引言

两相流扩散界面模型来源于流体力学中两相流体的运动界面研究，广泛应用于石油开采和提炼、化工、材料加工及生物工程等领域，对其数学理论进行研究具有重要的理论意义和应用科学背景。不同于将接触面当作是一条光滑曲面的两相流自由界面模型，两相流扩散界面模型是将两种流体的接触界面当作是具有一定界面厚度的两流体相互作用的层区域，通过引入相场变量和界面混合自由能来确定各流体的区域位置以及两相界面的变化，通常由描述流体流速、压力等变化的Navier-Stokes方程组与描述互不相溶两相流扩散界面运动的Cahn-Hilliard方程组或Allen-Cahn方程组耦合而来，即得到Navier-Stokes-Cahn-Hilliard (NSCH)方程组或Navier-Stokes-Allen-Cahn方程组。NSCH方程组首先由Lowengrub等^[1]在20世纪提出，Abels等^[2]、Kotschote等^[3]进一步研究了可压模型的适定性结果。Abels等^[2]得到了三维可压NSCH方程全局弱解的存在性，Kotschote等^[3]得到了局部强解的存在唯一性。Chen等^[4]研究了一维可压模型周期边值问题和混合边界问题强解的全局存在性和大时间行为。王暐翼等^[5]研究了带有van der Waals状态方程的一维可压方程组强解的存在唯一性。

由于不需要直接去处理复杂的两相界面，扩散界面模型在研究两相运动界面问题上有着很大的优势，但在模拟计算时对界面厚度有一定的要求。因此对于两相流扩散界面模型界面厚度趋于零的极限问题的研究，有助于进一步理解两类不同界面的两相流模型的内在联系，同时为界面运动的模拟计算提供理论基础。而关于两相流扩散界面模型的界面极限问题的研究主要集中在不可压缩NSCH方程组方面。Wang等^[6]、Xu等^[7]分别研究了二维和三维不可压模型在广义Navier边界条件下两相流界面厚度趋于零时的极限问题，采用渐近展开的方法推导出了相应的不同类型两相流自由界面条件和移动接触线条件。Abels等^[8]研究了非齐次不可压两相流扩散界面模型的界面极限，推导出了相应的自由界面问题。

本文主要研究三维空间中可压缩两相流扩散界面模型的界面极限问题，通过渐近匹配展开的方法，证明了在界面厚度趋于零时，可在NSCH方程组解的渐近极限中得到两相流自由边界问题的界面条件。本文工作的难点和创新之处在于，除了要克服界面厚度的小尺度而采用伸缩变换来进行内展开外，在研究可压缩两相流模型的界面极限时，由于密度分别与流体速度和相场变量耦合，其耦合项出现在界面方程的渐近展开式中，也将会以不同的形式出现在自由界面的条件中。

1 NSCH模型的建立及主要定理

在可压混合两相流中，用ϕ=ϕ₁－ϕ₂表示质量浓度差，ϕ_i=M_i/M(i=1, 2)分别表示两种流体的质量浓度，M_i为体积为V的混合流体中各流体的质量。总密度ρ=ρ₁+ρ₂，ρ_i=M_i/V为此时流体i的质量密度。平均速度u由ρu=ρ₁u₁+ρ₂u₂给出，u_i为流体i的速度。混合流体的化学势为μ。可压混合两相流扩散界面模型可由如下方程组表示。

NSCH模型

$ \left\{ {\begin{array}{*{20}{l}} {{\partial _t}\rho + {\mathop{\rm div}\nolimits} (\rho \mathit{\boldsymbol{u}}) = 0}\\ {{\partial _t}(\rho \mathit{\boldsymbol{u}}) + {\mathop{\rm div}\nolimits} (\rho \mathit{\boldsymbol{u}} \otimes \mathit{\boldsymbol{u}}) = {\mathop{\rm div}\nolimits} T}\\ {{\partial _t}(\rho \phi ) + {\mathop{\rm div}\nolimits} (\rho \phi \mathit{\boldsymbol{u}}) = \Delta \mu }\\ {\rho \mu = \frac{1}{\varepsilon }\left( {{\phi ^3} - \phi } \right) - \varepsilon \Delta \phi } \end{array}} \right. $

(1)

式中，t>0，ρ>0，－1≤ϕ≤1。自由能密度为

$ f(\phi ) = \frac{{{\phi ^4}}}{4} - \frac{{{\phi ^2}}}{2} $

T为Cauchy应力张量，满足

$ \mathit{\boldsymbol{T}} = \mathit{\boldsymbol{S}} - p\mathit{\boldsymbol{I}} - \varepsilon \nabla \phi \otimes \nabla \phi $

其中I是单位矩阵，S为牛顿黏性应力，满足

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{S}} = \lambda (\rho ,\phi )\left( {\left( {\nabla u + {\nabla ^{\rm{T}}}\mathit{\boldsymbol{u}}} \right) - \frac{2}{3}{\mathop{\rm div}\nolimits} \mathit{\boldsymbol{uI}}} \right) + \\ {\lambda ^\prime }(\rho ,\phi ){\mathop{\rm div}\nolimits} \mathit{\boldsymbol{uI}} \end{array} $

式中λ(ρ, ϕ)＞0，λ′(ρ, ϕ)>0均为黏性系数；ε>0为混合流体的扩散界面厚度；压力p由下式给出

$ p = a{\rho ^\gamma } - \frac{1}{\varepsilon }\left( {\frac{{{\phi ^4}}}{4} - \frac{{{\phi ^2}}}{2}} \right) - \frac{\varepsilon }{2}|\nabla \phi {|^2},\gamma > 1 $

接下来介绍两相流自由界面模型及其界面条件。设未知的两相流间的自由界面为

$ \varGamma:=\left\{(t, x) \mid \phi_{\varepsilon}(t, x)=0\right\} $

式中，x∈Ω⊂$ \mathbb{R} $³，对于∀t>0，将区域Ω分为Ω=Ω⁺∪Γ∪Ω^－。V_n为界面法向速度，κ为界面的曲率，n为界面的单位法向量，σ为表面张力，Ω⁺和Ω^-分别表示两种流体各自占据的区域

