Existence of weak solutions to a diffuse interface model involving magnetic fluids with unmatched densities

TL;DR

This paper proves the global existence of weak solutions for a complex diffuse interface model describing magnetic fluids with unmatched densities, coupling Navier-Stokes, magnetization, and Cahn-Hilliard equations in 2D and 3D.

Contribution

It establishes the existence of weak solutions for a novel coupled PDE model involving magnetic fluids with unmatched densities, extending previous models.

Findings

Proved global existence of weak solutions in 2D and 3D.

Coupled Navier-Stokes, magnetization, and Cahn-Hilliard equations.

Model accounts for partial mixing and density dependence.

Abstract

In this article we prove the global existence of weak solutions for a diffuse interface model in a bounded domain (both in 2D and 3D) involving incompressible magnetic fluids with unmatched densities. The model couples the incompressible Navier-Stokes equations, gradient flow of the magnetization vector and the Cahn-Hilliard dynamics describing the partial mixing of two fluids. The density of the mixture depends on an order parameter and the modelling, specifically the density dependence, is inspired from Abels, Garcke and Gr\"{u}n 2011.