Well posedness of an isothermal diffusive model for binary mixtures of incompressible fluids

TL;DR

This paper proves the existence and uniqueness of solutions for a thermodynamically consistent isothermal diffusive model describing binary mixtures of incompressible fluids, incorporating a velocity-dependent chemical potential.

Contribution

It introduces a new model with a velocity-dependent chemical potential and establishes its well-posedness, including existence and uniqueness of solutions.

Findings

Model is thermodynamically consistent

Existence of solutions proved

Uniqueness of solutions established

Abstract

We consider a model describing the behavior of a mixture of two incompressible fluids with the same density in isothermal conditions. The model consists of three balance equations: continuity equation, Navier-Stokes equation for the mean velocity of the mixture, and diffusion equation (Cahn-Hilliard equation). We assume that the chemical potential depends upon the velocity of the mixture in such a way that an increase of the velocity improves the miscibility of the mixture. We examine the thermodynamic consistence of the model which leads to the introduction of an additional constitutive force in the motion equation. Then, we prove existence and uniqueness of the solution of the resulting differential problem.