Homogenization of MHD flows in porous media

The paper is concerned with the homogenization of a nonlinear differential system describing flow an electrically conducting, incompressible and viscous Newtonian fluid through periodic porous medium, in presence magnetic field. We introduce variational formulation equipped boundary conditions. show existence solution problem, derive uniform estimates solutions depending on characteristic parameters flow. Using two-scale convergence method, we rigorously equation for current density, two-pressure Stokes system. derive, case constant permeability, explicit relation expressing macroscopic velocity as function Lorentz force, pressure gradient, external body via two permeability filtration tensors. When field absent, this reduces to Darcy law.