Result: published experiments demonstrate optical cavitation of PFC droplets inside hydrogel. This study verifies a conditional finite-PFC model. Target laser threshold, channel formation, collapse jets and impact peaks are not established. Missing: the spatial models and measurements listed below.

结果：已发表实验表明，水凝胶中的 PFC 液滴可以发生光致空化。本研究验证一个有条件的有限 PFC 模型。目标样品的激光阈值、通道形成、塌缩射流和冲击峰值尚未确立。缺少：下文列出的空间模型与实测数据。

The earlier ideal-gas control calculates gel deformation and discharge through an existing outlet. Its open-gel peaks are 1.625 m/s outlet speed and 9.143 kPa elastic resistance. Its sealed control emits zero liquid. These results belong to that outlet model. PFC vaporization and fracture require additional equations.

此前的理想气体模型计算凝胶变形及已有出口中的排液。开口凝胶的峰值为出口速度 1.625 m/s、弹性阻力 9.143 kPa。密封对照的排液量为零。这些结果属于该出口模型。PFC 汽化和断裂需要额外方程。

Two native COMSOL 6.4 models include finite real-fluid PFC mass, a vapor core, a surviving liquid shell and intact gel. Each retains separate inner and outer interfaces. The stationary model solves four phase, mass and traction equations. The transient uses three differential states and four thermodynamic algebraic unknowns. Native Helmholtz EOS functions are checked against independent CoolProp calculations. The transient starts from a supplied post-nucleation seed. Seed formation remains outside the solved model.

两个原生 COMSOL 6.4 模型包含有限真实流体 PFC 质量、蒸气核、剩余液体壳层和完整凝胶。内外界面分别处理。稳态模型求解四个相态、质量与牵引方程。瞬态模型包含三个微分状态及四个热力学代数未知量。原生 Helmholtz 状态方程与独立 CoolProp 计算比较。瞬态从预设的成核后种子开始。种子形成仍在已求解模型范围之外。

S1: embedded-droplet experiments establish feasibility in published specimens. Their bubble model includes water vapor and noncondensable gas. These PFC models omit both. Read the complete derivation and source ledger.

S1：嵌入液滴实验确立了已发表样品中的可行性。其气泡模型含水蒸气和不凝气体。当前 PFC 模型省略两者。参阅完整推导和来源台账。

Equal stationary stretch compares equilibrium state energies. Required state energy varies between those cases. The viscosity sweep holds retained seed energy fixed within each size. Optical design comparisons must match local incident fluence and absorption law. Mechanical comparisons must match absorbed energy and geometry. Retained seed energy includes storage and ambient work relative to the all-liquid reference. Preparation is lossless here. Measured absorption must also account for pre-seed conduction, viscosity, acoustics and water/coating heating.

同稳态伸长比比较平衡状态能量。各算例所需的状态能量不同。黏度扫描在同尺寸内固定保留种子能量。光学设计比较须匹配局部入射能流密度与吸收规律。力学比较须匹配吸收能量和几何。保留种子能量包含相对全液态参考的储能与环境功。这里假定制备无损。实测吸收还须计入成核前导热、黏性、声学及水／涂层加热。

At stretch 2.1, shear modulus 20 kPa and the same stipulated 54 mN/m outer interface:

条件：伸长比 2.1、剪切模量 20 kPa、预设外界面张力 54 mN/m。

PFP requires lower state temperature and energy than PFH here. The 0.6 µm droplet has stronger capillary confinement. Its initial Laplace excess is 360 kPa. The 6 µm droplet has 36 kPa. Their elastic-to-capillary ratios at 20 kPa modulus are 0.111 and 1.111. Smaller droplets therefore face stronger curvature effects despite lower total energy. These temperatures describe finite-mass curved equilibrium states. Optical activation temperatures are not established.

这些对照中，PFP 的状态温度和能量低于 PFH。0.6 µm 液滴受到更强的毛细约束。其初始 Laplace 超压为 360 kPa。6 µm 液滴为 36 kPa。在 20 kPa 模量下，两者弹性／毛细比为 0.111 和 1.111。小液滴总能量较低，但曲率效应更强。这些温度描述有限质量的曲面平衡状态。光学激活温度尚未确立。

The liquid shell survives substantial expansion. At outer stretch 2.1, vapor contains about 20.2% of the smaller PFP mass. The larger case has about 9.2%. A pre-existing gas approximation removes this finite inventory constraint. Reproducing these states requires phase-specific sensible and phase energy.

显著膨胀后，液壳仍然存活。外伸长比 2.1 时，小 PFP 算例约 20.2% 的质量为蒸气。大算例约为 9.2%。预存气体近似会移除有限物质量约束。重现这些状态需要各相的显热及相变能量。

PFP equilibria at matched stretch 2.1 and outer tension 54 mN/m:

PFP 平衡条件：同伸长比 2.1、外界面张力 54 mN/m。

Higher modulus raises gel resistance and equilibrium state energy. Modulus matters more when capillary confinement is weaker. The 20 kPa, stretch-2.1 state has 30.438 kPa elastic resistance. The earlier outlet model reaches a separate 9.143 kPa peak. Strength and fracture toughness require independent measurements.

模量越高，凝胶阻力和平衡状态能量越大。毛细约束较弱时，模量影响更明显。20 kPa、伸长比 2.1 的状态具有 30.438 kPa 弹性阻力。此前出口模型的峰值为另一状态下的 9.143 kPa。强度与断裂韧性须独立实测。

For 0.6 µm-diameter PFP at 20 kPa modulus and stretch 2.1:

条件：PFP 直径 0.6 µm、模量 20 kPa、伸长比 2.1。

Interface chemistry strongly affects the smaller droplets. Inner liquid/vapor tension sets the core pressure jump. Outer liquid/aqueous tension sets initial compression and gel traction. Use values matching interface, coating, composition and temperature. Shell disappearance requires a new gas/aqueous-gel interface and phase closure.

界面化学显著影响小液滴。内侧液／汽张力控制蒸气核压差。外侧液／水张力控制初始压缩与凝胶牵引。取值须匹配界面、涂层、组分及温度。液壳消失后，需要新的气体／水性凝胶界面规律与相态闭合。

PFP, 20 kPa shear modulus, the specified seed, and no hydraulic outlet:

条件：PFP、剪切模量 20 kPa、指定种子、无液压出口。

At 0.01 Pa s, the 0.6 µm case reaches stretch 2.1 in 0.3384 µs. Its peak radial wall speed is 0.9596 m/s. The 6 µm case reaches it in 1.2022 µs. Its peak is 2.7231 m/s. All six cases emit zero exterior mass. The small, high-viscosity case still grows at its 1.5 µs cutoff. Its maximum radius is not established beyond that window.

