Worked theory and symbol glossary / 完整理论推导与符号表

1 · Problem, status and conventions

1 · 问题、状态与约定

The objective is to derive exactly the benchmark and conditional source/gel equations implemented in the course, and to identify their restrictions. The two accepted native calculations were independently reviewed. This companion makes the algebra explicit; it does not supply a new laser, fracture or jet validation.

目的为完整推导本课程基准及有条件源／凝胶方程，并说明限制。两套已验收原生计算已接受独立核查。本文补全代数过程，不提供新的激光、断裂或射流验证。

Exact geometric/conservation identities are distinguished below from constitutive laws and closure choices. The scalar radius source assumes a sphere. Model A solves a linear potential wave in the exterior 3D water and couples it to the nonlinear source. Model B solves a finite-strain 3D gel at stationary equilibrium, with a finite-mass PFP phase state. Neither resolves a moving nonspherical free interface.

下文区分几何／守恒恒等式、本构规律、闭合选择。标量半径源假设球形；A 在外部三维水域求线性势波并耦合非线性源。B 求有限应变三维凝胶稳态平衡和有限质量 PFP 相态；二者都不解析移动的非球形自由界面。

Pressure symbols without an “ac” or “f” subscript are absolute unless a pressure difference is explicitly shown. A positive radial speed means expansion. At an inner hole/cavity the outward normal of the surrounding water/gel points inward toward the centre. In a source component, COMSOL r1 is normalized radius, not a spatial coordinate; r1t and r1tt are its time derivatives. In the gel, COMSOL u,v,w are displacements, while specific internal energy is written with a phase subscript to avoid ambiguity.

无 ac、f 下标的压力符号默认绝对压力，明确写出的压差除外。径向速度为正表示膨胀；水／凝胶内孔处的域外法向朝向孔中心。源组件中 r1 是归一化半径，不是空间坐标，r1t／r1tt 是时间导数。凝胶中 u,v,w 为位移，比内能带相下标以避免混淆。

2 · P0: incompressible spherical collapse

2 · P0：不可压球形塌陷

Assumptions: unbounded, inviscid, incompressible, initially still water; a spherical empty cavity; constant Δp. Boundary data are v(R,t)=Ṙ, v→0 and p→p∞ as s→∞; initial data are R(0)=a and Ṙ(0)=0. These assumptions deliberately exclude shocks, thermal gas, surface tension and a jet.

假设为无限、无黏、不可压、初始静止水域；球形空腔；恒定压差。边界为 v(R,t)=Ṙ、无穷远 v→0、p→p∞；初值 R(0)=a、Ṙ(0)=0。此基准未包含激波、热气体、表面张力或射流。

P1 · Integrate mass conservation

P1 · 积分质量守恒

The integration constant in s is fixed by the no-penetration velocity at the moving spherical boundary.

对空间积分的常量由移动球面的流速边界条件确定。

P2 · Integrate radial momentum and apply both pressure boundaries

P2 · 积分径向动量，施加两端压力条件

Substitute the velocity in Eq. (P1) into Euler momentum ∂t v+v∂s v=−(∂s p)/ρ. The lower-limit evaluation is performed after computing the partial derivative at fixed s.

采用欧拉动量方程 ∂t v+v∂s v=−(∂s p)/ρ。先在固定 s 处求偏时间导数，再代入积分下限。

P3 · Integrate the first integral without crossing zero radius

P3 · 求第一积分，不穿越零半径

Multiplying Eq. (P2) by R²Ṙ produces Eq. (P3). Its integration constant is fixed by R(0)=a and Ṙ(0)=0. The negative square root in Eq. (P3) describes collapse; separation of variables gives the time integral in Eq. (P4). Checks: both terms of the radius equation have units m²/s²; the initial acceleration is inward; the energy ratio is one. For the supplied a=50 μm, ρ=1000 kg/m³ and Δp=80 kPa, τ=5.590170 μs and speed at R/a=0.2 is −81.3224 m/s. The integral reaches t/τ≈0.914681 at zero radius, where this ideal model becomes singular. The native stop at R/a≈0.05 avoids that singular endpoint.

