粘性液体—气体两相流模型解的存在性研究
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摘要
本文主要研究了一维空间中粘性液体-气体两相流简化模型
其中n=αgρg,m=αlρl分别代表气体和液体的质量；ρl,ρg,ull和ug分别代表液体、气体的密度和速度;P表示流体的公共压力；q代表外力,像重力、摩擦力等；μ表示粘性系数。未知量αg,αl分别代表气体与液体所占的体积比例,满足以下的关系式：ag+αl=1.
模型(0.0.1)是一种“漂流”模型,用于模拟管道中不稳定的可压缩液体-气体流。这里我们假设两种流体的速度是一样的,有公共的压强,同时考虑了粘性效应,忽略气体在混合动量方程中的效应和外力的作用。
本文主要研究模型(0.0.1)在适当条件下弱解的存在性,唯一性和经典解的存在性和唯一性等问题,分为以下两个方面：
·模型(0.0.1)在1维空间中的自由边界问题,其中粘性依赖于液体和气体的质量,即μ=Cnβ/(ρl-m)β+1,在自由边界处流体连续连接到真空,得到了当0<β<1时弱解的存在性和唯一性。
·模型(0.0.1)在1维空间中的固定边界问题(初始可含真空),即在边界处u=0,粘性依赖液体的质量,满足μ∈C2[0,∞),μ(m)≥M1>0,得到了经典解的存在唯一性。
We study the existence of the solutions to a viscous liquid-gas two-phase flow model in1dimensional space:
where n=αgρg, m=αlρl denote gas mass and liquid mass respectively; ρl-,ρg, ul and ug denote density and velocity of liquid and gas respectively; P is common pressure for both phases; q represents external forces like gravity and friction;μ is viscosity coefficient。The unknown vari-ables αg,αl∈[0,1] denote gas and liquid volume fractions, satisfying the fundamental relation: αg+αl=1.
The model (0.0.2) can be viewed to be a kind of drifting model, which are widely used to describe unstable compressible liquid-gas stream in pipeline, and where we assumed the fluids velocities are equal and have the common pressure, considered the viscous effect, neglected the effect of gas in the mixture momentum equation and the external force.
For the model (0.0.2), we study the existence and uniqueness of the weak solution, as well as the existence and uniqueness of the classical solution, under approximate conditions, which have be divided into the following areas:
· The free boundary problem about model (0.0.2) in one dimensional space has been in-vestigated, where the viscosity coefficient depends on the masses of fluid and gas, i.e. and both the two fluids connect to vacuum state continuously at the free boundaries. We get the existence and uniqueness of the weak solution, as0<β<1.
· The fixed boundary problem (i.e. u=0) about model (0.0.2) in one dimensional space has been investigated, where the initial condition contains vacuum, and the viscosity co-efficient depends on the mass of liquid and satisfies μ∈C2[0,∞),μ(m)≥Mi>0. We obtain the existence and uniqueness of the classical solution.
其中n=αgρg,m=αlρl分别代表气体和液体的质量；ρl,ρg,ull和ug分别代表液体、气体的密度和速度;P表示流体的公共压力；q代表外力,像重力、摩擦力等；μ表示粘性系数。未知量αg,αl分别代表气体与液体所占的体积比例,满足以下的关系式：ag+αl=1.
模型(0.0.1)是一种“漂流”模型,用于模拟管道中不稳定的可压缩液体-气体流。这里我们假设两种流体的速度是一样的,有公共的压强,同时考虑了粘性效应,忽略气体在混合动量方程中的效应和外力的作用。
本文主要研究模型(0.0.1)在适当条件下弱解的存在性,唯一性和经典解的存在性和唯一性等问题,分为以下两个方面：
·模型(0.0.1)在1维空间中的自由边界问题,其中粘性依赖于液体和气体的质量,即μ=Cnβ/(ρl-m)β+1,在自由边界处流体连续连接到真空,得到了当0<β<1时弱解的存在性和唯一性。
·模型(0.0.1)在1维空间中的固定边界问题(初始可含真空),即在边界处u=0,粘性依赖液体的质量,满足μ∈C2[0,∞),μ(m)≥M1>0,得到了经典解的存在唯一性。
We study the existence of the solutions to a viscous liquid-gas two-phase flow model in1dimensional space:
where n=αgρg, m=αlρl denote gas mass and liquid mass respectively; ρl-,ρg, ul and ug denote density and velocity of liquid and gas respectively; P is common pressure for both phases; q represents external forces like gravity and friction;μ is viscosity coefficient。The unknown vari-ables αg,αl∈[0,1] denote gas and liquid volume fractions, satisfying the fundamental relation: αg+αl=1.
