Strong Well-Posedness of a Diffuse Interface Model for a Viscous, Quasi-Incompressible Two-Phase Flow

TL;DR

This paper proves the strong well-posedness of a diffuse interface model describing the flow of two viscous, quasi-incompressible fluids with different densities, incorporating diffusion and coupling Navier-Stokes and Cahn-Hilliard systems.

Contribution

It establishes the existence and uniqueness of strong solutions for a complex inhomogeneous two-phase flow model with different densities, extending previous models.

Findings

Existence of unique strong solutions for small times.

Model accommodates different densities and partial mixing.

Couples Navier-Stokes with Cahn-Hilliard system in a novel way.

Abstract

We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. In contrast to previous works, we study a model for the general case that the fluids have different densities due to Lowengrub and Truskinovski. This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard system, where the density of the mixture depends on the concentration, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. We prove existence of unique strong solutions for the non-stationary system for sufficiently small times.