Multiscale modeling of magnetorheological suspensions

We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. By upscaling the quasistatic Maxwell equations coupled with the Stokes’ equations, we are able to capture the magnetorheological effect. The model we obtain generalizes the one introduced by Nueringer and Rosensweig (Phys Fluids 7:1927–1937, 1964) and Rosensweig (Ferrohydrodynamics, Dover Publications, Mineola, 2014) for quasistatic phenomena. We derive the macroscopic constitutive properties explicitly in terms of the solutions of local problems. The effective coefficients have a nonlinear dependence on the volume fraction when chain structures are present. The velocity profiles computed for some simple flows exhibit an apparent yield stress, and the flow profile resembles a Bingham fluid flow