Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results

TL;DR

This paper introduces a symmetrization technique for inviscid two-fluid flow equations, leading to new insights on shock waves and vortex sheets, including local existence results and stability conditions.

Contribution

It provides a novel symmetric form for two-fluid flow models, enabling immediate results on shock wave existence and vortex sheet stability, improving prior regularity requirements.

Findings

All compressive shock waves exist locally in time.

Vortex sheets in 2D are locally stable under a 'supersonic' condition.

Enhanced regularity results for 2D vortex sheets compared to previous studies.

Abstract

We consider two models of a compressible inviscid isentropic two-fluid flow. The first one describes the liquid-gas two-phase flow. The second one can describe the mixture of two fluids of different densities or the mixture of fluid and particles. Introducing an entropy-like function, we reduce the equations of both models to a symmetric form which looks like the compressible Euler equations written in the nonconservative form in terms of the pressure, the velocity and the entropy. Basing on existing results for the Euler equations, this gives a number of instant results for both models. In particular, we conclude that all compressive shock waves in these models exist locally in time. For the 2D case, we make the conclusion about the local-in-time existence of vortex sheets under a "supersonic" stability condition. In the sense of a much lower regularity requirement for the initial data, our result for 2D vortex sheets essentially improves a recent result for vortex sheets in the liquid-gas two-phase flow.