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

Abstract

We developed the rigorous periodic homogenization for a non-linear strongly coupled system, which models 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, our result show that the suspension's behavior is governed by a generalized magnetohydrodynamic system. The fluid is modeled by a stationary Navier-Stokes system, while the magnetic field is model by an approximation of 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 usual div-curl lemma is not applicable.