黏度 0.01 Pa s 时，0.6 µm 算例在 0.3384 µs 达到伸长比 2.1。其径向壁速峰值为 0.9596 m/s。6 µm 算例在 1.2022 µs 达到该值。其峰值为 2.7231 m/s。全部六个算例的外排质量为零。小尺寸高黏度算例在 1.5 µs 截止时仍在膨胀。窗口之后的最大半径尚未确立。

At 0.01 Pa s, dimensionless viscous coefficients are 3.333 and 0.333. They correspond to the smaller and larger radii. Higher viscosity slows radial growth in these matched-seed cases. An outlet adds another route for energy and fluid motion. Compare radial motion and outlet flow with their own observables.

在 0.01 Pa s 下，无量纲黏性系数为 3.333 和 0.333，分别对应小半径与大半径。匹配种子条件下，黏度增加会减慢径向膨胀。出口提供另一条能量和流体运动路径。径向运动与出口流动须分别比较各自观测量。

All 132 stationary states have finite inventory and positive shell thickness. Phase densities are ordered. Both phase pressure-density slopes are positive. Every state passes the internal core/phase Hessian test at fixed temperature, volume and mass. The minimum bulk-to-capillary curvature ratio is 20.4526. Nucleation accessibility and general shape stability are not established.

全部 132 个稳态具有有限物质量和正液壳厚度。相密度顺序正确。两相的压力－密度斜率均为正。每个状态均通过固定温度、体积与质量的内部蒸气核／相态 Hessian 检验。最小体积／毛细曲率比为 20.4526。成核可达性与一般形状稳定性尚未确立。

36 closed-energy states and 94 fixed-temperature states have non-restoring radial slopes. Thermal constraints change the stability result. Halving the perturbation changes closed-energy slopes by at most 3.205e-05 relatively. No sign changes occur. Test branch stability separately from phase-root convergence.

36 个闭能量状态和 94 个定温状态具有非恢复径向斜率。热约束会改变稳定性结果。扰动减半时，闭能量斜率的最大相对变化为 3.205e-05。符号均未变化。分支稳定性须与相平衡根收敛分开检验。

Use an illustrative local incident fluence of 20 mJ/cm² (200 J/m²). Projected-disk energies are 56.55 pJ and 5.655 nJ for 0.6 and 6 µm diameters. PFP stretch-2.1 state-energy ratios are 0.217 and 0.968. These are required effective cross-section ratios under lossless energy allocation. Effective absorption cross section divided by geometric area can exceed one. Measured absorption and activation thresholds are not established. Missing: absorber/coating heating, depth attenuation, nucleation sites and preparation losses. S7 uses different size, formulation, temperature and exposure history.

采用示例局部入射能流密度 20 mJ/cm²（200 J/m²）。0.6 和 6 µm 直径的投影圆盘能量为 56.55 pJ 和 5.655 nJ。PFP 在伸长比 2.1 的状态能量比为 0.217 和 0.968。这是假定无损分配时所需的有效截面比。有效吸收截面除以几何面积的比值可以超过一。实测吸收与激活阈值尚未确立。缺少：吸收体／涂层加热、深度衰减、成核位点与制备损失。S7使用不同尺寸、配方、温度及照射历史。

A stipulated thermal diffusivity of 1.4 × 10^-7 m²/s and 5 ns pulse gives Fourier numbers 0.00778 and 0.0000778. Heating remains spatially confined during the pulse. Uniform PFC temperature is not established. Activation requires spatial heat transfer and a nucleation-rate model. Apply equilibrium closure after a seed forms under compatible conditions.

预设热扩散率 1.4 × 10^-7 m²/s 和 5 ns 脉冲得到 Fourier 数 0.00778 与 0.0000778。脉冲期间加热局限于局部空间。PFC 温度均匀性尚未确立。激活阶段需要空间传热与成核速率模型。种子形成后，在相容条件下应用平衡闭合。

The slit-cost sweep uses covers 1/5/20/50 µm, half-widths 0.5/1/5 µm and toughnesses 0.1/1/10/100 J/m². These values define hypothetical cracks. A 20 µm cover, 0.5 µm half-width and 1 J/m² toughness costs 20 pJ. Small and large PFP seed energies are 13.50 pJ and 5.857 nJ. Under retained-seed allocation, the smaller case falls below this cost. Fracture impossibility is not established. Cooling, interface changes, prestress and other reference-energy changes alter available work. Exceeding the cost also leaves initiation and channel connectivity unresolved.

狭缝功扫描采用覆盖厚度 1/5/20/50 µm、半宽 0.5/1/5 µm、韧性 0.1/1/10/100 J/m²。这些值定义假想裂纹。20 µm 覆盖、0.5 µm 半宽及 1 J/m² 韧性的功为 20 pJ。小、大 PFP 种子能量为 13.50 pJ 与 5.857 nJ。分配保留种子能量时，小算例低于该成本。断裂不可能性尚未确立。冷却、界面变化、预应力及其他参考能量变化会改变可用功。超过该成本仍无法确定起裂与通道连通。

Finite depth requires a separate geometry check. Center-to-surface distance equals initial radius plus cover. A 3 µm radius under 1 µm cover encounters the initial surface plane at stretch 1.333. This occurs before the 2.1 screen. The infinite-host trajectory then loses its intact-interior interpretation. The CSV records standoff and this gate for every case. Finite-domain convergence remains necessary for other depths.

有限深度须单独检验几何。中心到表面的距离等于初始半径加覆盖厚度。半径 3 µm、覆盖 1 µm 的液滴在伸长比 1.333 接触初始表面平面。这早于 2.1 的筛查值。此后，无限基质轨迹不再代表完整内部膨胀。CSV 为每个算例记录距离及该门槛。其他深度仍须检验有限域收敛。

The implementation plan lists boundary conditions, tolerances, calibration and held-out comparisons. Begin with one 6 µm PFP droplet and a defined free surface. Retain a 0.6 µm case to test curvature and viscosity. This sequence supports validation. An optimal size is not established. Measure absorption, high-rate gel properties, toughness, strength and depth. Couple solid damage to an actual fluid-filled path. A prescribed outlet tests discharge after connection exists.