$ \left\{ {\begin{array}{*{20}{l}} {{\varOmega ^ + } = \left\{ {(t,x)\mid {\phi _\varepsilon }(t,x) > 0} \right\}}\\ {{\varOmega ^ - } = \left\{ {(t,x)\mid {\phi _\varepsilon }(t,x) < 0} \right\}} \end{array}} \right. $

在t>0, x∈Ω^±中，令(ρ^±, u^±)满足可压的Navier-Stokes方程组

$ \left\{ {\begin{array}{*{20}{l}} {{\partial _t}{\rho ^ \pm } + {\mathop{\rm div}\nolimits} \left( {{\rho ^ \pm }{\mathit{\boldsymbol{u}}^ \pm }} \right) = 0}\\ {{\partial _t}\left( {{\rho^ \pm }{\mathit{\boldsymbol{u}}^ \pm }} \right) + {\mathop{\rm div}\nolimits} \left( {{\rho^ \pm }{\mathit{\boldsymbol{u}}^ \pm } \otimes {\mathit{\boldsymbol{u}}^ \pm }} \right) = {\mathop{\rm div}\nolimits} {T^ \pm }} \end{array}} \right. $

(2)

$ \phi=\pm 1 \quad x \in \varOmega^{\pm} $

(3)

应力张量表示为T^±=－p^±I+S^±，其中

$ p^{\pm}=a\left(\rho^{\pm}\right)^{\gamma} $

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{S}} = \lambda (\rho ,\phi )\left( {\left( {\nabla \mathit{\boldsymbol{u}} + {\nabla ^{\rm{T}}}\mathit{\boldsymbol{u}}} \right) - \frac{2}{3}{\mathop{\rm div}\nolimits} \mathit{\boldsymbol{uI}}} \right) + \\ {\lambda ^\prime }(\rho ,\phi ){\mathop{\rm div}\nolimits} \mathit{\boldsymbol{uI}} \end{array} $

且在自由边界Γ上满足跳跃条件

$ {{{[\mathit{\boldsymbol{u}}]}_\varGamma } = 0} $

(4)

$ {{{[\mathit{\boldsymbol{Tn}}]}_\varGamma } = \sigma \kappa \mathit{\boldsymbol{n}}} $

(5)

其中[h]_Γ=h⁺－h^－是函数h⁺和h^－在界面Γ上的差。即在界面上，流体速度是连续的，应力张量存在跳跃性，且与界面的表面张力和曲率有关。设集合S¹={(t, x)∈Γ|V_n－un=0}，即在S¹上界面速度与流体速度相同

$ {V_{\rm{n}}} = \mathit{\boldsymbol{un}} $

(6)

在此界面上密度ρ可以是任意的，即密度的跳跃不是必需的。在Γ\S¹上，满足如下条件

$ \left\{ {\begin{array}{*{20}{l}} {{{[\rho ]}_\varGamma } = 0}\\ {\rho \left( {{V_{\rm{n}}} - \mathit{\boldsymbol{un}}} \right) = - \frac{1}{2}\mathit{\boldsymbol{n}}{{[\nabla \mu ]}_\varGamma }}\\ {\rho \mu = - \frac{1}{2}\sigma \kappa } \end{array}} \right. $

(7)

即在Γ\S¹上，密度ρ是连续的，界面速度由u和化学势梯度的法向跳跃同时决定。

定理1  设(ρ, u, ϕ, μ)为NSCH方程组(1)的解，且关于ρ、u、ϕ、μ在远离界面的区域存在外渐近展开，靠近界面的区域存在内渐近展开，展开式分别为

外展开

$ \left\{ {\begin{array}{*{20}{l}} {\rho = {\rho _0} + \varepsilon {\rho _1} + {\varepsilon ^2}{\rho _2} \cdots }\\ {\mathit{\boldsymbol{u}} = {\mathit{\boldsymbol{u}}_0} + \varepsilon {\mathit{\boldsymbol{u}}_1} + {\varepsilon ^2}{\mathit{\boldsymbol{u}}_2} \cdots }\\ {\phi = {\phi _0} + \varepsilon {\phi _1} + {\varepsilon ^2}{\phi _2} \cdots }\\ {\mu = {\mu _0} + \varepsilon {\mu _1} + {\varepsilon ^2}{\mu _2} \cdots } \end{array}} \right. $

(8)

内展开

$ \left\{ {\begin{array}{*{20}{l}} {\rho = {{\tilde \rho }_0} + \varepsilon {{\tilde \rho }_1} + {\varepsilon ^2}{{\tilde \rho }_2} \cdots }\\ {\mathit{\boldsymbol{u}} = {{\mathit{\boldsymbol{\tilde u}}}_0} + \varepsilon {{\mathit{\boldsymbol{\tilde u}}}_1} + {\varepsilon ^2}{{\mathit{\boldsymbol{\tilde u}}}_2} \cdots }\\ {\phi = {\widetilde \phi _0} + \varepsilon {\rm{ }}{\widetilde \phi _1} + {\varepsilon ^2}{{\widetilde \phi }_2} \cdots }\\ {\mu = {{\tilde \mu }_0} + \varepsilon {{\tilde \mu }_1} + {\varepsilon ^2}{{\tilde \mu }_2} \cdots } \end{array}} \right. $

(9)

并假设ρ₀>0和$ \tilde{\rho } $₀>0，则当ε→0时，两相流扩散界面模型NSCH方程组(1)收敛到两相流自由界面问题(2)~(3), 且在界面Γ上满足自由界面条件(4)~(7)。

需注意，与不可压两相流的界面极限分析结果不同的是，由于密度分别与流体速度和相场变量耦合，当界面厚度趋于零时，自由界面条件将会出现不同的形式。特别是当界面速度与流体速度不一致时，即在Γ\S¹上，流体密度在界面上不发生跳跃，且满足Gibbs-Thomson条件，即式(7)。

2 定理的证明

本文的证明思路为：在远离界面的区域采用外渐近展开，推导两种流体在各自流体区域内满足的可压缩流体方程组，在接触界面附近利用伸缩变换进行内渐近展开，并结合匹配条件推导出在接触界面上的自由边界条件。

在远离界面的区域，用h^±表示在Ω⁺和Ω^－内的函数h，则可将式(8)改写为

$ \left\{ {\begin{array}{*{20}{l}} {{\rho ^ \pm } = \rho _0^ \pm + \varepsilon \rho _1^ \pm + {\varepsilon ^2}\rho _2^ \pm \cdots }\\ {{\mathit{\boldsymbol{u}}^ \pm } = \mathit{\boldsymbol{u}}_0^ \pm + \varepsilon \mathit{\boldsymbol{u}}_1^ \pm + {\varepsilon ^2}\mathit{\boldsymbol{u}}_2^ \pm \cdots }\\ {{\phi ^ \pm } = \phi _0^ \pm + \varepsilon \phi _1^ \pm + {\varepsilon ^2}\phi _2^ \pm \cdots }\\ {{\mu ^ \pm } = \mu _0^ \pm + \varepsilon \mu _1^ \pm + {\varepsilon ^2}\mu _2^ \pm \cdots } \end{array}} \right. $

(10)

根据ρ₀>0，设ρ₀^±>0。将式(10)代入式(1)中第一式得到

$ O(1):{\partial _t}\rho _0^ \pm + {\mathop{\rm div}\nolimits} \left( {\rho _0^ \pm \mathit{\boldsymbol{u}}_0^ \pm } \right) = 0 $

(11)

将式(10)代入式(1)中第二式，得到

$ O\left( {{\varepsilon ^{ - 1}}} \right):\nabla \left( {f\left( {\phi _0^ \pm } \right)} \right) = 0 $

(12)

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O(1):{\partial _t}\left( {\rho _0^ \pm \mathit{\boldsymbol{u}}_0^ \pm } \right) + {\mathop{\rm div}\nolimits} \left( {\rho _0^ \pm \mathit{\boldsymbol{u}}_0^ \pm \otimes \mathit{\boldsymbol{u}}_0^ \pm } \right) = {\mathop{\rm div}\nolimits} \left( {S_0^ \pm - } \right.\\ \left. {p_0^ \pm \mathit{\boldsymbol{I}}} \right) + {\mathop{\rm div}\nolimits} \left( {6{{\left( {\phi _0^ \pm } \right)}^3}\phi _1^ \pm - \phi _0^ \pm \phi _1^ \pm } \right)\mathit{\boldsymbol{I}} \end{array} $

(13)

其中，

$ f\left( {\phi _0^ \pm } \right) = \frac{{{{\left( {\phi _0^ \pm } \right)}^4}}}{4} - \frac{{{{\left( {\phi _0^ \pm } \right)}^2}}}{2} $

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} S_0^ \pm = \lambda \left( {\rho _0^ \pm ,\phi _0^ \pm } \right)\left( {\left( {\nabla \mathit{\boldsymbol{u}}_0^ \pm + {\nabla ^{\rm{T}}}\mathit{\boldsymbol{u}}_0^ \pm } \right) - \frac{2}{3} \cdot } \right.\\ \left. {{\mathop{\rm div}\nolimits} \mathit{\boldsymbol{u}}_0^ \pm \mathit{\boldsymbol{I}}} \right) + {\lambda ^\prime }\left( {\rho _0^ \pm ,\phi _0^ \pm } \right){\mathop{\rm div}\nolimits} \mathit{\boldsymbol{u}}_0^ \pm \mathit{\boldsymbol{I}} \end{array} $

$ p_{0}^{\pm}=a\left(\rho_{0}^{\pm}\right)^{\gamma} $

将式(10)代入式(1)第三式，得到

$ O(1):{\partial _t}\left( {\rho _0^ \pm \phi _0^ \pm } \right) + {\mathop{\rm div}\nolimits} \left( {\rho _0^ \pm \phi _0^ \pm \mathit{\boldsymbol{u}}_0^ \pm } \right) = \Delta \mu _0^ \pm $

(14)

将式(10)代入式(1)第四式，得到

$ O\left(\varepsilon^{-1}\right): f^{\prime}\left(\phi_{0}^{\pm}\right)=0 $

(15)

由式(12)和式(15)得，在区域Ω^±中

$ \phi _0^ \pm = \pm 1 $

(16)

由式(13)和式(16)得

$ {\partial _t}\left( {\rho _0^ \pm \mathit{\boldsymbol{u}}_0^ \pm } \right) + {\mathop{\rm div}\nolimits} \left( {\rho _0^ \pm \mathit{\boldsymbol{u}}_0^ \pm \otimes \mathit{\boldsymbol{u}}_0^ \pm } \right) = {\mathop{\rm div}\nolimits} S_0^ \pm - {\mathop{\rm div}\nolimits} p_0^ \pm \mathit{\boldsymbol{I}} $

(17)

由式(11)和式(16)得

$ \Delta \mu _0^ \pm = 0 $

(18)

结合式(11)、(16)、(18)得到引理如下。

引理1  假设ρ₀^±>0，令方程组(10)中ε→0，在远离过渡层的区域，可通过式(11)、(12)、(14)、(15)得到式(3)，其中，在Ω⁺内，ϕ=1；在Ω^－内，ϕ=－1；在Ω^±内，Δμ₀^±=0。即当界面厚度ε→0时，NSCH模型(1)在远离界面区域收敛到方程组(2)~(3)。

为了方便分析界面Γ附近的区域，用d(t, x)描述界面附近区域内的点到界面Γ的距离，则正向朝Ω⁺的界面Γ的法向量n=$ \nabla $d，界面Γ的法向速度V_n=－∂_td。引入新变量ξ=d(t, x)/ε，那么对于任意函数h(t, x)(例如h=ρ, u, ϕ, μ)可重新写为

$ h(t, x)=\tilde{h}(t, \xi) $

则关于h有下列推导，其中$\nabla $n=κ为界面的平均曲率。

$ \left\{ {\begin{array}{*{20}{l}} {\nabla h = {\varepsilon ^{ - 1}}{\partial _\xi }\tilde h\mathit{\boldsymbol{n}}}\\ {\Delta h = {\varepsilon ^{ - 2}}{\partial _{\xi \xi }}\tilde h + {\varepsilon ^{ - 1}}{\partial _\xi }\tilde h\kappa }\\ {{\partial _t}h = {\partial _t}\tilde h - {\varepsilon ^{ - 1}}{\partial _\xi }\tilde h{V_{\rm{n}}}} \end{array}} \right. $

(19)

