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. ε [Lt ]
Ha [Lt ] ε−1,
and finite load factor, i.e. ε [Lt ] [Kscr ] [Lt ] ε−1,
where ε 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.
Filtration Law in Porous Media with Poor Separation of Scales