We study a two phase flow with interactions of liquid and rarefied gas inside the gas phase. The gas phase is modeled by the BGK model of the Boltzmann equation. The liquid phase is modeled by the incompressible Navier-Stokes equations. In this paper we have excluded the heat transfer between two phases. The interface boundary conditions for the liquid and gas phases is presented. The BGK model is solved by a semi-Lagrangian scheme with a meshfree reconstruction since the droplet moves inside the gas phase and its interface deforms with respect to time and space. Therefore, the similar meshfree particle method is also suitable to solve the incompressible Navier-Stokes equations for liquid phase. To validate the coupled solutions of the BGK model and the incompressible Navier-Stokes equations, we have compared the results of the BGK model and the incompressible Navier-Stokes equations, with those of the Boltzmann and the incompressible Navier-Stokes equations, where the Boltzmann equation is solved by a DSMC method. Results in $1D$ and $2D$ physical spaces are presented.
翻译：我们研究的是气相中液态和稀有气体相互作用的两阶段流。 气相由BGK模型模拟的Boltzmann方程式。 液相由不压缩的纳维- Stokes方程式模拟。 在本文中, 我们排除了两个阶段之间的热传输。 液和气相的界面边界条件被展示。 BGK模型通过半Lagrangian方案解决,自气相中滴液相移动及其在时间和空间方面的界面变形后,以网状重建为模范。 因此, 类似的无网相粒法也适合解决液相中不可压缩的纳维- Stokes方程式。 为了验证BGK模型和不可压缩的纳维- Stokes方程式的结合解决方案, 我们比较了BGK模型和可压缩的纳维- Stokes方程式的结果, 与Boltzmann 和 D- D- D- D- 等价的可压缩的纳维- Stokes 方程式, 其中Boltzmann 和 Bolzmann 等式以 $ 和 $ D 等价解。
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