The model (0.0.2) can be viewed to be a kind of drifting model, which are widely used to describe unstable compressible liquid-gas stream in pipeline, and where we assumed the fluids velocities are equal and have the common pressure, considered the viscous effect, neglected the effect of gas in the mixture momentum equation and the external force.
For the model (0.0.2), we study the existence and uniqueness of the weak solution, as well as the existence and uniqueness of the classical solution, under approximate conditions, which have be divided into the following areas:
· The free boundary problem about model (0.0.2) in one dimensional space has been in-vestigated, where the viscosity coefficient depends on the masses of fluid and gas, i.e. and both the two fluids connect to vacuum state continuously at the free boundaries. We get the existence and uniqueness of the weak solution, as0<β<1.
· The fixed boundary problem (i.e. u=0) about model (0.0.2) in one dimensional space has been investigated, where the initial condition contains vacuum, and the viscosity co-efficient depends on the mass of liquid and satisfies μ∈C2[0,∞),μ(m)≥Mi>0. We obtain the existence and uniqueness of the classical solution.
引文
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[3]D. Bresch, B. Desjardins, J.M. Ghidaglia, E. Grenier. The low mach number limit and the generic two-phase model [EB/01]. Preprint,2007.
[4]C.H. Chang, M.S. Liou. A robust and accurate approach to compressible multiphase flow:Stratified flow model and AUSM+-up scheme [J]. J. Comput. Phys.,2007,225:840-873.
[5]P. Chen, D.Y. Fang, T. Zhang. Free boundary problem for compressible flows with density dependent viscosity coefficients [J]. Communications on Pure and Applied Analysis,2011,10:459-478.
[6]H.J. Choe, H. Kim. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids [J]. J. Differential Equations,2003,190:504-523.
[7]Y. Cho, H.J. Choe, H. Kim. Unique solvability of the initial boundary value problems for compressible viscous fluids [J]. J. Math. Pures Appl.,2004,83:243-275.
[8]Y. Cho, B.J. Jin. Blow-up of viscous heat-conducting compressible flows [J]. J. Math. Anal. Appl.,2006, 320:819-826.
[9]Y. Cho, H. Kim. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities [J]. Manuscripta Math.,2006,120:91-129.
[10]J.Cortes. On the construction of upwind schemes for non-equilibrium transient two-phase flow [J]. Com-put. Fluids,2002,31:159-182.
[11]H.B. Cui, H.Y. Wen, H.Y. Yin. Global classical solutions of viscous liquid-gas two-phase flow model 2012, arXiv:1201.1560v1.
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[14]S.J. Ding, H.Y. Wen, L. Yao, C.J. Zhu. Global classical spherically symmetric solution to compressible Navier-Stokes equations with large initial data and vacuum [EB/01].2011, arXiv:1103.1421vl.
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[19]S. Evje, K.K. Fjelde. On a rough AUSM scheme for a one dimensional two-phase model [J]. Comput. Fluids, 2003,32:1497-1530.
[20]S. Evje,T.Flatten. Hybrid flux-splitting schemes for a common two-fluid model [J]. J. comput. Phys., 2003,192:175-210.
[21]S. Evje, T. Flatten. On the wave structure of two-phase flow models [J]. SIAM J. Appl. Math.,2006/07, 67:487-511.
[22]S. Evje, K.H. Karlsen. Global existence of weak solutions for a viscous two-phase model [J]. J. Differ. Equations,2008,245:2660-2703.
[23]S. Evje, K.H. Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law [J]. Commun. Pure Appl. Anal.,2009,8:1867-1894.
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