PDE Models for Ferrofluids and Their Numerical Analysis

Ignacio Tomas

Department of Mathematics

University of Maryland

A ferrofluid is a liquid which becomes strongly magnetized in the presence of applied magnetic fields. In this talk we will survey some models for ferrofluids: their physical origins, PDE models, and related numerics. There are two generally accepted ferrofluid models which we will call by the name of their developers: the Rosensweig and Shliomis model. We will start by developing a numerical scheme for the Rosensweig model and carefully track the requirements to devise of an energy-stable scheme. Both the Rosensweig and Shliomis models deal with one-phase flows, which is the case of many technological applications. However, many applications arise naturally in the form of a two-phase flow: one of the phases has magnetic properties and the other one does not (e.g. magnetic manipulation of microchannel flows, microvalves, magnetically guided transport, etc). We have also developed a matching-density two-phase ferrofluid model starting from the simplified framework of the Shliomis model and the Cahn-Hilliard equation. This model satisfies an energy law, and with the lessons learned from the Rosensweig model, we were able to devise an energy-stable scheme. In addition, with some simplifications of the two-phase model, it is possible to prove convergence of the scheme, and as a by product, existence of solutions of the simplified PDE system. Finally, I will illustrate the capabilities of the numerical schemes with some numerical simulations.