Global existence of weak solutions to a diffuse interface model for magnetic fluids

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 Grün 2011.

Existence of weak solutions of diffuse interface models for magnetic fluids

“…To the best of our knowledge there are only a few articles so far dealing with diffuse interface magnetic fluids. The models we present here (see also [2,3]) differ from the ones introduced in [5,6] for modeling diffuse interface for magnetic fluids. Concerning the fluids undergoing partial mixing, we consider two cases: (a) the fluids have identical densities and (b) the fluids have unmatched specific densities.…”

“…In this article we present a summary of some recent results on the global existence of weak solutions of diffuse interface models involving two incompressible fluids with different magnetic properties. The results presented here are based on our recent research detailed in [2,3]. To the best of our knowledge there are only a few articles so far dealing with diffuse interface magnetic fluids.…”

“…We consider two different kinds of potentials Ψ(ϕ) and we will specify them later. The derivation of the model from the total energy of the system can be found in [2] (the matched density case) and [3] (the case of unmatched density). We specify that the energetic variational approach used in our modeling is motivated from [4] and the density dependence is inspired from [1].…”