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

The paper is concerned with the homogenization of the equations describing the flow of a ferrofluid through a heterogeneous porous medium 12 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 = lambda 0 1 omega fH in 12, where lambda 0 is a positive constant and 1 omega f is the characteristic function of 12f (the pore space). In the second model, M and H satisfy the Langevin magnetization law M = Ms L(b1 |H|) |H| 1 omega f H, where L is the Langevin function given by L(x) = 1 tanh 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.(c) 2023 Elsevier Inc. All rights reserved.