检验：半径方程各项单位 m²/s²，初始加速度向内，归一化能量为 1。供给 a=50 μm、ρ=1000 kg/m³、Δp=80 kPa 时，τ=5.590170 μs，R/a=0.2 的速度为 −81.3224 m/s。积分在 t/τ≈0.914681 到达零半径奇点；原生在 R/a≈0.05 停止以避开它。

3 · Derive the incompressible Neo-Hookean reference law

3 · 推导不可压 Neo-Hookean 参考规律

This reference is infinite, incompressible and spherical, with zero far-field stress relative to ambient. It is not the finite nearly incompressible 3D FE solution. For Neo-Hookean material the difference σθ−σr=G(λθ²−λr²); the arbitrary incompressibility pressure cancels from this difference. This constitutive choice is the reference used in the cited PAAm cavitation work, not a formulation-specific fracture model.

参考规律假设无限、不可压、球形，远场相对环境应力为零；不同于有限近不可压三维 FE 解。Neo-Hookean 材料满足 σθ−σr=G(λθ²−λr²)，不可压压力在应力差中抵消。此为引文 PAAm 空化参考本构，不是特定配方断裂模型。

G1 · Integrate incompressibility with the cavity boundary

G1 · 结合空腔边界积分不可压约束

G2 · Integrate radial stress balance with an explicit change of variable

G2 · 显式换元积分径向应力平衡

The change of variable in Eq. (G2) differentiates the mapping in Eq. (G1). The factorization uses (ℓ⁶−1)=(ℓ³−1)(ℓ³+1). The apparent removable singularity at ℓ=1 therefore has a finite limit. The pressure is positive for an expanded cavity, with pel(1)=0, slope 4G at λ=1 and limiting pressure 5G/2 as λ→∞.

约分使用 (ℓ⁶−1)=(ℓ³−1)(ℓ³+1)，因此 ℓ=1 的表观奇点可去。膨胀腔的压力为正，pel(1)=0、初始斜率为 4G，λ→∞ 时压力趋于 5G/2。

G3 · Integrate reversible pressure work

G3 · 积分可逆压力功

Checks: pressure is in Pa, G a³ is in J, Ws(1)=0, and dWs/dV=pel. Finite outer radius and finite compressibility produce a small FE discrepancy rather than exact equality. Movahed et al. (2016) provides the PAAm comparison and the limited uniaxial stretch proxy.

检验：压力单位 Pa，G a³ 为 J，Ws(1)=0，dWs/dV=pel。有限外半径及可压性使 FE 有小差异，而非严格相等。Movahed 等（2016）提供 PAAm 比较与有限适用的单轴伸长替代判据。

4 · H0: nonlinear balance and its linearization

4 · H0：非线性平衡与线性化

For H0 assume the same spherical incompressible inertia, add a Newtonian radial loss, constant surface coefficient, polytropic gas and the gel reference law Eq. (G2). Initial conditions are R=R0(1+xinit), Ṙ=0; the far pressure is p0+pd. The equilibrium reference has pg0=p0+2σ/R0 and pel(1)=0.

H0 保留球形不可压惯性，添加径向牛顿损耗、常界面系数、多方气体与 Eq. (G2) 凝胶参考。初值 R=R0(1+xinit)、Ṙ=0；远压 p0+pd。参考平衡为 pg0=p0+2σ/R0、pel(1)=0。

H2 · Expand every first-order term about equilibrium

H2 · 在平衡附近展开每个一阶项

Subtracting the zeroth-order balance in Eq. (H1) and retaining first-order terms gives Eq. (H2). All inertial products of perturbations are discarded only in this linearization. The gas-energy formula in Eq. (H3) requires κ≠1; here κ=1.4. The energy identity assumes pd is constant during each run; a time-varying applied pressure would require its external work explicitly. For the H0 values, Ks=506080 Pa and ω0≈2.249622×10⁶ s⁻¹. G=0 recovers the fluid reference; η=0 removes the energy loss. A zero perturbation with pd=0 must remain at equilibrium. The linear approximation is a small-perturbation check, not the full collapse law.

