Impact is locally one-dimensional and linear; equivalent duration assumes zero outgoing axial momentum and no added force source.", "symbols_zh": "原始公式：$p_{\\mathrm{early}}$ 为早期压缩冲击尺度，$p_{\\mathrm{obs}}$ 为带符号有限观察均值（Pa），$Z_l,Z_r>0$ 为液体／接收体阻抗（Pa s m⁻¹），$\\rho$ 为载液密度（kg m⁻³），$c$ 为声速、$U_j$ 为入射速度（m s⁻¹），$L_j$ 为有限射流长度（m），$\\tau_{\\mathrm{eq}}$ 为刚性停止冲量等效时长（s）。$A_o$ 为观察面积、$dA$ 为面积微元（m²），$\\tau_o$ 为平均时长，$t$ 为当前时间、$\\xi$ 为积分时间（s），$p,p_{\\mathrm{ref}}$ 为局部／参考压力（Pa）。$\\int$ 将局部压力 $p(\\boldsymbol x,\\xi)$ 在载荷区域／窗口积分，位置 $\\boldsymbol x$（m）为隐含自变量。冲击局部一维且线性；等效时长假设出射轴向动量为零且无额外力源。", "kind": "Reference answer: original impact and observer formulas", "kind_zh": "参考答案：原始冲击与观测量公式", "diagram": { "type": "impact", "labels": [ "U_j, L_j finite incoming jet", "ρ, c; Z_l liquid impedance", "Z_r receiver impedance", "p_early early contact scale", "τ_eq = L_j/c", "A_o, τ_o finite observer", "p(x,ξ), p_ref, p_obs(t), dA" ], "notes": [ "Location, footprint and time window must stay fixed", "Early compressive pressure is not buried release-interface traction" ] }, "caption_en": "A finite impact scale is an input to load-path analysis, not proof of intact release.", "caption_zh": "有限冲击尺度是传力路径分析的输入，并非完整释放的证明。" }, { "id": "C4-E01", "tex": "\\boldsymbol D_u=\\frac{\\nabla\\boldsymbol u+(\\nabla\\boldsymbol u)^{\\mathsf T}}{2},\\qquad \\boldsymbol T_l=-p\\boldsymbol I+2\\mu\\boldsymbol D_u,\\qquad \\boldsymbol t_l=\\boldsymbol T_l\\boldsymbol n_f", "symbols_en": "$\\boldsymbol u(\\boldsymbol x,t)$ is liquid velocity (m s⁻¹), $\\boldsymbol x$ position (m), and $t$ time (s). $\\nabla$ is the spatial gradient (m⁻¹); superscript $\\mathsf T$ transposes a tensor. $\\boldsymbol D_u$ is strain-rate tensor (s⁻¹), $\\boldsymbol T_l$ liquid Cauchy stress (Pa), $p$ liquid pressure (Pa), $\\mu\\ge0$ dynamic viscosity (Pa s), and $\\boldsymbol I$ the dimensionless identity tensor. $\\boldsymbol n_f$ is the unit solid-to-liquid normal; $\\boldsymbol t_l$ is traction on the solid (Pa). Subscripts u, l and f label velocity strain, liquid and film surface.", "symbols_zh": "$\\boldsymbol u(\\boldsymbol x,t)$为液体速度（m s⁻¹），$\\boldsymbol x$为位置（m），$t$为时间（s）。$\\nabla$为空间梯度（m⁻¹）；上标$\\mathsf T$表示张量转置。$\\boldsymbol D_u$为应变率张量（s⁻¹），$\\boldsymbol T_l$为液体柯西应力（Pa），$p$为液体压力（Pa），$\\mu\\ge0$为动力黏度（Pa s），$\\boldsymbol I$为无量纲单位张量。$\\boldsymbol n_f$为固体指向液体的单位法向；$\\boldsymbol t_l$为固体上的牵引（Pa）。下标u、l、f分别标识速度应变、液体及薄膜表面。", "kind": "Constitutive assumption and traction identity", "kind_zh": "本构假设与牵引恒等式", "diagram": { "type": "film", "labels": [ "film surface", "n_f ↓ solid → liquid", "p → compressive load", "u, x, t in liquid", "μ, D_u — shear", "T_l, t_l at exposed face", "I — identity tensor" ], "notes": [ "Positive opening direction ↑", "The release interface is a separate surface." ] }, "caption_en": "Pressure and shear act at the exposed film face; the drawn normal fixes the traction sign.", "caption_zh": "压力与剪切作用于薄膜外露面；所画法向决定牵引符号。" }, { "id": "C4-E02", "tex": "\\begin{aligned}\\boldsymbol F_l(t)&=\\int_{A_f(t)}\\boldsymbol t_l\\,dA,\\\\ \\boldsymbol I_l&=\\int_{t_a}^{t_b}\\boldsymbol F_l(t)\\,dt,\\\\ \\mathcal W_l&=\\int_{t_a}^{t_b}\\int_{A_f(t)}\\boldsymbol t_l\\cdot\\boldsymbol v_f\\,dA\\,dt.\\end{aligned}", "symbols_en": "$A_f(t)$ is the loaded solid surface (m²), $dA$ its surface-area element (m²), and $\\int$ denotes integration over the indicated surface or time. $\\boldsymbol t_l$ is applied liquid traction (Pa), $\\boldsymbol F_l$ its resultant (N), $\\boldsymbol I_l$ its impulse (N s), and $\\mathcal W_l$ work delivered to the solid (J). $\\boldsymbol v_f$ is local solid surface velocity (m s⁻¹). $t_a0$ is film areal mass (kg m⁻²), $D_f>0$ bending stiffness (N m), $\\rho_f>0$ constant density (kg m⁻³), $h_f>0$ film thickness (m), $E_f>0$ Young’s modulus (Pa), and $-1<\\nu_f<1/2$ Poisson’s ratio (dimensionless). $z$ is thickness coordinate (m), zero at the neutral midplane; $dz$ is its integration element (m). $\\int$ is definite thickness integration. Subscripts A and f label areal quantity and film.", "symbols_zh": "$m_A>0$为薄膜面密度（kg m⁻²），$D_f>0$为弯曲刚度（N m），$\\rho_f>0$为恒定密度（kg m⁻³），$h_f>0$为薄膜厚度（m），$E_f>0$为杨氏模量（Pa），$-1<\\nu_f<1/2$为泊松比（无量纲）。$z$为厚度坐标（m），中性面处为零；$dz$为其积分微元（m）。$\\int$表示沿厚度定积分。下标A与f标识面量与薄膜。", "kind": "Derived plate parameters under constitutive assumptions", "kind_zh": "在本构假设下推导的板参数", "diagram": { "type": "film", "labels": [ "h_f — thickness", "z = 0 — neutral midplane", "ρ_f → m_A", "E_f, ν_f → D_f", "z² dz — bending moment weight" ], "notes": [ "Density integrates once; bending weights distance squared." ] }, "caption_en": "Areal inertia and bending stiffness come from different thickness integrals.", "caption_zh": "面惯性与弯曲刚度来自不同的厚度积分。" }, { "id": "C4-E04", "tex": "\\begin{aligned}K_f&=\\frac{m_A}{2}\\int_{\\Omega_f}\\dot w^2\\,dA,\\\\U_b&=\\frac{D_f}{2}\\int_{\\Omega_f}\\left[w_{xx}^2+w_{yy}^2+2\\nu_fw_{xx}w_{yy}+2(1-\\nu_f)w_{xy}^2\\right]dA,\\\\U_T&=\\frac{T_0}{2}\\int_{\\Omega_f}|\\nabla_\\parallel w|^2\\,dA.\\end{aligned}", "symbols_en": "$\\Omega_f$ is the reference film plane (m²), $dA$ its area element (m²), $w(x,y,t)$ vertical displacement (m), $x,y$ in-plane coordinates (m), and $t$ time (s). A dot differentiates in time; subscripts $xx,yy,xy$ differentiate twice in the indicated coordinates, giving curvatures (m⁻¹). $\\nabla_\\parallel$ is the in-plane gradient; $|\\ |$ denotes Euclidean magnitude. $m_A$ is areal mass (kg m⁻²), $D_f$ bending stiffness (N m), $\\nu_f$ Poisson’s ratio, and $T_0\\ge0$ prescribed isotropic tensile force per edge length (N m⁻¹). $K_f,U_b,U_T$ are kinetic, bending and pretension energies (J); $\\int$ denotes area integration. Labels f, b, T identify film, bending and tension.", "symbols_zh": "$\\Omega_f$为薄膜参考平面（m²），$dA$为面积微元（m²），$w(x,y,t)$为竖直位移（m），$x,y$为面内坐标（m），$t$为时间（s）。上点表示时间导数；下标$xx,yy,xy$表示对所示坐标求二阶导数，得到曲率（m⁻¹）。$\\nabla_\\parallel$为面内梯度；$|\\ |$为欧氏模长。$m_A$为面密度（kg m⁻²），$D_f$为弯曲刚度（N m），$\\nu_f$为泊松比，$T_0\\ge0$为给定各向同性单位边长拉力（N m⁻¹）。$K_f,U_b,U_T$分别为动能、弯曲能及预张力能（J）；$\\int$表示面积积分。标签f、b、T分别指薄膜、弯曲与张力。", "kind": "Linear-plate energy model", "kind_zh": "线性板的能量模型", "diagram": { "type": "film", "labels": [ "Ω_f, dA — film plane", "x, y — in-plane directions", "w, ẇ, t — film motion", "w_xx, w_yy, w_xy — curvature", "D_f, ν_f — bending", "m_A → K_f", "T₀ → U_T", "U_b — shape energy" ], "notes": [ "Clamped edge: displacement and normal slope fixed." ] }, "caption_en": "Bending energy is stored by curvature; inertia depends on velocity.", "caption_zh": "曲率储存弯曲能，速度决定惯性动能。" }, { "id": "C4-E40", "tex": "\\begin{aligned}B_{\\alpha\\beta}&=D_f[(1-\\nu_f)w_{\\alpha\\beta}+\\nu_f\\delta_{\\alpha\\beta}\\nabla_\\parallel^2w],\\\\\\delta U_b&=\\sum_{\\alpha,\\beta=1}^{2}\\int_{\\Omega_f}B_{\\alpha\\beta}\\partial_\\alpha\\partial_\\beta(\\delta w)\\,dA\\\\&=\\sum_{\\alpha,\\beta=1}^{2}\\oint_{\\partial\\Omega_f}n_\\alpha B_{\\alpha\\beta}\\partial_\\beta(\\delta w)\\,d\\ell-\\sum_{\\alpha,\\beta=1}^{2}\\int_{\\Omega_f}(\\partial_\\alpha B_{\\alpha\\beta})\\partial_\\beta(\\delta w)\\,dA\\\\&=\\sum_{\\alpha,\\beta=1}^{2}\\oint_{\\partial\\Omega_f}[n_\\alpha B_{\\alpha\\beta}\\partial_\\beta(\\delta w)-n_\\beta(\\partial_\\alpha B_{\\alpha\\beta})\\delta w]\\,d\\ell+\\sum_{\\alpha,\\beta=1}^{2}\\int_{\\Omega_f}(\\partial_\\beta\\partial_\\alpha B_{\\alpha\\beta})\\delta w\\,dA,\\\\\\delta U_T&=T_0\\oint_{\\partial\\Omega_f}\\delta w\\,\\partial_nw\\,d\\ell-T_0\\int_{\\Omega_f}\\delta w\\nabla_\\parallel^2w\\,dA,\\\\\\delta w=0,\\quad\\partial_n\\delta w=0&\\ \\Longrightarrow\\ \\nabla_\\parallel\\delta w=\\boldsymbol0\\quad\\text{on the fixed clamp}.\\end{aligned}", "symbols_en": "$\\Omega_f$ is the smooth reference film plane (m²), $\\partial\\Omega_f$ its boundary, $dA$ area element (m²), and $d\\ell$ boundary arc-length element (m). $w$ is displacement (m); $\\delta w$ an admissible infinitesimal variation (m); $\\delta U_b,\\delta U_T$ the corresponding bending/pretension energy variations (J). $\\alpha,\\beta=1,2$ index the two Cartesian in-plane directions, with $\\sum$ summing all four index pairs. $\\partial_\\alpha$ is coordinate differentiation (m⁻¹); $w_{\\alpha\\beta}=\\partial_\\alpha\\partial_\\beta w$ is curvature (m⁻¹). $D_f$ is bending stiffness (N m), $\\nu_f$ Poisson’s ratio, $T_0$ pretension (N m⁻¹), and $B_{\\alpha\\beta}$ the bending-energy tensor conjugate to curvature (N). $\\delta_{\\alpha\\beta}$ is the dimensionless Kronecker identity, distinct from variation notation. $n_\\alpha,n_\\beta$ are components of the outward in-plane unit boundary normal; $\\partial_n$ its derivative. $\\nabla_\\parallel$ and $\\nabla_\\parallel^2$ are in-plane gradient and Laplacian; $\\boldsymbol0$ is zero gradient. $\\int,\\oint$ denote area and closed-boundary integrals. $\\Rightarrow$ uses zero tangential derivative of an identically zero boundary variation plus the prescribed zero normal derivative.", "symbols_zh": "$\\Omega_f$为光滑薄膜参考平面（m²），$\\partial\\Omega_f$为其边界，$dA$为面积微元（m²），$d\\ell$为边界弧长微元（m）。$w$为位移（m）；$\\delta w$为容许无穷小变分（m）；$\\delta U_b,\\delta U_T$为相应弯曲／预张力能变分（J）。$\\alpha,\\beta=1,2$标识两个笛卡尔面内方向，$\\sum$对四种指标组合求和。$\\partial_\\alpha$为坐标求导（m⁻¹）；$w_{\\alpha\\beta}=\\partial_\\alpha\\partial_\\beta w$为曲率（m⁻¹）。$D_f$为弯曲刚度（N m），$\\nu_f$为泊松比，$T_0$为预张力（N m⁻¹），$B_{\\alpha\\beta}$为与曲率共轭的弯曲能张量（N）。$\\delta_{\\alpha\\beta}$为无量纲Kronecker单位张量，与变分记号不同。$n_\\alpha,n_\\beta$为面内边界外单位法向分量；$\\partial_n$为沿该法向求导。$\\nabla_\\parallel$及$\\nabla_\\parallel^2$为面内梯度及拉普拉斯算子；$\\boldsymbol0$为零梯度。$\\int,\\oint$分别表示面积积分与闭合边界积分。$\\Rightarrow$使用恒为零的边界变分具有零切向导数，并结合给定零法向导数。", "kind": "Explicit integration-by-parts boundary terms", "kind_zh": "明确写出的分部积分边界项", "diagram": { "type": "film", "labels": [ "Ω_f — plate interior", "∂Ω_f, dℓ — fixed clamped edge", "w, δw — displacement and variation", "B_αβ, D_f, ν_f — curvature moments", "n_α, n_β — in-plane boundary normal", "δU_b, δU_T — virtual energy", "T₀ — pretension", "boundary: δw = 0, ∂ₙδw = 0" ], "notes": [ "The first bending boundary term contains the slope variation.", "The second contains the displacement variation; both vanish at the clamp." ] }, "caption_en": "Two integrations by parts expose the boundary moment and shear terms before they are removed.", "caption_zh": "两次分部积分先明确显示边界弯矩与剪力项，再依据条件将其消去。" }, { "id": "C4-E05", "tex": "\\begin{aligned}\\delta U_b&=D_f\\int_{\\Omega_f}\\left[w_{xxxx}+\\{2\\nu_f+2(1-\\nu_f)\\}w_{xxyy}+w_{yyyy}\\right]\\delta w\\,dA\\\\&=D_f\\int_{\\Omega_f}(w_{xxxx}+2w_{xxyy}+w_{yyyy})\\delta w\\,dA=D_f\\int_{\\Omega_f}\\nabla_\\parallel^4w\\,\\delta w\\,dA,\\\\\\delta U_T&=-T_0\\int_{\\Omega_f}\\nabla_\\parallel^2w\\,\\delta w\\,dA.\\end{aligned}", "symbols_en": "$\\delta$ denotes an infinitesimal admissible variation, not physical crack opening here; $\\delta w$ is a displacement variation (m). $U_b,U_T$ are bending and pretension energy (J); $D_f$ is bending stiffness (N m); $\\nu_f$ is dimensionless Poisson’s ratio; $T_0$ is uniform pretension (N m⁻¹). $w$ is transverse displacement (m); $x,y$ are in-plane coordinates (m). Subscripts $xxxx,xxyy,yyyy$ denote the indicated fourth derivatives (m⁻³). $\\nabla_\\parallel^2=\\partial_x^2+\\partial_y^2$ is the in-plane Laplacian; $\\nabla_\\parallel^4$ is that Laplacian applied twice. $\\Omega_f$ is film area (m²); $dA$ is its element (m²); $\\int$ is area integration. The variations and their normal slopes vanish at a clamped boundary.", "symbols_zh": "$\\delta$在此表示无穷小容许变分，而非物理裂纹张开；$\\delta w$为位移变分（m）。$U_b,U_T$为弯曲能及预张力能（J）；$D_f$为弯曲刚度（N m）；$\\nu_f$为无量纲泊松比；$T_0$为均匀预张力（N m⁻¹）。$w$为横向位移（m）；$x,y$为面内坐标（m）。下标$xxxx,xxyy,yyyy$表示所示四阶导数（m⁻³）。$\\nabla_\\parallel^2=\\partial_x^2+\\partial_y^2$为面内拉普拉斯算子；$\\nabla_\\parallel^4$为将该算子作用两次。$\\Omega_f$为薄膜面积（m²）；$dA$为面积微元（m²）；$\\int$表示面积积分。夹持边界处变分及其法向斜率均为零。", "kind": "Derived variational identity for fixed clamps", "kind_zh": "固定夹持条件下推导的变分恒等式", "diagram": { "type": "film", "labels": [ "Ω_f, x, y — film plane", "w, δw — displacement and variation", "D_f, ν_f → ∇⁴w restoring load", "T₀ → −∇²w restoring load", "U_b, U_T — stored energy", "clamp: δw = 0, slope variation = 0" ], "notes": [ "Four spatial derivatives arise from two integrations by parts." ] }, "caption_en": "A fixed clamp removes the boundary virtual-work terms.", "caption_zh": "固定夹持边界使边界虚功项消失。" }, { "id": "C4-E06", "tex": "\\begin{aligned}m_A\\ddot w+D_f\\nabla_\\parallel^4w-T_0\\nabla_\\parallel^2w&=p_{\\rm load}-t_{\\rm coh},\\\\w(\\boldsymbol x,0)=0,\\quad\\dot w(\\boldsymbol x,0)&=0,\\\\w=0,\\quad\\partial_n w&=0\\quad\\text{on a clamped edge}.\\end{aligned}", "symbols_en": "$w(\\boldsymbol x,t)$ is film displacement in the positive departure direction (m); $\\boldsymbol x=(x,y)$ is in-plane position (m), and $t$ time (s). $\\dot w,\\ddot w$ are time derivatives (m s⁻¹, m s⁻²). $m_A$ is areal mass (kg m⁻²), $D_f$ bending stiffness (N m), and $T_0\\ge0$ pretension (N m⁻¹). $\\nabla_\\parallel^2$ is in-plane Laplacian (m⁻²), and $\\nabla_\\parallel^4$ its square. $p_{\\rm load}$ is net applied transverse traction projected positively (Pa), whereas $t_{\\rm coh}\\ge0$ is an opening-resisting cohesive traction magnitude (Pa). $\\partial_n$ differentiates along the outward in-plane boundary normal. Initial zero is displacement or velocity according to its row. Labels load and coh identify applied and cohesive forces.", "symbols_zh": "$w(\\boldsymbol x,t)$为沿正离开方向的薄膜位移（m）；$\\boldsymbol x=(x,y)$为面内位置（m），$t$为时间（s）。$\\dot w,\\ddot w$为时间导数（m s⁻¹、m s⁻²）。$m_A$为面密度（kg m⁻²），$D_f$为弯曲刚度（N m），$T_0\\ge0$为预张力（N m⁻¹）。$\\nabla_\\parallel^2$为面内拉普拉斯算子（m⁻²），$\\nabla_\\parallel^4$为其平方。$p_{\\rm load}$为沿正方向投影的净外加横向牵引（Pa），$t_{\\rm coh}\\ge0$为抵抗张开的内聚牵引幅值（Pa）。$\\partial_n$表示沿面内边界外法向求导。初始零值按对应行分别表示位移或速度。标签load和coh表示外载与内聚力。", "kind": "Reduced transient momentum balance with explicit initial/boundary data", "kind_zh": "含明确初始与边界数据的简化瞬态动量平衡", "diagram": { "type": "film", "labels": [ "w, ẇ, ẅ ↑ positive departure", "x, t — location and time", "m_A — film inertia", "D_f — bending stiffness", "T₀ — pretension", "p_load ↑ exposed load", "t_coh ↓ opening resistance", "clamp: w = 0, ∂ₙw = 0" ], "notes": [ "The film starts at rest.", "The release interface supplies resistance." ] }, "caption_en": "Film acceleration balances applied traction, elastic restoring forces and cohesive resistance.", "caption_zh": "薄膜加速度由外加牵引、弹性恢复力与内聚阻力的平衡决定。" }, { "id": "C4-E41", "tex": "\\begin{aligned}\\int_{\\Omega_f}m_A\\ddot w\\dot w\\,dA&=\\frac{dK_f}{dt},\\qquad \\int_{\\Omega_f}D_f\\nabla_\\parallel^4w\\dot w\\,dA=\\frac{dU_b}{dt},\\\\-\\int_{\\Omega_f}T_0\\nabla_\\parallel^2w\\dot w\\,dA&=\\frac{dU_T}{dt},\\\\\\frac{d}{dt}(K_f+U_b+U_T)&=\\int_{\\Omega_f}p_{\\rm load}\\dot w\\,dA-\\int_{\\Omega_f}t_{\\rm coh}\\dot w\\,dA.\\end{aligned}", "symbols_en": "$\\Omega_f$ is reference film area (m²), $dA$ its area element (m²), $w$ displacement (m), and $\\dot w,\\ddot w$ velocity and acceleration (m s⁻¹, m s⁻²). $m_A$ is areal mass (kg m⁻²), $D_f$ bending stiffness (N m), $T_0$ constant pretension (N m⁻¹), $p_{\\rm load}$ applied transverse traction (Pa), and $t_{\\rm coh}$ resisting cohesive traction (Pa). $K_f,U_b,U_T$ are kinetic, bending and pretension energies (J); $t$ is time (s); $d/dt$ is its derivative. $\\nabla_\\parallel^2$ is planar Laplacian and $\\nabla_\\parallel^4$ its square; $\\int$ is area integration. The derivatives are valid for the stated constant parameters and sufficiently regular fields at a fixed clamp with zero velocity and normal velocity slope. Every row has power units W; f, b, T, load and coh label film, bending, tension, applied and cohesive.", "symbols_zh": "$\\Omega_f$为薄膜参考面积（m²），$dA$为面积微元（m²），$w$为位移（m），$\\dot w,\\ddot w$为速度及加速度（m s⁻¹、m s⁻²）。$m_A$为面密度（kg m⁻²），$D_f$为弯曲刚度（N m），$T_0$为恒定预张力（N m⁻¹），$p_{\\rm load}$为外加横向牵引（Pa），$t_{\\rm coh}$为抵抗内聚牵引（Pa）。$K_f,U_b,U_T$为动能、弯曲能及预张力能（J）；$t$为时间（s）；$d/dt$为时间导数。$\\nabla_\\parallel^2$为平面拉普拉斯算子，$\\nabla_\\parallel^4$为其平方；$\\int$表示面积积分。导数适用于所述恒定参数、足够正则的场，以及速度和法向速度斜率均为零的固定夹持边界。各行单位均为功率W；f、b、T、load、coh分别表示薄膜、弯曲、张力、外加及内聚。", "kind": "Transient plate power identity under fixed clamps", "kind_zh": "固定夹持条件下瞬态板的功率恒等式", "diagram": { "type": "energy", "labels": [ "p_load ẇ → incoming solid power", "t_coh ẇ → interface power", "m_A, ẇ, ẅ → K_f storage", "D_f, ∇⁴w → U_b storage", "T₀, ∇²w → U_T storage", "Ω_f, dA — reference plane", "t — event time" ], "notes": [ "Fixed clamps perform no boundary work.", "Recoverable cohesion and fracture loss must be counted once." ] }, "caption_en": "The transient plate equation conserves power between applied work, motion, shape and interface response.", "caption_zh": "瞬态薄板方程在外加功、运动、形状与界面响应之间保持功率守恒。" }, { "id": "C4-E07", "tex": "\\begin{aligned}m_{A,P}\\ddot w_P+D_P\\nabla_\\parallel^4w_P-T_P\\nabla_\\parallel^2w_P&=p_{\\rm liq}-t_{P\\to f},\\\\m_A\\ddot w+D_f\\nabla_\\parallel^4w-T_0\\nabla_\\parallel^2w&=t_{P\\to f}-t_{\\rm coh}.