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

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.

Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

“…The idea in our paper can be also extended to study the nonlinear stability of non-isentropic two-phase flow. We remark that in the recent preprint [33], Ruan and Trakhinin introduced a different symmetrization for the two-phase flow models to prove the local existence of compressible shock waves and 2D compressible vortex sheets by the result in [28].…”

Shock waves and characteristic discontinuities in ideal compressible two-fluid MHD

“…It is still unclear whether neutrally stable nonplanar Alfvén discontinuities do exist locally in time. However, the linear results in [7] are automatically carried over two-fluid MHD because the free boundary problem (20), (21) has absolutely the same form as that for Alfvén discontinuities in classical MHD.…”

“…An essential improvement of the last results for the liquid-gas two-phase model was done in our very recent work [20], where vortex sheets were also considered for model (1) without magnetic field. In [20] we also prove the local-in-time existence of all compressive shock waves in the mentioned liquid-gas model as well as, for example, in model (1) without magnetic field.…”

“…In this paper, using the crucial simple idea from our recent study in [20] and introducing an entropy-like function, we reduce the equations (1) of two-fluid MHD to a symmetric form which looks like the classical MHD system written in the nonconservative form in terms of the pressure, the velocity, the magnetic field and the entropy. This gives a number of instant results.…”

The asymptotic limits of Riemann solutions for the isentropic drift‐flux model of compressible two‐phase flows

“…For the case u r < u l , if k is required adequately small to satisfy (3.7), then the Riemann solution of (1.1) and (1.2) is represented by the notation S 1 ∪ J 2 ∪ S 3 (see Figure 2B), which is made up of 1-shockwave S 1 , 2-contact discontinuity J 2 , and 3-shockwave S 3 . In what follows, the two lemmas given below are provided to interpret the limiting k → 0 behavior of Riemann solution of (1.1) and (1.2) for the case u r < u l before the major theorem is drawn.…”