在界面附近区域可将式(9)写为

$ \left\{ {\begin{array}{*{20}{l}} {\tilde \rho = {{\tilde \rho }_0} + \varepsilon {{\tilde \rho }_1} + {\varepsilon ^2}{{\tilde \rho }_2} \cdots }\\ {\mathit{\boldsymbol{\tilde u}} = {{\mathit{\boldsymbol{\tilde u}}}_0} + \varepsilon {{\mathit{\boldsymbol{\tilde u}}}_1} + {\varepsilon ^2}{{\mathit{\boldsymbol{\tilde u}}}_2} \cdots }\\ {\tilde \phi = {{\tilde \phi }_0} + \varepsilon {\rm{ }}{{\tilde \phi }_1} + {\varepsilon ^2}{\rm{ }}{{\tilde \phi }_2} \cdots }\\ {\tilde \mu = {{\tilde \mu }_0} + \varepsilon {{\tilde \mu }_1} + {\varepsilon ^2}{{\tilde \mu }_2} \cdots } \end{array}} \right. $

(20)

将式(19)代入式(1)第一式，整理可得

$ {\partial _t}\tilde \rho - {\varepsilon ^{ - 1}}{\partial _\xi }\tilde \rho {V_{\rm{n}}} + {\mathop{\rm div}\nolimits} (\tilde \rho \mathit{\boldsymbol{\tilde u}}) = 0 $

(21)

再将式(20)代入式(21)，可得关于ε的同阶项的等式

$ O\left( {{\varepsilon ^{ - 1}}} \right): - {\partial _\xi }{\tilde \rho _0}{V_{\rm{n}}} + {\partial _\xi }\left( {{{\tilde \rho }_0}{{\mathit{\boldsymbol{\tilde u}}}_0}} \right)\mathit{\boldsymbol{n}} = 0 $

(22)

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O\left( {{\varepsilon ^0}} \right):{\partial _t}{{\tilde \rho }_0} - {\partial _\xi }{{\tilde \rho }_1}{V_{\rm{n}}} + {\mathop{\rm div}\nolimits} \left( {{{\tilde \rho }_1}{{\mathit{\boldsymbol{\tilde u}}}_0}} \right) + \\ {\partial _\xi }\left( {{{\tilde \rho }_0}{{\mathit{\boldsymbol{\tilde u}}}_1}} \right)\mathit{\boldsymbol{n}} = 0 \end{array} $

(23)

将式(19)代入式(1)第二式整理可得

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\tilde u}}{\partial _t}\tilde \rho + \tilde \rho {\partial _t}\mathit{\boldsymbol{\tilde u}} - {\varepsilon ^{ - 1}}\mathit{\boldsymbol{\tilde u}}{\partial _\xi }\tilde \rho {V_{\rm{n}}} - {\varepsilon ^{ - 1}}\tilde \rho {\partial _\xi }\mathit{\boldsymbol{\tilde u}}{V_{\rm{n}}} + \mathit{\boldsymbol{\tilde u}} \cdot \\ {\mathop{\rm div}\nolimits} (\tilde \rho \mathit{\boldsymbol{\tilde u}}) + \tilde \rho \mathit{\boldsymbol{\tilde u}}{\mathop{\rm div}\nolimits} \mathit{\boldsymbol{\tilde u}} = {\mathop{\rm div}\nolimits} T \end{array} $

式中，

$ {\mathop{\rm div}\nolimits} T = {\mathop{\rm div}\nolimits} S - \nabla p - \varepsilon {\mathop{\rm div}\nolimits} \left( {\nabla \phi \otimes \nabla \phi - |\nabla \phi {|^2}\mathit{\boldsymbol{I}}/2} \right) $

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathop{\rm div}\nolimits} S = {\mathop{\rm div}\nolimits} \left( {{\lambda _0}\left( {\nabla \mathit{\boldsymbol{\tilde u}} + {\nabla ^{\rm{T}}}\mathit{\boldsymbol{\tilde u}}} \right)} \right) + {\mathop{\rm div}\nolimits} \left( {\left( {\lambda _0^\prime - } \right.} \right.\\ \left. {\left. {\frac{2}{3}{\lambda _0}} \right){\mathop{\rm div}\nolimits} \mathit{\boldsymbol{\tilde uI}}} \right) = {\varepsilon ^{ - 2}}{\partial _\xi }\left[ {\left( {\lambda _0^\prime + \frac{1}{3}\lambda } \right){\partial _\xi }\mathit{\boldsymbol{\tilde un}}} \right]\mathit{\boldsymbol{n}} + {\varepsilon ^{ - 2}} \cdot \\ \begin{array}{*{20}{l}} {{\partial _\xi }\left( {{\lambda _0}{\partial _\xi }\mathit{\boldsymbol{\tilde u}}} \right) + {\varepsilon ^{ - 1}}{\mathop{\rm div}\nolimits} \left( {{\lambda _0}{\partial _\xi }\mathit{\boldsymbol{\tilde u}} \otimes \mathit{\boldsymbol{n}}) + {\varepsilon ^{ - 1}}{\lambda _0}{\partial _\xi }\mathit{\boldsymbol{\tilde u}}\kappa + } \right.}\\ {{\varepsilon ^{ - 1}}{\partial _\xi }\left[ {\left( {\lambda _0^\prime - \frac{2}{3}{\lambda _0}} \right){\mathop{\rm div}\nolimits} \mathit{\boldsymbol{\tilde u}}} \right]\mathit{\boldsymbol{n}}} \end{array} \end{array} $

其中，λ₀=λ($ \tilde{\rho } $₀, $ \tilde{\phi } $₀)，λ′₀=λ′($ \tilde{\rho } $₀, $ \tilde{\phi } $₀), $ \nabla $p=$ \nabla $(aρ^γ－ε^－1f(ϕ))=ε^－1∂_ξpn－ε^－2∂_ξf($ \tilde{\phi } $)n

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varepsilon {\mathop{\rm div}\nolimits} \left( {\nabla \phi \otimes \nabla \phi - |\nabla \phi {|^2}\mathit{\boldsymbol{I}}/2} \right) = \varepsilon {\mathop{\rm div}\nolimits} (\nabla \phi \otimes \\ \nabla \phi ) - \varepsilon \nabla \left( {|\nabla \phi {|^2}\mathit{\boldsymbol{I}}/2} \right) = - \frac{1}{2}{\varepsilon ^{ - 2}}{\partial _\xi }{\left| {{\partial _\xi }\tilde \phi } \right|^2}\mathit{\boldsymbol{n}} + \\ {\varepsilon ^{ - 1}}{\left| {{\partial _\xi }\tilde \phi } \right|^2}\kappa \mathit{\boldsymbol{n}} \end{array} $

将式(20)、(21)代入式(1)第二式，并取关于ε的同阶项的等式可得

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O\left( {{\varepsilon ^{ - 2}}} \right):{\partial _\xi }\left[ {\left( {\lambda _0^\prime + \frac{1}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right]\mathit{\boldsymbol{n}} + {\partial _\xi }\left( {{\lambda _0}} \right. \cdot \\ \left. {{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}} \right) + {\partial _\xi }f\left( {{{\tilde \phi }_0}} \right)\mathit{\boldsymbol{n}} - \frac{1}{2}{\partial _\xi }{\left| {{\partial _\xi }{{\tilde \phi }_0}} \right|^2}\mathit{\boldsymbol{n}} = 0 \end{array} $

(24)

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O\left( {{\varepsilon ^{ - 1}}} \right): - {{\tilde \rho }_0}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}{V_{\rm{n}}} + {{\tilde \rho }_0}{{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0} + {\partial _\xi }p\mathit{\boldsymbol{n}} = \\ {\mathop{\rm div}\nolimits} \left( {{\lambda _0}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0} \otimes \mathit{\boldsymbol{n}}} \right) + {\lambda _0}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{\kappa }} + {\partial _\xi }\left[ {\left( {\lambda _0^\prime - \frac{2}{3}{\lambda _0}} \right) \cdot } \right.\\ \left. {{\mathop{\rm div}\nolimits} {{\mathit{\boldsymbol{\tilde u}}}_0}} \right]\mathit{\boldsymbol{n}} + {\partial _\xi }\left[ {\left( {\lambda _1^\prime + \frac{1}{3}{\lambda _1}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right]\mathit{\boldsymbol{n}} + {\partial _\xi }\left( {{\lambda _1}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}} \right) + \\ {\partial _\xi }\left[ {\left( {\lambda _0^\prime + \frac{1}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}\mathit{\boldsymbol{n}}} \right]\mathit{\boldsymbol{n}} + {\partial _\xi }\left( {{\lambda _0}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}} \right) + \\ {\partial _\xi }\left( {{f^\prime }\left( {{{\tilde \phi }_0}} \right){{\tilde \phi }_1}} \right)\mathit{\boldsymbol{n}} - {\left| {{\partial _\xi }{\rm{ }}{{\tilde \phi }_0}} \right|^2}\kappa \mathit{\boldsymbol{n}} + {\partial _\xi }\left( {{\partial _\xi }{\rm{ }}{{\tilde \phi }_0}{\partial _\xi }{{\tilde \phi }_1}} \right)\mathit{\boldsymbol{n}} \end{array} $

(25)

其中λ₁=$ \nabla $λ($ \tilde{\rho } $₀, $ \tilde{\phi } $₀)($ \tilde{\rho } $₁, $ \tilde{\phi } $₁), λ′₁=$ \nabla $λ′($ \tilde{\rho } $₀, $ \tilde{\phi } $₀)·($ \tilde{\rho } $₁, $ \tilde{\phi } $₁)。

将式(18)、(19)代入式(1)第三式中可得

$ - {\varepsilon ^{ - 1}}\tilde \rho {\partial _\xi }\tilde \phi {V_{\rm{n}}} + {\varepsilon ^{ - 1}}\tilde \rho \mathit{\boldsymbol{\tilde un}}{\partial _\xi }\tilde \phi = {\varepsilon ^{ - 2}}{\partial _{\xi \xi }}\tilde \mu + {\varepsilon ^{ - 1}}{\partial _\xi }{\tilde \mu _0}\kappa $

比较关于ε同阶项得

$ O\left( {{\varepsilon ^{ - 2}}} \right):{\partial _{\xi \xi }}{\tilde \mu _0} = 0 $

(26)

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O\left( {{\varepsilon ^{ - 1}}} \right): - {{\tilde \rho }_0}{\partial _\xi }{{\tilde \phi }_0}{V_{\rm{n}}} + {{\tilde \rho }_0}{{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}{\partial _\xi }{{\tilde \phi }_0} = {\partial _{\xi \xi }}{{\tilde \mu }_1} + \\ {\partial _\xi }{{\tilde \mu }_0}\kappa \end{array} $

(27)

将式(18)、(19)代入式(1)第四式可得到

$ \tilde \rho \tilde \mu = \frac{1}{\varepsilon }\left( {{{\tilde \phi }^3} - \tilde \phi } \right) - {\varepsilon ^{ - 1}}{\partial _{\xi \xi }}\tilde \phi - {\partial _\xi }\tilde \phi \kappa $

比较关于ε同阶项得

$ {O\left( {{\varepsilon ^{ - 1}}} \right):{f^\prime }\left( {{{\tilde \phi }_0}} \right) = {\partial _{\xi \xi }}{\rm{ }}{{\tilde \phi }_0}} $

(28)

$ {O\left( {{\varepsilon ^0}} \right):{{\tilde \rho }_0}{{\tilde \mu }_0} = {f^{\prime \prime }}\left( {{{\tilde \phi }_0}} \right){\rm{ }}{{\tilde \phi }_1} - {\partial _{\xi \xi }}{\rm{ }}{{\tilde \phi }_1} - {\partial _\xi }{\rm{ }}{{\tilde \phi }_0}\kappa } $

(29)

关于外展开和内展开所需的匹配条件为

$ \mathop {\lim }\limits_{\xi \to \pm \infty } {\tilde h_0}(\xi ) = h_0^ \pm (x),\mathop {\lim }\limits_{\xi \to \pm \infty } \left( {{\partial _\xi }{h_1}(\xi )\mathit{\boldsymbol{n}}} \right) = \nabla h_0^ \pm (x) $

考虑内渐近展开得到的零阶近似方程组(22)、(24)、(26)、(28)，边界条件为

$ \left\{ {\begin{array}{*{20}{l}} {{{\tilde \rho }_0}(t, \pm \infty ) = \rho _0^ \pm (t,x)}\\ {{{\mathit{\boldsymbol{\tilde u}}}_0}(t, \pm \infty ) = \mathit{\boldsymbol{u}}_0^ \pm (t,x)}\\ {{{\tilde \phi }_0}(t, \pm \infty ) = \pm 1}\\ {{{\tilde \mu }_0}(t, \pm \infty ) = \mu _0^ \pm (t,x)} \end{array}} \right. $