\\end{aligned}", "symbols_en": "$w_P,w$ are PVC and film displacements in the same positive direction (m); double dots are accelerations (m s⁻²). $m_{A,P},m_A$ are their areal masses (kg m⁻²); $D_P,D_f$ their bending stiffnesses (N m); $T_P,T_0$ their prescribed pretensions (N m⁻¹). $\\nabla_\\parallel^2$ is the in-plane Laplacian and $\\nabla_\\parallel^4$ its square. $p_{\\rm liq}$ is net liquid traction on PVC (Pa), $t_{P\\to f}$ transmitted PVC-to-film traction projected positively (Pa), and $t_{\\rm coh}$ film-release resistance (Pa). P labels PVC, f film, A areal, liq liquid and coh cohesion; the arrow labels the transmission direction. Each sheet has its actual initial and support conditions. No value of transmitted traction is supplied without an additional contact/bond closure.", "symbols_zh": "$w_P,w$为PVC与薄膜沿同一正方向的位移（m）；双上点为加速度（m s⁻²）。$m_{A,P},m_A$为两者面密度（kg m⁻²）；$D_P,D_f$为弯曲刚度（N m）；$T_P,T_0$为给定预张力（N m⁻¹）。$\\nabla_\\parallel^2$为面内拉普拉斯算子，$\\nabla_\\parallel^4$为其平方。$p_{\\rm liq}$为液体作用于PVC的净牵引（Pa），$t_{P\\to f}$为沿正方向投影的PVC传向薄膜牵引（Pa），$t_{\\rm coh}$为薄膜释放阻力（Pa）。P表示PVC，f表示薄膜，A表示面量，liq表示液体，coh表示内聚；箭头标识传递方向。每层均具有自身实际初始及支承条件。若无附加接触／粘接闭合关系，传递牵引就尚未确定。", "kind": "Conditional coupled-sheet model", "kind_zh": "有条件的耦合薄片模型", "diagram": { "type": "film", "labels": [ "p_liq → PVC", "w_P, m_A,P, D_P, T_P — PVC response", "t_P→f → contact or bond", "w, m_A, D_f, T₀ — film response", "t_coh → release interface" ], "notes": [ "No liquid passes through intact PVC.", "Transmitted traction is an unknown, not a copied pressure." ] }, "caption_en": "The PVC-to-film load follows intermediate-sheet motion and contact.", "caption_zh": "PVC传向薄膜的载荷由中间片层运动与接触决定。" }, { "id": "C4-E08", "tex": "m_A[\\dot w(\\boldsymbol x,t_b)-\\dot w(\\boldsymbol x,t_a)]=J_A-\\int_{t_a}^{t_b}[D_f\\nabla_\\parallel^4w-T_0\\nabla_\\parallel^2w+t_{\\rm coh}]\\,dt,\\qquad J_A=\\int_{t_a}^{t_b}p_{\\rm load}\\,dt", "symbols_en": "$m_A$ is film areal mass (kg m⁻²), $w(\\boldsymbol x,t)$ vertical displacement (m), $\\boldsymbol x$ in-plane position (m), and $\\dot w$ velocity (m s⁻¹). $t_a,t_b,t,dt$ are pulse endpoints, integration time and time element (s). $J_A$ is local applied impulse per area (Pa s), $p_{\\rm load}$ net applied traction (Pa), $t_{\\rm coh}$ cohesive resistance (Pa), $D_f$ bending stiffness (N m), and $T_0$ pretension (N m⁻¹). $\\nabla_\\parallel^2$ and $\\nabla_\\parallel^4$ are the in-plane Laplacian and its square. $\\int$ integrates the actual local load/response during the pulse. Labels A, f, load and coh mean areal, film, applied and cohesive.", "symbols_zh": "$m_A$为薄膜面密度（kg m⁻²），$w(\\boldsymbol x,t)$为竖直位移（m），$\\boldsymbol x$为面内位置（m），$\\dot w$为速度（m s⁻¹）。$t_a,t_b,t,dt$分别为脉冲起止时刻、积分时间及时间微元（s）。$J_A$为局部单位面积外加冲量（Pa s），$p_{\\rm load}$为净外加牵引（Pa），$t_{\\rm coh}$为内聚阻力（Pa），$D_f$为弯曲刚度（N m），$T_0$为预张力（N m⁻¹）。$\\nabla_\\parallel^2$与$\\nabla_\\parallel^4$为面内拉普拉斯算子及其平方。$\\int$表示对脉冲期间实际局部载荷／响应积分。标签A、f、load、coh分别表示面量、薄膜、外加及内聚。", "kind": "Exact time integral within the reduced plate model", "kind_zh": "简化板模型内的精确时间积分", "diagram": { "type": "impulse", "labels": [ "t_a → t_b — pulse interval", "p_load(t) → J_A", "m_A → velocity change", "ẇ(x,t_a), ẇ(x,t_b)", "D_f∇⁴w, −T₀∇²w, t_coh — reaction impulses" ], "notes": [ "Applied impulse and reaction impulse both enter." ] }, "caption_en": "Pulse integration exposes the reactions that can invalidate a free velocity jump.", "caption_zh": "脉冲积分明确显示哪些反作用冲量会使自由速度跃变失效。" }, { "id": "C4-E09", "tex": "\\Delta\\dot w\\simeq\\frac{J_A}{m_A},\\qquad \\mathcal E_A\\simeq\\frac12m_A(\\Delta\\dot w)^2=\\frac{J_A^2}{2m_A}", "symbols_en": "$\\Delta\\dot w$ is the film velocity change from rest (m s⁻¹); $\\Delta$ denotes final-minus-initial difference, and a dot is a time derivative. $J_A$ is local applied impulse per area (Pa s), $m_A>0$ film areal mass (kg m⁻²), and $\\mathcal E_A$ resulting kinetic energy per area (J m⁻²). $\\simeq$ marks negligible restoring/cohesive impulse during the pulse, not guaranteed transmission through an adhered stack. Subscript A labels quantities per area.", "symbols_zh": "$\\Delta\\dot w$为薄膜从静止开始的速度变化（m s⁻¹）；$\\Delta$表示终值减初值，上点为时间导数。$J_A$为局部单位面积外加冲量（Pa s），$m_A>0$为薄膜面密度（kg m⁻²），$\\mathcal E_A$为所得单位面积动能（J m⁻²）。$\\simeq$表示脉冲期间恢复力／内聚力冲量可忽略，而非保证粘接叠层中的载荷传递。下标A表示单位面积量。", "kind": "Short-pulse free-response approximation", "kind_zh": "短脉冲后的自由响应近似", "diagram": { "type": "film", "labels": [ "J_A ↑ pulse impulse", "m_A — areal mass", "Δẇ ↑ initial response", "E_A — motion energy per area" ], "notes": [ "Reaction impulses must be negligible." ] }, "caption_en": "Impulse creates film kinetic energy that may subsequently feed deformation and fracture.", "caption_zh": "冲量产生薄膜动能，之后才可能转入变形与断裂。" }, { "id": "C4-E10", "tex": "t_b^{\\rm scale}=a_f^2\\sqrt{\\frac{m_A}{D_f}},\\qquad t_T^{\\rm scale}=a_f\\sqrt{\\frac{m_A}{T_0}},\\qquad t_n^{\\rm scale}=\\sqrt{\\frac{m_A}{K_n}}", "symbols_en": "$t_b^{\\rm scale},t_T^{\\rm scale},t_n^{\\rm scale}$ are bending, pretension and normal-cohesion response scales (s), not exact modal periods; superscript scale labels estimates and subscripts b, T, n identify the restoring mechanisms. $a_f>0$ is lateral deformation length (m), $m_A>0$ areal mass (kg m⁻²), $D_f>0$ bending stiffness (N m), $T_0>0$ pretension (N m⁻¹), and $K_n>0$ small-opening normal cohesive stiffness (Pa m⁻¹). $\\sqrt{\\ }$ is the positive root. If pretension or cohesion is absent, its corresponding timescale is omitted rather than divided by zero.", "symbols_zh": "$t_b^{\\rm scale},t_T^{\\rm scale},t_n^{\\rm scale}$分别为弯曲、预张力及法向内聚响应尺度（s），而非精确模态周期；上标scale表示估计，下标b、T、n标识恢复机制。$a_f>0$为横向变形长度（m），$m_A>0$为面密度（kg m⁻²），$D_f>0$为弯曲刚度（N m），$T_0>0$为预张力（N m⁻¹），$K_n>0$为小张开法向内聚刚度（Pa m⁻¹）。$\\sqrt{\\ }$取正根。若预张力或内聚机制不存在，则不使用对应时间尺度，不能除以零。", "kind": "Derived term-balance estimates", "kind_zh": "通过各项平衡推导的估算", "diagram": { "type": "timescale", "labels": [ "a_f — lateral deformation length", "m_A — inertia", "D_f → t_b,scale", "T₀ → t_T,scale", "K_n → t_n,scale", "pulse duration → compare response times" ], "notes": [ "PVC also has a finite through-thickness wave time." ] }, "caption_en": "Pulse duration must be compared with the response scales of the actual load path.", "caption_zh": "脉冲持续时间须与实际传力路径的响应尺度比较。" }, { "id": "C4-E11", "tex": "G=-\\left.\\frac{\\partial\\mathcal P}{\\partial A_c}\\right|_{\\mathcal C},\\qquad G\\ge\\Gamma(\\psi,T_i,v_c)", "symbols_en": "$G$ is quasistatic energy-release rate (J m⁻²), $\\mathcal P$ total mechanical potential including the external loading system (J), $A_c$ crack area (m²), and $\\partial/\\partial A_c$ a partial derivative. $\\mathcal C$ labels the held loading control, such as pressure or displacement; it is not a material coefficient. $\\Gamma$ is fracture resistance (J m⁻²), $\\psi$ mode-mixture parameter (dimensionless), $T_i$ interface temperature (K), and $v_c$ crack-front speed (m s⁻¹). Subscripts c and i label crack and interface. The inequality is an energetic propagation criterion for the specified fracture model, not a sufficient transfer criterion.", "symbols_zh": "$G$为准静态能量释放率（J m⁻²），$\\mathcal P$为包含外部加载系统的总力学势能（J），$A_c$为裂纹面积（m²），$\\partial/\\partial A_c$为偏导数。$\\mathcal C$标识保持不变的加载控制量，例如压力或位移，而非材料系数。$\\Gamma$为断裂阻力（J m⁻²），$\\psi$为模态混合参数（无量纲），$T_i$为界面温度（K），$v_c$为裂纹前沿速度（m s⁻¹）。下标c与i表示裂纹及界面。此不等式是指定断裂模型的能量扩展判据，而非充分转印判据。", "kind": "Definition with quasistatic fracture criterion", "kind_zh": "定义与准静态断裂判据", "diagram": { "type": "film", "labels": [ "A_c → advancing release area", "G → available energy per new area", "Γ(ψ,T_i,v_c) → resistance", "P — system potential", "C — held source control" ], "notes": [ "Strength has units Pa; fracture resistance has units J/m²." ] }, "caption_en": "A crack is driven by energy per newly separated area, under a specified load control.", "caption_zh": "裂纹由每单位新分离面积的可用能量驱动，且加载控制必须明确。" }, { "id": "C4-E12", "tex": "P_{\\rm load}=\\frac{d}{dt}(K_f+U_f)+\\int_{\\mathcal L_c}\\Gamma(\\psi,T_i,v_c)v_c\\,d\\ell+P_{\\rm other}", "symbols_en": "$P_{\\rm load}$ is mechanical power delivered to the solid (W), $K_f$ its kinetic energy (J), $U_f$ its recoverable elastic energy including any recoverable interface energy (J), and $d/dt$ the time derivative with $t$ in s. $\\mathcal L_c$ is the active crack-front line; $d\\ell$ is its arc-length element (m). $v_c\\ge0$ is front speed (m s⁻¹), $\\Gamma$ fracture energy (J m⁻²), $\\psi$ dimensionless mode mixture, and $T_i$ interface temperature (K). $P_{\\rm other}\\ge0$ is additional nonfracture dissipation (W). $\\int$ integrates along the front. Labels load, f, c, i and other denote applied, solid film, crack, interface and remaining dissipative processes.", "symbols_zh": "$P_{\\rm load}$为传入固体的力学功率（W），$K_f$为固体动能（J），$U_f$为可恢复弹性能，含任何可恢复界面能（J），$d/dt$为时间导数，$t$单位s。$\\mathcal L_c$为活跃裂纹前沿曲线；$d\\ell$为弧长微元（m）。$v_c\\ge0$为前沿速度（m s⁻¹），$\\Gamma$为断裂能（J m⁻²），$\\psi$为无量纲模态混合参数，$T_i$为界面温度（K）。$P_{\\rm other}\\ge0$为其他非断裂耗散（W）。$\\int$表示沿前沿积分。标签load、f、c、i、other分别表示外加、固体薄膜、裂纹、界面及其余耗散过程。", "kind": "Conditional dynamic energy balance", "kind_zh": "有条件的动态能量平衡", "diagram": { "type": "energy", "labels": [ "P_load → solid power", "K_f — motion storage", "U_f — elastic storage", "L_c, dℓ, v_c → new crack area/time", "Γ(ψ,T_i,v_c) → fracture consumption", "P_other → other losses" ], "notes": [ "Crack dissipation is counted once." ] }, "caption_en": "Dynamic loading can store kinetic energy before or during crack advance.", "caption_zh": "动态载荷可在裂纹扩展前或扩展中储存动能。" }, { "id": "C4-E13", "tex": "\\mathscr L_rg=\\frac{p_0}{D_f},\\qquad g=\\mathscr L_rw,\\qquad \\mathscr L_r f=\\frac1r\\frac{d}{dr}\\left(r\\frac{df}{dr}\\right),\\qquad w(b)=0,\\quad w^{\\prime}(b)=0", "symbols_en": "$r\\in[0,b]$ is radius in the plate plane (m), $b>0$ delamination radius (m), $w(r)$ opening displacement (m), $p_0>0$ externally maintained pressure difference (Pa), and $D_f>0$ bending stiffness (N m). $\\mathscr L_r$ is the axisymmetric planar Laplacian (m⁻²); $f$ is an arbitrary smooth radial function, and $g=\\mathscr L_rw$ has units m⁻¹. $d/dr$ and a prime denote radial differentiation; the operator at $r=0$ is its regular limit. Labels r, f and 0 denote radial operator, film/arbitrary function according to context, and maintained load. The two zero values prescribe edge displacement and slope.", "symbols_zh": "$r\\in[0,b]$为板平面内半径（m），$b>0$为脱层半径（m），$w(r)$为张开位移（m），$p_0>0$为外部保持的压差（Pa），$D_f>0$为弯曲刚度（N m）。$\\mathscr L_r$为轴对称平面拉普拉斯算子（m⁻²）；$f$为任意光滑径向函数，$g=\\mathscr L_rw$的单位为m⁻¹。$d/dr$及撇号表示径向求导；$r=0$处算子取正则极限。标签r、f、0分别表示径向算子、薄膜／任意函数（依语境）及恒定载荷。两个零值指定边缘位移与斜率。", "kind": "Quasistatic plate boundary-value model", "kind_zh": "准静态板的边值模型", "diagram": { "type": "film", "labels": [ "r = 0 — regular centre", "r = b — clamped crack front", "p₀ ↑ uniform maintained pressure", "D_f — plate stiffness", "w(r) ↑ opening", "g = L_r w — curvature sum", "f — radial test function" ], "notes": [ "No point force at the centre.", "External pressure remains fixed as b changes." ] }, "caption_en": "Uniform pressure loads a clamped circular delamination with regular central fields.", "caption_zh": "均匀压力作用于边缘夹持的圆形脱层，中心场必须正则。" }, { "id": "C4-E14", "tex": "\\begin{aligned}\\frac{d}{dr}(rg^{\\prime})&=\\frac{p_0r}{D_f},\\\\rg^{\\prime}&=\\frac{p_0r^2}{2D_f}+C_1,\\\\g^{\\prime}&=\\frac{p_0r}{2D_f}+\\frac{C_1}{r},\\quad C_1=0,\\\\g(r)&=\\frac{p_0r^2}{4D_f}+C_2.\\end{aligned}", "symbols_en": "$r\\in(0,b]$ is radial coordinate (m); $b$ is delamination radius (m). $g(r)=\\mathscr L_rw$ is the curvature sum (m⁻¹), with a prime and $d/dr$ denoting radial differentiation. $p_0$ is maintained pressure difference (Pa); $D_f$ bending stiffness (N m). $C_1,C_2$ are integration constants with units m⁻¹. $C_1=0$ follows from regular central curvature gradient and absence of a point load. $\\mathscr L_r$ is the radial Laplacian, $w$ displacement (m); the subscript 0 labels the maintained pressure, not the initial liquid pressure.", "symbols_zh": "$r\\in(0,b]$为径向坐标（m）；$b$为脱层半径（m）。$g(r)=\\mathscr L_rw$为曲率和（m⁻¹），撇号及$d/dr$表示径向求导。$p_0$为恒定压差（Pa）；$D_f$为弯曲刚度（N m）。$C_1,C_2$为单位m⁻¹的积分常数。$C_1=0$来自中心曲率梯度正则及无点载荷条件。$\\mathscr L_r$为径向拉普拉斯算子，$w$为位移（m）；下标0标识恒定压力，而非液体初始压力。", "kind": "Derived first two radial integrations", "kind_zh": "推导中的前两次径向积分", "diagram": { "type": "film", "labels": [ "p₀ → curvature forcing", "D_f — resistance", "r, b — radial domain", "g(r), g′ — curvature fields", "C₁ = 0 — centre regularity", "C₂ — remaining curvature offset" ], "notes": [ "An inverse-radius curvature gradient is rejected at the centre." ] }, "caption_en": "Regularity removes the singular integration constant before edge conditions are used.", "caption_zh": "在使用边缘条件之前，正则性先排除奇异积分常数。" }, { "id": "C4-E15", "tex": "\\begin{aligned}\\frac{d}{dr}(rw^{\\prime})&=rg=\\frac{p_0r^3}{4D_f}+C_2r,\\\\rw^{\\prime}&=\\frac{p_0r^4}{16D_f}+\\frac{C_2r^2}{2}+C_3,\\quad C_3=0,\\\\w^{\\prime}&=\\frac{p_0r^3}{16D_f}+\\frac{C_2r}{2},\\\\w(r)&=\\frac{p_0r^4}{64D_f}+\\frac{C_2r^2}{4}+C_4.\\end{aligned}", "symbols_en": "$r\\in(0,b]$ is radial position (m), $b$ delamination radius (m), $w$ displacement (m), and $g$ curvature sum (m⁻¹). A prime and $d/dr$ are radial derivatives; $w^{\\prime}$ is dimensionless slope. $p_0$ is maintained pressure (Pa), $D_f$ bending stiffness (N m), $C_2$ curvature integration constant (m⁻¹), and $C_3,C_4$ displacement integration constants (m). $C_3=0$ excludes a central inverse-radius slope. All rows are regular-limit solutions at the centre. Subscripts 2,3,4 index distinct constants, not derivative orders.", "symbols_zh": "$r\\in(0,b]$为径向位置（m），$b$为脱层半径（m），$w$为位移（m），$g$为曲率和（m⁻¹）。撇号及$d/dr$为径向导数；$w^{\\prime}$为无量纲斜率。$p_0$为恒定压力（Pa），$D_f$为弯曲刚度（N m），$C_2$为曲率积分常数（m⁻¹），$C_3,C_4$为位移积分常数（m）。$C_3=0$排除中心反比于半径的斜率。各行在中心均取正则极限。下标2、3、4标识不同常数，而非导数阶数。", "kind": "Derived second pair of radial integrations", "kind_zh": "推导中的后两次径向积分", "diagram": { "type": "film", "labels": [ "r, b — radial geometry", "w(r), w′ — displacement and slope", "g — curvature sum", "p₀, D_f — forcing and stiffness", "C₂ — curvature offset", "C₃ = 0 — regular slope", "C₄ — height offset" ], "notes": [ "The centre carries no singular force." ] }, "caption_en": "The regular displacement contains quartic, quadratic and constant contributions.", "caption_zh": "正则位移含四次项、二次项与常数项。" }, { "id": "C4-E16", "tex": "\\begin{aligned}0=w^{\\prime}(b)&=\\frac{p_0b^3}{16D_f}+\\frac{C_2b}{2}\\quad\\Rightarrow\\quad C_2=-\\frac{p_0b^2}{8D_f},\\\\0=w(b)&=\\frac{p_0b^4}{64D_f}-\\frac{p_0b^4}{32D_f}+C_4\\quad\\Rightarrow\\quad C_4=\\frac{p_0b^4}{64D_f},\\\\w(r)&=\\frac{p_0}{64D_f}(r^4-2b^2r^2+b^4)=\\frac{p_0}{64D_f}(b^2-r^2)^2.\\end{aligned}", "symbols_en": "$w(r)$ is displacement (m), $w^{\\prime}$ radial slope, $r\\in[0,b]$ radial position (m), and $b>0$ delamination radius (m), allowing division by $b$. $p_0>0$ is maintained pressure (Pa), $D_f>0$ bending stiffness (N m), $C_2$ curvature constant (m⁻¹), and $C_4$ displacement constant (m). The prime denotes $d/dr$; $\\Rightarrow$ denotes the algebraic consequence of each clamped boundary condition. Subscripts 2 and 4 label constants, and f labels film.", "symbols_zh": "$w(r)$为位移（m），$w^{\\prime}$为径向斜率，$r\\in[0,b]$为径向位置（m），$b>0$为脱层半径（m），因此可以除以$b$。$p_0>0$为恒定压力（Pa），$D_f>0$为弯曲刚度（N m），$C_2$为曲率常数（m⁻¹），$C_4$为位移常数（m）。撇号表示$d/dr$；$\\Rightarrow$表示各夹持边界条件的代数结果。下标2与4标识常数，f表示薄膜。", "kind": "Exact solution of the stated linear-plate benchmark", "kind_zh": "所述线性板基准问题的精确解", "diagram": { "type": "film", "labels": [ "r = 0 → w_max", "r = b → w = 0, slope = 0", "p₀ ↑ held pressure", "D_f — bending resistance", "C₂, C₄ — boundary-fixed constants", "w(r) — quartic profile" ], "notes": [ "The bonded outside region supplies the clamp." ] }, "caption_en": "Applying both edge conditions fixes the unique regular blister profile.", "caption_zh": "同时应用两个边缘条件，确定唯一正则鼓泡轮廓。" }, { "id": "C4-E17", "tex": "\\mathscr L_r(r^2)=4,\\qquad \\mathscr L_r(r^4)=16r^2,\\qquad D_f\\mathscr L_r^2w=p_0,\\qquad w^{\\prime}(r)=\\frac{p_0r(r^2-b^2)}{16D_f}", "symbols_en": "$\\mathscr L_r$ is the radial planar Laplacian (m⁻²), and superscript 2 on the operator means composition twice; superscripts on $r$ are powers. $r$ is radial position (m), $b$ delamination radius (m), $w$ blister displacement (m), and its prime is radial slope. $D_f$ is bending stiffness (N m); $p_0$ maintained pressure (Pa). Literal 4 and 16 are numerical coefficients; $\\mathscr L_r(r^2)$ is dimensionless and $\\mathscr L_r(r^4)$ has units m². The slope expression vanishes at $r=0$ and $r=b$.", "symbols_zh": "$\\mathscr L_r$为径向平面拉普拉斯算子（m⁻²），算子上的上标2表示连续作用两次；$r$上的上标表示幂。$r$为径向位置（m），$b$为脱层半径（m），$w$为鼓泡位移（m），其撇号为径向斜率。$D_f$为弯曲刚度（N m）；$p_0$为恒定压力（Pa）。数字4与16是数值系数；$\\mathscr L_r(r^2)$无量纲，$\\mathscr L_r(r^4)$单位为m²。斜率式在$r=0$及$r=b$均为零。", "kind": "Governing-equation and boundary verification", "kind_zh": "控制方程与边界条件核验", "diagram": { "type": "film", "labels": [ "r — radial coordinate", "b — crack edge", "w, w′ — opening and slope", "D_f L_r²w = p₀ — equation residual", "r = 0 and b → zero slope" ], "notes": [ "L_r r² = 4", "L_r r⁴ = 16r²" ] }, "caption_en": "Direct radial differentiation verifies the pressure balance and regular clamped profile.", "caption_zh": "直接径向求导可检验压力平衡及正则夹持轮廓。" }, { "id": "C4-E18", "tex": "\\begin{aligned}V_{\\rm bl}&=2\\pi\\int_0^b w(r)r\\,dr=\\frac{\\pi p_0}{32D_f}\\int_0^b(b^4r-2b^2r^3+r^5)\\,dr,\\\\&=\\frac{\\pi p_0}{32D_f}\\left[\\frac{b^4r^2}{2}-\\frac{b^2r^4}{2}+\\frac{r^6}{6}\\right]_0^b=\\frac{\\pi p_0b^6}{192D_f}.