Magnetic Fluid Flows In Porous Media

We investigate the filtration law in rigid porous media for steady-state slow?ow of an incompressible and viscous Newtonian magnetic ?uid in the presenceof the external magnetic field and the intensity of magnetization. The seepagelaw under a magnetic field is obtained by upscaling the ?ow at the pore scale.The macroscopic equivalent model for the magnetic field is also obtained. We usethe method of multiple-scale expansions which gives rigorously the macroscopicbehavior without any preconditions on the form of the macroscopic equations.The macroscopic mass ?ow depends on the macroscopic magnetic force and thegradient of pressure and of magnetic pressure, as Markus Zahn and Rold E.Rosensweig had described in their experiments. In particular, the filtration lawis shown to resemble Darcy's law of common ?uid but with some additional termsproportional to the macroscopic gradient of the magnetic pressure and magneticforce. The permeability tensor is symmetric and positive.