能量恒等式假设每次运行 pd 恒定；随时间变化的外压需显式计入外功。H0 数值为 Ks=506080 Pa、ω0≈2.249622×10⁶ s⁻¹。G=0 恢复流体参考，η=0 去掉损耗；零扰动且 pd=0 必须保持平衡。线性近似只是小扰动检验，不替代完整塌陷方程。

To verify Eq. (H3), differentiate the gas law at fixed entropy, differentiate K from Eq. (P4) and use dWs/dV from Eq. (G3). Substitution of Eq. (H1) into the kinetic-energy derivative in Eq. (H4) cancels gas, ambient, capillary and elastic pressure work; only the nonnegative viscous loss remains. The period from Eq. (H3) is 2π/ω0≈2.792996 μs for the supplied inputs.

检验 Eq. (H3)：对定熵气体规律求导、用 Eq. (P4) 的动能及 Eq. (G3) 的压力功导数。将 Eq. (H1) 代入 Eq. (H4) 动能导数，气体、环境、毛细和弹性压力功抵消，仅余非负黏性耗散。供给参数的周期约 2.792996 μs。

5 · Model A: conservative source/wave closure

5 · 模型 A：守恒源／波闭合

This closure uses a spherical incompressible near-field kinetic energy truncated at fixed b, plus a linear 3D wave potential outside b. It is a declared energy-consistent matched model, not an exact compressible Navier–Stokes/free-interface solution. R must remain below b and source Mach number must remain small. Initial R=a, Ṙ=0, φ=0 and φt=0; gas or phase initial states are supplied in M1/M3.

此闭合将球形不可压近场动能截断于固定 b，并在 b 外求线性三维势波。它是声明的能量一致匹配模型，不是完整可压 Navier–Stokes／自由界面精确解。要求 R<b、源马赫数小；初值 R=a、Ṙ=0、φ=φt=0，相态初值见 M1／M3。

A1 · Integrate the near kinetic energy and derive its generalized inertia

A1 · 积分近场动能，推导广义惯性

The generalized source force is chosen as 4πR²(pB−p∞+ρ⟨φt⟩b). Dividing Eq. (A1) by 4πρR² gives the implemented source equation. Using an exact finite-shell Bernoulli condition instead would add a matching-velocity term; that is a different closure. The conserved ledger verifies the implemented coupling algebra, not every omitted fluid mechanism.

广义源力选择为 4πR²(pB−p∞+ρ⟨φt⟩b)，将 Eq. (A1) 除以 4πρR² 即得实现方程。若改用精确有限壳层伯努利条件，会增加匹配处速度项，属于不同闭合。守恒账检验的是实现耦合代数，不证明所有省略的流体现象。

A3 · Prove source/wave pressure-work cancellation

A3 · 证明源／波压力功抵消

Integration by parts uses the wave PDE in Eq. (A2) to cancel the displayed volume terms in Eq. (A3). The inner boundary uses the inward water-domain normal and the fixed-flux average. For a polytropic gas U is its gas energy. For the finite PFP state, the thermodynamic argument in section 7 gives dU=−pL dV after including the internal interface.

分部积分结合波 PDE 抵消体积分项。内边界使用朝孔内部的水域法向和固定通量平均。多方气体时 U 为气体能；有限 PFP 时，计入内界面后第 7 节热力学推导给出 dU=−pL dV。

The equal and opposite source pressure work in Eq. (A3) and Eq. (A4) cancels. Here pl is the PFP liquid pressure and σLL its declared outer coefficient; in the gas branch replace pl by pg and σLL by the gas/water coefficient. All energy terms use the same reference offsets as the native tables. The wave already takes source pressure work; adding a second radiation-loss term to the radius equation would alter this ledger.