实施计划列出边界条件、容差、标定与留出比较。先采用一个 6 µm PFP 液滴及明确自由表面。保留 0.6 µm 算例检验曲率与黏度。该顺序用于验证。最优尺寸尚未确立。须测量吸收、高速凝胶性能、韧性、强度与深度。固体损伤须耦合实际充液通道。预设出口检验连通后的排液。

Maximum native/reference stationary relative difference is 4.516e-09. Maximum normalized transient waveform difference is 2.958e-05. Maximum native normalized energy residual is 4.971e-09. Independent DOP853/Radau and tighter-step calculations agree. Entropy drift is checked separately from the energy equation. Fresh-process reload and refinement evidence is in verification/. The prior independent review corrected seed-energy wording, fracture bounds, depth applicability, branch guards, stability scope and notation.

原生／参考稳态最大相对差为 4.516e-09。瞬态归一化波形最大差为 2.958e-05。原生归一化能量最大残差为 4.971e-09。独立 DOP853/Radau 与更小步长计算一致。熵漂移独立于能量方程检验。新进程重载和细化证据位于 verification/。此前独立审查修正了种子能量表述、断裂上界、深度适用性、分支限制、稳定性范围与符号。

The offline reader includes a computed-state selector. Download the native curved-equilibrium model and post-nucleation model. The package includes Java/Python sources, parameter and native/reference CSVs, LaTeX, the accepted English PDF and verification records. README.md gives reproduction commands.

离线阅读器包含计算状态选择器。可下载原生曲面平衡模型和成核后模型。包中包含 Java/Python 源码、参数与原生／参考 CSV、LaTeX、已接受的英文 PDF 和验证记录。README.md提供复现命令。

Status: the conditional finite-PFC comparison is complete and numerically verified. Target laser-to-fracture-to-jet validation is not established. Missing: specimen-specific spatial models and measured mechanism checks.

状态：有条件的有限 PFC 比较已完成并通过数值验证。目标激光－断裂－射流验证尚未确立。缺少：样品专属空间模型及机制实测检验。

Determine the state energy needed to expand a PFC vapor core inside intact hydrogel. Calculate gel load and energy. Identify the additional conditions for exterior ejection and collapse pressure.

确定完整水凝胶内 PFC 蒸气核膨胀所需的状态能量。计算凝胶载荷与能量。明确外部排液和塌缩压力所需的额外条件。

This conditional study solves finite PFC mass, concentric vapor and liquid phases, real-fluid thermodynamics, separate interfaces, gel elasticity and host viscosity. The transient starts from a supplied post-nucleation seed. Its retained energy is calculated relative to an all-liquid reference. Laser heating and seed formation remain unresolved.

本有条件研究求解有限 PFC 质量、同心气液相、真实流体热力学、独立界面、凝胶弹性和基质黏性。瞬态从预设的成核后种子开始。其保留能量相对全液态参考计算。激光加热与种子形成仍未解决。

Two models are retained. The first solves prescribed-radius curved equilibrium and local stability. The second solves insulated radial expansion from a prescribed seed. It stops at first turning, the declared stretch screen or the time cutoff. Both have zero outlet flow. The effectively infinite, undrained gel has radial inertia. Omitted effects are liquid-shell inertia, thermal gradients, phase-transfer kinetics, damage, water vapor, gas diffusion and acoustic radiation. Inner traction is quasistatic. These assumptions bound every result.

保留两类模型。第一类求解指定半径的曲面平衡与局部稳定性。第二类求解指定种子的绝热径向膨胀。首次转折、预设伸长比或时间截止时停止。两者出口流量均为零。有效无限、不排水凝胶保留径向惯性。省略液壳惯性、温度梯度、相变动力学、损伤、水蒸气、气体扩散及声辐射。内界面牵引为准静态。这些假设限定全部结果。

Use spherical coordinates directed outward from the inclusion center. Both radii are positive. Expansion has positive radial speed. Thermodynamic pressures are absolute. Resistance and pressure differences are signed excesses. The initial liquid has outer capillary traction and ambient loading. At infinity, displacement and speed vanish; stress approaches minus ambient pressure. PFC mass is closed and exterior mass flux is zero. Both phases have uniform temperature and equal chemical potential. This is a post-activation equilibrium approximation. Nanosecond thermal homogenization is not established.

采用从液滴中心向外的球坐标。两种半径均为正。膨胀时径向速度为正。热力学压力为绝对压力。阻力和压差为带符号超压。初始液体受到外界面毛细牵引与环境载荷。无穷远处位移和速度消失，应力趋于负环境压力。PFC 质量封闭，外排质量通量为零。两相温度均匀且化学势相等。这是激活后的平衡近似。纳秒内热均匀化尚未确立。

The all-liquid reference starts at room temperature, prescribed radius and Laplace pressure. The seed starts at stretch 1.2 with zero speed and zero accumulated loss. Its liquid pressure exceeds static host traction by 50 kPa. Include seed storage and ambient work. Preparation is lossless here; pre-seed heat, viscous and acoustic losses are omitted. Gel elasticity remains reversible up to the selected screen. Viscosity heats the host. Stretch 2.1 is a declared stopping screen. Measured failure strain is not established.

全液态参考从室温、指定半径和 Laplace 压力开始。种子的初始伸长比为 1.2，速度为零，累积损耗为零。液压超过静态基质牵引 50 kPa。计入种子储能与环境功。这里假定无损制备；省略成核前热、黏性和声学损失。选定筛查值以内的凝胶弹性可逆。黏性加热基质。伸长比 2.1 是预设停止值。实测破坏应变尚未确立。

Foundations: mass conservation; the first law; Helmholtz EOS S2; continuum interface energy and traction; incompressible neo-Hookean elasticity; Kelvin–Voigt viscosity; spherical inertia S6, S8. The core/shell closure and comparisons are original reduced-model calculations. Physical validation requires activation data, radius histories, damage, exterior mass flux and pressure at a defined sensor.

基础：质量守恒；热力学第一定律；Helmholtz 状态方程 S2；连续介质界面能和牵引；不可压缩 neo-Hookean 弹性；Kelvin–Voigt 黏性；球形惯性 S6, S8。核／壳闭合及比较为原创降阶计算。物理验证需要激活数据、半径历史、损伤、外排质量通量及指定传感器处的压力。

All scalars are real. Pi is the dimensionless circle constant. Superscripts on lengths are ordinary powers; a dot denotes differentiation with time. Subscripts 0, l, v, o, i, el, amb, gel, and host label initial, liquid, vapor, outer, inner, elastic, ambient, gel and effective host quantities. A partial derivative holds the displayed conditioning variable fixed. Integrals use the stated coordinate, area, volume or time measure. Logs and exponentials act on dimensionless arguments. Ordinary arithmetic, inequalities and limits have their usual real meanings.

所有标量为实数。圆周率无量纲。长度上标表示幂；点表示时间导数。下标 0、l、v、o、i、el、amb、gel、host 分别表示初始、液体、蒸气、外、内、弹性、环境、凝胶及有效基质。偏导数保持指定条件变量不变。积分使用注明的坐标、面积、体积或时间测度。对数和指数的自变量无量纲。算术、不等式与极限遵循通常的实数约定。

Gel modulus, viscosity, optical cross section, cover, cohesive strength and toughness are stipulated. They are unmeasured for the target formulation. The source ledger states the support for interface and EOS inputs.

凝胶模量、黏度、光截面、覆盖厚度、黏聚强度及韧性均为预设值。目标配方尚未实测。来源台账说明界面与状态方程输入的依据。

Symbols before Eq. (E01): $a_0,R,b$ are initial liquid, current outer and vapor-core radii (m), with $0<b<R$; $V_0,V,V_b,V_l$ are corresponding volumes (m³); pi is dimensionless. These are geometric identities for concentric spheres.

式 (E01) 前的符号：$a_0,R,b$ 为初始液体、当前外侧及蒸气核半径（m），满足 $0<b<R$；$V_0,V,V_b,V_l$ 为对应体积（m³）；圆周率无量纲。这些是同心球的几何恒等式。

The last identity subtracts the vapor volume from the whole inclusion. The surviving liquid shell is a separate domain; neither outer volume nor core volume is a PFC mass by itself.

末式从液滴总体积减去蒸气体积。剩余液体壳层是独立区域。体积须乘相密度才能得到质量。

Symbols before Eq. (E02): $m,m_l,m_v$ are fixed total, liquid and vapor PFC masses (kg); $\rho_{l0},\rho_l,\rho_v$ are initial liquid and current phase densities (kg/m³); $V_0,V_l,V_b$ are volumes (m³); $x_v$ is the dimensionless vapor mass fraction. Densities and total mass are strictly positive.

式 (E02) 前的符号：$m,m_l,m_v$ 为固定总 PFC、液相及汽相质量（kg）；$\rho_{l0},\rho_l,\rho_v$ 为初始液相及当前相密度（kg/m³）；$V_0,V_l,V_b$ 为体积（m³）；$x_v$ 为无量纲蒸气质量分数。密度与总质量严格为正。

The first row defines masses from EOS densities. The second imposes closed PFC inventory and divides by its positive initial mass. A root with negative shell volume or a vapor fraction outside the indicated interval is rejected, even if the nonlinear solver reports convergence.

首行用状态方程密度定义质量。次行施加封闭 PFC 质量守恒，并除以正初始质量。负液壳体积或越界蒸气分数的根均被拒绝，即使非线性求解器收敛。

Symbols before Eq. (E03): $p_{l0},p_\infty$ are initial liquid and ambient absolute pressures (Pa); $\gamma_o$ is constant outer tension (N/m); $a_0$ is initial radius (m); $T_0$ is initial temperature (K); $\rho_{l0}$ is initial liquid density (kg/m³); $\rho_l(T,p)$ denotes the liquid EOS density function. The initial gel is unstretched.

式 (E03) 前的符号：$p_{l0},p_\infty$ 为初始液体与环境绝对压力（Pa）；$\gamma_o$ 为恒定外张力（N/m）；$a_0$ 为初始半径（m）；$T_0$ 为初始温度（K）；$\rho_{l0}$ 为初始液体密度（kg/m³）；$\rho_l(T,p)$ 为液相状态方程密度函数。初始凝胶未伸长。

The first relation adopts the spherical capillary traction at the initially liquid outer interface. The second evaluates a liquid state at that pressure, rather than using an unrelated gas density. The jump has units N/m divided by m, or Pa, and decreases with increasing radius.

首式采用初始液态外界面的球形毛细牵引。次式在该压力下求液体状态。压差单位为 N/m 除以 m，即 Pa，并随半径增加而下降。

Symbols before Eq. (E04): $a_H(T,\rho)$ is specific Helmholtz energy (J/kg); $T$ is temperature (K); $\rho$ is a generic phase density (kg/m³); $p,u,s_H,g$ are pressure (Pa), specific internal energy (J/kg), entropy (J/(kg K)), and chemical potential (J/kg). Partial derivatives are at fixed subscripted variable. These thermodynamic identities are evaluated using the adopted EOS S2.

式 (E04) 前的符号：$a_H(T,\rho)$ 为比 Helmholtz 能（J/kg）；$T$ 为温度（K）；$\rho$ 为一般相密度（kg/m³）；$p,u,s_H,g$ 分别为压力（Pa）、比内能（J/kg）、比熵（J/(kg K)）及化学势（J/kg）。偏导保持下标变量不变。采用状态方程 S2求这些热力学恒等式。

The first row differentiates the Helmholtz representation at its stated independent variables. The second reconstructs internal and Gibbs energies. Pressure divided by density has unit J/kg, so the Gibbs expression is dimensionally consistent. Both phases use the same energy reference. No constant ideal-gas exponent or separately fitted pressure pulse appears.

首行按指定自变量对 Helmholtz 能求导。次行重建内能与 Gibbs 能。压力除以密度的单位是 J/kg，因此 Gibbs 表达式量纲一致。两相采用相同能量参考。压力由真实流体状态方程计算。

Symbols before Eq. (E05): $p_v,p_l$ are vapor and liquid absolute pressures (Pa); $g_v,g_l$ are vapor and liquid chemical potentials (J/kg); $\gamma_i$ is constant inner tension (N/m); $b$ is positive vapor-core radius (m); $T$ is uniform temperature (K). Equal temperature and instantaneous phase equilibrium are adopted constitutive approximations; interfaces are spherical.

式 (E05) 前的符号：$p_v,p_l$ 为蒸气与液体绝对压力（Pa）；$g_v,g_l$ 为两相化学势（J/kg）；$\gamma_i$ 为恒定内张力（N/m）；$b$ 为正蒸气核半径（m）；$T$ 为均匀温度（K）。等温与瞬时相平衡为采用的本构近似；界面为球形。

The capillary relation uses the inner liquid/gas interface. Chemical equilibrium equates chemical potential at the different phase pressures; it does not replace either pressure by a flat-interface saturation value. Using 0.072 N/m water/gas tension for every interface would change the problem. A very small core also raises the capillary jump; continuum curvature and constant tension become uncertain at molecular scales.