\\end{aligned}", "symbols_en": "$V_{\\rm bl}$ is added blister volume (m³), $w(r)$ displacement (m), $r\\in[0,b]$ radial integration coordinate (m), $dr$ its integration element (m), and $b$ delamination radius (m). $p_0$ is maintained pressure (Pa); $D_f$ bending stiffness (N m); $\\pi$ the dimensionless circle constant. $\\int$ is radial integration, and $[\\ ]_0^b$ denotes antiderivative at the upper endpoint minus its value at zero. Subscript bl labels the blister; superscripts on lengths are powers.", "symbols_zh": "$V_{\\rm bl}$为新增鼓泡体积（m³），$w(r)$为位移（m），$r\\in[0,b]$为径向积分坐标（m），$dr$为积分微元（m），$b$为脱层半径（m）。$p_0$为恒定压力（Pa）；$D_f$为弯曲刚度（N m）；$\\pi$为无量纲圆周率。$\\int$表示径向积分，$[\\ ]_0^b$表示原函数在上端点的值减去其零端点值。下标bl表示鼓泡；长度上的上标均表示幂。", "kind": "Exact geometric volume integral for the benchmark", "kind_zh": "基准模型的精确几何体积积分", "diagram": { "type": "film", "labels": [ "V_bl — volume below lifted plate", "w(r) ↑ local opening", "r dr → annular area weight", "b — circular crack edge", "p₀, D_f — compliance inputs" ], "notes": [ "Every annulus contributes its own displacement." ] }, "caption_en": "Blister volume sums the opening of all concentric annuli.", "caption_zh": "鼓泡体积等于各同心圆环张开贡献之和。" }, { "id": "C4-E19", "tex": "C_b=\\left.\\frac{\\partial V_{\\rm bl}}{\\partial p_0}\\right|_b=\\frac{\\pi b^6}{192D_f},\\qquad V_{\\rm bl}=C_bp_0,\\qquad U_b=\\int_0^{V_{\\rm bl}}\\frac{v}{C_b}\\,dv=\\frac{V_{\\rm bl}^2}{2C_b}=\\frac{p_0V_{\\rm bl}}2,\\qquad \\mathcal P=U_b-p_0V_{\\rm bl}=-\\frac{\\pi p_0^2b^6}{384D_f}", "symbols_en": "$C_b>0$ is blister volume compliance (m³ Pa⁻¹), defined by the pressure derivative at fixed radius, including the regular zero-load value. $V_{\\rm bl}\\ge0$ is added volume (m³), $p_0\\ge0$ maintained pressure (Pa), $b>0$ crack radius (m), and $D_f>0$ bending stiffness (N m). $\\partial/\\partial p_0$ is the pressure derivative; the vertical bar holds radius fixed. $U_b$ is stored plate bending energy (J), $\\mathcal P$ plate-plus-pressure-reservoir potential (J), and $v$ a dummy volume along the fixed-radius elastic loading curve (m³), with $dv$ its element (m³). $\\int$ integrates that loading curve. Subscripts b and bl label blister compliance/bending and blister volume, respectively; the meanings are explicitly fixed here. No crack-area variation is taken during the loading-curve integral. For positive pressure, compliance also equals volume divided by pressure; at zero pressure the derivative avoids division by zero.", "symbols_zh": "$C_b>0$为鼓泡体积柔度（m³ Pa⁻¹），由固定半径下的压力导数定义，包含正则零载荷值。$V_{\\rm bl}\\ge0$为新增体积（m³），$p_0\\ge0$为恒定压力（Pa），$b>0$为裂纹半径（m），$D_f>0$为弯曲刚度（N m）。$\\partial/\\partial p_0$为压力导数；竖线表示保持半径不变。$U_b$为板弯曲储能（J），$\\mathcal P$为板与恒压储库的总势能（J），$v$为固定半径弹性加载曲线上的虚拟体积变量（m³），$dv$为其微元（m³）。$\\int$表示该加载曲线积分。下标b与bl分别表示鼓泡柔度／弯曲及鼓泡体积，此处已明确其含义。在加载曲线积分过程中不改变裂纹面积。正压力时，柔度也等于体积除以压力；零压力时采用导数避免除以零。", "kind": "Derived potential under maintained-pressure control", "kind_zh": "维持压力控制条件下推导的势能", "diagram": { "type": "energy", "labels": [ "p₀ — maintained source pressure", "V_bl, v — cavity volume", "C_b — volume compliance", "U_b = ½p₀V_bl — plate storage", "−p₀V_bl — source potential", "P = U_b − p₀V_bl", "b, D_f — geometry and stiffness" ], "notes": [ "The external pressure reservoir supplies work as the crack grows." ] }, "caption_en": "Source work changes the sign of the relevant total potential.", "caption_zh": "压力源做功改变相关总势能的符号。" }, { "id": "C4-E20", "tex": "\\begin{aligned}A_c&=\\pi b^2,\\qquad \\left.\\frac{d\\mathcal P}{db}\\right|_{p_0}=-\\frac{6\\pi p_0^2b^5}{384D_f},\\qquad\\frac{dA_c}{db}=2\\pi b,\\\\G_{p_0}&=-\\frac{(d\\mathcal P/db)_{p_0}}{dA_c/db}=\\frac{p_0^2b^4}{128D_f}.\\end{aligned}", "symbols_en": "$A_c$ is delaminated area (m²), $b>0$ its radius (m), $\\pi$ the circle constant, $\\mathcal P$ total potential including maintained pressure (J), $p_0$ fixed pressure difference (Pa), and $D_f$ bending stiffness (N m). $d/db$ differentiates in radius; the vertical bar and subscript $p_0$ indicate held pressure. $G_{p_0}$ is energy-release rate under that control (J m⁻²). Subscript c labels crack area; all length superscripts are powers. Positive $b$ makes the area derivative nonzero.", "symbols_zh": "$A_c$为脱层面积（m²），$b>0$为其半径（m），$\\pi$为圆周率，$\\mathcal P$为包含恒压源的总势能（J），$p_0$为固定压差（Pa），$D_f$为弯曲刚度（N m）。$d/db$表示按半径求导；竖线及下标$p_0$表示保持压力不变。$G_{p_0}$为该控制方式下的能量释放率（J m⁻²）。下标c表示裂纹面积；长度上的上标均为幂。$b$为正保证面积导数非零。", "kind": "Derived quasistatic energy-release rate", "kind_zh": "推导得到的准静态能量释放率", "diagram": { "type": "film", "labels": [ "b → moving circular crack edge", "A_c = πb² → released area", "p₀ — fixed load", "D_f — plate stiffness", "P → differentiable system potential", "G_p₀ → energy per new area" ], "notes": [ "The derivative is taken at fixed pressure." ] }, "caption_en": "Geometry and compliance convert pressure into fracture energy per area.", "caption_zh": "几何与柔度将压力转换为单位面积断裂能。" }, { "id": "C4-E21", "tex": "p_{0,\\rm crit}=\\frac{\\sqrt{128D_f\\Gamma}}{b^2},\\qquad \\frac{G_{p_0}}{\\Gamma}=\\left(\\frac{p_0}{p_{0,\\rm crit}}\\right)^2", "symbols_en": "$p_{0,\\rm crit}>0$ is critical maintained pressure (Pa) for constant fracture resistance $\\Gamma>0$ (J m⁻²); $D_f>0$ is bending stiffness (N m), $b>0$ delamination radius (m), $p_0\\ge0$ applied maintained pressure (Pa), and $G_{p_0}$ the corresponding energy-release rate (J m⁻²). $\\sqrt{\\ }$ is the positive root; subscript crit labels the onset threshold. The quotient $G_{p_0}/\\Gamma$ is dimensionless. This onset criterion assumes an existing crack and the quasistatic bending model.", "symbols_zh": "$p_{0,\\rm crit}>0$为恒定断裂阻力$\\Gamma>0$（J m⁻²）对应的临界恒压（Pa）；$D_f>0$为弯曲刚度（N m），$b>0$为脱层半径（m），$p_0\\ge0$为外加恒压（Pa），$G_{p_0}$为相应能量释放率（J m⁻²）。$\\sqrt{\\ }$取正根；下标crit表示起始阈值。比值$G_{p_0}/\\Gamma$无量纲。此起始判据假设存在预裂纹，并采用准静态弯曲模型。", "kind": "Derived onset threshold within the benchmark", "kind_zh": "基准模型内推导的起始阈值", "diagram": { "type": "film", "labels": [ "p₀ → pressure loading", "p₀,crit → onset threshold", "D_f, Γ → stiffness and fracture resistance", "b → pre-existing crack size", "G_p₀/Γ → propagation screen" ], "notes": [ "Larger existing delamination lowers critical pressure." ] }, "caption_en": "The pressure threshold depends on crack geometry and fracture resistance, not pressure alone.", "caption_zh": "压力阈值取决于裂纹几何及断裂阻力，而非压力本身。" }, { "id": "C4-E22", "tex": "\\begin{aligned}U_b\\big|_{\\bar V}&=\\frac{\\bar V^2}{2C_b}=\\frac{96D_f\\bar V^2}{\\pi b^6},\\qquad p(b)=\\frac{192D_f\\bar V}{\\pi b^6},\\\\G_{\\bar V}&=-\\frac{dU_b/db}{2\\pi b}=\\frac{288D_f\\bar V^2}{\\pi^2b^8}=\\frac{p(b)^2b^4}{128D_f},\\\\G_{p_0}&\\propto b^4\\ \\text{at fixed }p_0,\\qquad G_{\\bar V}\\propto b^{-8}\\ \\text{at fixed }\\bar V.\\end{aligned}", "symbols_en": "$\\bar V>0$ is imposed fixed added volume (m³), $U_b$ plate bending energy (J), $C_b$ volume compliance (m³ Pa⁻¹), $D_f$ bending stiffness (N m), $b>0$ crack radius (m), and $p(b)$ the pressure required at that radius (Pa). $G_{\\bar V}$ and $G_{p_0}$ are energy-release rates under fixed volume and fixed pressure, respectively (J m⁻²); $p_0$ is maintained pressure (Pa). $d/db$ is radius differentiation holding $\\bar V$ fixed, $\\pi$ the circle constant, and $\\propto$ means proportional at fixed other parameters. The bar labels controlled volume, not averaging. The fixed-volume system here supplies no additional pressure-source work.", "symbols_zh": "$\\bar V>0$为强制固定的新增体积（m³），$U_b$为板弯曲能（J），$C_b$为体积柔度（m³ Pa⁻¹），$D_f$为弯曲刚度（N m），$b>0$为裂纹半径（m），$p(b)$为该半径处所需压力（Pa）。$G_{\\bar V}$与$G_{p_0}$分别为恒容及恒压控制下的能量释放率（J m⁻²）；$p_0$为恒定压力（Pa）。$d/db$表示保持$\\bar V$不变的半径求导，$\\pi$为圆周率，$\\propto$表示其他参数固定时成正比。横线标识受控体积，而非平均。此恒容系统不再提供额外压力源功。", "kind": "Derived loading-control comparison", "kind_zh": "推导得到的加载控制方式比较", "diagram": { "type": "energy", "labels": [ "V̄ — fixed added volume", "b ↑ advancing crack", "p(b) ↓ as compliance grows", "C_b, D_f — elastic compliance", "U_b — plate storage", "G_V̄ ∝ b⁻⁸", "G_p₀ ∝ b⁴ under fixed p₀" ], "notes": [ "Same instantaneous state does not imply the same crack trajectory." ] }, "caption_en": "Fixed pressure and fixed volume lead to opposite changes in driving force as the crack grows.", "caption_zh": "裂纹扩展时，恒压与恒容控制使驱动力发生相反变化。" }, { "id": "C4-E23", "tex": "\\begin{aligned}D_f&=\\frac{(2.00\\times10^9\\ {\\rm Pa})(10.0\\times10^{-6}\\ {\\rm m})^3}{12(1-0.35^2)}=1.899335\\times10^{-7}\\ {\\rm N\\,m},\\\\p_{0,\\rm crit}&=\\frac{\\sqrt{128(1.899335\\times10^{-7}\\ {\\rm N\\,m})(0.10\\ {\\rm J\\,m^{-2}})}}{(100\\times10^{-6}\\ {\\rm m})^2}=155.921\\ {\\rm kPa}.\\end{aligned}", "symbols_en": "$D_f$ is film bending stiffness (N m) and $p_{0,\\rm crit}$ the positive critical maintained pressure (Pa). The substituted teaching Young’s modulus is 2.00×10⁹ Pa, thickness 10.0×10⁻⁶ m, Poisson’s ratio 0.35, crack radius 100×10⁻⁶ m, and fracture resistance 0.10 J m⁻². Pa, N, m, J and kPa denote pascal, newton, metre, joule and kilopascal. $\\sqrt{\\ }$ is the positive root; superscripts are powers. Subscript f labels the film and crit the onset threshold.", "symbols_zh": "$D_f$为薄膜弯曲刚度（N m），$p_{0,\\rm crit}$为正临界恒压（Pa）。代入的教学杨氏模量为2.00×10⁹ Pa，厚度10.0×10⁻⁶ m，泊松比0.35，裂纹半径100×10⁻⁶ m，断裂阻力0.10 J m⁻²。Pa、N、m、J及kPa分别为帕、牛顿、米、焦耳及千帕。$\\sqrt{\\ }$取正根；上标表示幂。下标f表示薄膜，crit表示起始阈值。", "kind": "Checked teaching substitution", "kind_zh": "已核验的教学参数代入", "diagram": { "type": "film", "labels": [ "h_f = 10 μm — plate thickness", "b = 100 μm — existing crack", "E_f = 2 GPa, ν_f = 0.35", "D_f = 1.8993×10⁻⁷ N m", "Γ = 0.10 J/m²", "p₀,crit = 155.921 kPa" ], "notes": [ "This is not the 1 μm array-loaded payload." ] }, "caption_en": "The stated plate and interface yield a conditional 156 kPa maintained-pressure threshold.", "caption_zh": "所给薄板与界面产生约156 kPa的条件性恒压阈值。" }, { "id": "C4-E24", "tex": "\\begin{aligned}w(0)&=\\frac{p_{0,\\rm crit}b^4}{64D_f}=1.28270\\ \\mu{\\rm m},\\qquad \\frac{w(0)}{h_f}=0.12827,\\\\V_{\\rm bl}&=\\frac{\\pi p_{0,\\rm crit}b^6}{192D_f}=1.34324\\times10^{-14}\\ {\\rm m^3},\\\\|M_r(b)|&=\\frac{p_{0,\\rm crit}b^2}{8},\\qquad |\\sigma_{rr}|_{\\rm edge}=\\frac{6|M_r(b)|}{h_f^2}=11.6941\\ {\\rm MPa}.\\end{aligned}", "symbols_en": "$w(0)$ is centre displacement (m), $h_f=10$ μm thickness (m), $b=100$ μm crack radius (m), $D_f=1.899335$×10⁻⁷ N m bending stiffness, and $p_{0,\\rm crit}=155.921$ kPa maintained threshold. $V_{\\rm bl}$ is blister volume (m³), $M_r(b)$ radial bending moment per edge length (N), and $\\sigma_{rr}$ radial normal bending stress (Pa). Subscript edge labels its outer-surface value at the clamped edge; r labels radial direction. $|\\ |$ denotes magnitude, $\\pi$ the circle constant; μm and MPa are micrometre and megapascal. This stress result uses the same linear isotropic plate assumptions.", "symbols_zh": "$w(0)$为中心位移（m），$h_f=10$ μm为厚度（m），$b=100$ μm为裂纹半径（m），$D_f=1.899335$×10⁻⁷ N m为弯曲刚度，$p_{0,\\rm crit}=155.921$ kPa为恒压阈值。$V_{\\rm bl}$为鼓泡体积（m³），$M_r(b)$为单位边长径向弯矩（N），$\\sigma_{rr}$为径向法向弯曲应力（Pa）。下标edge表示夹持边缘厚度外表面的值；r表示径向。$|\\ |$表示幅值，$\\pi$为圆周率；μm与MPa分别为微米及兆帕。应力结果采用相同线性各向同性薄板假设。", "kind": "Magnitude, geometric-validity and competing-failure checks", "kind_zh": "数量级、几何适用性与竞争失效模式检验", "diagram": { "type": "film", "labels": [ "w(0) = 1.283 μm — centre lift", "h_f = 10 μm — thickness", "b = 100 μm — clamp", "V_bl = 13.4324 pL", "p₀,crit, D_f — input state", "M_r, σ_rr — edge bending", "|σ_rr| = 11.694 MPa" ], "notes": [ "Small deflection relative to thickness is checked.", "Payload allowable strength is still missing." ] }, "caption_en": "A fracture threshold must be checked against deflection validity and payload bending stress.", "caption_zh": "断裂阈值必须同时检验挠度适用性与对象弯曲应力。" }, { "id": "C4-E25", "tex": "\\delta_0=\\frac{T_{\\max}}{K_n},\\qquad t_n(\\delta)=\\begin{cases}K_n\\delta,&0\\le\\delta\\le\\delta_0,\\\\T_{\\max}\\dfrac{\\delta_c-\\delta}{\\delta_c-\\delta_0},&\\delta_0<\\delta<\\delta_c,\\\\0,&\\delta\\ge\\delta_c.\\end{cases}", "symbols_en": "$\\delta\\ge0$ now denotes physical relative tensile opening (m), not variational notation. $t_n(\\delta)$ is tensile cohesive traction magnitude (Pa), $K_n>0$ initial normal stiffness (Pa m⁻¹), $T_{\\max}>0$ peak tensile traction (Pa), $\\delta_0=T_{\\max}/K_n$ peak-traction opening (m), and $\\delta_c>\\delta_0$ complete-separation opening (m). Subscripts n, 0 and c label normal, peak onset and final separation; max labels the maximum. The law assumes monotonically increasing opening and a strictly positive softening interval. Compression and unloading are not specified by this formula.", "symbols_zh": "$\\delta\\ge0$此处表示物理相对拉伸张开（m），不再是变分符号。$t_n(\\delta)$为拉伸内聚牵引幅值（Pa），$K_n>0$为初始法向刚度（Pa m⁻¹），$T_{\\max}>0$为峰值拉伸牵引（Pa），$\\delta_0=T_{\\max}/K_n$为峰值牵引处张开（m），$\\delta_c>\\delta_0$为完全分离张开（m）。下标n、0、c分别表示法向、峰值起始及最终分离；max表示最大值。关系假设张开单调增大，且软化区间严格为正。此式未指定压缩与卸载行为。", "kind": "Adopted monotonic tensile cohesive law", "kind_zh": "采用的单调拉伸内聚定律", "diagram": { "type": "cohesive", "labels": [ "δ → interface opening", "t_n(δ) ↑ tensile resistance", "K_n — initial curve slope", "δ₀ → peak opening", "T_max → peak traction", "δ_c → complete separation" ], "notes": [ "Compression is a different branch.", "Tensile damage must not be inferred from pressure magnitude." ] }, "caption_en": "The traction–separation curve contains strength, stiffness and a finite separation distance.", "caption_zh": "牵引—分离曲线包含强度、刚度及有限分离距离。" }, { "id": "C4-E26", "tex": "\\begin{aligned}\\Gamma&=\\int_0^{\\delta_c}t_n(\\delta)\\,d\\delta\\\\&=\\frac{K_n\\delta_0^2}{2}+\\frac{T_{\\max}}{\\delta_c-\\delta_0}\\left[\\delta_c\\delta-\\frac{\\delta^2}{2}\\right]_{\\delta_0}^{\\delta_c}\\\\&=\\frac{T_{\\max}\\delta_0}{2}+\\frac{T_{\\max}(\\delta_c-\\delta_0)}{2}=\\frac{T_{\\max}\\delta_c}{2},\\\\\\delta_c&=\\frac{2\\Gamma}{T_{\\max}},\\qquad K_n>\\frac{T_{\\max}^2}{2\\Gamma}\\quad\\text{for }\\delta_0<\\delta_c.\\end{aligned}", "symbols_en": "$\\Gamma>0$ is the fracture energy of the adopted monotonic law (J m⁻²), $\\delta$ tensile opening integration variable (m), and $d\\delta$ its element (m). $t_n$ is tensile cohesive traction (Pa), $K_n>0$ initial stiffness (Pa m⁻¹), $T_{\\max}>0$ peak traction (Pa), $\\delta_0$ opening at the peak (m), and $\\delta_c>\\delta_0$ final separation opening (m). $\\int$ integrates traction against opening; endpoint brackets mean upper minus lower value. Subscripts n, 0, c and max label normal, peak onset, final separation and maximum. The last inequality enforces a nonzero softening branch; it is a model-admissibility condition.", "symbols_zh": "$\\Gamma>0$为所采用单调关系的断裂能（J m⁻²），$\\delta$为拉伸张开积分变量（m），$d\\delta$为其微元（m）。$t_n$为拉伸内聚牵引（Pa），$K_n>0$为初始刚度（Pa m⁻¹），$T_{\\max}>0$为峰值牵引（Pa），$\\delta_0$为峰值处张开（m），$\\delta_c>\\delta_0$为最终分离张开（m）。$\\int$表示牵引对张开的积分；端点方括号表示上端值减下端值。下标n、0、c、max分别表示法向、峰值起始、最终分离与最大值。末行不等式保证非零软化分支，是模型的可接受条件。", "kind": "Exact work integral of the adopted cohesive law", "kind_zh": "所采用内聚定律的精确功积分", "diagram": { "type": "cohesive", "labels": [ "t_n(δ) — tensile traction curve", "δ, dδ — separation distance", "δ₀, δ_c — branch endpoints", "T_max — curve peak", "K_n — ascending slope", "Γ — full curve area" ], "notes": [ "Reaching the peak does not pay the entire curve area." ] }, "caption_en": "Complete separation needs the entire traction–opening area, not merely the peak value.", "caption_zh": "完全分离需要完整牵引—张开曲线面积，而不仅仅达到峰值。" }, { "id": "C4-E27", "tex": "\\Gamma=0.005\\ {\\rm J\\,m^{-2}},\\quad T_{\\max}=0.10\\ {\\rm MPa},\\quad K_n=10^{13}\\ {\\rm Pa\\,m^{-1}}\\quad\\Rightarrow\\quad\\delta_0=10\\ {\\rm nm},\\quad\\delta_c=100\\ {\\rm nm}", "symbols_en": "$\\Gamma$ is assumed fracture energy (J m⁻²), $T_{\\max}$ assumed peak tensile traction (Pa), $K_n$ assumed initial normal stiffness (Pa m⁻¹), $\\delta_0$ peak-traction opening (m), and $\\delta_c$ final tensile separation (m). MPa and nm denote megapascal and nanometre. $\\Rightarrow$ denotes substitution into the triangular law. These are separate illustrative cohesive parameters, not measurements of the film release interface. Subscripts max, n, 0 and c identify peak, normal, peak opening and complete opening.", "symbols_zh": "$\\Gamma$为假设断裂能（J m⁻²），$T_{\\max}$为假设峰值拉伸牵引（Pa），$K_n$为假设初始法向刚度（Pa m⁻¹），$\\delta_0$为峰值牵引处张开（m），$\\delta_c$为最终拉伸分离（m）。MPa与nm表示兆帕与纳米。$\\Rightarrow$表示代入三角形关系。这些是独立内聚教学参数，而非薄膜释放界面的测量。下标max、n、0、c分别标识峰值、法向、峰值张开与完全张开。", "kind": "Cohesive admissibility teaching calculation", "kind_zh": "内聚模型容许条件的教学计算", "diagram": { "type": "cohesive", "labels": [ "T_max = 0.10 MPa — curve peak", "K_n = 10¹³ Pa/m — initial slope", "δ₀ = 10 nm", "δ_c = 100 nm", "Γ = 0.005 J/m² — curve area" ], "notes": [ "10 nm < 100 nm: the softening interval is admissible." ] }, "caption_en": "The example has distinct peak and final separation openings.", "caption_zh": "本例具有不同的峰值张开与最终分离张开。" }, { "id": "C4-E28", "tex": "\\begin{aligned}m_j&=\\frac{\\rho\\pi d_j^2L_j}{4}=3.926990817\\times10^{-12}\\ {\\rm kg},\\\\U_j&=\\sqrt{\\frac{2E_j}{m_j}}=31.9153824\\ {\\rm m\\,s^{-1}},\\\\E_{\\rm in}&=NE_j=50.0\\ {\\rm nJ},\\qquad I_{\\rm in}=Nm_jU_j=3.133285343\\times10^{-9}\\ {\\rm N\\,s}.