(30)

引理2  设($ \tilde{\rho } $₀, $ \boldsymbol{\tilde{u}} $₀, $ \tilde{\phi } $₀, $ \tilde{\mu } $₀)是方程组(22)、(24)、(26)、(28)的解，则有以下式子成立

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \tilde \rho (t,\xi )\left( {{V_{\rm{n}}} - {{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right)(t,\xi ) = \rho _0^ \pm (t,x)\left( {{V_{\rm{n}}} - \mathit{\boldsymbol{u}}_0^ \pm \mathit{\boldsymbol{n}}} \right) \cdot \\ (t,x) \end{array} $

(31)

$ {{{\tilde \mu }_0}(t,\xi ) = \mu _0^ \pm (t,x)} $

(32)

$ {{{\tilde \phi }_0}(t,\xi ) = \tan z(\xi /\sqrt 2 )} $

(33)

证明：根据式(22)可得

$ {\partial _\xi }\left( {{{\tilde \rho }_0}\left( {{V_{\rm{n}}} - {{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right)} \right) = 0 $

对式(22)积分可得式(31)。式(32)可直接根据式(26)和式(30)中第三式得出。将式(28)两边同乘∂_ξ$ \tilde{\phi } $₀可得

$ \frac{1}{2}{\partial _\xi }{\left| {{\partial _\xi }{{\tilde \phi }_0}} \right|^2} = {\partial _\xi }f\left( {{{\tilde \phi }_0}} \right) $

(34)

得到∂_ξ$ \tilde{\phi } $₀=(1－$ \tilde{\phi } $₀²)/$ \sqrt{2} $，则可得式(33)。引理2得证。

引理3  设($ \tilde{\rho } $₀, $ \boldsymbol{\tilde{u}} $₀, $ \tilde{\phi } $₀, $ \tilde{ \mu } $₀)是方程组(22)、(24)、(26)、(28)的解，则有以下两式成立

$ {{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0} = 0} $

(35)

$ {\left( {{V_{\rm{n}}} - {{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right){\partial _\xi }{{\tilde \rho }_0} = 0} $

(36)

证明：式(24)两边同乘向量n，得到

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\partial _\xi }\left[ {\left( {\lambda _0^\prime + \frac{4}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right] + {\partial _\xi }f\left( {{{\tilde \phi }_0}} \right) - \frac{1}{2} \cdot \\ {\partial _\xi }{\left| {{\partial _\xi }{{\tilde \phi }_0}} \right|^2} = 0 \end{array} $

对上式在(－∞, ξ)上积分，即为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {\lambda _0^\prime + \frac{4}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}} + f\left( {{{\tilde \phi }_0}} \right) - \frac{1}{2}{\left| {{\partial _\xi }{{\tilde \phi }_0}} \right|^2} = \\ \begin{array}{*{20}{l}} {{{\left. {\left( {\lambda _0^\prime + \frac{4}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_0}} \right|}_{\xi = - \infty }}\mathit{\boldsymbol{n}} + {{\left. {f\left( {{{\widetilde \phi }_0}} \right)} \right|}_{\xi = - \infty }} - }\\ {{{\left. {\frac{1}{2}{{\left| {{\partial _\xi },{\rm{ }}{{\widetilde \phi }_0}} \right|}^2}} \right|}_{\xi = - \infty }}} \end{array} \end{array} $

根据式(30)可得∂_ξ$ \boldsymbol{\tilde{u}} $₀(t, ±∞)=0, ∂_ξ$ \tilde{\phi } $₀(t, ±∞)=0, f($ \tilde{\phi } $₀(t, ±∞))=f(±1)=0，则

$ \left( {\lambda _0^\prime + \frac{4}{3}{\lambda _0}} \right){\partial _\xi }{\mathit{\boldsymbol{\tilde u}}_0}\mathit{\boldsymbol{n}} + f\left( {{{\widetilde \phi }_0}} \right) - \frac{1}{2}{\left| {{\partial _\xi }{{\widetilde \phi }_0}} \right|^2} = 0 $

将式(34)代入，且$ \left( {{{{\lambda }'}}_{0}}+\frac{4}{3}{{\lambda }_{0}} \right)>0 $，则有

$ {\partial _\xi }{\mathit{\boldsymbol{\tilde u}}_0}\mathit{\boldsymbol{n}} = 0 $

(37)

将式(24)两边同乘切向量τ，得到∂_ξ(λ₀∂_ξ$ \boldsymbol{\tilde{u}} $₀)·τ=0，对其在(－∞, ξ)上积分，即为λ₀∂_ξ$\boldsymbol{\tilde{u}}$₀τ=λ₀∂_ξ$ \boldsymbol{\tilde{u}} $₀τ|_ξ=-∞，则由λ₀>0及边界条件可得

$ {\lambda _0}{\partial _\xi }{\mathit{\boldsymbol{\tilde u}}_0}\mathit{\boldsymbol{\tau }} = 0 $

(38)

根据式(37)、(38)有∂_ξ$ \boldsymbol{\tilde{u}} $₀=0，可导出式(35)。根据式(22)可得$ \tilde{\rho } $₀∂_ξ$ \boldsymbol{\tilde{u}} $₀n=∂_ξ$ \tilde{\rho } $₀(V_n－ $ \boldsymbol{\tilde{u}} $₀n)，再结合式(35)可证得式(36)。引理3得证。

引理4  设($ \tilde{\rho } $₀, $\boldsymbol{\tilde{u}}$₀, $ \tilde{\phi } $₀, $ \tilde{\mu } $₀)是方程组(22)、(24)、(26)、(28)的解，则有式(39)、(40)成立。

$ {\left[ {{\mathit{\boldsymbol{u}}_0}} \right]_\varGamma } = 0,{\rm{ 在界面 }}\varGamma {\rm{ 上}} $

(39)

$ {\left[ {{\rho _0}} \right]_\varGamma } = 0,{\rm{ 在 }}\varGamma \backslash {S^1}{\rm{ 上}} $

