Sharp Interface Limit for Compressible Immiscible Two-Phase Dynamics with Relaxation

TL;DR

This paper establishes the sharp interface limit for a modified Jin-Xin relaxation scheme applied to compressible immiscible two-phase flow, demonstrating that shock interactions pass through interfaces unaffected, supported by theoretical proofs and simulations.

Contribution

It introduces a new relaxation scheme for two-phase flow and proves its convergence to the sharp interface limit, combining analytical and numerical methods.

Findings

Shock waves pass through interfaces without effect.

The relaxation scheme accurately captures the sharp interface limit.

Numerical simulations confirm theoretical results.

Abstract

In this paper, the compressible immiscible two-phase flow with relaxation is investigated, this model can be regarded as a natural modification of Jin-Xin relaxation scheme proposed and developed by S.Jin and Z.P.Xin([Comm.Pure Appl.Math., 48,1995]) in view of the numerical approximation of conservation laws. Given any entropy solution consists of two different families of shocks interacting at some positive time for the standard two-phase compressible Euler equations, it is proved that such entropy solution is the sharp interface limit for a family global strong solutions of the modified Jin-Xin relaxation scheme for Navier-Stokes/Allen-Cahn system, here the relaxation time is selected as the thickness of the interface, weighted estimation and improved antiderivative method are used in the proof. Moreover, the simulation results are given by this modified Jin-Xin relaxation scheme method. Both numerical and theoretical results show that, the interacting shock waves can pass through the interface without any effect.