\\end{aligned}", "symbols_en": "$m_j$ is carrier mass in one uniform cylindrical jet (kg), $\\rho=1000$ kg m⁻³ carrier density, $d_j=10$ μm jet diameter, $L_j=50$ μm jet length, and $\\pi$ the circle constant. $E_j=2.00$ nJ is assigned kinetic energy per jet, $U_j$ its uniform speed (m s⁻¹), $N=25$ independent identically supplied jets, $E_{\\rm in}$ total incoming kinetic energy (J), and $I_{\\rm in}$ aligned incoming momentum (N s). $\\sqrt{\\ }$ is the positive root. Subscripts j and in label single jet and incoming array; nJ is nanojoule. Simultaneous aligned arrival is assumed for the directional sum, but pressure fields are not added as N times a peak.", "symbols_zh": "$m_j$为单股均匀圆柱射流的载液质量（kg），$\\rho=1000$ kg m⁻³为载液密度，$d_j=10$ μm为射流直径，$L_j=50$ μm为长度，$\\pi$为圆周率。$E_j=2.00$ nJ为指定单射流动能，$U_j$为其均匀速度（m s⁻¹），$N=25$为独立且供给相同的射流数，$E_{\\rm in}$为总入射动能（J），$I_{\\rm in}$为同向入射动量（N s）。$\\sqrt{\\ }$取正根。下标j、in分别标识单射流与入射阵列；nJ为纳焦耳。方向求和假设同时同向到达，但不将压力场按N倍峰值相加。", "kind": "Finite incoming state inherited from the teaching source", "kind_zh": "沿用教学源模型的有限入射状态", "diagram": { "type": "jet", "labels": [ "N = 25 — independent cells", "d_j = 10 μm", "L_j = 50 μm", "ρ = 1000 kg/m³ → m_j", "E_j = 2 nJ → U_j = 31.915 m/s", "E_in = 50 nJ", "I_in = 3.1333×10⁻⁹ N s" ], "notes": [ "Finite carrier mass, not PFC vapor mass.", "No additional cluster amplification factor." ] }, "caption_en": "Finite jet energy and aligned momentum provide the input to solid coupling.", "caption_zh": "有限射流能量与同向动量提供固体耦合输入。" }, { "id": "C4-E29", "tex": "\\begin{aligned}m_f&=\\rho_fA_fh_f=(2330)(1.00\\times10^{-6})(1.00\\times10^{-6})\\ {\\rm kg}=2.33\\times10^{-9}\\ {\\rm kg},\\\\E_{\\rm solid}&=\\eta_EE_{\\rm in}=0.20(50.0\\ {\\rm nJ})=10.0\\ {\\rm nJ},\\\\I_{\\rm net}&=\\eta_II_{\\rm in}=0.50(3.133285343\\times10^{-9}\\ {\\rm N\\,s})=1.566642672\\times10^{-9}\\ {\\rm N\\,s}.\\end{aligned}", "symbols_en": "$m_f$ is payload mass (kg), $\\rho_f=2330$ kg m⁻³ density, $A_f=1.00$ mm² = 1.00×10⁻⁶ m² release area, and $h_f=1.00$ μm = 1.00×10⁻⁶ m thickness. $E_{\\rm solid}$ is assigned useful solid energy (J), $E_{\\rm in}$ incoming jet kinetic energy (J), and $\\eta_E=0.20$ a dimensionless energy-allocation assumption. $I_{\\rm net}$ is assigned net opening-direction payload impulse after all load-path reactions (N s), $I_{\\rm in}$ aligned incoming momentum (N s), and $\\eta_I=0.50$ a dimensionless impulse assumption. Labels f, in, solid, net, E and I identify film, incoming, useful solid allocation, net, energy and impulse. The numerical first-row factors represent density, area and thickness in SI units.", "symbols_zh": "$m_f$为对象质量（kg），$\\rho_f=2330$ kg m⁻³为密度，$A_f=1.00$ mm² = 1.00×10⁻⁶ m²为释放面积，$h_f=1.00$ μm = 1.00×10⁻⁶ m为厚度。$E_{\\rm solid}$为指定有效固体能量（J），$E_{\\rm in}$为入射射流动能（J），$\\eta_E=0.20$为无量纲能量分配假设。$I_{\\rm net}$为考虑全部传力路径反作用后的净张开方向对象冲量（N s），$I_{\\rm in}$为同向入射动量（N s），$\\eta_I=0.50$为无量纲冲量假设。标签f、in、solid、net、E、I分别标识薄膜、入射、有效固体分配、净量、能量与冲量。第一行数值因子分别是SI制密度、面积及厚度。", "kind": "Explicit conditional solid-coupling allocation", "kind_zh": "明确给定且有条件的固体耦合分配", "diagram": { "type": "film", "labels": [ "A_f = 1 mm² — intended release", "h_f = 1 μm — payload thickness", "ρ_f = 2330 kg/m³ → m_f = 2.33 μg", "η_E = 0.20 → E_solid = 10 nJ", "η_I = 0.50 → I_net = 1.5666×10⁻⁹ N s", "E_in, I_in — incoming array" ], "notes": [ "Load-path coupling fractions are assumptions.", "This payload differs from the blister benchmark." ] }, "caption_en": "Declared coupling converts incoming jet budgets into conditional payload budgets.", "caption_zh": "明确声明的耦合将入射射流预算转换为条件性对象预算。" }, { "id": "C4-E30", "tex": "E_{\\rm req}(v_f)=\\Gamma A_f+\\frac12m_fv_f^2,\\qquad I_{\\rm req}(v_f)=m_fv_f,\\qquad E_{\\rm solid}\\ge E_{\\rm req}(v_f),\\qquad I_{\\rm net}\\ge I_{\\rm req}(v_f)", "symbols_en": "$E_{\\rm req}(v_f)$ is minimum fracture-plus-translation energy (J), $I_{\\rm req}(v_f)$ required net impulse for the minimum departure speed (N s), $v_f\\ge0$ that required speed (m s⁻¹), $m_f>0$ payload mass (kg), $A_f$ full intended release area (m²), and $\\Gamma\\ge0$ assumed uniform fracture energy (J m⁻²). $E_{\\rm solid}$ is assigned available solid energy (J); $I_{\\rm net}$ is net opening-direction impulse after interface/support reactions (N s). Subscripts req, f, solid and net identify required, payload, useful solid and net quantities. The inequalities are necessary under the stated zero-initial-energy/no-additional-source assumptions; they are not spatial fracture solutions.", "symbols_zh": "$E_{\\rm req}(v_f)$为最低断裂加平动能量（J），$I_{\\rm req}(v_f)$为达到最低离开速度所需净冲量（N s），$v_f\\ge0$为所需速度（m s⁻¹），$m_f>0$为对象质量（kg），$A_f$为全部目标释放面积（m²），$\\Gamma\\ge0$为假设均匀断裂能（J m⁻²）。$E_{\\rm solid}$为指定可用固体能量（J）；$I_{\\rm net}$为考虑界面／支承反作用后的净张开方向冲量（N s）。下标req、f、solid、net分别标识所需、对象、有效固体及净量。在所述零初始能量／无额外源假设下，不等式是必要条件，而非空间断裂解。", "kind": "Necessary global energy and momentum screens", "kind_zh": "全局能量与动量的必要条件筛查", "diagram": { "type": "energy", "labels": [ "E_solid → available budget", "ΓA_f → full release cost", "½m_f v_f² → departure kinetic energy", "I_net → net opening impulse", "m_f v_f → required momentum", "E_req, I_req — minimum screens" ], "notes": [ "Passing is necessary, not sufficient.", "Support reactions are already included in net impulse." ] }, "caption_en": "The full intended release area, not a pressure peak, determines the fracture-energy cost.", "caption_zh": "决定断裂能消耗的是全部目标释放面积，而非压力峰值。" }, { "id": "C4-E31", "tex": "\\Gamma=0.020\\ {\\rm J\\,m^{-2}}\\quad\\Rightarrow\\quad \\Gamma A_f=(0.020)(1.00\\times10^{-6})\\ {\\rm J}=20.0\\ {\\rm nJ}>E_{\\rm solid}=10.0\\ {\\rm nJ}", "symbols_en": "$\\Gamma$ is assumed uniform release fracture energy (J m⁻²), $A_f=1.00$×10⁻⁶ m² intended full release area, and $E_{\\rm solid}=10.0$ nJ assigned available solid energy. nJ is nanojoule; $\\Rightarrow$ denotes numerical substitution. The comparison assumes no initially stored recoverable energy and no additional post-impact source. Subscript f labels film area, and solid labels the assigned useful energy.", "symbols_zh": "$\\Gamma$为假设均匀释放断裂能（J m⁻²），$A_f=1.00$×10⁻⁶ m²为全部目标释放面积，$E_{\\rm solid}=10.0$ nJ为指定可用固体能量。nJ为纳焦耳；$\\Rightarrow$表示数值代入。比较假设没有初始储存的可恢复能量，也没有额外冲击后源。下标f表示薄膜面积，solid表示指定有效能量。", "kind": "Failed release-energy screen", "kind_zh": "未通过的释放能量筛查", "diagram": { "type": "energy", "labels": [ "Γ = 0.020 J/m²", "A_f = 1 mm²", "ΓA_f = 20 nJ — required release", "E_solid = 10 nJ — available", "early rigid impact ≈ 47.23 MPa" ], "notes": [ "High local pressure coexists with insufficient total release work." ] }, "caption_en": "The assigned source cannot release the whole interface even before departure energy is added.", "caption_zh": "即使尚未计入离开动能，所指定源也无法释放整个界面。" }, { "id": "C4-E32", "tex": "\\begin{aligned}\\Gamma A_f&=5.00\\ {\\rm nJ},\\qquad \\frac12m_fv_f^2=\\frac12(2.33\\times10^{-9})(0.50)^2\\ {\\rm J}=0.29125\\ {\\rm nJ},\\\\E_{\\rm req}(0.50\\ {\\rm m\\,s^{-1}})&=5.29125\\ {\\rm nJ}<10.0\\ {\\rm nJ},\\\\I_{\\rm req}(0.50\\ {\\rm m\\,s^{-1}})&=(2.33\\times10^{-9})(0.50)\\ {\\rm N\\,s}=1.165\\times10^{-9}\\ {\\rm N\\,s}0$ normal payload flight gap (m), and $\\theta\\in(-\\pi/2,\\pi/2)$ departure angle relative to the target normal. $\\tan$ is tangent; the numerical angle is converted from degrees to radians for evaluation. μm is micrometre and $\\Rightarrow$ denotes substitution. Subscript f labels film/payload flight, not liquid-jet travel; $\\Delta$ marks position difference. The relation assumes straight flight without significant drag, gravity or rotation-dependent aerodynamic force.", "symbols_zh": "$\\Delta x$为对象横向偏移（m），$H_f>0$为对象法向飞行间隙（m），$\\theta\\in(-\\pi/2,\\pi/2)$为相对目标法向的离开角。$\\tan$为正切；数值角度由度转换为弧度计算。μm为微米，$\\Rightarrow$表示代入。下标f标识薄膜／对象飞行，而非液体射流运动；$\\Delta$表示位置差。关系假设直线飞行，且无显著阻力、重力或随转动变化的气动力。", "kind": "Ballistic placement estimate", "kind_zh": "弹道式定位估算", "diagram": { "type": "film", "labels": [ "H_f — payload normal flight gap", "θ — departure angle", "Δx — lateral placement error", "target normal ↓ receiver", "H_f = 100 μm, θ = 1°" ], "notes": [ "A 1° departure error gives about 1.75 μm offset in this example." ] }, "caption_en": "A finite receiving gap converts angular error into placement error.", "caption_zh": "有限接收间隙将角度误差转换为落位误差。" }, { "id": "C4-E35", "tex": "\\boldsymbol t_l=\\boldsymbol T_l\\boldsymbol n_f,\\qquad \\boldsymbol I_l=\\int_{t_a}^{t_b}\\int_{A_f(t)}\\boldsymbol t_l\\,dA\\,dt,\\qquad \\mathcal W_l=\\int_{t_a}^{t_b}\\int_{A_f(t)}\\boldsymbol t_l\\cdot\\boldsymbol v_f\\,dA\\,dt", "symbols_en": "Original formulas: $\\boldsymbol t_l$ is fluid traction on the solid (Pa), $\\boldsymbol T_l$ liquid stress tensor (Pa), $\\boldsymbol n_f$ solid-to-liquid unit normal, $\\boldsymbol I_l$ applied liquid impulse (N s), and $\\mathcal W_l$ delivered work (J). $A_f(t)$ is loaded surface (m²), $dA$ its element (m²), $\\boldsymbol v_f$ surface velocity (m s⁻¹), and $t_a,t_b,t,dt$ event endpoints, integration time and time element (s). $\\int$ means surface/time integration; the dot denotes a vector projection. Subscripts l and f label liquid and film. The same identities first load PVC if it is the exposed layer.", "symbols_zh": "原公式：$\\boldsymbol t_l$为流体施加于固体的牵引（Pa），$\\boldsymbol T_l$为液体应力张量（Pa），$\\boldsymbol n_f$为固体指向液体的单位法向，$\\boldsymbol I_l$为液体外加冲量（N s），$\\mathcal W_l$为传递功（J）。$A_f(t)$为受载表面（m²），$dA$为其微元（m²），$\\boldsymbol v_f$为表面速度（m s⁻¹），$t_a,t_b,t,dt$分别为事件起止时间、积分时间及时间微元（s）。$\\int$表示表面／时间积分；点乘表示向量投影。下标l、f表示液体与薄膜。若PVC为外露层，同一恒等式首先加载PVC。", "kind": "Original load, impulse and work formulas", "kind_zh": "原始载荷、冲量与功公式", "diagram": { "type": "impact", "labels": [ "T_l, n_f → t_l at exposed face", "A_f(t), t_a → t_b — footprint and event", "I_l — force-time integral", "v_f → W_l requires motion" ], "notes": [ "Exposed PVC and buried film are different surfaces." ] }, "caption_en": "Reference answer: pressure, impulse and work describe different parts of the transmission path.", "caption_zh": "参考答案：压力、冲量与功描述传递路径中的不同部分。" }, { "id": "C4-E36", "tex": "E_{\\rm solid}\\ge\\Gamma A_f+\\frac12m_fv_f^2,\\qquad I_{\\rm net}\\ge m_fv_f", "symbols_en": "Original necessary screens: $E_{\\rm solid}$ is assigned available solid energy (J), $\\Gamma$ uniform release fracture energy (J m⁻²), $A_f$ intended full release area (m²), $m_f$ payload mass (kg), $v_f$ minimum required departure speed (m s⁻¹), and $I_{\\rm net}$ net opening impulse after all reactions (N s). The payload initially rests and has no additional recoverable source. Subscripts solid, f and net label useful solid allocation, payload and net impulse.", "symbols_zh": "原必要筛选：$E_{\\rm solid}$为指定可用固体能量（J），$\\Gamma$为均匀释放断裂能（J m⁻²），$A_f$为全部目标释放面积（m²），$m_f$为对象质量（kg），$v_f$为最低所需离开速度（m s⁻¹），$I_{\\rm net}$为考虑全部反作用后的净张开冲量（N s）。对象初始静止，且无额外可恢复源。下标solid、f、net标识有效固体分配、对象及净冲量。", "kind": "Original global screens", "kind_zh": "原始全局筛查公式", "diagram": { "type": "energy", "labels": [ "E_solid — available work budget", "ΓA_f — release cost", "m_f, v_f — finite payload translation", "I_net — actual net impulse" ], "notes": [ "Peak pressure supplies neither the area nor the energy budget." ] }, "caption_en": "Reference answer: finite energy and net impulse must reach the intended interface.", "caption_zh": "参考答案：有限能量与净冲量必须到达目标界面。" }, { "id": "C4-E37", "tex": "w(r)=\\frac{p_0}{64D_f}(b^2-r^2)^2,\\qquad G_{p_0}=\\frac{p_0^2b^4}{128D_f},\\qquad p_{0,\\rm crit}=\\frac{\\sqrt{128D_f\\Gamma}}{b^2}", "symbols_en": "Original blister formulas: $w(r)$ is opening displacement (m), $r\\in[0,b]$ radial position (m), $b>0$ existing circular crack radius (m), $p_0\\ge0$ uniform maintained pressure difference (Pa), $D_f>0$ linear plate bending stiffness (N m), $G_{p_0}$ fixed-pressure energy-release rate (J m⁻²), $\\Gamma>0$ constant fracture resistance (J m⁻²), and $p_{0,\\rm crit}$ positive onset pressure (Pa). $\\sqrt{\\ }$ is the positive root. Subscripts f, 0 and crit denote film, maintained load and threshold. These expressions require regular centre and a clamped crack edge, quasistatic bending, and negligible stretching/pretension.", "symbols_zh": "原鼓泡公式：$w(r)$为张开位移（m），$r\\in[0,b]$为径向位置（m），$b>0$为预存圆裂纹半径（m），$p_0\\ge0$为均匀恒定压差（Pa），$D_f>0$为线性薄板弯曲刚度（N m），$G_{p_0}$为恒压能量释放率（J m⁻²），$\\Gamma>0$为恒定断裂阻力（J m⁻²），$p_{0,\\rm crit}$为正起始压力（Pa）。$\\sqrt{\\ }$取正根。下标f、0、crit分别表示薄膜、恒定载荷与阈值。表达式要求中心正则、裂纹边缘夹持、准静态弯曲，且拉伸／预张力可忽略。", "kind": "Original pressure-controlled benchmark formulas", "kind_zh": "原始压力控制基准模型公式", "diagram": { "type": "film", "labels": [ "r, b — circular crack geometry", "w(r) ↑ clamped profile", "p₀ — uniform maintained pressure", "D_f — bending stiffness", "Γ — fracture resistance", "G_p₀, p₀,crit — onset quantities" ], "notes": [ "A finite short jet is not a maintained pressure reservoir." ] }, "caption_en": "Reference answer: this pressure threshold follows a particular geometry and loading control.", "caption_zh": "参考答案：此压力阈值源于特定几何及加载控制。" }, { "id": "C4-E38", "tex": "\\Gamma=\\int_0^{\\delta_c}t_n(\\delta)\\,d\\delta=\\frac12T_{\\max}\\delta_c,\\qquad G\\ge\\Gamma(\\psi,T_i,v_c)", "symbols_en": "Original interface formulas: $\\Gamma$ is tensile cohesive fracture work per area (J m⁻²), $\\delta$ physical opening integration variable (m), $d\\delta$ its element (m), $\\delta_c$ complete opening (m), $t_n$ tensile cohesive traction (Pa), and $T_{\\max}$ peak tensile traction (Pa). The half-product applies to the adopted triangular monotonic law. $G$ is energy available per new crack area (J m⁻²), $\\psi$ dimensionless mode mixture, $T_i$ interface temperature (K), and $v_c$ crack speed (m s⁻¹). $\\int$ denotes opening integration. Subscripts n, c, i and max identify normal, crack/final opening, interface and maximum.", "symbols_zh": "原界面公式：$\\Gamma$为单位面积拉伸内聚断裂功（J m⁻²），$\\delta$为物理张开积分变量（m），$d\\delta$为其微元（m），$\\delta_c$为完全张开（m），$t_n$为拉伸内聚牵引（Pa），$T_{\\max}$为峰值拉伸牵引（Pa）。一半乘积适用于所采用的三角形单调关系。$G$为单位新增裂纹面积可用能量（J m⁻²），$\\psi$为无量纲模态混合参数，$T_i$为界面温度（K），$v_c$为裂纹速度（m s⁻¹）。$\\int$表示张开积分。下标n、c、i、max分别表示法向、裂纹／最终张开、界面与最大值。", "kind": "Original cohesive work and fracture criteria", "kind_zh": "原始内聚功与断裂判据", "diagram": { "type": "cohesive", "labels": [ "δ → tensile opening", "t_n, T_max — strength curve", "δ_c — final separation", "Γ — full curve area", "G — crack energy supply", "ψ, T_i, v_c — actual resistance conditions" ], "notes": [ "Reaching strength does not establish complete separation." ] }, "caption_en": "Reference answer: successful release requires both initiation and separation work at the correct interface.", "caption_zh": "参考答案：成功释放需要在正确界面上同时满足起始与分离功要求。" }, { "id": "C4-E39", "tex": "E_{\\rm solid}\\ge\\Gamma A_f+\\frac12m_fv_f^2,\\qquad I_{\\rm net}\\ge m_fv_f,\\qquad \\Delta x=H_f\\tan\\theta", "symbols_en": "Original screens and placement relation: $E_{\\rm solid}$ is available useful solid energy (J), $\\Gamma$ fracture energy (J m⁻²), $A_f$ full release area (m²), $m_f$ payload mass (kg), $v_f$ minimum departure speed (m s⁻¹), and $I_{\\rm net}$ net opening impulse (N s). $\\Delta x$ is lateral payload offset (m), $H_f$ normal payload flight gap (m), $\\theta$ departure angle relative to receiver normal, and $\\tan$ tangent. The screens assume initial rest without another energy source; the placement relation assumes straight short flight. Subscripts solid, f and net label useful solid energy, payload and net impulse.", "symbols_zh": "原筛选与落位关系：$E_{\\rm solid}$为可用有效固体能量（J），$\\Gamma$为断裂能（J m⁻²），$A_f$为全部释放面积（m²），$m_f$为对象质量（kg），$v_f$为最低离开速度（m s⁻¹），$I_{\\rm net}$为净张开冲量（N s）。$\\Delta x$为对象横向偏移（m），$H_f$为对象法向飞行间隙（m），$\\theta$为相对接收面法向的离开角，$\\tan$为正切。筛选假设初始静止且无其他能量源；落位关系假设短时直线飞行。下标solid、f、net标识有效固体能量、对象及净冲量。", "kind": "Original necessary operating-window checks", "kind_zh": "原始工作窗口的必要条件检验", "diagram": { "type": "film", "labels": [ "ΓA_f — intended release", "m_f, v_f — outgoing intact object", "E_solid, I_net — transmitted budgets", "H_f, θ → Δx at receiver" ], "notes": [ "Full release, intact flight and repeatable reset are separate requirements." ] }, "caption_en": "Reference answer: passing global budgets starts, rather than completes, the transfer argument.", "caption_zh": "参考答案：通过整体预算是转印论证的起点，而非终点。" }, { "id": "C5-E01", "tex": "\\begin{aligned}p(\\rho)&=(p_j+\\Pi)\\left(\\frac{\\rho}{\\rho_j}\\right)^{\\gamma}-\\Pi,\\\\ c^{2}&=\\frac{dp}{d\\rho}=\\frac{\\gamma\\,[\\,p(\\rho)+\\Pi\\,]}{\\rho},\\qquad c_0=\\sqrt{\\frac{\\gamma\\,(p_j+\\Pi)}{\\rho_j}}.\\end{aligned}", "symbols_en": "$p$ — Absolute pressure; $\\rho$ — Liquid density; $\\rho_j$ — Reference density 998.1539919 at the reference pressure; $p_j$ — Reference pressure 101325; $\\Pi$ — Stiffness pressure 6.00 × 10⁸; $\\gamma$ — Stiffness exponent 4.4; $c$ — Local sound speed; $c_0$ — Reference sound speed; $d/d\\rho$ — Derivative operator with respect to density.", "symbols_zh": "$p$ — 绝对压力；$\\rho$ — 液体密度；$\\rho_j$ — 参考压力下的参考密度 998.1539919；$p_j$ — 参考压力 101325；$\\Pi$ — 刚度压力 6.00 × 10⁸；$\\gamma$ — 刚度指数 4.4；$c$ — 局部声速；$c_0$ — 参考声速；$d/d\\rho$ — 对密度的导数算符。", "kind": "Adopted barotropic stiffened-gas law and exact sound speed", "kind_zh": "采用的正压刚化气体律与精确声速", "diagram": { "type": "impact", "labels": [ "p(ρ): barotropic stiffened-gas law", "ρ_j, p_j: undisturbed jet state", "Π, γ: stiffness parameters", "c² = dp/dρ = γ(p+Π)/ρ", "c_0 = 1626.447 m/s" ], "notes": [ "Adopted constitutive law, not derived from first principles", "Both solvers use this same law" ] }, "caption_en": "One density-only law fixes pressure and sound speed for every later step.", "caption_zh": "一条仅含密度的状态律为后续每一步确定压力与声速。" }, { "id": "C5-E02", "tex": "\\begin{aligned}p_{\\mathrm{early}}&=\\frac{Z_lZ_r}{Z_l+Z_r}\\,U\\ \\xrightarrow{\\ Z_r\\to\\infty\\ }\\ Z_0U,\\qquad Z_0=\\rho_jc_0,\\\\ p_A&=p_j+Z_0U.