Mathematical justification of a compressible bi-fluid system with different pressure laws: A semi-discrete approach and numerical illustrations

Abstract

In this paper we obtain a one velocity macroscopic description of a mixture of two viscous compressible fluids by homogenization/averaging. The paper is written in two parts that can be read independently one of the other. In the first part, we propose two numerical schemes respectively for a two-fluid immiscible system, (i.e. two fluids separated by sharp interfaces) and for a bifluid mixture system (diffuse interface description via volume fractions). We present various simulations that suggest that the two systems of equations describe the same mixture at different scales and, as a consequence, that averaged velocity, density and pressure of the first model match with the respective velocity, density and pressure of the second model. In a second part, we show how to mathematically formalize the previous assertion via kinetic equations verified by the Young measures associated to oscillatory solutions of the model describing the finer scale.