Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

TL;DR

This paper proves the existence of weak solutions for a novel diffuse interface model describing two incompressible fluids with different densities, involving a coupled Navier-Stokes and Cahn-Hilliard system.

Contribution

It establishes the existence of weak solutions for a new model of two-phase flows with different densities, featuring a solenoidal velocity field and a non-homogeneous Navier-Stokes system.

Findings

Existence of weak solutions in 2D and 3D domains.

Application to a new model with solenoidal velocity field.

Coupled Navier-Stokes and Cahn-Hilliard system analysis.

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 in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels, Garcke, and Gr\"un for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order parameter.