\\end{aligned}", "symbols_en": "$p_{\\mathrm{early}}$ — Early compressive contact pressure from Chapter 3 matching; $Z_l$ — Liquid longitudinal impedance; $Z_r$ — Receiver longitudinal impedance; $U$ — Prescribed incoming speed 99.027; $Z_0$ — Reference acoustic impedance $ρ_{j}$ $c_{0}$; $p_A$ — Absolute acoustic wall pressure; $p_j$ — Reference pressure; $\\rho_j$ — Reference density; $c_0$ — Reference sound speed.", "symbols_zh": "$p_{\\mathrm{early}}$ — 第三章匹配得到的早期压缩接触压力；$Z_l$ — 液体纵向阻抗；$Z_r$ — 接收体纵向阻抗；$U$ — 给定入射速度 99.027；$Z_0$ — 参考声阻抗 $ρ_{j}$ $c_{0}$；$p_A$ — 绝对声学壁面压力；$p_j$ — 参考压力；$\\rho_j$ — 参考密度；$c_0$ — 参考声速。", "kind": "Rigid-wall limit of the Chapter 3 acoustic matching", "kind_zh": "第三章声学匹配的刚性壁面极限", "diagram": { "type": "impact", "labels": [ "U: incoming liquid toward the wall", "Z_r → ∞: rigid wall, v_i → 0", "Z_0 = ρ_j c_0", "p_A = p_j + Z_0 U", "Z_0 U = 160.765 MPa" ], "notes": [ "Linear estimate: valid only for U/c_0 ≪ 1", "Chapter 3 Eq. (C3-E24) is the starting point" ] }, "caption_en": "Letting the receiver impedance grow without bound turns the Chapter 3 contact pressure into ρc times speed.", "caption_zh": "令接收体阻抗趋于无穷，第三章的接触压力变为 ρc 乘速度。" }, { "id": "C5-E03", "tex": "\\begin{aligned}\\rho_1\\,(U_s-U)&=\\rho_j\\,U_s,\\\\ p_1-p_j&=\\rho_j\\,U_s\\,U,\\\\ x&\\equiv\\frac{U}{U_s}=1-\\frac{\\rho_j}{\\rho_1}.\\end{aligned}", "symbols_en": "$\\rho_1$ — Density behind the shock; $\\rho_j$ — Density of the incoming liquid; $U_s$ — Shock speed relative to the incoming liquid; $U$ — Incoming liquid speed toward the wall; $p_1$ — Absolute pressure behind the shock; $p_j$ — Absolute pressure ahead of the shock; $x$ — Velocity ratio U/$U_{s}$.", "symbols_zh": "$\\rho_1$ — 激波后密度；$\\rho_j$ — 入射液体密度；$U_s$ — 相对入射液体的激波速度；$U$ — 入射液体朝壁面的速度；$p_1$ — 激波后绝对压力；$p_j$ — 激波前绝对压力；$x$ — 速度比 U/$U_{s}$。", "kind": "Mass and momentum jump across the stopping shock", "kind_zh": "停止激波两侧的质量与动量跳跃", "diagram": { "type": "impact", "labels": [ "Wall at z = 0; liquid moves toward −z", "Shock front travels toward +z", "Ahead: ρ_j, p_j, speed U", "Behind: ρ_1, p_1, at rest", "U_s: front speed relative to incoming liquid" ], "notes": [ "Exact for a planar steady shock", "Closure for ρ_1 and p_1 comes in Eq. (C5-E04)" ] }, "caption_en": "A shock that stops the liquid obeys mass and momentum balance across its thin front.", "caption_zh": "使液体停止的激波，其薄波前两侧服从质量与动量平衡。" }, { "id": "C5-E04", "tex": "\\begin{aligned}e(p,\\rho)&=\\frac{p+\\gamma\\Pi}{(\\gamma-1)\\,\\rho},\\qquad e_1-e_j=\\tfrac12\\,(p_1+p_j)\\left(\\frac1{\\rho_j}-\\frac1{\\rho_1}\\right),\\\\ &x^{2}+\\frac{(\\gamma+1)M^{2}}{2}\\,x-M^{2}=0\\ \\Longrightarrow\\ x=-k+\\sqrt{k^{2}+M^{2}},\\\\ &k=\\frac{(\\gamma+1)M^{2}}{4},\\qquad M=\\frac{U}{c_0},\\qquad p_H=p_j+\\frac{\\rho_jU^{2}}{x}.\\end{aligned}", "symbols_en": "$e$ — Specific internal energy of the stiffened-gas liquid; $e_1$ — Specific internal energy behind the shock; $e_j$ — Specific internal energy of the incoming liquid; $k$ — Auxiliary constant (γ+1)M²/4; $M$ — Impact Mach number U/$c_{0}$; $x$ — Velocity ratio U/$U_{s}$ from Eq. (C5-E03); $p_H$ — Hugoniot wall pressure; $\\gamma$ — Stiffness exponent 4.4; $\\Pi$ — Stiffness pressure 6.00 × 10⁸; $c_0$ — Reference sound speed; $U$ — Prescribed impact speed.", "symbols_zh": "$e$ — 刚化气体液体的比内能；$e_1$ — 激波后比内能；$e_j$ — 入射液体比内能；$k$ — 辅助常数 (γ+1)M²/4；$M$ — 撞击马赫数 U/$c_{0}$；$x$ — 来自式（C5-E03）的速度比 U/$U_{s}$；$p_H$ — Hugoniot 壁面压力；$\\gamma$ — 刚度指数 4.4；$\\Pi$ — 刚度压力 6.00 × 10⁸；$c_0$ — 参考声速；$U$ — 给定撞击速度。", "kind": "Energy-conserving Hugoniot pressure in closed form", "kind_zh": "能量守恒的 Hugoniot 压力封闭形式", "diagram": { "type": "impact", "labels": [ "e(p,ρ): stiffened-gas internal energy", "Energy jump: e₁ − e_j = ½(p₁+p_j)(1/ρ_j − 1/ρ₁)", "x: positive root of the quadratic", "M = U/c_0 = 0.0609", "p_H = 174.623 MPa" ], "notes": [ "Reference value for every later comparison", "Closed form agrees with a numerical root of the energy residual" ] }, "caption_en": "Energy conservation closes the jump and gives one closed-form reference pressure.", "caption_zh": "能量守恒使跳跃条件封闭，并给出唯一的封闭形式参照压力。" }, { "id": "C5-E05", "tex": "\\begin{aligned}\\rho_1&=\\rho_j\\,\\frac{U+S}{S},\\qquad U_s=U+S,\\\\ p(\\rho_1)&=p_j+\\rho_j\\,(U+S)\\,U,\\qquad p_B=p_j+\\rho_j\\,(U+S_B)\\,U.\\end{aligned}", "symbols_en": "$S$ — Lab-frame shock speed toward +z; $S_B$ — Root of the barotropic condition; $\\rho_1$ — Density behind the shock; $\\rho_j$ — Incoming density; $U$ — Incoming speed; $U_s$ — Shock speed relative to the incoming liquid; $p(\\rho_1)$ — Pressure from the law of Eq. (C5-E01) at density $ρ_{1}$; $p_j$ — Incoming pressure; $p_B$ — Barotropic shock pressure.", "symbols_zh": "$S$ — 实验室参考系中朝 +z 的激波速度；$S_B$ — 正压条件的根；$\\rho_1$ — 激波后密度；$\\rho_j$ — 入射密度；$U$ — 入射速度；$U_s$ — 相对入射液体的激波速度；$p(\\rho_1)$ — 由式（C5-E01）在密度 $ρ_{1}$ 处给出的压力；$p_j$ — 入射压力；$p_B$ — 正压激波压力。", "kind": "Barotropic shock pressure used by both solvers", "kind_zh": "两个求解器所用的正压激波压力", "diagram": { "type": "impact", "labels": [ "S: lab-frame shock speed", "U_s = U + S", "ρ₁ = ρ_j (U+S)/S", "p(ρ₁) from Eq. (C5-E01)", "p_B = 174.155 MPa" ], "notes": [ "p_B lies 0.268% below p_H", "The gap is the energy equation that the barotropic model omits" ] }, "caption_en": "Replacing the energy equation by the density-only law shifts the planar pressure by a fraction of a percent.", "caption_zh": "以仅含密度的定律替代能量方程，使平面压力改变不足百分之一。" }, { "id": "C5-E06", "tex": "\\begin{aligned}\\bar p_R(t)&=\\frac{2}{R^{2}}\\int_0^{R}p(r,0,t)\\,r\\,dr,\\qquad R\\in\\left\\{a,\\ \\tfrac a2\\right\\},\\\\ \\tau&=\\frac{c_0\\,t}{a},\\qquad \\Lambda=\\frac{\\max_t\\,\\bar p_a(t)}{p_H}.\\end{aligned}", "symbols_en": "$\\bar p_R$ — Area-weighted mean absolute pressure on the wall disk; $R$ — Sensor radius, a or a/2; $p(r,0,t)$ — Wall pressure at radius r, z = 0; $r$ — Radial coordinate; $a$ — Jet radius; $t$ — Time from first contact; $\\tau$ — Dimensionless time $c_{0}$ t / a; $c_0$ — Reference sound speed; $\\Lambda$ — Finite-area peak factor; $p_H$ — Hugoniot wall pressure.", "symbols_zh": "$\\bar p_R$ — 壁面圆盘上的面积加权平均绝对压力；$R$ — 传感区半径，取 a 或 a/2；$p(r,0,t)$ — 半径 r、z = 0 处的壁面压力；$r$ — 径向坐标；$a$ — 射流半径；$t$ — 自首次接触起的时间；$\\tau$ — 无量纲时间 $c_{0}$ t / a；$c_0$ — 参考声速；$\\Lambda$ — 有限面积峰值因子；$p_H$ — Hugoniot 壁面压力。", "kind": "Area-weighted wall observer and dimensionless time", "kind_zh": "面积加权壁面观测量与无量纲时间", "diagram": { "type": "impact", "labels": [ "Wall z = 0, disk r ≤ R", "R = a: full sensor; R = a/2: centre", "Weight 2r/R² dr", "τ = c_0 t / a", "Λ = peak p̄_a / p_H" ], "notes": [ "Observer is fixed before comparing solvers", "Fixed footprint and window follow Chapter 3 Eq. (C3-E26)" ] }, "caption_en": "The reported wall pressure is a mean over a fixed disk with area weights.", "caption_zh": "所报告的壁面压力是固定圆盘上带面积权重的平均值。" }, { "id": "C5-E07", "tex": "\\begin{aligned}f_{\\mathrm{rel}}(\\tau)&=1-(1-\\tau)^{2},\\qquad 0\\le\\tau\\le1,\\\\ \\frac{\\bar p_a(\\tau)}{p_H}&\\approx 1-f_{\\mathrm{rel}}(\\tau)=(1-\\tau)^{2}\\quad\\text{if the released annulus keeps no excess pressure.}\\end{aligned}", "symbols_en": "$f_{\\mathrm{rel}}$ — Fraction of the wall disk reached by edge release; $\\tau$ — Dimensionless time $c_{0}$ t / a; $\\bar p_a$ — Full-sensor mean wall pressure; $p_H$ — Hugoniot wall pressure; $a$ — Jet radius; $c_0$ — Sound speed carrying the release inward.", "symbols_zh": "$f_{\\mathrm{rel}}$ — 边缘释放已到达的壁面圆盘面积分数；$\\tau$ — 无量纲时间 $c_{0}$ t / a；$\\bar p_a$ — 全传感区平均壁面压力；$p_H$ — Hugoniot 壁面压力；$a$ — 射流半径；$c_0$ — 向内传播释放的声速。", "kind": "Edge-release scale estimate for the finite sensor", "kind_zh": "有限传感区的边缘释放尺度估计", "diagram": { "type": "impact", "labels": [ "Release front from the jet edge", "Annulus r > a(1−τ) has relaxed", "Core r < a(1−τ) still at p_H", "f_rel = 1 − (1−τ)²", "a/c_0 = 0.615 ns" ], "notes": [ "Free side of the jet exists at the first instant of contact", "Estimate fixes the sign and size of the gap, not its value" ] }, "caption_en": "Because the jet edge is free from first contact, an annulus relaxes while the core holds p_H.", "caption_zh": "射流边缘自首次接触起即为自由面，环带松弛而核心仍保持 p_H。" }, { "id": "C5-E08", "tex": "\\begin{aligned}A_j&=\\pi a^{2},\\qquad \\dot m=\\rho_j\\,U\\,A_j,\\qquad \\dot K=\\tfrac12\\,\\rho_j\\,U^{3}A_j,\\\\ K(t)&=\\dot K\\,t\\quad\\text{for }t\\le t_w,\\qquad t_w<\\frac{2a}{U_s-U}.\\end{aligned}", "symbols_en": "$A_j$ — Jet cross-section area; $a$ — Jet radius; $\\dot m$ — Incident mass flow rate; $\\dot K$ — Incident kinetic-energy flux; $K$ — Kinetic energy crossing the plane up to time t; $\\rho_j$ — Incoming density; $U$ — Incoming speed; $t_w$ — Incident-flux window; $U_s$ — Shock speed relative to the incoming liquid.", "symbols_zh": "$A_j$ — 射流横截面积；$a$ — 射流半径；$\\dot m$ — 入射质量流率；$\\dot K$ — 入射动能通量；$K$ — 截至时间 t 穿过平面的动能；$\\rho_j$ — 入射密度；$U$ — 入射速度；$t_w$ — 入射通量时间窗；$U_s$ — 相对入射液体的激波速度。", "kind": "Incident mass and kinetic-energy flux through the observation plane", "kind_zh": "穿过观察平面的入射质量与动能通量", "diagram": { "type": "energy", "labels": [ "Plane z = 2a above the wall", "A_j = π a²", "ṁ = ρ_j U A_j", "K̇ = ½ ρ_j U³ A_j", "K̇ = 1.5226 mW; K(1 ns) = 1.5226 pJ" ], "notes": [ "Compare with Chapter 3 Eq. (C3-E27) for energy accounting", "The reflected shock reaches z = 2a at about 1.20 ns for a = 1 μm" ] }, "caption_en": "The energy delivered to the plane is a kinetic-energy flux times the window, which bounds nothing at the wall.", "caption_zh": "到达平面的能量等于动能通量乘以时间窗，它并不约束壁面。" }, { "id": "C5-E09", "tex": "\\begin{aligned}M&=\\frac{U}{c_0}\\in[0.031,\\ 0.092],\\qquad t_a=\\frac{a}{c_0},\\\\ \\frac{\\rho_1}{\\rho_j}&=\\frac{1}{1-x}=1.0594,\\qquad \\frac{p_H-p_B}{p_H}=0.268\\%.\\end{aligned}", "symbols_en": "$M$ — Impact Mach number; $U$ — Prescribed speed, 50 to 150; $c_0$ — Reference sound speed; $t_a$ — Time for sound to cross the jet radius; $a$ — Jet radius, 0.5 to 2 μm; $\\rho_1/\\rho_j$ — Compression ratio behind the shock; $x$ — Velocity ratio at U = 99.027 m s⁻¹; $p_H$ — Hugoniot pressure; $p_B$ — Barotropic pressure.", "symbols_zh": "$M$ — 撞击马赫数；$U$ — 给定速度，50 至 150；$c_0$ — 参考声速；$t_a$ — 声波穿过射流半径的时间；$a$ — 射流半径，0.5 至 2 μm；$\\rho_1/\\rho_j$ — 激波后压缩比；$x$ — U = 99.027 m s⁻¹ 时的速度比；$p_H$ — Hugoniot 压力；$p_B$ — 正压压力。", "kind": "Dimensionless groups and validity range of the model", "kind_zh": "模型的无量纲数与适用范围", "diagram": { "type": "timescale", "labels": [ "M = U / c_0 = 0.0609 at the nominal case", "Tested M from 0.031 to 0.092", "t_a = a / c_0 = 0.615 ns for a = 1 μm", "ρ₁/ρ_j = 1.0594", "(p_H − p_B)/p_H = 0.268%" ], "notes": [ "Outside the tested M range the conclusions are not claimed", "Liquid-only: no gas, no elastic wall, no cavitation" ] }, "caption_en": "Small Mach number and small compression are what let a barotropic liquid stand in for the full energy law.", "caption_zh": "小马赫数与小压缩，使正压液体可以替代完整能量律。" }, { "id": "C5-E10", "tex": "\\begin{aligned}\\partial_t\\rho+\\partial_z(\\rho u)&=0,\\\\ \\partial_t(\\rho u)+\\partial_z\\!\\left(\\rho u^{2}+p(\\rho)\\right)&=0,\\\\ u(0,t)=0,&\\qquad (\\rho,u)\\big|_{z=H}=(\\rho_j,\\,-U).\\end{aligned}", "symbols_en": "$\\rho$ — Liquid density; $u$ — Axial velocity, negative toward the wall; $p(\\rho)$ — Pressure from Eq. (C5-E01); $z$ — Axial coordinate from the wall; $t$ — Time; $H$ — Column length 4 μm; $\\rho_j$ — Inflow density; $U$ — Inflow speed; $\\partial_t$ — Partial derivative with respect to time; $\\partial_z$ — Partial derivative with respect to axial position.", "symbols_zh": "$\\rho$ — 液体密度；$u$ — 轴向速度，朝壁面为负；$p(\\rho)$ — 来自式（C5-E01）的压力；$z$ — 距壁面的轴向坐标；$t$ — 时间；$H$ — 液柱长度 4 μm；$\\rho_j$ — 入流密度；$U$ — 入流速度；$\\partial_t$ — 对时间的偏导数；$\\partial_z$ — 对轴向位置的偏导数。", "kind": "One-dimensional conservative equations of the reference solver", "kind_zh": "参考求解器的一维守恒方程", "diagram": { "type": "impact", "labels": [ "Wall at z = 0: u = 0 (mirror ghost state)", "Inflow at z = H: ρ_j, −U", "Mass: ∂ₜρ + ∂_z(ρu) = 0", "Momentum: ∂ₜ(ρu) + ∂_z(ρu² + p) = 0", "H = 4 μm" ], "notes": [ "Equations are conservative so shocks keep the right speed", "Same law p(ρ) as the COMSOL model" ] }, "caption_en": "Mass and momentum in conservative form let a shock form and travel at the correct speed.", "caption_zh": "守恒形式的质量与动量方程使激波得以形成并以正确速度传播。" }, { "id": "C5-E11", "tex": "\\begin{aligned}\\boldsymbol Q&=\\begin{pmatrix}\\rho\\\\ \\rho u\\end{pmatrix},\\quad \\boldsymbol F=\\begin{pmatrix}\\rho u\\\\ \\rho u^{2}+p\\end{pmatrix},\\quad \\Delta z=\\frac{H}{N},\\\\ \\boldsymbol F_{i+\\frac12}&=\\tfrac12(\\boldsymbol F_L+\\boldsymbol F_R)-\\tfrac12\\,\\alpha\\,(\\boldsymbol Q_R-\\boldsymbol Q_L),\\quad \\alpha=\\max(|u_L|+c_L,\\ |u_R|+c_R),\\\\ \\boldsymbol Q^{*}&=\\boldsymbol Q^{n}+\\Delta t\\,\\mathcal L(\\boldsymbol Q^{n}),\\quad \\boldsymbol Q^{n+1}=\\boldsymbol Q^{n}+\\tfrac12\\Delta t\\,[\\mathcal L(\\boldsymbol Q^{n})+\\mathcal L(\\boldsymbol Q^{*})].\\end{aligned}", "symbols_en": "$\\boldsymbol Q$ — Vector of conserved variables ρ and ρu; $\\boldsymbol F$ — Physical flux vector; $N$ — Number of cells, 4000; $H$ — Column length 4 μm; $\\Delta z$ — Cell size H/N, 1 nm; $\\boldsymbol F_{i+\\frac12}$ — Numerical flux at the face between cells i and i+1; $\\boldsymbol Q_L$ — State left of the face; $\\boldsymbol Q_R$ — State right of the face; $\\alpha$ — Largest local wave speed (Rusanov); $c_L$ — Sound speed of Eq. (C5-E01) left of the face; $c_R$ — Sound speed of Eq. (C5-E01) right of the face; $\\mathcal L$ — Spatial operator −∂_z F discretized by the face fluxes; $\\Delta t$ — Time step 0.4 Δz / (U + 1.2 $c_{wall}$); $\\boldsymbol Q^{*}$ — Predictor state; $\\boldsymbol Q^{n}$ — Old state; $\\boldsymbol Q^{n+1}$ — New state.", "symbols_zh": "$\\boldsymbol Q$ — 守恒变量向量 ρ 与 ρu；$\\boldsymbol F$ — 物理通量向量；$N$ — 网格单元数，4000；$H$ — 液柱长度 4 μm；$\\Delta z$ — 单元尺寸 H/N，1 nm；$\\boldsymbol F_{i+\\frac12}$ — 单元 i 与 i+1 之间界面的数值通量；$\\boldsymbol Q_L$ — 界面左侧状态；$\\boldsymbol Q_R$ — 界面右侧状态；$\\alpha$ — 最大局部波速（Rusanov）；$c_L$ — 界面左侧由式（C5-E01）给出的声速；$c_R$ — 界面右侧由式（C5-E01）给出的声速；$\\mathcal L$ — 由界面通量离散的空间算子 −∂_z F；$\\Delta t$ — 时间步 0.4 Δz / (U + 1.2 $c_{wall}$)；$\\boldsymbol Q^{*}$ — 预测态；$\\boldsymbol Q^{n}$ — 旧状态；$\\boldsymbol Q^{n+1}$ — 新状态。", "kind": "Rusanov flux with Heun time stepping", "kind_zh": "Rusanov 通量与 Heun 时间推进", "diagram": { "type": "impact", "labels": [ "4000 cells of 1 nm between wall and inflow", "Face flux: average minus α/2 jump", "α: largest |u| + c on either side", "Heun: predictor then average of two operators", "Wall: mirror ghost state" ], "notes": [ "Conserves mass and momentum exactly in the interior", "Wall pressure history is read from the cell next to the wall" ] }, "caption_en": "A simple dissipative flux and a two-stage time step are enough to carry the shock to the wall.", "caption_zh": "简单的耗散通量与两步时间推进足以把激波推进到壁面。" }, { "id": "C5-E12", "tex": "\\begin{aligned}\\partial_t\\rho+\\nabla\\!\\cdot(\\rho\\boldsymbol u)&=0,\\\\ \\rho\\left(\\partial_t\\boldsymbol u+\\boldsymbol u\\!\\cdot\\!\\nabla\\boldsymbol u\\right)&=-\\nabla p+\\nabla\\!\\cdot\\!\\left[\\mu\\left(\\nabla\\boldsymbol u+\\nabla\\boldsymbol u^{\\mathsf T}\\right)-\\tfrac23\\mu\\,(\\nabla\\!\\cdot\\boldsymbol u)\\,\\boldsymbol I\\right],\\\\ \\rho&=\\rho_j\\left(\\frac{p+\\Pi}{p_j+\\Pi}\\right)^{1/\\gamma}.\\end{aligned}", "symbols_en": "$\\rho$ — Liquid density; $\\boldsymbol u$ — Velocity vector with components ($u_{r}$, w); $p$ — Absolute pressure; $\\mu$ — Dynamic viscosity 1 mPa s; $\\boldsymbol I$ — Identity tensor; $\\rho_j$ — Reference density; $p_j$ — Reference pressure; $\\Pi$ — Stiffness pressure of Eq. (C5-E01); $\\gamma$ — Stiffness exponent of Eq. (C5-E01); $\\nabla$ — Gradient in axisymmetric (r, z) coordinates; $\\mathsf T$ — Transpose.", "symbols_zh": "$\\rho$ — 液体密度；$\\boldsymbol u$ — 速度向量，分量为 ($u_{r}$, w)；$p$ — 绝对压力；$\\mu$ — 动力黏度 1 mPa s；$\\boldsymbol I$ — 单位张量；$\\rho_j$ — 参考密度；$p_j$ — 参考压力；$\\Pi$ — 式（C5-E01）的刚度压力；$\\gamma$ — 式（C5-E01）的刚度指数；$\\nabla$ — 轴对称 (r, z) 坐标中的梯度；$\\mathsf T$ — 转置。", "kind": "Axisymmetric compressible flow equations of the COMSOL model", "kind_zh": "COMSOL 模型的轴对称可压缩流动方程", "diagram": { "type": "impact", "labels": [ "Mass: ∂ₜρ + ∇·(ρu) = 0", "Momentum with Newtonian viscous stress", "ρ(p): inverse of the stiffened-gas law", "Axisymmetric (r, z) coordinates", "μ = 1 mPa·s" ], "notes": [ "Physics interface: laminar flow, compressible", "The same liquid law as the C++ reference" ] }, "caption_en": "The finite-element model solves the same liquid with radial motion and a small viscosity added.", "caption_zh": "有限元模型求解同一种液体，并增加径向运动与很小的黏性。" }, { "id": "C5-E13", "tex": "\\begin{aligned}&\\text{wall }z=0:\\ \\boldsymbol u\\!\\cdot\\!\\boldsymbol n=0\\ (\\text{slip}),\\qquad \\text{axis }r=0:\\ \\partial_ru_z=0,\\\\ &\\text{inflow }z=4a:\\ \\boldsymbol u=-U\\,\\boldsymbol e_z,\\qquad \\text{side }r=a:\\ p=p_j\\ (\\text{release}),\\\\ &\\text{initial: }\\ w(r,z,0)=-U\\tanh\\frac{z}{\\delta},\\quad p(r,z,0)=p_j,\\quad \\delta=0.02a.