毛细关系使用内侧液／气界面。化学平衡令不同相压力下的化学势相等。平面饱和压力不能直接替代它们。将全部界面张力设为水／气的 0.072 N/m 会改变问题。小蒸气核提高毛细压差；分子尺度下连续曲率和恒定张力近似变得不确定。

Symbols before Eq. (E06): $s,r$ are positive reference and current host radial coordinates (m); $R,a_0$ are current and initial inclusion radii (m); $C$ is the volume-offset parameter (m³); $\lambda_r,\lambda_\theta$ are dimensionless radial/circumferential stretches; $d r/d s$ is the radial derivative. Incompressible spherical motion and $r\ge R$ are assumed.

式 (E06) 前的符号：$s,r$ 为正参考与当前基质径向坐标（m）；$R,a_0$ 为当前与初始液滴半径（m）；$C$ 为体积偏移参数（m³）；$\lambda_r,\lambda_\theta$ 为无量纲径向／周向伸长比；$d r/d s$ 为径向导数。假设球形不可压缩运动，且 $r\ge R$。

The first row conserves host volume between inner and any outer material sphere. The second collects constants and differentiates with respect to reference radius. Division by positive squared radius yields the third row, which verifies unit volume ratio explicitly.

首行保持内界面与任意外侧材料球间的基质体积。次行合并常量并对参考半径求导。除以正半径平方得到第三行，明确验证体积比为一。

Symbols before Eq. (E07): $\sigma_{rr},\sigma_{\theta\theta},\chi,G$ are elastic radial stress, tangential stress, incompressibility multiplier and positive shear modulus (Pa); $\lambda_r,\lambda_\theta$ are dimensionless stretches from Eq. (E06); $r$ is current radius (m); $d/dr$ is spatial differentiation. This is the incompressible neo-Hookean constitutive control and elastic radial equilibrium.

式 (E07) 前的符号：$\sigma_{rr},\sigma_{\theta\theta},\chi,G$ 为弹性径向应力、切向应力、不可压缩乘子及正剪切模量（Pa）；$\lambda_r,\lambda_\theta$ 为式 (E06) 的无量纲伸长比；$r$ 为当前半径（m）；$d/dr$ 表示空间求导。采用不可压缩 neo-Hookean 本构与弹性径向平衡。

The first two expressions declare the host material law. Their subtraction eliminates the unknown multiplier. The last relation is the spherically symmetric divergence of elastic stress, used to obtain the elastic traction component; inertia and viscosity are added separately below.

前两式定义基质材料律。相减消去未知乘子。末式为球对称弹性应力散度，用于计算弹性牵引。惯性与黏性在下文分别加入。

Symbols before Eq. (E08): $p_{el}$ is elastic wall resistance (Pa); $G$ is shear modulus (Pa); $R,a_0,r,s$ are radii (m); $\beta=s/r$ is dimensionless; $C=R^3-a_0^3$ is positive during expansion (m³); $\int$ denotes definite spatial integration. Upper endpoint infinity means an effectively infinite host. At exactly $R=a_0$, resistance follows by continuity.

式 (E08) 前的符号：$p_{el}$ 为弹性壁阻力（Pa）；$G$ 为剪切模量（Pa）；$R,a_0,r,s$ 为半径（m）；$\beta=s/r$ 无量纲；$C=R^3-a_0^3$ 在膨胀时为正（m³）；$\int$ 表示空间定积分。上限无穷表示有效无限基质。$R=a_0$ 时取连续极限。

Integrating Eq. (E07) with the far stress condition gives the first row. Eq. (E06) gives the variable transformation and its Jacobian. At the inner boundary the ratio is the initial/current inclusion radius; at infinity it tends to one. Factoring the numerator as the product of one minus the cube and one plus the cube permits cancellation for every finite radius in expansion; the endpoint integral has a removable limiting value. The final row evaluates both endpoints. This derivation makes the geometry assumption visible; it is not a planar-layer fracture law.

结合远场应力条件积分式 (E07)，得到首行。式 (E06) 给出变量变换及 Jacobian。内边界比值为初始／当前液滴半径；无穷远处趋于一。分子分解为一减三次方与一加三次方的乘积。在有限膨胀半径处约去非零因子。端点采用可去极限。末行计算两个端点。该结果依赖球形几何。平面层断裂需要另一模型。

Symbols before Eq. (E09): $E_{gel}$ is recoverable gel strain energy (J); $p_{el}$ is resistance (Pa); $R,a_0,\xi$ are current, initial and dummy radii (m); $G$ is shear modulus (Pa); $dE_{gel}/dR$ is the radius derivative. All integrals are along the reversible intact branch. $\pi$ is the dimensionless circle constant.

式 (E09) 前的符号：$E_{gel}$ 为可恢复凝胶应变能（J）；$p_{el}$ 为阻力（Pa）；$R,a_0,\xi$ 为当前、初始与积分半径（m）；$G$ 为剪切模量（Pa）；$dE_{gel}/dR$ 为半径导数。积分沿可逆完整分支进行。 $\pi$ 为无量纲圆周率。

The first row defines work as pressure times incremental cavity volume. The second substitutes Eq. (E08) and expands. Integrating each power gives the third row; evaluating the initial endpoint yields the constant in the fourth. Differentiation verifies the last identity independently and is also checked symbolically by the audit script.

首行以压力乘腔体体积增量定义功。次行代入式 (E08) 并展开。逐项积分各幂得到第三行。初始端点给出第四行常量。求导独立验证末式；审查脚本也作符号检验。

Symbols before Eq. (E10): $\lambda=R/a_0$ is stretch; $\epsilon$ is small dimensionless stretch excess; $p_{el}$ is resistance (Pa); $E_{gel}$ is strain energy (J); $G$ is modulus (Pa); $a_0$ is radius (m); $O$ denotes the asymptotic remainder order as excess tends to zero. $\pi$ is the dimensionless circle constant.

式 (E10) 前的符号：$\lambda=R/a_0$ 为伸长比；$\epsilon$ 为小无量纲伸长增量；$p_{el}$ 为阻力（Pa）；$E_{gel}$ 为应变能（J）；$G$ 为模量（Pa）；$a_0$ 为半径（m）；$O$ 表示增量趋零时的渐近余项阶数。 $\pi$ 为无量纲圆周率。

The zero-stretch results follow by direct substitution into Eqs. (E08)–(E09). Letting the radius ratios tend to zero gives the large-stretch elastic limit. Expanding reciprocal stretch and its fourth power to first order gives the pressure expansion; integrating its product with the leading cavity-volume derivative gives the energy expansion. The large-stretch pressure limit is a property of this intact neo-Hookean model; it is not a rupture pressure for a real hydrogel.

