From kinetic mixtures to compressible two-phase flow: A BGK-type model and rigorous derivation

TL;DR

This paper introduces a BGK-type kinetic model for binary gas mixtures that captures compressible two-phase flow, rigorously derives the Euler equations, and confirms the model's accuracy through numerical experiments.

Contribution

It presents a novel BGK-type model with species-dependent exponents, derives the two-phase Euler equations, and provides rigorous convergence analysis with numerical validation.

Findings

The model accurately captures compressible two-phase flow dynamics.

Rigorous derivation of Euler equations from the kinetic model.

Numerical experiments confirm asymptotic preserving property.

Abstract

We propose a BGK-type kinetic model for a binary gas mixture, designed to serve as a kinetic formulation of compressible two-phase fluid dynamics. The model features species-dependent adiabatic exponents, and the relaxation operator is constructed by solving an entropy minimization problem under moments constraints. Starting from this model, we derive the compressible two-phase Euler equations via a formal Chapman--Enskog expansion and identify dissipative corrections of Navier--Stokes type. We then rigorously justify the Euler limit using the relative entropy method, establishing quantitative convergence estimates under appropriate regularity assumptions. Finally, we present numerical experiments based on an implicit-explicit Runge--Kutta method, which confirm the asymptotic preserving property and demonstrate the convergence from the BGK model to the isentropic two-phase Euler system in the hydrodynamic regime.