Incompressible limit for a fluid mixture

TL;DR

This paper investigates the incompressible limit of multicomponent ideal fluids, establishing convergence of densities and velocities using relative energy inequalities, and highlighting the importance of the mobility tensor in pressure convergence.

Contribution

It extends incompressible limit analysis to multicomponent fluids, addressing the challenges posed by non-divergence-free velocity fields and pressure convergence.

Findings

Convergence of densities and velocity fields under certain conditions.

Pressure convergence in L1 achieved through specific mobility tensor configurations.

Relative energy inequality effectively used for limit passage.

Abstract

In this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to single-component flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergence-free. We show that certain constellations of the mobility tensor allow to control gradients of the entropic variables and yield the convergence of the pressure in L1.