(40)

证明：由式(35)可直接得出式(39)。则在Γ\S¹上，根据式(36)可得∂_ξ$ \tilde{\rho } $₀=0，导出式(40)。引理4得证。

考虑内渐近展开得到的一阶近似方程组(23)、(25)、(27)、(29)及边界条件式(30)。

引理5  设($ \tilde{\rho } $₀, $ {\boldsymbol{\tilde{u}}} $₀, $ \tilde{\phi } $₀, $ \tilde{ \mu } $₀)是方程组(23)、(25)、(27)、(29)的解，则有式(41)成立。

$ {\left[ {{S_0}} \right]_\varGamma }\mathit{\boldsymbol{n}} - {\left[ {{p_0}} \right]_\varGamma }\mathit{\boldsymbol{n}} = \sigma \kappa \mathit{\boldsymbol{n}} $

(41)

证明：因为$ \tilde{\phi } $₀为方程(27)、(29)的解，根据式(33)及∂_ξ$ {\boldsymbol{\tilde{u}}} $₀=0，则式(25)可化简为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\partial _\xi }{p_0}\mathit{\boldsymbol{n}} = {\partial _\xi }\left[ {\left( {\lambda _0^\prime + \frac{1}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}\mathit{\boldsymbol{n}}} \right]\mathit{\boldsymbol{n}} + {\partial _\xi }\left( {{\lambda _0}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}} \right) + \\ {\partial _\xi }\left( {{f^\prime }\left( {{{\tilde \phi }_0}} \right){{\tilde \phi }_1}} \right)\mathit{\boldsymbol{n}} - {\left| {{\partial _\xi }{{\tilde \phi }_0}} \right|^2}\kappa \mathit{\boldsymbol{n}} + {\partial _\xi }\left( {{\partial _\xi }{{\tilde \phi }_0}{\partial _\xi }{{\tilde \phi }_1}} \right)\mathit{\boldsymbol{n}} \end{array} $

将其两边在(－∞, +∞)上积分，左边为

$ \int_{ - \infty }^{ + \infty } {{\partial _\xi }} {p_0}\mathit{\boldsymbol{n}}{\rm{d}}\xi = \left( {p_0^ + - p_0^ - } \right)\mathit{\boldsymbol{n}} = {\left[ {{p_0}} \right]_\varGamma }\mathit{\boldsymbol{n}} $

将等式右边第一行积分得到

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_{ - \infty }^{ + \infty } {{\partial _\xi }} \left[ {\left( {\left( {{\lambda ^\prime }_0 + \frac{1}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}\mathit{\boldsymbol{n}}} \right)\mathit{\boldsymbol{n}} + {\partial _\xi }\left( {{\lambda _0}} \right.} \right. \cdot \\ \begin{array}{*{20}{l}} {\left. {\left. {{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}} \right)} \right]{\rm{d}}\xi = \left[ {\left( {\left( {{\lambda ^\prime }_0 + \frac{1}{3}{\lambda _0}} \right){\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}\mathit{\boldsymbol{n}}} \right)\mathit{\boldsymbol{n}} + {\lambda _0} \cdot } \right.}\\ {\left. {\left. {{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}} \right]} \right|_{\xi = - \infty }^{\xi = + \infty } = \left[ {\left( {{\lambda ^\prime }_0 - \frac{2}{3}{\lambda _0}} \right){\mathop{\rm div}\nolimits} {\mathit{\boldsymbol{u}}_0}\mathit{\boldsymbol{n}} + {\lambda _0}\left( {\nabla {\mathit{\boldsymbol{u}}_0} + } \right.} \right.} \end{array}\\ \begin{array}{*{20}{l}} {{{\left. {\left. {{\nabla ^{\rm{T}}}{\mathit{\boldsymbol{u}}_0}} \right)\mathit{\boldsymbol{n}}} \right]}_\varGamma } = \left[ {{\lambda ^\prime }_0{\mathop{\rm div}\nolimits} {\mathit{\boldsymbol{u}}_0}\mathit{\boldsymbol{I}} + {\lambda _0}\left( {\nabla {\mathit{\boldsymbol{u}}_0} + {\nabla ^{\rm{T}}}{\mathit{\boldsymbol{u}}_0} - } \right.} \right.}\\ {{{\left. {\left. {\frac{2}{3}{\mathop{\rm div}\nolimits} {\mathit{\boldsymbol{u}}_0}\mathit{\boldsymbol{I}}} \right)} \right]}_\varGamma }\mathit{\boldsymbol{n}} = {{\left[ {{S_0}} \right]}_\varGamma }\mathit{\boldsymbol{n}}} \end{array} \end{array} $

将等式右边第一行积分并结合边界条件可得

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_{ - \infty }^{ + \infty } {\left[ {{\partial _\xi }\left( {{f^\prime }\left( {{{\tilde \phi }_0}} \right){{\tilde \phi }_1}} \right)n - {{\left| {{\partial _\xi }{{\tilde \phi }_0}} \right|}^2}\kappa \mathit{\boldsymbol{n}} - {\partial _\xi }\left( {{\partial _\xi }{{\tilde \phi }_0}} \right.} \right.} \cdot \\ \left. {\left. {{\partial _\xi }{{\tilde \phi }_1}} \right)\mathit{\boldsymbol{n}}} \right]{\rm{d}}\xi = - \int_{ - \infty }^{ + \infty } {{{\left| {{\partial _\xi }{{\widetilde \phi }_0}} \right|}^2}} \kappa \mathit{\boldsymbol{n}}{\rm{d}}\xi = - \sigma \kappa \mathit{\boldsymbol{n}} \end{array} $

式中，$ \sigma =\int_{-\infty }^{+\infty }{{{\left| {{\partial }_{\xi }}{{{\tilde{\phi }}}_{0}} \right|}^{2}}\text{d}\xi } $为常数。

