Homogenization of ferrofluid flow models in porous media with Langevin magnetization law

Abstract

The paper is concerned with the homogenization of the equations describing the flow of a ferrofluid through a heterogeneous porous medium Ω in the presence of an applied magnetic field. We discuss two models where the magnetization M is parallel to the magnetic field H. In the first one M and H satisfy the relation M=λ01ΩfH in Ω, where λ0 is a positive constant and 1Ωf is the characteristic function of Ωf (the pore space). In the second model, M and H satisfy the Langevin magnetization law M=MsL(b1|H|)|H|1ΩfH, where L is the Langevin function given by L(x)=1tanh⁡x−1x, Ms is the saturation magnetization and b1 is a positive physical constant. The velocity and the pressure satisfy the Stokes equation with a Kelvin magnetic force. We perform the homogenization of the equations of each of the two models. Using the two-scale convergence method, we rigorously derive the homogenized equation for the magnetic potential and determine the asymptotic limit of the magnetization. Then we rigorously derive a Darcy law.