\\end{aligned}", "symbols_en": "$\\boldsymbol n$ — Unit normal to the boundary; $\\boldsymbol u$ — Velocity vector; $u_z$ — Axial velocity component; $w$ — Axial velocity w(r,z,t); $\\boldsymbol e_z$ — Unit vector along +z; $U$ — Inflow speed; $a$ — Jet radius; $p_j$ — Release pressure and initial pressure; $\\delta$ — Onset thickness 0.02 a; $\\tanh$ — Hyperbolic tangent.", "symbols_zh": "$\\boldsymbol n$ — 边界单位法向量；$\\boldsymbol u$ — 速度向量；$u_z$ — 轴向速度分量；$w$ — 轴向速度 w(r,z,t)；$\\boldsymbol e_z$ — 沿 +z 的单位向量；$U$ — 入流速度；$a$ — 射流半径；$p_j$ — 释放压力与初始压力；$\\delta$ — 起始过渡厚度 0.02 a；$\\tanh$ — 双曲正切。", "kind": "Boundary and initial conditions of the axisymmetric model", "kind_zh": "轴对称模型的边界条件与初始条件", "diagram": { "type": "impact", "labels": [ "Rigid slip wall at z = 0", "Axis r = 0: symmetry", "Inflow at z = 4a: speed U", "Side r = a: pressure released to p_j", "Initial w = −U tanh(z/δ), δ = 0.02a" ], "notes": [ "Liquid starts at first contact with a thin onset layer", "The side condition is what creates the edge release" ] }, "caption_en": "A free side and a rigid slip wall are the only boundaries that change the planar picture.", "caption_zh": "自由侧边与刚性滑移壁面是仅有的改变平面图景的边界。" }, { "id": "C5-E14", "tex": "\\begin{aligned}\\dot m_p&=\\int_0^{a}\\rho\\,\\max(-w,0)\\,2\\pi r\\,dr\\Big|_{z=2a},\\qquad \\dot K_p=\\int_0^{a}\\tfrac12\\rho\\,(u_r^{2}+w^{2})\\max(-w,0)\\,2\\pi r\\,dr\\Big|_{z=2a},\\\\ U_{\\mathrm{inc}}&=\\frac{\\int_0^{t_w}\\!\\!\\int_0^a\\rho\\,\\max(-w,0)\\,(-w)\\,2\\pi r\\,dr\\,dt}{\\int_0^{t_w}\\dot m_p\\,dt},\\qquad \\bar p_R=\\frac{\\int_0^R p\\,2\\pi r\\,dr}{\\pi R^{2}}\\Big|_{z=0}.\\end{aligned}", "symbols_en": "$\\dot m_p$ — Computed mass flow through the plane; $\\dot K_p$ — Computed kinetic-energy flux through the plane; $U_{\\mathrm{inc}}$ — Mass-flux-weighted incident speed; $\\bar p_R$ — Area-weighted wall pressure of Eq. (C5-E06); $w$ — Axial velocity; $u_r$ — Radial velocity; $\\rho$ — Density; $\\max(-w,0)$ — Positive part of the inward axial speed; $t_w$ — Incident-flux window 1 ns; $a$ — Jet radius; $R$ — Sensor radius; $r$ — Radial coordinate.", "symbols_zh": "$\\dot m_p$ — 计算得到的穿过平面的质量流率；$\\dot K_p$ — 计算得到的穿过平面的动能通量；$U_{\\mathrm{inc}}$ — 质量通量加权的入射速度；$\\bar p_R$ — 式（C5-E06）的面积加权壁面压力；$w$ — 轴向速度；$u_r$ — 径向速度；$\\rho$ — 密度；$\\max(-w,0)$ — 朝壁面轴向速度的正部；$t_w$ — 入射通量时间窗 1 ns；$a$ — 射流半径；$R$ — 传感区半径；$r$ — 径向坐标。", "kind": "Observers evaluated in the COMSOL model", "kind_zh": "COMSOL 模型中计算的观测量", "diagram": { "type": "impact", "labels": [ "Plane z = 2a: ṁ_p, K̇_p, U_inc", "Wall z = 0: p̄_a over r ≤ a", "Centre disk r ≤ a/2", "Weight 2πr dr", "t_w = 1 ns" ], "notes": [ "Same observers for every mesh and every case", "Observer definitions are fixed before the comparison" ] }, "caption_en": "One set of fixed integrals defines every number reported from the finite-element model.", "caption_zh": "一组固定的积分定义了有限元模型报告的每个数值。" }, { "id": "C5-E15", "tex": "\\begin{aligned}\\varepsilon(X;X_{\\mathrm{ref}})&=\\frac{|X-X_{\\mathrm{ref}}|}{|X_{\\mathrm{ref}}|},\\qquad \\Delta_k=\\varepsilon\\!\\left(\\bar P_k;\\ \\bar P_{k-1}\\right),\\\\ \\bar P_k&=\\max_{0\\le t\\le0.3\\,\\mathrm{ns}}\\bar p_a(t)\\ \\text{on band size }h_k,\\qquad h_k=\\frac{a}{40},\\frac{a}{80},\\frac{a}{160},\\frac{a}{320}.\\end{aligned}", "symbols_en": "$\\varepsilon$ — Relative error; $X$ — Computed quantity; $X_{\\mathrm{ref}}$ — Stated reference value; $\\Delta_k$ — Successive change on refinement step k; $\\bar P_k$ — Peak full-sensor wall pressure on band size $h_{k}$; $\\bar p_a$ — Full-sensor mean wall pressure; $h_k$ — Maximum element size in the wall band r ≤ a, z ≤ 0.15 a; $k$ — Refinement index, 1 to 4.", "symbols_zh": "$\\varepsilon$ — 相对误差；$X$ — 被评价的计算量；$X_{\\mathrm{ref}}$ — 所注明的参考值；$\\Delta_k$ — 第 k 次加密的相邻变化；$\\bar P_k$ — 带宽 $h_{k}$ 上的全传感区壁面压力峰值；$\\bar p_a$ — 全传感区平均壁面压力；$h_k$ — 壁面加密带 r ≤ a、z ≤ 0.15 a 内的最大单元尺寸；$k$ — 加密序号，取 1 至 4。", "kind": "Relative error and successive-refinement change", "kind_zh": "相对误差与逐次加密变化量", "diagram": { "type": "timescale", "labels": [ "ε = |X − X_ref| / |X_ref|", "Δ_k compares each level with the previous level", "Band h = a/40, a/80, a/160, a/320", "Peak window 0 to 0.3 ns", "Elements 16,176 → 57,668" ], "notes": [ "Denominator is the previous (coarser) level", "The peak forms about 0.03 ns after contact" ] }, "caption_en": "Mesh convergence is judged by how little the peak moves when only the wall band is refined.", "caption_zh": "网格收敛以仅加密壁面带时峰值的变化量大小来判断。" }, { "id": "C5-E16", "tex": "\\begin{aligned}\\Delta P_z(t)&=P_z(t)-P_z(0)=\\int_0^{t}\\left(F_{\\mathrm{wall}}-F_{\\mathrm{top}}\\right)dt',\\qquad P_z=\\int_V\\rho w\\,dV,\\\\ \\Delta M(t)&=M(t)-M(0)=-\\int_0^{t}\\dot m_{\\mathrm{top}}\\,dt',\\qquad M=\\int_V\\rho\\,dV,\\\\ e_P&=\\frac{\\max_t|\\Delta P_z-\\int(F_{\\mathrm{wall}}-F_{\\mathrm{top}})dt'|}{\\max_t|\\Delta P_z|},\\quad e_M\\ \\text{defined likewise.}\\end{aligned}", "symbols_en": "$P_z$ — Axial momentum of the liquid in the domain; $F_{\\mathrm{wall}}$ — Pressure force on the wall, ∫ p dA; $F_{\\mathrm{top}}$ — Inflow pressure force plus momentum flux, ∫(p + ρw²) dA; $M$ — Liquid mass in the domain; $\\dot m_{\\mathrm{top}}$ — Net mass flow into the domain through the top; $V$ — Domain volume (axisymmetric); $e_P$ — Momentum balance residual; $e_M$ — Mass balance residual; $t$ — Current time; $t\\prime$ — Integration variable; $w$ — Axial velocity.", "symbols_zh": "$P_z$ — 计算域内液体的轴向动量；$F_{\\mathrm{wall}}$ — 壁面压力，∫ p dA；$F_{\\mathrm{top}}$ — 入流压力加动量通量，∫(p + ρw²) dA；$M$ — 计算域内液体质量；$\\dot m_{\\mathrm{top}}$ — 经顶部净流入计算域的质量流率；$V$ — 计算域体积（轴对称）；$e_P$ — 动量平衡残差；$e_M$ — 质量平衡残差；$t$ — 当前时间；$t\\prime$ — 积分变量；$w$ — 轴向速度。", "kind": "Time-integrated momentum and mass balance of the confined model", "kind_zh": "受限模型的时间积分动量与质量平衡", "diagram": { "type": "energy", "labels": [ "Confined column: slip side wall", "ΔP_z = ∫(F_wall − F_top) dt", "ΔM = −∫ṁ_top dt", "e_P = 0.106%", "e_M = 0.018%" ], "notes": [ "Both residuals compare a change of the state with its own flux integral", "Residuals test the solver, not the physics" ] }, "caption_en": "Integrating forces and flows over the event must reproduce the change in momentum and mass.", "caption_zh": "对整个事件的力与流量积分，必须再现动量与质量的变化。" }, { "id": "C5-E17", "tex": "\\begin{aligned}\\delta_{2}&=\\sqrt{\\frac{\\sum_{m=1}^{N_s}\\left(p^{C}_m-p^{X}_m\\right)^{2}}{\\sum_{m=1}^{N_s}\\left(p^{X}_m\\right)^{2}}},\\qquad \\delta_{\\infty}=\\max_m\\frac{|p^{C}_m-p^{X}_m|}{p^{X}_m},\\\\ \\bar p_{\\mathrm{pl}}&=\\frac{1}{N_s}\\sum_{m=1}^{N_s}p_m,\\qquad t_m\\in[0.2,\\ 2.0]\\ \\mathrm{ns},\\quad N_s=361.\\end{aligned}", "symbols_en": "$\\delta_2$ — Relative L2 difference of the two wall histories; $\\delta_\\infty$ — Largest pointwise relative difference; $p^{C}_m$ — COMSOL wall pressure at sample m; $p^{X}_m$ — C++ wall pressure interpolated to the same time; $N_s$ — Number of common samples, 361; $m$ — Sample index; $\\bar p_{\\mathrm{pl}}$ — Plateau pressure, mean over the window; $t_m$ — Sample time spaced by 5 ps.", "symbols_zh": "$\\delta_2$ — 两条壁面压力历程的相对 L2 差；$\\delta_\\infty$ — 最大逐点相对差；$p^{C}_m$ — 样本 m 处的 COMSOL 壁面压力；$p^{X}_m$ — 线性插值到同一时刻的 C++ 壁面压力；$N_s$ — 共同样本数，361；$m$ — 样本序号；$\\bar p_{\\mathrm{pl}}$ — 压力平台，窗口内平均；$t_m$ — 间隔 5 ps 的样本时间。", "kind": "Cross-solver comparison metrics for the confined column", "kind_zh": "受限液柱的跨求解器比较指标", "diagram": { "type": "impact", "labels": [ "C++ finite volume: 4000 cells", "COMSOL: confined axisymmetric column", "Window 0.2 to 2.0 ns, 361 samples", "δ₂ = 0.034%, δ_∞ = 0.096%", "Plateau: 174.205 vs 174.148 MPa" ], "notes": [ "Linear interpolation of the C++ history onto COMSOL times", "Differences are below the engineering tolerance by a wide margin" ] }, "caption_en": "Two codes with different numerics reach the same plateau history, which checks the implementation.", "caption_zh": "两个数值方法不同的程序得到同一平台历程，这检验了实现的正确性。" }, { "id": "C5-E18", "tex": "\\begin{aligned}\\Lambda_i&=\\frac{\\max_t\\,\\bar p_{a,i}(t)}{p_{H,i}},\\qquad i\\in\\{\\text{five cases}\\},\\\\ \\sigma_\\Lambda&=\\frac{\\max_i\\Lambda_i-\\min_i\\Lambda_i}{\\overline{\\Lambda}},\\qquad \\frac{\\bar p_{a}(\\tau)}{p_H}=g(\\tau)\\ \\text{for all cases, }\\tau\\le0.49.\\end{aligned}", "symbols_en": "$\\Lambda_i$ — Peak factor of case i; $\\bar p_{a,i}$ — Full-sensor wall pressure of case i; $p_{H,i}$ — Hugoniot pressure of case i from Eq. (C5-E04); $i$ — Case index over the five converged cases; $\\sigma_\\Lambda$ — Relative spread of the peak factor; $\\overline{\\Lambda}$ — Mean peak factor over the cases; $g(\\tau)$ — Common normalized pressure history; $\\tau$ — Dimensionless time $c_{0}$ t / a.", "symbols_zh": "$\\Lambda_i$ — 工况 i 的峰值因子；$\\bar p_{a,i}$ — 工况 i 的全传感区壁面压力；$p_{H,i}$ — 工况 i 由式（C5-E04）得到的 Hugoniot 压力；$i$ — 五个收敛工况的序号；$\\sigma_\\Lambda$ — 峰值因子的相对离散度；$\\overline{\\Lambda}$ — 各工况峰值因子的平均；$g(\\tau)$ — 共同的无量纲压力历程；$\\tau$ — 无量纲时间 $c_{0}$ t / a。", "kind": "Scaling collapse and spread of the finite-area factor", "kind_zh": "有限面积因子的尺度坍缩与离散度", "diagram": { "type": "timescale", "labels": [ "U = 50, 99.03, 150 m/s at a = 1 μm", "a = 0.5, 1, 2 μm at U = 99.03 m/s", "Λ_i from 0.9303 to 0.9314", "σ_Λ = 0.12%", "One curve g(τ) for all five cases" ], "notes": [ "Collapse supports a single dimensionless result for this model", "Not tested outside the stated U and a ranges" ] }, "caption_en": "When pressure is scaled by p_H and time by a/c_0, all five cases fall on one history.", "caption_zh": "当压力以 p_H 无量纲化、时间以 a/c_0 无量纲化后，五个工况落在同一历程上。" }, { "id": "C5-E19", "tex": "\\begin{aligned}\\max_t\\bar p_{a/2}&=p_H(U)\\,(1\\pm0.0043),\\qquad \\max_t\\bar p_{a}=\\Lambda\\,p_H(U),\\quad \\Lambda=0.931,\\\\ t_{\\mathrm{hi}}&\\simeq\\frac{a}{c_0},\\qquad \\frac{p_H-p_j}{Z_0U}=\\frac{U_s}{c_0}=1.0856.\\end{aligned}", "symbols_en": "$\\bar p_{a/2}$ — Centre-disk wall pressure, r ≤ a/2; $\\bar p_a$ — Full-sensor wall pressure, r ≤ a; $p_H(U)$ — Hugoniot pressure at the given speed; $\\Lambda$ — Finite-area peak factor 0.931; $t_{\\mathrm{hi}}$ — Duration of the high-pressure interval; $a$ — Jet radius; $c_0$ — Reference sound speed; $Z_0$ — Reference acoustic impedance; $U_s$ — Shock speed relative to the incoming liquid; $p_j$ — Reference pressure.", "symbols_zh": "$\\bar p_{a/2}$ — 中心圆盘壁面压力，r ≤ a/2；$\\bar p_a$ — 全传感区壁面压力，r ≤ a；$p_H(U)$ — 给定速度下的 Hugoniot 压力；$\\Lambda$ — 有限面积峰值因子 0.931；$t_{\\mathrm{hi}}$ — 高压阶段的持续时间；$a$ — 射流半径；$c_0$ — 参考声速；$Z_0$ — 参考声阻抗；$U_s$ — 相对入射液体的激波速度；$p_j$ — 参考压力。", "kind": "Delivered rigid-wall relations and link to the Chapter 3 ceiling", "kind_zh": "交付的刚性壁面关系及其与第三章上限的联系", "diagram": { "type": "impact", "labels": [ "Centre r ≤ a/2: p_H within 0.43%", "Full sensor r ≤ a: 0.931 p_H", "High-pressure duration ≈ a / c_0 = 0.61 ns", "Nonlinear factor over the linear ceiling: U_s / c_0 = 1.0856", "Chapter 3 ceiling Z_0 U = 160.765 MPa" ], "notes": [ "Conditional on the prescribed jet speed U", "U is not predicted by this chapter" ] }, "caption_en": "The result is a speed-to-pressure map; whether a given source produces that speed belongs to Chapters 1 to 3.", "caption_zh": "所得结果是“速度到压力”的映射；某一能源能否产生该速度属于第一至第三章。" }, { "id": "C5-E20", "tex": "\\begin{aligned}p_{\\mathrm{wall}}&\\le p_j+\\frac{U_s}{c_0}\\,Z_0\\,U\\quad(\\text{centre disk}),\\qquad p_{\\mathrm{wall}}\\approx\\Lambda\\,p_H\\quad(\\text{full sensor}),\\\\ U&\\le\\sqrt{\\frac{2E_{\\mathrm{av}}}{m_j}}\\quad(\\text{Chapter 3 energy bound}).\\end{aligned}", "symbols_en": "$p_{\\mathrm{wall}}$ — Mean wall pressure over the stated disk; $p_j$ — Reference pressure; $U_s/c_0$ — Nonlinear factor 1.0856; $Z_0$ — Reference acoustic impedance; $U$ — Impact speed; $\\Lambda$ — Finite-area peak factor; $p_H$ — Hugoniot pressure; $E_{\\mathrm{av}}$ — Kinetic energy available to the jet (Chapter 3); $m_j$ — Jet mass (Chapter 3).", "symbols_zh": "$p_{\\mathrm{wall}}$ — 所述圆盘上的平均壁面压力；$p_j$ — 参考压力；$U_s/c_0$ — 非线性因子 1.0856；$Z_0$ — 参考声阻抗；$U$ — 撞击速度；$\\Lambda$ — 有限面积峰值因子；$p_H$ — Hugoniot 压力；$E_{\\mathrm{av}}$ — 可用于射流的动能（第三章）；$m_j$ — 射流质量（第三章）。", "kind": "Reference formulas: speed bound and pressure map together", "kind_zh": "参考公式：速度上界与压力映射合并", "diagram": { "type": "impact", "labels": [ "Energy bound: U ≤ √(2E_av/m_j)", "Pressure map: p_wall = f(U)", "Centre: Hugoniot level", "Full sensor: Λ p_H", "Both are conditional on stated inputs" ], "notes": [ "A bound is not a prediction", "The map is not a measurement" ] }, "caption_en": "A speed bound and a pressure map are two separate claims joined only by the prescribed speed.", "caption_zh": "速度上界与压力映射是两个独立结论，仅通过给定速度相连。" }, { "id": "C5-E21", "tex": "\\begin{aligned}\\Lambda&=\\frac{\\bar P_{\\max}}{p_H}\\approx 1-\\kappa\\,f_{\\mathrm{rel}}(\\tau_{\\mathrm{pk}}),\\quad f_{\\mathrm{rel}}(\\tau)=1-(1-\\tau)^{2},\\\\ \\tau_{\\mathrm{pk}}&=\\frac{c_0t_{\\mathrm{pk}}}{a},\\qquad t_{\\mathrm{pk}}=0.029\\ \\mathrm{ns},\\quad \\kappa=\\frac{1-\\Lambda}{f_{\\mathrm{rel}}(\\tau_{\\mathrm{pk}})}.\\end{aligned}", "symbols_en": "$\\Lambda$ — Finite-area peak factor; $\\bar P_{\\max}$ — Peak full-sensor pressure; $p_H$ — Hugoniot pressure; $f_{\\mathrm{rel}}$ — Released area fraction from Eq. (C5-E07); $\\kappa$ — Fraction of the excess lost inside the released annulus; $\\tau_{\\mathrm{pk}}$ — Dimensionless time of the peak; $t_{\\mathrm{pk}}$ — Time of the peak, 0.029 ns; $c_0$ — Reference sound speed; $a$ — Jet radius.", "symbols_zh": "$\\Lambda$ — 有限面积峰值因子；$\\bar P_{\\max}$ — 全传感区压力峰值；$p_H$ — Hugoniot 压力；$f_{\\mathrm{rel}}$ — 来自式（C5-E07）的已释放面积分数；$\\kappa$ — 释放环带内损失的超压比例；$\\tau_{\\mathrm{pk}}$ — 峰值出现的无量纲时间；$t_{\\mathrm{pk}}$ — 峰值时刻，0.029 ns；$c_0$ — 参考声速；$a$ — 射流半径。", "kind": "Reference formulas: edge-release explanation of the finite-area factor", "kind_zh": "参考公式：用边缘释放解释有限面积因子", "diagram": { "type": "impact", "labels": [ "Λ = 0.931 from the converged band meshes", "τ_pk = c_0 t_pk / a = 0.0472", "f_rel(τ_pk) = 0.0921", "κ = (1 − Λ) / f_rel = 0.75", "Annulus keeps about a quarter of its excess" ], "notes": [ "κ between 0 and 1 is consistent with partial release", "Full release would give Λ = 0.908" ] }, "caption_en": "A released annulus that keeps part of its excess pressure accounts for a peak factor between 0.908 and 1.", "caption_zh": "释放环带保留部分超压，可解释峰值因子介于 0.908 与 1 之间。" }, { "id": "C5-E22", "tex": "\\begin{aligned}\\text{valid for}\\quad &M\\in[0.031,\\,0.092],\\ \\ a\\in[0.5,\\,2]\\ \\mu\\mathrm m,\\ \\ \\tau\\le 1,\\\\ &\\text{liquid only, rigid flat wall, barotropic law of Eq. (C5-E01)},\\\\ \\delta_2&=0.034\\%\\ \\Longrightarrow\\ \\text{implementation agrees;}\\\\ &\\text{the model is not tested against experiment.}\\end{aligned}", "symbols_en": "$M$ — Impact Mach number; $a$ — Jet radius; $\\tau$ — Dimensionless time $c_{0}$ t / a; $\\delta_2$ — Relative L2 difference between the two solvers; $\\mu\\mathrm m$ — Micrometre, 10⁻⁶ m.", "symbols_zh": "$M$ — 撞击马赫数；$a$ — 射流半径；$\\tau$ — 无量纲时间 $c_{0}$ t / a；$\\delta_2$ — 两个求解器间的相对 L2 差；$\\mu\\mathrm m$ — 微米，10⁻⁶ m。", "kind": "Reference formulas: validity range and limits of the verification", "kind_zh": "参考公式：适用范围与核验的局限", "diagram": { "type": "timescale", "labels": [ "M from 0.031 to 0.092", "a from 0.5 to 2 μm", "First arrival only: τ ≤ 1", "Liquid-only barotropic model", "δ₂ = 0.034% checks the implementation" ], "notes": [ "No experiment enters this chapter", "Missing: measured wall pressure for a real jet" ] }, "caption_en": "The verification covers one model over a stated range; it does not reach beyond that range.", "caption_zh": "核验只覆盖所述范围内的单一模型，不超出该范围。" }, { "id": "A-E01", "tex": "\\begin{aligned}r^3-R^3&=r_0^3-R_{\\mathrm{ref}}^3,\\\\3r^2\\frac{\\partial r}{\\partial r_0}&=3r_0^2,\\\\\\lambda_r:=\\frac{\\partial r}{\\partial r_0}&=\\frac{r_0^2}{r^2},\\qquad\\lambda_\\theta=\\lambda_\\phi=\\frac{r}{r_0},\\\\\\lambda_r\\lambda_\\theta\\lambda_\\phi&=1.\\end{aligned}", "symbols_en": "$r_0\\geq R_{\\mathrm{ref}}>0$ and $r\\geq R>0$ are reference/current material radii (m); $R_{\\mathrm{ref}}$, $R$ are cavity radii (m). $\\lambda_r$, $\\lambda_\\theta$, $\\lambda_\\phi$ are dimensionless principal stretches; $\\theta,\\phi$ are tangential directions. $\\partial r/\\partial r_0$ is the derivative with respect to the reference material coordinate at fixed time.", "symbols_zh": "$r_0\\geq R_{\\mathrm{ref}}>0$ 与 $r\\geq R>0$ 为参考／当前材料半径（m）；$R_{\\mathrm{ref}}$、$R$ 为腔体半径（m）。$\\lambda_r$、$\\lambda_\\theta$、$\\lambda_\\phi$ 为无量纲主伸长；$\\theta,\\phi$ 为切向方向。$\\partial r/\\partial r_0$ 表示构形映射中固定时刻的材料坐标导数。", "kind": "Exact kinematics under incompressibility", "kind_zh": "不可压缩条件下的精确运动学关系", "diagram": { "type": "gel", "labels": [ "Rref: reference wall", "R: current wall", "r₀ → r: material shell", "λr = r₀²/r²", "λθ = λφ = r/r₀" ], "notes": [ "Network material volume is unchanged", "Radial direction points outward" ] }, "caption_en": "The same network shell moves outward while thinning radially and stretching tangentially.", "caption_zh": "同一网络球壳向外移动，同时径向变薄、切向伸长。" }, { "id": "A-E02", "tex": "\\begin{aligned}\\mathbf T^e&=-\\chi\\mathbf I+G_g\\mathbf B,\\qquad\\mathbf B=\\mathbf F\\mathbf F^{\\mathsf T},\\\\T^e_{rr}&=-\\chi+G_g\\lambda_r^2,\\qquad T^e_{\\theta\\theta}=T^e_{\\phi\\phi}=-\\chi+G_g\\lambda_\\theta^2,\\\\T^e_{\\theta\\theta}-T^e_{rr}&=G_g\\left[\\left(\\frac r{r_0}\\right)^2-\\left(\\frac{r_0}r\\right)^4\\right].