Low-Mach-number limit of a compressible two-phase flow system with algebraic closure

TL;DR

This paper rigorously analyzes the low Mach number limit of a two-phase compressible flow system, showing convergence to an incompressible non-homogeneous fluid system with preserved volume fractions.

Contribution

It introduces a novel approach using modulated quantities and two relative entropy functionals tailored to the two-phase structure, advancing the mathematical understanding of such limits.

Findings

Partial densities converge to constants as Mach number tends to zero.

Velocity field converges to a divergence-free vector field.

Volume fractions are transported by the limit flow and remain nontrivial.

Abstract

We analyse a bi-fluid isentropic compressible Navier-Stokes system with barotropic pressure laws in a two-phase framework with equal pressure and single velocity. We focus on the rigorous analysis of the low Mach number limit under well-prepared initial data. Our main result shows that, as the Mach number tends to zero, the partial densities converge to constant states while the velocity field converges to a divergence-free vector field, and we recover the incompressible non-homogenous fluid system. The volume fractions remain nontrivial and are transported by the limit flow. Our method relies on the introduction of suitable modulated quantities and on two relative entropy functionals adapted to the two-phase structure: a standard entropy commonly used in the literature, and a logarithmic entropy, which is essential here as the former is not sufficient due to the structure of the underlying two-phase system.