Magnetohydrodynamic flow through porous media

We investigate the filtration law in rigid porous media for steady-state slow flow of an electrically conducting, incompressible and viscous Newtonian fluid in the presence of magnetic field. The seepage law under magnetic field is obtained by upscaling the flow at the pore scale by using the method of multiple scale expansions. The macroscopic magnetic field and electric flux are also obtained. For finite Hartmann number, i.e. $\text{ε⪡Ha⪡ε}{}^{\mathrm{-1}}$ where $\text{ε}$ characterizes the separation of scale, the filtration law is shown to resemble a 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 verifies the filtration analog of the Hall effect. Mass and electric fluxes are coupled.