直接代入式 (E08)–(E09)，得到零伸长结果。半径比趋零给出大伸长弹性极限。将倒数伸长比及四次幂展开到一阶，得到压力展开。积分该压力与主阶体积导数的乘积，得到能量展开。大伸长压力极限属于完整 neo-Hookean 模型。实际凝胶破裂压力尚未确立。

Symbols before Eq. (E11): $u_h(r,t)$ is radial host speed (m/s); $R,r$ are inclusion and host radii (m); $W$ is inner-wall speed (m/s); $\rho_h$ is host density (kg/m³); $K_h$ is host kinetic energy (J); $\int$ is spatial integration over an effectively infinite incompressible host. $\pi$ is the dimensionless circle constant.

式 (E11) 前的符号：$u_h(r,t)$ 为基质径向速度（m/s）；$R,r$ 为液滴与基质半径（m）；$W$ 为内壁速度（m/s）；$\rho_h$ 为基质密度（kg/m³）；$K_h$ 为基质动能（J）；$\int$ 在有效无限不可压缩基质中作空间积分。 $\pi$ 为无量纲圆周率。

Incompressibility gives the first flux relation; division by positive area gives velocity. Substituting velocity squared in kinetic energy cancels two radius powers. The antiderivative of inverse squared radius is minus inverse radius, whose far endpoint is zero and inner endpoint contributes inverse inclusion radius. This retains host inertia but omits liquid-shell inertia, which must be tested in the full spatial model.

不可压缩条件给出首个通量关系。除以正面积得到速度。将速度平方代入动能，约去两个半径幂。反平方半径的原函数为负反半径。无穷远端点为零；内端点贡献液滴半径倒数。保留基质惯性，省略液壳惯性。完整空间模型须检验此省略。

Symbols before Eq. (E12): $D_{rr},D_{\theta\theta}$ are host radial/tangential deformation rates (1/s); $u_h$ is radial speed (m/s); $r,R$ are radii (m); $W$ is wall speed (m/s); $\eta$ is effective viscosity (Pa s); $P_{dis}$ is viscous-loss power (W); $p_{vis}$ is viscous wall resistance (Pa). There are two equal tangential components. $\pi$ is the dimensionless circle constant.

式 (E12) 前的符号：$D_{rr},D_{\theta\theta}$ 为基质径向／切向形变率（1/s）；$u_h$ 为径向速度（m/s）；$r,R$ 为半径（m）；$W$ 为壁速（m/s）；$\eta$ 为有效黏度（Pa s）；$P_{dis}$ 为黏性损耗功率（W）；$p_{vis}$ 为黏性壁阻力（Pa）。两个切向分量相等。 $\pi$ 为无量纲圆周率。

Differentiate Eq. (E11) and divide its velocity by radius for the first row. Squaring and summing the three deformation-rate components yields six equal factors; the integral of inverse fourth power contributes inverse cubed radius divided by three. Dividing loss power by signed volume flux gives viscous traction when speed is nonzero; its continuous zero-speed limit avoids division by zero. Loss power stays nonnegative in either motion direction.

对式 (E11) 求导，并将速度除以半径，得到首行。三个形变率分量平方求和，得到六倍共同因子。反四次幂积分贡献反三次方半径除以三。非零壁速时，损耗功率除以带符号体积通量得到黏性牵引。零壁速采用连续极限，避免除零。两种运动方向的损耗功率均非负。

Symbols before Eq. (E13): $R,W$ are radius and speed (m, m/s); $\rho_h$ is density (kg/m³); $p_l,p_\infty,p_{el},p_{vis},p_{need}$ are pressures (Pa); $\gamma_o$ is outer tension (N/m); dots are time derivatives. The radial momentum equation is the adopted incompressible host approximation.

式 (E13) 前的符号：$R,W$ 为半径与速度（m、m/s）；$\rho_h$ 为密度（kg/m³）；$p_l,p_\infty,p_{el},p_{vis},p_{need}$ 为压力（Pa）；$\gamma_o$ 为外张力（N/m）；点为时间导数。径向动量方程采用不可压缩基质近似。

The first row collects static ambient, elastic and outer capillary resistance. The second is spherical inertia driven by the remaining traction. Dividing by positive density and radius, and substituting Eq. (E12), gives the third. Each side of the momentum equation is Pa. A positive excess traction gives positive acceleration at zero initial speed. With zero elasticity and viscosity the equation reduces to the capillary spherical-inertia control, not a directional-jet model.

首行合并环境、弹性和外侧毛细静态阻力。次行由剩余牵引驱动球形惯性。除以正密度和半径，再代入式 (E12)，得到第三行。动量方程两侧单位均为 Pa。初速为零时，正超额牵引产生正加速度。弹性与黏性为零时，退化为毛细球形惯性对照。定向射流须另行求解。

Symbols before Eq. (E14): $U_{PFC},E_{surf},E_{amb}$ are internal energy, interface-energy change and ambient work (J); $m_l,m_v$ are phase masses (kg); $u_l,u_v$ are phase internal energies (J/kg); $\gamma_o,\gamma_i$ are distinct constant tensions (N/m); $R,a_0,b$ are radii (m); $p_\infty$ is ambient pressure (Pa); $V,V_0$ are inclusion volumes (m³). Initial internal energy is referenced separately below. $\pi$ is the dimensionless circle constant.

式 (E14) 前的符号：$U_{PFC},E_{surf},E_{amb}$ 为内能、界面能变化与环境功（J）；$m_l,m_v$ 为相质量（kg）；$u_l,u_v$ 为各相比内能（J/kg）；$\gamma_o,\gamma_i$ 为两个独立恒定张力（N/m）；$R,a_0,b$ 为半径（m）；$p_\infty$ 为环境压力（Pa）；$V,V_0$ 为液滴体积（m³）。初始内能参考在下文定义。 $\pi$ 为无量纲圆周率。

Internal energy includes phase conversion through the actual phase energies. Adding a second independent latent-heat term would double count it. The initial state has the outer liquid surface but no vapor core; the inner surface energy is therefore all new. Both surface terms have units N/m times m², or J. Constant tension assumes negligible interface entropy and ignores shell/coating rearrangement.