\\end{aligned}", "symbols_en": "$\\mathbf T^e$ and its $rr$, $\\theta\\theta$, $\\phi\\phi$ principal components are elastic Cauchy stresses (Pa), tensile positive. $\\chi$ is the incompressibility multiplier (Pa); $\\mathbf I$ is the dimensionless identity tensor; $G_g>0$ is network shear modulus (Pa). $\\mathbf F$ is the dimensionless deformation gradient, $\\mathbf B$ its left Cauchy–Green tensor, and $\\mathsf T$ denotes transpose. $\\lambda_r$, $\\lambda_\\theta$ are stretches; $r,r_0>0$ are current/reference material radii (m).", "symbols_zh": "$\\mathbf T^e$ 及其 $rr$、$\\theta\\theta$、$\\phi\\phi$ 主分量为弹性 Cauchy 应力（Pa），拉伸取正。$\\chi$ 为不可压缩约束乘子（Pa）；$\\mathbf I$ 为无量纲单位张量；$G_g>0$ 为网络剪切模量（Pa）。$\\mathbf F$ 为无量纲变形梯度，$\\mathbf B$ 为左 Cauchy–Green 张量，$\\mathsf T$ 表示转置。$\\lambda_r$、$\\lambda_\\theta$ 为伸长；$r,r_0>0$ 为当前／参考材料半径（m）。", "kind": "Constitutive assumption and exact subtraction", "kind_zh": "本构假设与精确相减关系", "diagram": { "type": "gel", "labels": [ "Gg: polymer network", "Tᵉrr: radial traction", "Tᵉθθ: tangential stress", "r₀ → r", "λr, λθ, F, B, χ, I" ], "notes": [ "Equal χ cancels in the stress difference", "Tensile stress is positive" ] }, "caption_en": "Radial and tangential stretches produce different network stresses around the cavity.", "caption_zh": "空腔周围的径向与切向伸长产生不同的网络应力。" }, { "id": "A-E03", "tex": "\\begin{aligned}\\frac{dT^e_{rr}}{dr}&=\\frac{2}{r}(T^e_{\\theta\\theta}-T^e_{rr}),\\\\T^e_{rr}(\\infty)&=-p_\\infty,\\qquad T^e_{rr}(R)=-p_b+\\frac{2\\sigma}{R},\\\\p_{\\mathrm{el}}(R):=p_b-p_\\infty-\\frac{2\\sigma}{R}&=2G_g\\int_R^\\infty\\left[\\left(\\frac r{r_0}\\right)^2-\\left(\\frac{r_0}r\\right)^4\\right]\\frac{dr}{r}.\\end{aligned}", "symbols_en": "$T^e_{rr}$, $T^e_{\\theta\\theta}$ are radial/tangential elastic stresses (Pa) in this static control; $r$ is current radius, $r_0=r_0(r)$ the reference coordinate, $R>0$ the cavity radius (m). $p_b$, $p_\\infty$ are inner/far-field pressures (Pa); $\\sigma\\geq0$ is constant tension (N m⁻¹); $G_g>0$ is shear modulus (Pa); $p_{\\mathrm{el}}$ is the signed elastic resistance (Pa). $d/dr$ differentiates in $r$, $\\int_R^\\infty$ integrates over exterior material, and $\\infty$ labels the infinite far field.", "symbols_zh": "$T^e_{rr}$、$T^e_{\\theta\\theta}$ 为该静态控制模型的径向／切向弹性应力（Pa）；$r$ 为当前半径，$r_0=r_0(r)$ 为参考坐标，$R>0$ 为腔体半径（m）。$p_b$、$p_\\infty$ 为内部／远场压力（Pa）；$\\sigma\\geq0$ 为恒定张力（N m⁻¹）；$G_g>0$ 为剪切模量（Pa）；$p_{\\mathrm{el}}$ 为带符号的弹性阻力（Pa）。$d/dr$ 表示对 $r$ 求导，$\\int_R^\\infty$ 对外部材料积分，$\\infty$ 标记无限远场。", "kind": "Static balance defining the elastic resistance", "kind_zh": "定义弹性阻力的静态平衡", "diagram": { "type": "gel", "labels": [ "R: cavity surface", "pb: cavity pressure", "2σ/R: interface jump", "p∞: far field", "pel: integrated network resistance", "r, r₀, Tᵉrr, Tᵉθθ, Gg" ], "notes": [ "Integrate from the wall to the far field", "Dynamic inertia is introduced later" ] }, "caption_en": "The wall pressure supports capillary traction and the accumulated elastic resistance.", "caption_zh": "壁面压力同时支撑毛细牵引与积分后的弹性阻力。" }, { "id": "A-E04", "tex": "\\begin{aligned}q^3&=1-\\frac{R^3-R_{\\mathrm{ref}}^3}{r^3},\\qquad q(R)=\\frac{R_{\\mathrm{ref}}}{R},\\qquad q(\\infty)=1,\\\\3q^2\\,dq&=3(1-q^3)\\frac{dr}{r},\\qquad \\frac{dr}{r}=\\frac{q^2\\,dq}{1-q^3},\\\\\\left(q^{-2}-q^4\\right)\\frac{dr}{r}&=\\frac{1-q^6}{1-q^3}\\,dq=(1+q^3)\\,dq.\\end{aligned}", "symbols_en": "$q=r_0/r$ is a positive dimensionless coordinate ratio; $r_0,r$ are reference/current material radii (m), and $R>R_{\\mathrm{ref}}>0$ are current/reference cavity radii (m). $q(R)$ and $q(\\infty)$ denote endpoint values. $dq,dr$ are coordinate differentials; $\\infty$ denotes the far-field limit. The factorization $1-q^6=(1-q^3)(1+q^3)$ applies for $q<1$ before taking the endpoint limit.", "symbols_zh": "$q=r_0/r$ 为正的无量纲坐标比；$r_0,r$ 为参考／当前材料半径（m），$R>R_{\\mathrm{ref}}>0$ 为当前／参考腔体半径（m）。$q(R)$、$q(\\infty)$ 表示端点值。$dq,dr$ 为坐标微分；$\\infty$ 表示远场极限。因式分解 $1-q^6=(1-q^3)(1+q^3)$ 先在 $q<1$ 时使用，再取端点极限。", "kind": "Exact change of variable for expansion", "kind_zh": "膨胀过程的精确变量代换", "diagram": { "type": "gel", "labels": [ "qwall = Rref/R", "q → 1 far away", "r₀/r = q", "R, Rref, r, r₀", "dr/r ↔ q²dq/(1−q³)" ], "notes": [ "Mapped limits retain the wall and infinity", "Factorization removes the apparent endpoint singularity" ] }, "caption_en": "The material coordinate ratio converts a spatial stress integral into a regular polynomial integral.", "caption_zh": "材料坐标比将空间应力积分转化为正则的多项式积分。" }, { "id": "A-E05", "tex": "\\begin{aligned}p_{\\mathrm{el}}(R)&=2G_g\\int_{R_{\\mathrm{ref}}/R}^{1}(1+q^3)\\,dq\\\\&=2G_g\\left[q+\\frac{q^4}{4}\\right]_{R_{\\mathrm{ref}}/R}^{1}\\\\&=2G_g\\left[\\frac54-\\frac{R_{\\mathrm{ref}}}{R}-\\frac14\\left(\\frac{R_{\\mathrm{ref}}}{R}\\right)^4\\right]\\\\&=\\frac{G_g}{2}\\left[5-4\\frac{R_{\\mathrm{ref}}}{R}-\\left(\\frac{R_{\\mathrm{ref}}}{R}\\right)^4\\right].\\end{aligned}", "symbols_en": "$p_{\\mathrm{el}}$ is elastic cavity resistance (Pa); $G_g>0$ is network shear modulus (Pa); $R\\geq R_{\\mathrm{ref}}>0$ are current/reference cavity radii (m). $q$ is the dimensionless integration variable. $\\int$ is a definite integral, and $[f(q)]_a^b$ means upper-endpoint minus lower-endpoint evaluation of the bracketed function.", "symbols_zh": "$p_{\\mathrm{el}}$ 为弹性腔体阻力（Pa）；$G_g>0$ 为网络剪切模量（Pa）；$R\\geq R_{\\mathrm{ref}}>0$ 为当前／参考腔体半径（m）。$q$ 为无量纲积分变量。$\\int$ 为定积分，$[f(q)]_a^b$ 表示括号中函数的上端点值减去下端点值。", "kind": "Derived neo-Hookean elastic resistance", "kind_zh": "推导得到的新胡克弹性阻力", "diagram": { "type": "gel", "labels": [ "pel(R): inward resistance", "Gg: network modulus", "Rref/R: wall stretch inverse", "q: stress-integral coordinate" ], "notes": [ "Integrate 1 and q³ term by term", "Constitutive result for an intact infinite network" ] }, "caption_en": "Elastic resistance follows from the full stretch field, not from a guessed constant pressure.", "caption_zh": "弹性阻力来自完整伸长场，而不是猜测的恒定压力。" }, { "id": "A-E06", "tex": "\\begin{aligned}p_{\\mathrm{el}}(R_{\\mathrm{ref}})&=\\frac{G_g}{2}(5-4-1)=0,\\\\\\frac{dp_{\\mathrm{el}}}{dR}&=2G_g\\left(\\frac{R_{\\mathrm{ref}}}{R^2}+\\frac{R_{\\mathrm{ref}}^4}{R^5}\\right)>0,\\\\\\lim_{R/R_{\\mathrm{ref}}\\to\\infty}p_{\\mathrm{el}}(R)&=\\frac52G_g.\\end{aligned}", "symbols_en": "$p_{\\mathrm{el}}$ is pressure resistance (Pa); $G_g>0$ is shear modulus (Pa); $R\\geq R_{\\mathrm{ref}}>0$ are cavity radii (m). $d/dR$ is the radius derivative (giving Pa m⁻¹ here); $\\lim$ takes the positive large-radius-ratio limit; $\\infty$ denotes an unbounded ratio.", "symbols_zh": "$p_{\\mathrm{el}}$ 为压力阻力（Pa）；$G_g>0$ 为剪切模量（Pa）；$R\\geq R_{\\mathrm{ref}}>0$ 为腔体半径（m）。$d/dR$ 表示半径导数（此处单位为 Pa m⁻¹）；$\\lim$ 取正的大半径比极限；$\\infty$ 表示比值无界。", "kind": "Limiting-state and sign checks", "kind_zh": "极限状态与符号检验", "diagram": { "type": "gel", "labels": [ "Rref: zero resistance", "R increasing → pel increasing", "large R → 5Gg/2", "dpel/dR > 0" ], "notes": [ "Expansion requires positive additional pressure", "Pressure plateau does not cap stored work" ] }, "caption_en": "Zero deformation gives zero elastic resistance, and positive expansion raises it monotonically.", "caption_zh": "零变形给出零弹性阻力，正向膨胀使阻力单调增大。" }, { "id": "A-E07", "tex": "\\begin{aligned}W_g(R)&=4\\pi\\int_{R_{\\mathrm{ref}}}^{R}p_{\\mathrm{el}}(\\xi)\\xi^2\\,d\\xi\\\\&=2\\pi G_g\\int_{R_{\\mathrm{ref}}}^{R}\\left(5\\xi^2-4R_{\\mathrm{ref}}\\xi-R_{\\mathrm{ref}}^4\\xi^{-2}\\right)d\\xi\\\\&=2\\pi G_g\\left[\\frac{5\\xi^3}{3}-2R_{\\mathrm{ref}}\\xi^2+\\frac{R_{\\mathrm{ref}}^4}{\\xi}\\right]_{R_{\\mathrm{ref}}}^{R}\\\\&=2\\pi G_g\\left[\\frac{5R^3}{3}-2R_{\\mathrm{ref}}R^2+\\frac{R_{\\mathrm{ref}}^4}{R}-\\frac{2R_{\\mathrm{ref}}^3}{3}\\right].\\end{aligned}", "symbols_en": "$W_g$ is stored elastic work relative to the unstretched state (J); $p_{\\mathrm{el}}(\\xi)$ is signed elastic resistance (Pa); $G_g>0$ is modulus (Pa). $R\\geq R_{\\mathrm{ref}}>0$ and dummy integration radius $\\xi$ are lengths (m); $d\\xi$ is its differential. $\\pi$ is the circle constant; $[\\ ]_{R_{\\mathrm{ref}}}^{R}$ denotes endpoint subtraction.", "symbols_zh": "$W_g$ 为相对未伸长状态的储存弹性功（J）；$p_{\\mathrm{el}}(\\xi)$ 为带符号弹性阻力（Pa）；$G_g>0$ 为模量（Pa）。$R\\geq R_{\\mathrm{ref}}>0$ 与虚拟积分半径 $\\xi$ 为长度（m）；$d\\xi$ 为其微分。$\\pi$ 为圆周率；$[\\ ]_{R_{\\mathrm{ref}}}^{R}$ 表示端点相减。", "kind": "Exact work integral within the elastic model", "kind_zh": "弹性模型内的精确功积分", "diagram": { "type": "energy", "labels": [ "Rref → R", "pel(ξ): cavity work pressure", "4πξ²dξ: volume increment", "Wg: network energy", "Gg, π" ], "notes": [ "Use pressure times displaced volume", "Elastic work can return during recoil" ] }, "caption_en": "The area factor converts radius increments into cavity volume increments before integrating work.", "caption_zh": "先用面积因子将半径增量转化为腔体体积增量，再积分功。" }, { "id": "A-E08", "tex": "\\begin{aligned}\\frac{dW_g}{dR}&=2\\pi G_g\\left(5R^2-4R_{\\mathrm{ref}}R-\\frac{R_{\\mathrm{ref}}^4}{R^2}\\right)=4\\pi R^2p_{\\mathrm{el}}(R),\\\\R=R_{\\mathrm{ref}}+\\delta R\\colon\\qquad p_{\\mathrm{el}}&=\\frac{4G_g\\delta R}{R_{\\mathrm{ref}}}+O\\!\\left(G_g\\frac{\\delta R^2}{R_{\\mathrm{ref}}^2}\\right),\\\\W_g&=8\\pi G_gR_{\\mathrm{ref}}\\delta R^2+O(G_g\\delta R^3).\\end{aligned}", "symbols_en": "$W_g$ is elastic work (J), $p_{\\mathrm{el}}$ resistance (Pa), $G_g>0$ shear modulus (Pa), $R,R_{\\mathrm{ref}}>0$ cavity radii (m), and $\\delta R=R-R_{\\mathrm{ref}}$ a small radius change (m), with $|\\delta R|/R_{\\mathrm{ref}}\\ll1$. $d/dR$ is differentiation in radius; $O$ denotes terms bounded by a constant times the indicated scale as that ratio tends to zero; $\\pi$ is dimensionless.", "symbols_zh": "$W_g$ 为弹性功（J），$p_{\\mathrm{el}}$ 为阻力（Pa），$G_g>0$ 为剪切模量（Pa），$R,R_{\\mathrm{ref}}>0$ 为腔体半径（m），$\\delta R=R-R_{\\mathrm{ref}}$ 为小半径变化（m），满足 $|\\delta R|/R_{\\mathrm{ref}}\\ll1$。$d/dR$ 表示对半径求导；$O$ 表示当该比值趋于零时，被某常数乘所示尺度界定的项；$\\pi$ 为无量纲量。", "kind": "Work-conjugacy and small-deformation checks", "kind_zh": "功共轭关系与小变形检验", "diagram": { "type": "gel", "labels": [ "δR = R−Rref", "small δR: pel ∝ δR", "Wg ∝ δR²", "4πR²pel = dWg/dR", "Gg, O" ], "notes": [ "Zero reference work and zero initial slope", "Elastic energy is nonnegative for expansion" ] }, "caption_en": "Near the reference cavity, pressure is linear in radius change and stored work is quadratic.", "caption_zh": "在参考腔体附近，压力与半径变化呈线性关系，储存功则呈二次关系。" }, { "id": "A-E09", "tex": "\\begin{aligned}\\mathbf T&=\\mathbf T^e+2\\eta_g\\mathbf D,\\qquad u_r(r,t)=\\frac{R^2\\dot R}{r^2},\\\\D_{rr}&=\\frac{\\partial u_r}{\\partial r}=-\\frac{2R^2\\dot R}{r^3},\\qquad D_{\\theta\\theta}=D_{\\phi\\phi}=\\frac{u_r}{r}=\\frac{R^2\\dot R}{r^3},\\\\T_{\\theta\\theta}-T_{rr}&=T^e_{\\theta\\theta}-T^e_{rr}+\\frac{6\\eta_gR^2\\dot R}{r^3}.\\end{aligned}", "symbols_en": "$\\mathbf T$, $\\mathbf T^e$ and their radial/tangential components are total/elastic Cauchy stresses (Pa). $\\eta_g\\geq0$ is the chosen medium viscosity (Pa s); $\\mathbf D$ and its components are strain rates (s⁻¹). $u_r$ is radial material speed (m s⁻¹); $r\\geq R(t)>0$ is current position, $R$ wall radius (m), $t$ time (s), and $\\dot R$ wall speed (m s⁻¹). $\\partial/\\partial r$ is the fixed-time spatial derivative.", "symbols_zh": "$\\mathbf T$、$\\mathbf T^e$ 及其径向／切向分量为总／弹性 Cauchy 应力（Pa）。$\\eta_g\\geq0$ 为所选介质黏度（Pa s）；$\\mathbf D$ 及其分量为应变率（s⁻¹）。$u_r$ 为径向材料速度（m s⁻¹）；$r\\geq R(t)>0$ 为当前位置，$R$ 为壁面半径（m），$t$ 为时间（s），$\\dot R$ 为泡壁速度（m s⁻¹）。$\\partial/\\partial r$ 为固定时刻的空间导数。", "kind": "Kelvin–Voigt closure and exact radial kinematics", "kind_zh": "Kelvin–Voigt 闭合关系与精确径向运动学", "diagram": { "type": "gel", "labels": [ "Ṙ: wall material speed", "ur = R²Ṙ/r²", "ηg: viscous network/medium contribution", "Drr, Dθθ, Dφφ", "T, Tᵉ, r, R, t" ], "notes": [ "Dashpot stress follows the velocity gradient", "Use one effective viscosity for each represented resistance" ] }, "caption_en": "The viscous term derives from the surrounding-medium strain rate, not from bubble mass.", "caption_zh": "黏性项来自周围介质的应变率，而不是气泡质量。" }, { "id": "A-E10", "tex": "\\begin{aligned}a_r&:=\\frac{\\partial u_r}{\\partial t}+u_r\\frac{\\partial u_r}{\\partial r}=\\frac{2R\\dot R^2+R^2\\ddot R}{r^2}-\\frac{2R^4\\dot R^2}{r^5},\\\\\\rho_g a_r&=\\frac{\\partial T_{rr}}{\\partial r}+\\frac{2}{r}(T_{rr}-T_{\\theta\\theta}),\\\\\\rho_g\\int_R^\\infty a_r\\,dr&=\\rho_g\\left[R\\ddot R+2\\dot R^2-\\frac{\\dot R^2}{2}\\right]=\\rho_g\\left(R\\ddot R+\\frac32\\dot R^2\\right),\\\\2\\int_R^\\infty\\frac{6\\eta_gR^2\\dot R}{r^4}\\,dr&=12\\eta_gR^2\\dot R\\left(\\frac{1}{3R^3}\\right)=\\frac{4\\eta_g\\dot R}{R}.\\end{aligned}", "symbols_en": "$a_r$ is material radial acceleration (m s⁻²); $u_r$ radial speed (m s⁻¹); $r\\geq R>0$ position and cavity radius (m); $t$ time (s); $\\dot R$, $\\ddot R$ wall speed/acceleration (m s⁻¹, m s⁻²). $\\rho_g>0$ is density (kg m⁻³), $\\eta_g\\geq0$ viscosity (Pa s), and $T_{rr},T_{\\theta\\theta}$ total stresses (Pa). $\\partial$ denotes fixed-other-coordinate derivatives and the integrals run through the exterior to infinity. Integrated acceleration times density has units Pa.", "symbols_zh": "$a_r$ 为材料径向加速度（m s⁻²）；$u_r$ 为径向速度（m s⁻¹）；$r\\geq R>0$ 为位置与腔体半径（m）；$t$ 为时间（s）；$\\dot R$、$\\ddot R$ 为泡壁速度／加速度（m s⁻¹、m s⁻²）。$\\rho_g>0$ 为密度（kg m⁻³），$\\eta_g\\geq0$ 为黏度（Pa s），$T_{rr},T_{\\theta\\theta}$ 为总应力（Pa）。$\\partial$ 表示保持其他坐标不变的偏导，积分经外部材料到无穷远。积分后的加速度乘以密度的单位为 Pa。", "kind": "Momentum law and explicitly evaluated integrals", "kind_zh": "动量定律与明确求值的积分", "diagram": { "type": "gel", "labels": [ "R, Ṙ, R̈: wall motion", "ar: material acceleration at r", "ρg: exterior inertia", "Trr, Tθθ: total stresses", "4ηgṘ/R: signed resistance", "ur, t" ], "notes": [ "Both Eulerian and convective acceleration are retained", "The infinite exterior supplies effective inertia" ] }, "caption_en": "Integrating the material acceleration and the viscous stress difference gives their distinct radial contributions.", "caption_zh": "对材料加速度与黏性应力差积分，得到各自不同的径向贡献。" }, { "id": "A-E11", "tex": "\\rho_g\\left(R\\ddot R+\\frac32\\dot R^2\\right)=p_b-p_\\infty-\\frac{2\\sigma}{R}-p_{\\mathrm{el}}(R)-\\frac{4\\eta_g\\dot R}{R}.", "symbols_en": "$\\rho_g$ is effective medium density (kg m⁻³); $R>0$ is the cavity radius (m); $\\dot R$, $\\ddot R$ are wall speed/acceleration (m s⁻¹, m s⁻²). $p_b$, $p_\\infty$, $p_{\\mathrm{el}}(R)$ are cavity, far-field and signed elastic pressures (Pa); $\\sigma$ is tension (N m⁻¹); $\\eta_g$ is the represented medium viscosity (Pa s). Each term has units Pa.", "symbols_zh": "$\\rho_g$ 为有效介质密度（kg m⁻³）；$R>0$ 为腔体半径（m）；$\\dot R$、$\\ddot R$ 为泡壁速度／加速度（m s⁻¹、m s⁻²）。$p_b$、$p_\\infty$、$p_{\\mathrm{el}}(R)$ 为腔内、远场及带符号弹性压力（Pa）；$\\sigma$ 为张力（N m⁻¹）；$\\eta_g$ 为所描述介质的黏度（Pa s）。每一项的单位均为 Pa。", "kind": "Derived intact-medium radial control", "kind_zh": "推导得到的完整介质径向对照模型", "diagram": { "type": "gel", "labels": [ "pb: inner driving pressure", "p∞: exterior pressure", "2σ/R: capillary resistance", "pel(R): network resistance", "−4ηgṘ/R: viscous term", "ρg, R, Ṙ, R̈" ], "notes": [ "Spherical motion has no directional jet", "A liquid pocket requires a separate layered problem" ] }, "caption_en": "The spherical control quantifies cavity work and wall motion before any directional-flow calculation.", "caption_zh": "球形控制模型先量化腔体功与泡壁运动，再进行定向流动计算。" }, { "id": "A-E12", "tex": "\\begin{aligned}K_g&=2\\pi\\rho_gR^3\\dot R^2,\\qquad E_\\sigma=4\\pi\\sigma R^2,\\qquad \\dot V=4\\pi R^2\\dot R,\\\\\\frac{d}{dt}(K_g+W_g+E_\\sigma)&=(p_b-p_\\infty)\\dot V-P_{\\mathrm{dis}},\\\\P_{\\mathrm{dis}}&=16\\pi\\eta_gR\\dot R^2\\geq0.\\end{aligned}", "symbols_en": "$K_g,W_g,E_\\sigma$ are medium kinetic energy, stored network work and interface energy (J); $\\rho_g$ density (kg m⁻³); $R>0$ radius (m); $\\dot R$ wall speed (m s⁻¹); $\\sigma$ tension (N m⁻¹). $\\dot V$ is cavity volume-change rate (m³ s⁻¹), $p_b,p_\\infty$ pressures (Pa), $P_{\\mathrm{dis}}$ nonnegative viscous power (W), $\\eta_g\\geq0$ viscosity (Pa s), $t$ time (s); $d/dt$ is the full time derivative and $\\pi$ the circle constant.", "symbols_zh": "$K_g,W_g,E_\\sigma$ 为介质动能、储存网络功与界面能（J）；$\\rho_g$ 为密度（kg m⁻³）；$R>0$ 为半径（m）；$\\dot R$ 为泡壁速度（m s⁻¹）；$\\sigma$ 为张力（N m⁻¹）。$\\dot V$ 为腔体体积变化率（m³ s⁻¹），$p_b,p_\\infty$ 为压力（Pa），$P_{\\mathrm{dis}}$ 为非负黏性功率（W），$\\eta_g\\geq0$ 为黏度（Pa s），$t$ 为时间（s）；$d/dt$ 为全时间导数，$\\pi$ 为圆周率。", "kind": "Derived mechanical energy balance", "kind_zh": "推导得到的机械能平衡", "diagram": { "type": "energy", "labels": [ "(pb−p∞)V̇: pressure work", "Kg: moving exterior", "Wg: network storage", "Eσ: cavity surface", "Pdis ≥ 0: irreversible loss", "R, Ṙ, ρg, ηg, σ, t" ], "notes": [ "Network work can return; dashpot loss cannot", "Volume rate fixes work signs" ] }, "caption_en": "Pressure work divides into inertia, recoverable network energy, surface energy and viscous loss.", "caption_zh": "压力功分配到惯性、可恢复的网络能、表面能和黏性损耗。" }, { "id": "A-E13", "tex": "\\begin{aligned}p_{\\mathrm{el}}&=\\frac{20000}{2}\\left[5-4\\left(\\frac5{30}\\right)-\\left(\\frac5{30}\\right)^4\\right]=43325.6173\\ \\mathrm{Pa},\\\\W_g&=2\\pi(20000)\\left[\\frac53(30\\times10^{-6})^3-2(5\\times10^{-6})(30\\times10^{-6})^2\\right.\\\\&\\hspace{30mm}\\left.+\\frac{(5\\times10^{-6})^4}{30\\times10^{-6}}-\\frac23(5\\times10^{-6})^3\\right]\\\\&=4.51603944\\times10^{-9}\\ \\mathrm J.