此处 pL 为 PFP 液压，σLL 为声明外界面系数；气体分支用 pg 和气／水系数替代。各能量项使用原生表一致的参考零点。波已接收源压力功，若半径方程再加独立辐射损耗，会改变能量账。

Eout in Eq. (A3) contains both the positive time integral of ρφt²/cL over the outer surface and the reactive boundary storage ρ∫Γoφ²dA/(2Lout), because φφt=(∂tφ²)/2 and the initial wave is zero. Therefore cumulative Eout is nonnegative, although its instantaneous derivative need not be. It is not interchangeable with a pure propagating-mode radiation integral at an arbitrary observation surface.

Eq. (A3) 的 Eout 含外表面 ρφt²/cL 的正时间积分及反应边界储能 ρ∫Γoφ²dA/(2Lout)，因为 φφt=(∂tφ²)/2、初始波场为零。累计 Eout 非负，瞬时导数却可为负；不能等同于任意观测面的纯传播辐射积分。

A5 · Convert the dimensional closure to COMSOL fields

A5 · 将有量纲闭合转换为 COMSOL 变量

Insert R=a xR and φ=(a²/τ)χ into Eq. (A2), divide by p∞/ρ and use τ²=ρa²/p∞ to obtain Eq. (A5). Thus the Coefficient Form PDE has ea=1/cL², c=1 and zero distributed forcing outside source populations. COMSOL Flux/Source uses ∂nχ+qχ=g; the last boundary line sets q=1/Lout and g=−χt/cL. The inner source sets q=0 and g=−τQ/(a²Ab). These are the exact fields used in M1 and M3.

向 Eq. (A2) 代入 R=a xR、φ=(a²/τ)χ，除以 p∞/ρ 并使用 τ²=ρa²/p∞ 得到 Eq. (A5)。系数形式 PDE 的 ea=1/cL²、c=1；源群以外分布源为零。Flux/Source 约定 ∂nχ+qχ=g；外边界 q=1/Lout、g=−χt/cL；内源 q=0、g=−τQ/(a²Ab)，对应 M1／M3 精确输入。

A6 · Separate discrete sources from a shared uniform population closure

A6 · 区分离散源与共享均匀群闭合

For N separate holes apply Eq. (A2) on each matching surface and let each radius equation use its own surface average of the common wave. The first two lines of Eq. (A6) follow from summing the boundary contributions in Eq. (A3). For the uniform population, replace holes by volumetric forcing −NQ/Vpop in the filled domain, and use its volume-average feedback for a single shared source state. The wave energy rate follows by multiplying that PDE by ρφt and integrating. N shared sources provide the opposite pressure work, preserving the ledger algebra. A common state is an additional closure assumption. It cannot be accepted from an energy residual alone: the refined case has a large local pressure spread and a negative absolute local pressure.

N 个分离孔分别施加 Eq. (A2)，各半径用共同波场的自身面平均。对 Eq. (A3) 边界贡献求和得到 Eq. (A6) 前两行。均匀群则在填充域施加 −NQ/Vpop，并用体平均反馈至共享状态。PDE 乘 ρφt 后积分给出波能率；N 个共享源提供相反压力功，保证能量代数。共享状态是附加闭合，不能仅凭能量残差验收；细化例有大局部压差和负绝对局部压力。

Checks: Qρφt has units W; all radius terms are m²/s²; expanding flux points outward; b→∞ recovers the unbounded incompressible radius coefficients. The accepted strongest gas control has maximum Mach≈0.03026 and energy residual≈0.10289%. Finite-b sensitivity is a modeling sensitivity, not a statistical confidence interval. Shock, jet focusing, wall stagnation and film release are unresolved.