实际相内能已包含相变能量。另加独立潜热项会重复计数。初始状态有外侧液体表面，无蒸气核。因此内表面能全部新增。两种表面项的单位均为 N/m 乘 m²，即 J。恒定张力假设界面熵可忽略，并省略壳／涂层重排。

Symbols before Eq. (E15): $U_{PFC}$ is PFC internal energy (J); $T$ is common phase temperature (K); $S$ is total PFC entropy (J/K); $g_l,g_v$ are phase chemical potentials (J/kg); $m_l,m_v$ are masses (kg); $p_l,p_v$ are pressures (Pa); $V_l,V_b,V$ are phase/total volumes (m³); $\gamma_i$ is constant inner tension (N/m); $A_i=4\pi b^2$ is core area (m²); $d$ is a differential along an admissible equilibrium trajectory. $b$ is the positive core radius (m).

式 (E15) 前的符号：$U_{PFC}$ 为 PFC 内能（J）；$T$ 为共同相温度（K）；$S$ 为 PFC 总熵（J/K）；$g_l,g_v$ 为化学势（J/kg）；$m_l,m_v$ 为质量（kg）；$p_l,p_v$ 为压力（Pa）；$V_l,V_b,V$ 为相／总体积（m³）；$\gamma_i$ 为恒定内张力（N/m）；$A_i=4\pi b^2$ 为蒸气核面积（m²）；$d$ 为沿可容许平衡轨迹的微分。 $b$ 为正蒸气核半径（m）。

The first row applies the thermodynamic fundamental relation to each phase and sums them. Closed mass and Eq. (E05) cancel the chemical terms; volume subtraction yields the third row. Differentiating core area and volume gives their ratio of two divided by core radius; Eq. (E05) then cancels the inner traction work with surface-energy change. The last relation explains why a curved core's interface energy belongs in the conserved total. For reversible, insulated phase adjustment, the temperature–entropy term is zero. The audit checks entropy drift without imposing an entropy equation.

首行分别应用热力学基本关系并求和。封闭质量与式 (E05) 消去化学项。体积相减给出第三行。对蒸气核面积和体积求导，其比值为二除以核半径。式 (E05) 使内牵引功与表面能变化相消。末式说明曲面核界面能必须计入总能量。可逆绝热调相时，温度－熵项为零。审查在未施加熵方程时检验熵漂移。

Symbols before Eq. (E16): $E_{seed}$ is retained seed energy relative to the all-liquid reference (J), under lossless seed preparation; $m$ is fixed PFC mass (kg); $u_{l0}$ is initial liquid internal energy (J/kg); $U_{PFC},E_{gel},E_{surf},E_{amb},K_h,D_h$ are PFC, gel, interface, ambient, kinetic and loss energies (J); $t,t'$ are end and dummy time (s), with the prime a label; $P_{dis}$ is loss power (W); $\epsilon_E$ is dimensionless. This is an insulated post-seed control; no later heat source or outflow exists.

式 (E16) 前的符号：$E_{seed}$ 为无损种子制备下相对全液态参考的保留能量（J）；$m$ 为固定 PFC 质量（kg）；$u_{l0}$ 为初始液体比内能（J/kg）；$U_{PFC},E_{gel},E_{surf},E_{amb},K_h,D_h$ 为 PFC、凝胶、界面、环境、动能与损耗能量（J）；$t,t'$ 为结束与积分时间（s），撇号是标签；$P_{dis}$ 为损耗功率（W）；$\epsilon_E$ 无量纲。种子后的系统绝热；无后续热源或外流。

The first row defines cumulative host loss. The second is the retained-seed-energy balance relative to the all-liquid reference. The last defines its numerical residual for positive retained seed energy. Equality with experimentally absorbed laser energy would additionally require accounting for pre-seed conduction, host dissipation, acoustic emission, and heat delivered to other materials. They are absent from the preparation control; the calculated budget is not an activation threshold. Gel dissipation is included as energy transferred into unrecovered host heating; it is not also fed back into PFC heat. These controls neglect heat conduction between host and PFC.

首行定义累积基质损耗。次行相对全液态参考核算保留种子能量。末行在正种子能量下定义数值残差。与实测激光吸收能量对应时，还须核算成核前导热、基质耗散、声辐射及其他材料受热。这里未计算这些制备过程，因此预算无法给出激活阈值。凝胶耗散成为不可恢复的基质热，未重复反馈给 PFC。模型省略基质与 PFC 间导热。

Symbols before Eq. (E17): $K_h$ is host kinetic energy (J); $\rho_h$ is density (kg/m³); $R,W$ are radius and speed (m, m/s); $p_l,p_\infty,p_{el},p_{vis}$ are pressures (Pa); $\gamma_o$ is outer tension (N/m); dots indicate time derivatives. The power calculation applies along Eq. (E13). $\pi$ is the dimensionless circle constant.

式 (E17) 前的符号：$K_h$ 为基质动能（J）；$\rho_h$ 为密度（kg/m³）；$R,W$ 为半径与速度（m、m/s）；$p_l,p_\infty,p_{el},p_{vis}$ 为压力（Pa）；$\gamma_o$ 为外张力（N/m）；点为时间导数。功率计算沿式 (E13) 进行。 $\pi$ 为无量纲圆周率。

Differentiate Eq. (E11) using the product rule and Eq. (E13)'s radius derivative. Factor the common volume flux and substitute momentum. Eq. (E09) removes elastic power, differentiation of Eq. (E14) removes ambient and outer surface power, and Eq. (E12) removes viscous power in the derivative of Eq. (E16). Eq. (E15) removes the remaining PFC/core-interface power. Thus the adopted dynamics are consistent with the stated first law. Symbolic differentiation verifies the first two rows; native/reference comparisons verify implementation, not the omitted physics.

对式 (E11) 应用乘积法则与式 (E13) 的半径导数。提取共同体积通量并代入动量。对式 (E16) 求导时，式 (E09) 消去弹性功率；式 (E14) 的导数消去环境与外表面功率；式 (E12) 消去黏性功率。式 (E15) 消去剩余 PFC／内界面功率。因此采用的动力学与第一定律一致。符号求导检验前两行；原生／参考比较检验实现。省略物理仍待验证。

Symbols before Eq. (E18): $R,b$ are prescribed outer/unknown core radii (m); $T$ is common unknown temperature (K); $\rho_l,\rho_v$ are unknown densities (kg/m³); $p(T,\rho),g(T,\rho)$ are EOS pressure and chemical-potential functions (Pa, J/kg); $p_{need}$ is static required liquid pressure (Pa); $\gamma_i$ is inner tension (N/m); $V,V_b$ are volumes (m³); $m$ is fixed mass (kg). The root must satisfy positive phase compressibility and valid inventory.

