Homogenization of a non-linear strongly coupled model of magnetorheological fluids

TL;DR

This paper rigorously derives a homogenized model for a suspension of magnetizable particles in a viscous fluid, revealing a coupled magnetohydrodynamic system as the particles' size diminishes.

Contribution

It introduces a novel homogenization approach for a non-linear, strongly coupled magnetorheological fluid model, including a new corrector result bypassing the div-curl lemma.

Findings

The suspension's behavior converges to a generalized magnetohydrodynamic system.

The fluid is modeled by stationary Navier-Stokes equations.

The magnetic field is described by Maxwell's equations.

Abstract

This paper concerns the rigorous periodic homogenization for a non-linear strongly coupled system, which models a suspension of magnetizable rigid particles in a non-conducting carrier viscous Newtonian fluid. The fluid drags the particles, thus alters the magnetic field. Vice versa, the magnetic field acts on the particles, which in turn affect the fluid via the no-slip boundary condition. As the size of the particles approaches zero, it is shown that the suspension's behavior is governed by a generalized magnetohydrodynamic system, where the fluid is modeled by a stationary Navier-Stokes system, while the magnetic field is modeled by Maxwell equations. A corrector result from the theory of two-scale convergence allows us to obtain the limit of the product of several weakly convergent sequences, where the div-curl lemma, which is a typical tool in these types of problems, is not applicable.