检验：Qρφt 为 W，半径项均为 m²/s²，膨胀通量向外，b→∞ 恢复无限不可压系数。最强气体对照最大马赫数约 0.03026，能量残差约 0.10289%。有限 b 敏感性是建模敏感性，不是统计置信区间；激波、射流聚焦、壁面停滞及薄膜脱离未解。

6 · Receiver and a restricted energy bound

6 · 接收器与受限能量上界

The receiver is a cylinder of radius 50 μm and thickness 5 μm centred 0.5 mm from the source. It averages gauge pressure and then applies a causal first-order low-pass filter with fc=1 MHz and zero initial state. No point-pressure or differently filtered peak is interchangeable with this measured quantity.

接收器为距源中心 0.5 mm、半径 50 μm、厚 5 μm 的圆柱。先平均表压，再施加 fc=1 MHz、零初态的因果一阶低通；点压力或不同滤波峰值不是相同观测量。

For a single outgoing spherical propagating mode with positive radiated energy, its energy flux gives the first line below. Cauchy–Schwarz applied to convolution then gives the bound; using the nearest distance is conservative for the declared finite receiver. This is a restricted propagating-mode bound, not an arbitrary near field, focused array or jet bound.

对单个球形传播出射模式及正辐射能，能流给出下面第一行；对卷积应用柯西–施瓦茨得到上界。对所声明有限接收器采用最近距离作保守估计。此上界只针对受限传播模式，不是任意近场、聚焦阵列或射流上界。

Eq. (O2) uses the kernel norm in Eq. (O1) and assumes the radiated propagating-mode energy does not exceed E0. Checks: the radicand has units Pa²; increasing the distance or filter time reduces the bound. For the declared receiver and highest sampled source energy it is ≈318.16 kPa, while the strongest sampled converged native pressure is ≈89.61 kPa. The bound is not attained by the sampled sources and does not prove a global physical maximum.

检验：根号内单位 Pa²，距离或滤波时间增大会降低上界。所声明接收器与最高采样源能量对应约 318.16 kPa，而收敛原生采样峰值约 89.61 kPa。采样源未达到此界，不能据此证明全局物理最大值。

7 · Finite PFP mass, curvature, chemical equilibrium and energy

7 · 有限 PFP 质量、曲率、化学平衡与能量

Assume a finite PFP inventory, common temperature, bulk EOS in each phase, chemical equilibrium, and constant declared interface coefficients. The radius floor is numerical only. Accepted roots must satisfy positive densities, 0<fv<1 and Rg≫εg. The actual hot coated gel interfacial law and nucleation rate are unavailable. The EOS is from Gao et al., archived in the exact PFP HEOS functions. The capillary controls use room-temperature measurements from Kandadai et al. and the author thesis Table 5.2 (printed p.116): 9.41 mN/m is PFP/saturated-air at 24±1 °C, not measured hot pure-vapor/gel tension.

假设有限 PFP 存量、共同温度、两相体相 EOS、化学平衡、常声明界面系数。汽核下限只用于数值，验收解须密度正、0<fv<1、Rg 远大于 εg。实际高温涂层凝胶界面定律与成核速率未知。EOS 来自 Gao 等，归档为精确 PFP HEOS 函数。毛细对照来自 Kandadai 等及作者论文表 5.2（印刷第 116 页）；9.41 mN/m 是 24±1 °C 的 PFP／饱和空气值，不是测得的高温纯汽／凝胶界面张力。

Eq. (T0) defines the cold mass and energy reference. The source activation state and the undeformed gel reference can have different volumes; mass remains fixed. The bulk EOS is an empirical Helmholtz correlation with traceable coefficient inputs, not a newly derived molecular law. Its reported range is 148.21–500 K and pressures up to 10 MPa; phase roots outside that range or with ρl=ρv at the critical limit do not justify using the fraction formula below.

