On an Incompressible Navier-Stokes/Cahn-Hilliard System with Degenerate Mobility

TL;DR

This paper proves the existence of weak solutions for a complex fluid flow model combining Navier-Stokes and Cahn-Hilliard equations with degenerate mobility, ensuring thermodynamic consistency.

Contribution

It establishes the existence of weak solutions for a coupled Navier-Stokes/Cahn-Hilliard system with degenerate mobility, extending previous models to more realistic fluid behaviors.

Findings

Existence of weak solutions proven for the model.

Model maintains thermodynamic consistency.

Handles fluids with different densities.

Abstract

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Gr\"un for fluids with different densities and leads to a solenoidal velocity field. It is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent.