The equations of ferrohydrodynamics: modeling and numerical methods

TL;DR

This paper models ferrohydrodynamics based on R. Rosensweig's equations, introduces a stable numerical scheme, proves its convergence, and demonstrates its effectiveness through numerical experiments.

Contribution

It provides a new energy-stable numerical scheme for ferrohydrodynamics equations, with proven existence, convergence, and practical validation.

Findings

The scheme is energy stable and mimics the continuous energy estimate.

Solutions of the numerical scheme always exist.

Numerical experiments demonstrate the scheme's potential in real applications.

Abstract

We discuss the equations describing the motion of ferrofluids subject to an external magnetic field. We concentrate on the model proposed by R. Rosensweig, provide an appropriate definition for the effective magnetizing field, and explain the simplifications behind this definition. We show that this system is formally energy stable, and devise a numerical scheme that mimics the same stability estimate. We prove that solutions of the numerical scheme always exist and, under further simplifying assumptions, that the discrete solutions converge. We also discuss alternative formulations proposed in pre-existing work, primarily involving a regularization of the magnetization equation and supply boundary conditions which lead to an energy stable system. We present a series of numerical experiments which illustrate the potential of the scheme in the context of real applications.