Eq. (T0) 定义冷态质量及能量参考。激活源状态与未变形凝胶参考体积可不同，但质量固定。体相 EOS 是有可追溯系数的经验 Helmholtz 关联，不是新推导的分子规律。报告范围为 148.21—500 K、压力至 10 MPa；超范围相根或临界极限 ρl=ρv 不能使用下式的分数计算。

T1 · Solve the mass/volume identity for the vapor fraction

T1 · 由质量／体积恒等式解汽分数

T2 · Enforce phase equilibrium with curvature

T2 · 在曲率下施加相平衡

The density transforms enforce positivity; they do not enforce a valid vapor fraction. Flat saturation pressure imposed on both phases would violate the first equation. For Model A the phase closure fixes entropy to its supplied activated value. For B the stationary root fixes retained energy instead.

密度变换保证正密度，但不保证有效汽分数。两相同时用同一平面饱和压力会违反第一式。A 将熵固定在给定激活值，B 稳态根则固定保留能量。

T3 · Show why phase energy is counted once

T3 · 证明相变能只计一次

In Eq. (T3), gldml+gvdmv=0 follows from mass conservation and the chemical-equilibrium condition in Eq. (T2). The coefficient of dVv vanishes by the curved pressure balance in Eq. (T2). This identity uses equilibrium and constant interface energy. It explains the source-energy work cancellation in Eq. (A4). No extra latent-heat term is permitted on top of the phase internal energies. For B the full cold-to-final retained energy is:

此恒等式使用相平衡与常界面能，并解释 Eq. (A4) 的源能功抵消。相内能之外不能再添加独立潜热。B 从冷态到末态的完整保留能量为：

Checks: Eq. (T1) is exact mass/volume algebra with dimensionless fv; each Eq. (T4) term is J; surface tension times area is energy. At 11.65 pJ the native B root has T≈339.197 K, fv≈0.16813 and shell thickness≈24.12 nm. Thin shells, constant-tension extrapolation, omitted surfactant/disjoining forces and activation kinetics bound the applicability. The supplied activated state is an input, not a derived nucleation path.

检验：Eq. (T1) 是严格质量／体积代数，fv 无量纲；Eq. (T4) 各项都是 J，张力乘面积为能量。11.65 pJ 原生 B 根温度约 339.197 K、fv≈0.16813、液壳厚约 24.12 nm。薄壳、常张力外推、未计表面活性剂／分离压与激活动力学限制适用范围；给定激活态是输入，不是已推导成核路径。

8 · Model B: free finite-strain gel and consistent surface variation

8 · 模型 B：自由有限应变凝胶与一致表面变分

The reference gel shell has radii a=300 nm and B a=6 μm, B=20. Kbulk/G=BK=1000. The outer gel boundary has zero traction relative to ambient. Only rigid-body modes are suppressed. The inner boundary receives follower pressure pl−p0 and the surface-energy variation. No radius, shape or displacement is prescribed. The initial displacement in M2 is only a Newton iteration guess. Require an invertible, orientation-preserving deformation: J>0.

参考凝胶壳半径 a=300 nm、B a=6 μm，B=20、BK=1000。外边界相对环境零牵引，仅抑制刚体模式。内边界承受随动压力 pl−p0 与表面能变分，未规定半径、形状或位移。M2 初始位移只是牛顿迭代初猜。

The mixed pressure formulation is a numerical device for the nearly incompressible constitutive law. It does not turn the finite-K model into the exact incompressible analytic reference.

混合压力形式是近不可压本构的数值处理，并不把有限 K 模型变成严格不可压解析参考。

B2 · Transform area and compute the free cavity volume

B2 · 变换面积并计算自由空腔体积

The minus sign follows from using the gel-domain normal, opposite to the cavity-volume outward normal. The volume equation is the divergence theorem with div(x)=3. Material-frame integration is required because Js and C already include deformation. The native surface derivatives use mean(d(u,X)) on the boundary; copying displayed identity constants from a simplified report would remove the deformation physics.