\\end{aligned}", "symbols_en": "$p_{\\mathrm{el}}$ is elastic resistance (Pa); $W_g$ is stored elastic work (J). The displayed 20000 is $G_g$ in Pa, 5 and 30 in the dimensionless ratios are both radii in µm, and $5\\times10^{-6}$, $30\\times10^{-6}$ are $R_{\\mathrm{ref}},R$ in m. $\\pi$ is dimensionless, $\\mathrm{Pa}$ means pascal and $\\mathrm J$ joule; $\\times$ indicates multiplication.", "symbols_zh": "$p_{\\mathrm{el}}$ 为弹性阻力（Pa）；$W_g$ 为储存弹性功（J）。所示 20000 为以 Pa 表示的 $G_g$，无量纲比值中的 5 与 30 均为以 µm 表示的半径，$5\\times10^{-6}$、$30\\times10^{-6}$ 为以 m 表示的 $R_{\\mathrm{ref}},R$。$\\pi$ 为无量纲量，$\\mathrm{Pa}$ 表示帕斯卡，$\\mathrm J$ 表示焦耳；$\\times$ 表示乘法。", "kind": "Teaching-input substitution", "kind_zh": "教学输入参数的代入", "diagram": { "type": "gel", "labels": [ "Gg = 20 kPa", "Rref = 5 µm", "R = 30 µm", "pel = 43.3256 kPa", "Wg = 4.51604 nJ" ], "notes": [ "Sixfold wall stretch: intactness is unverified", "Values are not PFC formulation measurements" ] }, "caption_en": "This soft network still stores several nanojoules when the cavity radius expands sixfold.", "caption_zh": "腔体半径膨胀六倍时，即使这个柔软网络也会储存数纳焦耳能量。" }, { "id": "A-E14", "tex": "\\begin{aligned}\\Delta V&=\\frac{4\\pi}{3}(R^3-R_{\\mathrm{ref}}^3)=1.12573737\\times10^{-13}\\ \\mathrm{m^3},\\\\W_{100}&:=\\Delta p_w\\Delta V=11.2573737\\ \\mathrm{nJ},\\\\\\frac{W_g}{W_{100}}&=0.401162791,\\qquad W_{100}-W_g=6.74133424\\ \\mathrm{nJ}.\\end{aligned}", "symbols_en": "$\\Delta V$ is expanded cavity volume (m³), $R=30$ µm and $R_{\\mathrm{ref}}=5$ µm are radii; $\\Delta p_w=100000$ Pa is a constant illustrative net pressure-work input. $W_{100}$ is the resulting work scale (J), with subscript 100 labeling its 100 kPa input, and $W_g=4.51603944$ nJ is stored network work. $\\pi$ is dimensionless; nJ means $10^{-9}$ J.", "symbols_zh": "$\\Delta V$ 为腔体膨胀体积（m³），$R=30$ µm、$R_{\\mathrm{ref}}=5$ µm 为半径；$\\Delta p_w=100000$ Pa 为恒定的示例净压力功输入。$W_{100}$ 为由此得到的功尺度（J），下标 100 标记其 100 kPa 输入，$W_g=4.51603944$ nJ 为储存网络功。$\\pi$ 为无量纲量；nJ 表示 $10^{-9}$ J。", "kind": "Finite work-budget comparison", "kind_zh": "有限功预算的比较", "diagram": { "type": "energy", "labels": [ "Δpw = 100 kPa", "ΔV: same cavity expansion", "W100 = 11.2574 nJ", "Wg = 4.51604 nJ", "residual = 6.74133 nJ", "R, Rref" ], "notes": [ "Compare energy at the same displaced volume", "Residual is not automatically an emitted jet" ] }, "caption_en": "The elastic cost occupies a finite fraction of the work budget even though its pressure is below 100 kPa.", "caption_zh": "即使弹性压力低于 100 kPa，其能量代价也占据功预算的有限份额。" }, { "id": "A-E15", "tex": "\\begin{aligned}p_{\\mathrm{visc,RHS}}&=-\\frac{4\\eta_g\\dot R}{R}=-\\frac{4(0.010)(-10)}{30\\times10^{-6}}=13333.3333\\ \\mathrm{Pa},\\\\P_{\\mathrm{dis}}&=16\\pi\\eta_gR\\dot R^2=16\\pi(0.010)(30\\times10^{-6})(100)=1.50796447\\times10^{-3}\\ \\mathrm W.\\end{aligned}", "symbols_en": "$p_{\\mathrm{visc,RHS}}$ is the signed viscous pressure term on the radial equation's right-hand side (Pa); RHS labels that side. $P_{\\mathrm{dis}}$ is dissipated power (W), $\\eta_g=0.010$ Pa s is illustrative viscosity, $R=30\\times10^{-6}$ m radius, $\\dot R=-10$ m s⁻¹ wall speed, and $\\pi$ the circle constant. The number 100 is the squared speed in m² s⁻².", "symbols_zh": "$p_{\\mathrm{visc,RHS}}$ 为径向方程右侧的带符号黏性压力项（Pa）；RHS 标记右侧。$P_{\\mathrm{dis}}$ 为耗散功率（W），$\\eta_g=0.010$ Pa s 为示例黏度，$R=30\\times10^{-6}$ m 为半径，$\\dot R=-10$ m s⁻¹ 为泡壁速度，$\\pi$ 为圆周率。数值 100 为以 m² s⁻² 表示的速度平方。", "kind": "Signed-pressure and positive-loss check", "kind_zh": "有符号压力与正耗散检验", "diagram": { "type": "gel", "labels": [ "Ṙ = −10 m/s: inward", "R = 30 µm", "ηg = 0.010 Pa s", "pvisc,RHS = +13.3333 kPa", "Pdis = +1.50796 mW" ], "notes": [ "Viscosity opposes collapse", "Dissipation is positive for either speed sign" ] }, "caption_en": "Inward motion changes the pressure sign but leaves dissipated power positive.", "caption_zh": "向内运动改变压力项符号，但耗散功率仍为正。" }, { "id": "A-E16", "tex": "\\mathrm{De}:=\\frac{\\tau_{\\mathrm{rel}}}{\\tau_e}.", "symbols_en": "$\\mathrm{De}$ is the dimensionless event Deborah number; $\\tau_{\\mathrm{rel}}>0$ is an independently identified network stress-relaxation time (s), and $\\tau_e>0$ is the specified cavity/jet event duration (s). Subscripts rel and e label relaxation and event respectively.", "symbols_zh": "$\\mathrm{De}$ 为无量纲事件 Deborah 数；$\\tau_{\\mathrm{rel}}>0$ 为独立确定的网络应力松弛时间（s），$\\tau_e>0$ 为指定的腔体／射流事件持续时间（s）。下标 rel、e 分别表示松弛与事件。", "kind": "Response-time definition", "kind_zh": "响应时间的定义", "diagram": { "type": "timescale", "labels": [ "τe: cavity/jet duration", "τrel: stress memory", "De = τrel/τe" ], "notes": [ "Memory persistence differs from stiffness", "Use the actual transient event" ] }, "caption_en": "The event duration determines which part of the network response can act before the jet or load arrives.", "caption_zh": "事件持续时间决定网络响应的哪一部分能在射流或载荷到达前发挥作用。" }, { "id": "A-E17", "tex": "\\begin{aligned}T_{\\mathrm{sh}}&=G_g\\gamma+\\eta_g\\dot\\gamma,\\\\T_{\\mathrm{sh}}=0:\\quad\\dot\\gamma&=-\\frac{G_g}{\\eta_g}\\gamma,\\qquad\\gamma(t)=\\gamma_0\\exp(-t/\\tau_{\\mathrm{KV}}),\\qquad\\tau_{\\mathrm{KV}}:=\\frac{\\eta_g}{G_g}.\\end{aligned}", "symbols_en": "$T_{\\mathrm{sh}}$ is shear stress (Pa); $\\gamma$ is dimensionless small shear strain, $\\dot\\gamma$ its rate (s⁻¹), $G_g>0$ shear modulus (Pa), and $\\eta_g>0$ viscosity (Pa s). $t\\geq0$ is time after release (s), $\\gamma_0$ initial strain, $\\tau_{\\mathrm{KV}}$ the Kelvin–Voigt zero-stress recovery/retardation time (s), and $\\exp$ the exponential of a dimensionless argument. sh and KV are labels, not multiplication.", "symbols_zh": "$T_{\\mathrm{sh}}$ 为剪切应力（Pa）；$\\gamma$ 为无量纲小剪切应变，$\\dot\\gamma$ 为其速率（s⁻¹），$G_g>0$ 为剪切模量（Pa），$\\eta_g>0$ 为黏度（Pa s）。$t\\geq0$ 为卸载后的时间（s），$\\gamma_0$ 为初始应变，$\\tau_{\\mathrm{KV}}$ 为 Kelvin–Voigt 零应力恢复／迟滞时间（s），$\\exp$ 表示对无量纲参数取指数。sh 与 KV 是标签，不是乘法。", "kind": "Linear constitutive check for the chosen dashpot", "kind_zh": "所选黏性阻尼元件的线性本构检验", "diagram": { "type": "timescale", "labels": [ "Tsh = 0 after unloading", "γ₀ → γ(t)", "Gg, ηg", "τKV = ηg/Gg", "γ̇, t" ], "notes": [ "Zero-stress strain recovery is not held-strain stress relaxation", "No periodic forcing is assumed" ] }, "caption_en": "Kelvin–Voigt strain recovery illustrates what its parameter ratio means and what it does not mean.", "caption_zh": "Kelvin–Voigt 应变恢复说明其参数比值的意义及不能代表的意义。" }, { "id": "A-E18", "tex": "\\begin{aligned}c_s&\\simeq\\sqrt{G_g/\\rho_g}=4.47213595\\ \\mathrm{m\\,s^{-1}},\\qquad t_s:=L_g/c_s=6.70820393\\ \\mathrm{\\mu s},\\\\t_{c,\\mathrm{liq}}&=0.9146813565\\,R_{\\max}\\sqrt{\\rho_l/\\Delta p_c}=2.74404407\\ \\mathrm{\\mu s},\\qquad\\frac{t_s}{t_{c,\\mathrm{liq}}}=2.44464147.\\end{aligned}", "symbols_en": "$c_s$ is a small-strain shear speed estimate (m s⁻¹), $G_g=20000$ Pa modulus, $\\rho_g=1000$ kg m⁻³ medium density, $L_g=30$ µm comparison length, and $t_s$ its shear crossing time (s). $t_{c,\\mathrm{liq}}$ is ideal empty-liquid-cavity collapse time (s), $R_{\\max}=30$ µm maximum radius, $\\rho_l=1000$ kg m⁻³ liquid density, and $\\Delta p_c=100000$ Pa collapse pressure gap. The coefficient is the Rayleigh integral; liq labels the liquid control, $\\simeq$ is an estimate, and µs means $10^{-6}$ s.", "symbols_zh": "$c_s$ 为小应变剪切波速估计（m s⁻¹），$G_g=20000$ Pa 为模量，$\\rho_g=1000$ kg m⁻³ 为介质密度，$L_g=30$ µm 为比较长度，$t_s$ 为该长度上的剪切传播时间（s）。$t_{c,\\mathrm{liq}}$ 为理想空液体腔的塌缩时间（s），$R_{\\max}=30$ µm 为最大半径，$\\rho_l=1000$ kg m⁻³ 为液体密度，$\\Delta p_c=100000$ Pa 为塌缩压力差。系数来自 Rayleigh 积分；liq 标记液体控制模型，$\\simeq$ 表示估计，µs 表示 $10^{-6}$ s。", "kind": "Teaching response-time comparison", "kind_zh": "响应时间的教学比较", "diagram": { "type": "timescale", "labels": [ "Lg = Rmax = 30 µm", "Gg = 20 kPa", "ρg = ρl = 1000 kg/m³", "cs = 4.47214 m/s", "ts = 6.70820 µs", "tc,liq = 2.74404 µs", "Δpc = 100 kPa" ], "notes": [ "Compare two distinct controls, not an actual gel-collapse prediction", "Support deformation need not equilibrate during the event" ] }, "caption_en": "The gel communication estimate exceeds the ideal liquid-collapse reference for the declared inputs.", "caption_zh": "对已声明输入，凝胶传播时间估计大于理想液体塌缩参考时间。" }, { "id": "A-E19", "tex": "c_L\\simeq\\sqrt{\\frac{K_g^{\\mathrm{bulk}}+4G_g/3}{\\rho_g}},\\qquad t_L:=\\frac{L_g}{c_L},\\qquad M_w:=\\frac{|\\dot R|}{c_L}.", "symbols_en": "$c_L$ is longitudinal wave speed estimate (m s⁻¹) in a separate small-strain compressible extension; $K_g^{\\mathrm{bulk}}>0$ is bulk modulus (Pa), distinctly labeled from kinetic energy $K_g$; $G_g$ shear modulus (Pa), $\\rho_g$ density (kg m⁻³), $L_g$ length (m), $t_L$ longitudinal crossing time (s), $M_w$ dimensionless wall Mach number, and $\\dot R$ wall speed (m s⁻¹). $|\\ |$ takes absolute value; $\\simeq$ denotes an estimate.", "symbols_zh": "$c_L$ 为独立小应变可压缩扩展中的纵波波速估计（m s⁻¹）；$K_g^{\\mathrm{bulk}}>0$ 为体积模量（Pa），其明确标签使之区别于动能 $K_g$；$G_g$ 为剪切模量（Pa），$\\rho_g$ 为密度（kg m⁻³），$L_g$ 为长度（m），$t_L$ 为纵波传播时间（s），$M_w$ 为无量纲泡壁 Mach 数，$\\dot R$ 为泡壁速度（m s⁻¹）。$|\\ |$ 表示绝对值；$\\simeq$ 表示估计。", "kind": "Separate compressibility diagnostic", "kind_zh": "独立的可压缩性诊断", "diagram": { "type": "timescale", "labels": [ "cL: bulk and shear response", "Kgᵇᵘˡᵏ, Gg, ρg", "Lg/cL = tL", "|Ṙ|/cL = Mw" ], "notes": [ "Small shear stiffness does not establish small longitudinal impedance", "Do not compute shock load from the incompressible control" ] }, "caption_en": "Shear and longitudinal propagation test different aspects of a water-rich gel's transient response.", "caption_zh": "剪切与纵向传播检验富水凝胶瞬态响应的不同方面。" }, { "id": "A-E20", "tex": "\\begin{aligned}\\mathbf j_l&=-\\frac{k_{\\mathrm{perm}}}{\\mu_l}\\nabla p_{\\mathrm{pore}},\\qquad\\frac{1}{M_d}\\frac{\\partial p_{\\mathrm{pore}}}{\\partial t}+\\nabla\\cdot\\mathbf j_l=0,\\\\\\frac{\\partial p_{\\mathrm{pore}}}{\\partial t}&=D_{\\mathrm{poro}}\\nabla^2p_{\\mathrm{pore}},\\qquad D_{\\mathrm{poro}}:=\\frac{k_{\\mathrm{perm}}M_d}{\\mu_l},\\\\t_{\\mathrm{poro}}&\\sim\\frac{L_g^2}{D_{\\mathrm{poro}}}=\\frac{\\mu_lL_g^2}{k_{\\mathrm{perm}}M_d}.\\end{aligned}", "symbols_en": "$\\mathbf j_l$ is relative solvent volume flux (m s⁻¹); $k_{\\mathrm{perm}}>0$ permeability (m²), $\\mu_l>0$ solvent viscosity (Pa s), $p_{\\mathrm{pore}}$ incremental pore pressure (Pa), $M_d>0$ storage/constrained drained modulus adopted for this scalar approximation (Pa), $t$ time (s), $D_{\\mathrm{poro}}$ diffusion coefficient (m² s⁻¹), $L_g$ drainage length (m), and $t_{\\mathrm{poro}}$ drainage time estimate (s). $\\nabla$, $\\nabla\\cdot$, $\\nabla^2$ are gradient, divergence and Laplacian in space; the dot here is the vector divergence operator, not an equation separator; $\\sim$ means scaling estimate.", "symbols_zh": "$\\mathbf j_l$ 为相对溶剂体积通量（m s⁻¹）；$k_{\\mathrm{perm}}>0$ 为渗透率（m²），$\\mu_l>0$ 为溶剂黏度（Pa s），$p_{\\mathrm{pore}}$ 为增量孔隙压力（Pa），$M_d>0$ 为该标量近似采用的储存／受约束排水模量（Pa），$t$ 为时间（s），$D_{\\mathrm{poro}}$ 为扩散系数（m² s⁻¹），$L_g$ 为排水长度（m），$t_{\\mathrm{poro}}$ 为排水时间估计（s）。$\\nabla$、$\\nabla\\cdot$、$\\nabla^2$ 为空间梯度、散度和 Laplace 算子；此处点号属于向量散度算子，不是方程分隔符；$\\sim$ 表示尺度估计。", "kind": "Darcy/storage approximation and derived diffusion", "kind_zh": "达西流动与储存近似及所推导的扩散关系", "diagram": { "type": "gel", "labels": [ "ppore: solvent pressure", "jl: solvent relative to network", "kperm: network permeability", "μl: solvent resistance", "Md: storage modulus", "Lg, tporo, Dporo, t" ], "notes": [ "Drainage redistributes solvent over Lg", "Small perturbation and constant-coefficient assumptions are separate" ] }, "caption_en": "Pressure gradients drive solvent through the network, producing a length-dependent drainage time.", "caption_zh": "压力梯度驱动溶剂通过网络，产生依赖长度的排水时间。" }, { "id": "A-E21", "tex": "\\rho_g\\left(R\\ddot R+\\frac32\\dot R^2\\right)=p_b-p_\\infty-\\frac{2\\sigma}{R}-p_{\\mathrm{el}}(R)-\\frac{4\\eta_g\\dot R}{R}.", "symbols_en": "$\\rho_g$ is effective exterior density (kg m⁻³); $R>0$ cavity radius (m); $\\dot R,\\ddot R$ wall speed/acceleration (m s⁻¹, m s⁻²); $p_b,p_\\infty,p_{\\mathrm{el}}$ cavity, far-field and elastic pressures (Pa); $\\sigma$ tension (N m⁻¹); $\\eta_g$ represented medium viscosity (Pa s). This quoted original formula has the intact spherical, no-slip assumptions of Eq. (A-E11).", "symbols_zh": "$\\rho_g$ 为外部有效密度（kg m⁻³）；$R>0$ 为腔体半径（m）；$\\dot R,\\ddot R$ 为泡壁速度／加速度（m s⁻¹、m s⁻²）；$p_b,p_\\infty,p_{\\mathrm{el}}$ 为腔内、远场及弹性压力（Pa）；$\\sigma$ 为张力（N m⁻¹）；$\\eta_g$ 为所描述介质的黏度（Pa s）。这一引用的原始公式采用式（A-E11）的完整球形、无滑移假设。", "kind": "Original radial control recalled", "kind_zh": "回顾原始径向对照模型", "diagram": { "type": "gel", "labels": [ "pb → radial cavity motion", "R, Ṙ, R̈, ρg", "p∞, 2σ/R, pel, ηg" ], "notes": [ "Radial growth alone has no preferred jet direction", "An outlet or a deforming solid supplies another problem" ] }, "caption_en": "A radial model describes expansion and resistance; it does not manufacture an outlet.", "caption_zh": "径向模型描述膨胀及阻力，不会凭空产生出口。" }, { "id": "A-E22", "tex": "\\begin{aligned}p_{\\mathrm{el}}(R)&=\\frac{G_g}{2}\\left[5-4\\frac{R_{\\mathrm{ref}}}{R}-\\left(\\frac{R_{\\mathrm{ref}}}{R}\\right)^4\\right],\\\\W_g(R)&=2\\pi G_g\\left[\\frac{5R^3}{3}-2R_{\\mathrm{ref}}R^2+\\frac{R_{\\mathrm{ref}}^4}{R}-\\frac{2R_{\\mathrm{ref}}^3}{3}\\right].\\end{aligned}", "symbols_en": "$p_{\\mathrm{el}}$ is elastic resistance (Pa), $W_g$ stored work (J), $G_g>0$ shear modulus (Pa), and $R\\geq R_{\\mathrm{ref}}>0$ current/reference cavity radii (m); $\\pi$ is dimensionless. These quote Eqs. (A-E05) and (A-E07) for the intact infinite incompressible neo-Hookean medium.", "symbols_zh": "$p_{\\mathrm{el}}$ 为弹性阻力（Pa），$W_g$ 为储存功（J），$G_g>0$ 为剪切模量（Pa），$R\\geq R_{\\mathrm{ref}}>0$ 为当前／参考腔体半径（m）；$\\pi$ 为无量纲量。这些公式引用式（A-E05）与（A-E07），适用于完整、无限、不可压缩 neo-Hookean 介质。", "kind": "Original pressure and work formulas recalled", "kind_zh": "回顾原始压力与功公式", "diagram": { "type": "energy", "labels": [ "Gg, Rref, R", "pel → 5Gg/2 at large R", "Wg grows with displaced volume", "π: sphere geometry" ], "notes": [ "Pressure plateau is not a fracture threshold", "Stored work is not automatically lost heat" ] }, "caption_en": "Pressure resistance and elastic work answer different parts of the feasibility calculation.", "caption_zh": "压力阻力与弹性功回答可行性计算中的不同部分。" }, { "id": "A-E23", "tex": "\\mathrm{De}=\\frac{\\tau_{\\mathrm{rel}}}{\\tau_e},\\qquad t_s=\\frac{L_g}{\\sqrt{G_g/\\rho_g}},\\qquad t_{\\mathrm{poro}}\\sim\\frac{\\mu_lL_g^2}{k_{\\mathrm{perm}}M_d}.", "symbols_en": "$\\mathrm{De}$ is dimensionless event Deborah number, $\\tau_{\\mathrm{rel}}$ measured stress-memory time, $\\tau_e$ event duration and $t_s,t_{\\mathrm{poro}}$ shear/drainage estimates (s). $L_g$ is a specified length (m), $G_g$ shear modulus (Pa), $\\rho_g$ density (kg m⁻³), $\\mu_l$ solvent viscosity (Pa s), $k_{\\mathrm{perm}}$ permeability (m²), and $M_d$ adopted drained storage modulus (Pa). $\\sim$ denotes scaling and the square root is the small-strain shear-speed estimate. These recall Eqs. (A-E16), (A-E18) and (A-E20).", "symbols_zh": "$\\mathrm{De}$ 为无量纲事件 Deborah 数，$\\tau_{\\mathrm{rel}}$ 为实测应力记忆时间，$\\tau_e$ 为事件持续时间，$t_s,t_{\\mathrm{poro}}$ 为剪切／排水时间估计（s）。$L_g$ 为指定长度（m），$G_g$ 为剪切模量（Pa），$\\rho_g$ 为密度（kg m⁻³），$\\mu_l$ 为溶剂黏度（Pa s），$k_{\\mathrm{perm}}$ 为渗透率（m²），$M_d$ 为所采用的排水储存模量（Pa）。$\\sim$ 表示尺度关系，平方根为小应变剪切波速估计。这些公式引用式（A-E16）、（A-E18）、（A-E20）。", "kind": "Original event-response formulas recalled", "kind_zh": "回顾原始事件响应公式", "diagram": { "type": "timescale", "labels": [ "τrel/τe = De", "Lg√(ρg/Gg) = ts", "μlLg²/(kpermMd) ≈ tporo" ], "notes": [ "Match material response to the transient event", "Do not substitute slow stiffness for every rate" ] }, "caption_en": "Three timescale comparisons constrain three different physical assumptions.", "caption_zh": "三个时间尺度比较分别约束三个不同的物理假设。" }, { "id": "A-E24", "tex": "\\mathbf J_{\\mathrm{target}}=\\int_{t_0}^{t_1}\\int_{A_t}\\mathbf t_{\\mathrm{load}}\\,dA\\,dt,\\qquad W_{\\mathrm{sep,min}}=\\int_{A_{\\mathrm{rel}}}\\Gamma_c\\,dA.", "symbols_en": "$\\mathbf J_{\\mathrm{target}}$ is vector impulse delivered to the specified target (N s); $\\mathbf t_{\\mathrm{load}}$ is external load traction (Pa), $A_t$ target loading area (m²), and $t_0,t_1$ defined start/end times (s). $W_{\\mathrm{sep,min}}$ is necessary fracture-work budget (J), $A_{\\mathrm{rel}}$ intended released interface area (m²), and $\\Gamma_c$ relevant interface fracture energy (J m⁻²), conditional on mode and rate. $dA,dt$ are area/time measures and $\\int$ is definite integration. Neither integral alone proves crack onset or intact landing.", "symbols_zh": "$\\mathbf J_{\\mathrm{target}}$ 为传递到指定目标的向量冲量（N s）；$\\mathbf t_{\\mathrm{load}}$ 为外加载荷牵引（Pa），$A_t$ 为目标受载面积（m²），$t_0,t_1$ 为明确的起止时间（s）。$W_{\\mathrm{sep,min}}$ 为必要断裂功预算（J），$A_{\\mathrm{rel}}$ 为目标释放界面面积（m²），$\\Gamma_c$ 为相关界面断裂能（J m⁻²），取决于模态与速率。$dA,dt$ 为面积／时间测度，$\\int$ 为定积分。这两个积分都不能独自证明裂纹起始或完整落点。", "kind": "Original load and interface-work accounting", "kind_zh": "原始载荷与界面功的收支核算", "diagram": { "type": "film", "labels": [ "tload: target traction", "At: loaded footprint", "t₀ → t₁: observation interval", "Jtarget: delivered impulse", "Arel, Γc: intended release", "Wsep,min" ], "notes": [ "Observe the actual direct or PVC-transmitted load path", "Release work is necessary, not sufficient" ] }, "caption_en": "A fair comparison follows the delivered load into the intended interface and the final placement.", "caption_zh": "公平比较要跟踪实际载荷进入目标界面及最终定位。" } ]