式 (E18) 前的符号：$R,b$ 为指定外半径与未知核半径（m）；$T$ 为未知共同温度（K）；$\rho_l,\rho_v$ 为未知密度（kg/m³）；$p(T,\rho),g(T,\rho)$ 为状态方程压力及化学势函数（Pa、J/kg）；$p_{need}$ 为静态所需液压（Pa）；$\gamma_i$ 为内张力（N/m）；$V,V_b$ 为体积（m³）；$m$ 为固定质量（kg）。根须满足正相压缩性与有效质量分配。

These four equations determine temperature, densities and core radius at prescribed stretch. They establish a static phase root. Stability and nucleation-barrier crossing require separate tests. Eq. (E16) gives state energy with zero kinetic energy and loss. Native stationary solves use perturbed initial guesses and solve all four equations.

在指定伸长比下，这四式确定温度、密度及核半径。它们给出静态相平衡根。稳定性与越过成核势垒须分别检验。动能和损耗为零时，式 (E16) 给出状态能量。原生稳态模型从扰动初猜求解全部四式。

Symbols before Eq. (E19): $R,W,D_h$ are radius, speed and accumulated loss (m, m/s, J). Dots denote time derivatives. $T$ is temperature (K); $\rho_l,\rho_v,\rho_h$ are liquid, vapor and host densities (kg/m³); $b$ is core radius (m). $p_l,p_{need}$ are pressures (Pa); $\eta$ is viscosity (Pa s); $\gamma_i$ is inner tension (N/m). $p(T,\rho)$ and $g(T,\rho)$ are EOS pressure (Pa) and chemical potential (J/kg), with $\rho$ the density argument (kg/m³). $V,V_b$ are inclusion/core volumes (m³); $m$ is fixed mass (kg). $U_{PFC},E_{gel},E_{surf},E_{amb},K_h,E_{seed}$ are internal, gel, surface, ambient, kinetic and seed energies (J). $u_{l0}$ is initial liquid specific internal energy (J/kg). $\pi$ is the dimensionless circle constant. $R,b,\rho_h$ are positive; phases remain admissible.

式 (E19) 前的符号：$R,W,D_h$ 为半径、壁速与累积损耗（m、m/s、J）；$T,\rho_l,\rho_v,b$ 为四个代数未知量（K、kg/m³、kg/m³、m）；$p_l,p_{need}$ 为压力（Pa）；$\eta,\rho_h,\gamma_i$ 为黏度、基质密度及内张力（Pa s、kg/m³、N/m）；$p(T,\rho),g(T,\rho)$ 为状态方程压力与化学势函数（Pa、J/kg），$\rho$ 为密度自变量（kg/m³）；$V,V_b$ 为液滴与核体积（m³）；$m$ 为固定质量（kg）；$U_{PFC},u_{l0},E_{gel},E_{surf},E_{amb},K_h,E_{seed}$ 为此前定义的能量／参考比内能（J、J/kg）。点为时间导数。$\pi$ 为无量纲圆周率。$R,b,\rho_h$ 为正；相态须有效。

The first three equations are radius, momentum and cumulative loss. Inner traction, phase equilibrium, mass and total energy supply four algebraic conditions. The native model has seven normalized Global Equations variables. The Python solver independently eliminates the four algebraic unknowns with a thermodynamic root and integrates three differential states. This is a solver-formulation comparison on the same constitutive assumptions; it is not an independent empirical theory.

前三式分别计算半径、动量及累积损耗。内牵引、相平衡、质量与总能量给出四个代数条件。原生模型有七个归一化 Global Equations 变量。Python 独立求热力学根以消去四个代数未知量，并积分三个微分状态。比较的是相同本构假设下的求解实现。独立实验理论仍待建立。

Symbols before Eq. (E20): $p_s,U_s,\tau,E_s$ are positive pressure, speed, time and energy scales (Pa, m/s, s, J). $\rho_h,a_0,m$ are positive host density, initial radius and PFC mass (kg/m³, m, kg). $\theta,\lambda,\widehat W,\widehat D$ are dimensionless time, radius, speed and loss; hats label normalized quantities. $t,R,W,D_h$ are dimensional time, radius, speed and loss (s, m, m/s, J). $p_l,p_{need}$ are pressures (Pa); $\eta$ is viscosity (Pa s). $d/d\theta$ is differentiation in normalized time. The square root is the positive branch; $\lambda>0$ permits division.

式 (E20) 前的符号：$p_s,U_s,\tau,E_s$ 为正压力、速度、时间与能量尺度（Pa、m/s、s、J）；$\rho_h,a_0,m$ 为正基质密度、初始半径与质量（kg/m³、m、kg）；$\theta,\lambda,\widehat W,\widehat D$ 为归一化时间、半径、速度及损耗；$t,R,W,D_h$ 为有量纲时间、半径、速度及损耗（s、m、m/s、J）；$p_l,p_{need}$ 为压力（Pa）；$\eta$ 为黏度（Pa s）。帽号表示无量纲量。$d/d\theta$ 为归一化时间求导。平方根取正分支；$\lambda>0$ 允许相除。

The first two rows define scales and variables. The chain rule gives the third. Substitute dimensional radius and speed into Eq. (E13), multiply by the time/speed scale, and use the squared speed definition to divide all driving pressures by the pressure scale. The dimensionless coefficient of viscous resistance is therefore inversely proportional to initial radius at fixed viscosity and pressure scale. A square root of positive pressure/density is valid and gives speed units.

前两行定义尺度与变量。链式法则给出第三行。将有量纲半径与速度代入式 (E13)，乘以时间／速度尺度，再用速度平方定义将各驱动压力除以压力尺度。因此固定黏度与压力尺度时，无量纲黏性系数反比于初始半径。正压力／密度允许取平方根，所得单位为速度。

Symbols before Eq. (E21): $\lambda$ is stretch; $\epsilon$ is a small dimensionless perturbation; $p_l(\lambda;E)$ is the liquid pressure from closed total-energy phase closure with a fixed energy label $E$ (J); $p_{need}(\lambda)$ is static required pressure (Pa); $k_E$ is closed-energy pressure-restoration derivative (Pa per stretch). The center state is an admissible stationary root and the neighboring phase roots must remain admissible.

式 (E21) 前的符号：$\lambda$ 为伸长比；$\epsilon$ 为小无量纲扰动；$p_l(\lambda;E)$ 为闭总能量相态闭合的液压（Pa），固定能量标签 $E$ 单位为 J；$p_{need}(\lambda)$ 为静态所需压力（Pa）；$k_E$ 为闭能量压力恢复导数（Pa／单位伸长比）。中心状态为可容许稳态根；两侧相根也须可容许。