负号来自凝胶域法向与空腔体积外法向相反；体积公式由 div(x)=3 的散度定理得到。Js、C 已包含变形，故需材料坐标积分。原生边界导数使用 mean(d(u,X))；从简化报告复制单位阵常数会删除实际变形物理。

B3 · Differentiate the cofactor and surface area

B3 · 对余子式与面积求变分

Hold the reference surface fixed. Write q=C n0, so Js=|q| and ns=q/|q| by Eq. (B2). The determinant differential and inverse-matrix differential give:

固定参考面，令 q=C n0；由 Eq. (B2) 得 Js=|q|、ns=q/|q|。行列式及逆矩阵的微分给出：

The first line of Eq. (B4) differentiates the norm. Its third line substitutes Eq. (B3) and uses ns·C n0=Js. The last line writes the two scalar products as a double contraction, using C=JF^(-T). This derives the tangent projection instead of assuming its form.

Eq. (B4) 第一行对范数求导，第三行代入 Eq. (B3) 并使用 ns·C n0=Js，末行用 C=JF^(-T) 将标量点积写成双点积。由此推导切向投影，而非预设它的形式。

Integrating σLL times Eq. (B4) over the material surface gives Eq. (B5). The native Weak Contribution uses −Ps:∇t(test d) in COMSOL’s residual convention and expands all nine components. For a sphere, differentiating σLL times 4πR² gives inward capillary pressure 2σLL/R, counted once.

对材料面积积分 σLL 乘 Eq. (B4) 得到 Eq. (B5)。原生弱贡献按 COMSOL 残差约定使用 −Ps:∇t(test d)，展开全部九项。球面上对 σLL 乘 4πR² 求导，产生一次 2σLL/R 向内毛细压力。

B6 · State the complete weak mechanical equilibrium

B6 · 写出完整弱力学平衡

Eq. (B6) combines the bulk constitutive derivative in Eq. (B1) with the surface variation in Eq. (B5) and the follower-pressure boundary data. The outer surface contributes zero traction; virtual rigid translations/rotations are removed by Rigid Motion Suppression. Tangential surface differentiation may replace the full gradient because Ps annihilates the reference-normal direction: C n0=Js ns makes Ps n0=0. Consequently the nine tangential Weak Contribution terms represent the same surface energy, without adding a separate Laplace pressure.

Eq. (B6) 结合 Eq. (B1) 体相本构导数、Eq. (B5) 表面变分及随动压力边界条件。外面零牵引；Rigid Motion Suppression 移除刚体平移／转动。由 C n0=Js ns 得 Ps n0=0，可用切向导数代替完整梯度。故九项切向 Weak Contribution 表示相同表面能，不能再加独立 Laplace 压力。

Checks: Js,J and F are dimensionless; Ps has N/m, and Ps times virtual strain times reference area has J. Positive pl−p0 expands the hole because pressure traction is −(pl−p0)ns. The finite FE elastic work and pressure agree with Eq. (G2)/Eq. (G3) within the recorded small discrepancies. Equivalent stretch and local λmax differ, and the local maximum is more mesh-sensitive.

检验：Js、J、F 无量纲，Ps 为 N/m，乘虚应变与参考面积为 J。pl−p0 为正时，压力牵引 −(pl−p0)ns 使孔扩大。有限 FE 弹性功与压力在记录的小误差内符合 Eq. (G2)／Eq. (G3)。等效伸长与局部 λmax 不同，后者更受网格影响。

9 · Stability, timescale and unsupported conclusions

9 · 稳定性、时间尺度与尚未支持的结论

An infinitesimal outward displacement is radially restoring when the selected closure gives positive C. At 11.65 pJ the independent phase/energy perturbation calculation gives CQ≈+46.052 kPa per stretch and CT≈−103.744 kPa per stretch. These are radial stability checks of equilibrium closures, not a proof of stability to nonspherical modes or a nucleation trajectory.