综上可得式(41)。引理5得证。

在S¹上，由式(36)可知，密度函数$ \tilde{\rho } $₀是满足边界条件的任意函数，为了方便分析，将式(25)、(27)、(29)化简为

$ \left\{ {\begin{array}{*{20}{l}} {{\partial _t}{{\tilde \rho }_0} + {\mathop{\rm div}\nolimits} \left( {{{\tilde \rho }_0}{{\mathit{\boldsymbol{\tilde u}}}_0}} \right) + {\partial _\xi }{{\tilde \rho }_0}{{\mathit{\boldsymbol{\tilde u}}}_1}\mathit{\boldsymbol{n}} + {{\tilde \rho }_0}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}\mathit{\boldsymbol{n}} = 0}\\ {{\partial _{\xi \xi }}{{\tilde \mu }_1} = 0}\\ {{{\tilde \rho }_0}{{\tilde \mu }_0} = {f^{\prime \prime }}\left( {{{\tilde \phi }_0}} \right){\rm{ }}{{\tilde \phi }_1} - {\partial _{\xi \xi }}{\rm{ }}{{\tilde \phi }_1} - {\partial _\xi }{\rm{ }}{{\tilde \phi }_0}\kappa } \end{array}} \right. $

(42)

引理6  设($ \tilde{\rho } $₀, ${\boldsymbol{\tilde{u}}} $₀, $ \tilde{\phi } $₀, $ \tilde{ \mu } $₀)是方程组(22)、(24)、(26)、(28)的解，则在Γ上，$ {{{\tilde{\mu }}}_{0}}\int_{-\infty }^{+\infty }{{{{\tilde{\rho }}}_{0}}{{\partial }_{\xi }}{{{\tilde{\phi }}}_{0}}\text{d}\xi =-\sigma \kappa } $成立。

证明：式(42)中第三式两边同乘∂_ξ$ \tilde{\phi } $₀并在(－∞, +∞)上积分得到

$ \begin{array}{*{20}{c}} {{{\tilde \mu }_0}\int_{ - \infty }^\infty {{{\tilde \rho }_0}} {\partial _\xi }{{\tilde \phi }_0}{\rm{d}}\xi = - \kappa \int_{ - \infty }^\infty {{{\left| {{\partial _\xi }{{\tilde \phi }_0}} \right|}^2}} {\rm{d}}\xi - }\\ {\int_{ - \infty }^\infty {{\partial _{\xi \xi }}} {{\tilde \phi }_1}{\partial _\xi }{{\tilde \phi }_0}{\rm{d}}\xi + \int_{ - \infty }^\infty {{f^\prime }} \left( {{{\tilde \phi }_0}} \right){\partial _\xi }{\rm{ }}{{\tilde \phi }_0}{{\tilde \phi }_1}{\rm{d}}\xi = - \sigma \kappa } \end{array} $

引理6得证。

在Γ\S¹上，由式(36)可得，∂_ξ$ \tilde{\rho } $₀=0。为了方便分析，将式(25)、(27)、(29)化简为

$ \left\{ {\begin{array}{*{20}{l}} {{\partial _t}{{\tilde \rho }_0} - {\partial _\xi }{{\tilde \rho }_1}\left( {{V_{\rm{n}}} - {{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right) + {\mathop{\rm div}\nolimits} \left( {{{\tilde \rho }_0}{{\mathit{\boldsymbol{\tilde u}}}_0}} \right) + {{\tilde \rho }_0}{\partial _\xi }{{\mathit{\boldsymbol{\tilde u}}}_1}\mathit{\boldsymbol{n}} = 0}\\ {{{\tilde \rho }_0}{\partial _\xi }{\phi _0}\left( {{V_{\rm{n}}} - {{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right) = - {\partial _{\xi \xi }}{{\tilde \mu }_1}}\\ {{{\tilde \rho }_0}{{\tilde \mu }_0} = {f^{\prime \prime }}\left( {{{\tilde \phi }_0}} \right){\rm{ }}{{\tilde \phi }_1} - {\partial _{\xi \xi }}{\rm{ }}{{\tilde \phi }_1} - {\partial _\xi }{\rm{ }}{{\tilde \phi }_0}\kappa } \end{array}} \right. $

(43)

引理7  设($ \tilde{\rho } $₀, $ {\boldsymbol{\tilde{u}}}$₀, $ \tilde{\phi } $₀, $ \tilde{ \mu } $₀)是方程组(22)、(24)、(26)、(28)的解，则在Γ\S¹上，有式(44)、(45)成立。

$ {{\rho _0}\left( {{V_{\rm{n}}} - {\mathit{\boldsymbol{u}}_0}\mathit{\boldsymbol{n}}} \right) = - \frac{1}{2}\mathit{\boldsymbol{n}}{{\left[ {\nabla {\mu _0}} \right]}_\varGamma }} $

(44)

$ {{\rho _0}{\mu _0} = - \frac{1}{2}\sigma \kappa } $

(45)

证明：根据引理6及∂_ξ$ \tilde{\rho } $₀=0，可得

$ {\tilde \rho _0}{\mu _0}\int_{ - \infty }^{ + \infty } {{\partial _\xi }} {\tilde \phi _0}{\rm{d}}\xi = - \sigma \kappa $

结合边界条件，通过上式可得出式(45)。将式(43)在(－∞, +∞)上积分，得到

$ {\tilde \rho _0}\left( {{V_{\rm{n}}} - {{\mathit{\boldsymbol{\tilde u}}}_0}\mathit{\boldsymbol{n}}} \right)\int_{ - \infty }^{ + \infty } {{\partial _\xi }} {\widetilde \phi _0}{\rm{d}}\xi = - \left. {{\partial _\xi }{{\tilde \mu }_1}} \right|_{ - \infty }^{ + \infty } $

再结合匹配条件，可得2$ \tilde{\rho } $₀(V_n－$ {\boldsymbol{\tilde{u}}} $₀n)=－n[$ \nabla $$ {\tilde{\mu }} $₀]_Γ，引理7得证。

定理1的证明：把全区域分为远离界面和靠近界面两个区域，首先在远离界面的区域中，当ε→0时，通过外渐近展开证明引理1成立，即NSCH模型(1)收敛到方程组(2)、(3)；在界面附近的区域中，通过内渐近展开得到引理2~7，即可推出NSCH模型(1)收敛到自由界面模型(2)，且满足自由界面条件(4)~(7)。

3 结束语

本文研究了当三维可压缩两相流扩散界面模型的界面厚度趋于零时的极限问题。通过渐近匹配展开的方法证明了两相流扩散界面模型在界面厚度趋于零时，可以得到两相流自由界面模型，并推导出相应的自由界面条件。该结果可为界面运动的模拟计算提供理论基础。