Magnetohydrodynamic flows in porous media

The aim of this work is to investigate the tensorial filtration law in rigid porous media for steady-state slow flow of an electrically conducting, incompressible and viscous Newtonian fluid in the presence of a magnetic field. The seepage law under a magnetic field is obtained by upscaling the flow at the pore scale. The macroscopic magnetic field and electric flux are also obtained. We use the method of multiple-scale expansions which gives rigorously the macroscopic behaviour without any preconditions on the form of the macroscopic equations. For finite Hartmann number, i.e. epsilon much less than Ha much less than epsilon(-1), and finite load factor, i.e. epsilon much less than K much less than epsilon(-1), where epsilon characterizes the separation of scales, the macroscopic mass flow and electric current are coupled and both depend on the macroscopic gradient of pressure and the electric field. The effective coefficients satisfy the Onsager relations. In particular, the filtration law is shown to resemble Darcy's law but with an additional term proportional to the electric field. The permeability tensor, which strongly depends on the magnetic induction, i.e. Hartmann number, is symmetric, positive and satisfies the filtration analogue of the Hall effect.