若所选闭合的 C 为正，微小向外扰动产生径向恢复作用。11.65 pJ 的独立相态／能量扰动计算给出 CQ≈+46.052 kPa／伸长、CT≈−103.744 kPa／伸长；它们只检验平衡闭合的径向稳定性，不证明非球形模态稳定或成核轨迹。

Using declared water-like kth=0.6 W/(m K), ρgel=1006.5 kg/m³ and cp=4180 J/(kg K), tth≈0.631 μs and tin≈0.0952 μs. The draft’s 20 ms optical pulse is many orders longer; an insulated post-activation equilibrium is not a long-pulse heating calculation. Incident energy, absorbed energy and local retained Qdep cannot be equated without a spatial absorption/heat-transfer calibration.

采用声明水样 kth=0.6 W/(m K)、ρgel=1006.5 kg/m³、cp=4180 J/(kg K)，tth≈0.631 μs、tin≈0.0952 μs。初稿 20 ms 光脉冲长许多个数量级；绝热激活后平衡不等于长脉冲加热计算。没有空间吸收／传热标定，不能将入射、吸收和局部 Qdep 等同。

λf=2.1 is a uniaxial PAAm screening proxy. It is not the fracture stretch of this gel at a biaxially stretched cavity under rapid heating. 11.75 pJ remains borderline under local-stretch mesh uncertainty; coated low-tension native branches were unconverged. A sealed intact cavity has no specified outlet for a jet. Actual optical activation, formulation-specific rupture, venting and repeatable transfer remain unresolved.

λf=2.1 只是单轴 PAAm 筛查替代判据，不是此凝胶在快速加热双轴腔面中的断裂伸长。11.75 pJ 受局部伸长网格不确定性影响，仍属边缘；低张力涂层原生分支未收敛。完整密闭空腔没有指定射流出口；实际光激活、配方特定破裂、排气及重复转印尚未解决。

10 · Traceability and second-pass checks

10 · 可追溯性与第二遍审查

Parameter/expression definitions and numerical verification are linked in the rebuild recipe and signed native-model review. For EOS ranges use Gao et al.; for interfacial measurements use Kandadai et al.; for the gel reference/proxy use Movahed et al.. Li et al. (2024) supplies a water-vapor hydrogel printing precedent, not this embedded-PFP jet validation.

参数／表达式与数值核查见重建配置及已签署原生模型核查。EOS 范围见 Gao 等，界面测量见 Kandadai 等，凝胶参考／替代判据见 Movahed 等。Li 等（2024）为水汽凝胶转印先例，不验证本嵌入 PFP 射流。

Second-pass audit: the P0/H0 supplied inputs are kept distinct; radius versus diameter, absolute versus gauge pressure and material versus spatial derivatives are explicit. Eq. (G2) was rechecked by factorization and differentiation; Eq. (G3) differentiates to pressure work; Eq. (A3)/Eq. (A4) cancel the same signed pressure work; Eq. (T3)/Eq. (T4) count phase/surface energy once; Eq. (B2) uses the correct inner normal and material measure. Unproved activation, jet and fracture steps are left unresolved. The equations are available as complete LaTeX source.

第二遍审查：P0／H0 供给输入互相区分，半径／直径、绝对／表压、材料／空间导数明确。Eq. (G2) 用因式分解和求导复核，Eq. (G3) 求导恢复压力功；Eq. (A3)／Eq. (A4) 同符号压力功抵消；Eq. (T3)／Eq. (T4) 相变／界面能只计一次；Eq. (B2) 使用正确内法向与材料测度。未证明的激活、射流、断裂保持未解。方程提供完整LaTeX 源文件。