Publication at Faculty of Mathematics and Physics |

2019

We propose a simple model for a two-phase flow with a diffuse interface. The model couples the compressible Navier-Stokes system governing the evolution of the fluid density and the velocity field with the Allen-Cahn equation for the order parameter.

We show that the model is thermodynamically consistent, in particular, a variant of the relative energy inequality holds. As a corollary, we show the weak-strong uniqueness principle, meaning any weak solution coincides with the strong solution emanating from the same initial data on the life span of the latter.

Such a result plays a crucial role in the analysis of the associated numerical schemes. Finally, we perform the low Mach number limit obtaining the standard incompressible model.