https://uca.hal.science/hal-03995934Amirat, YoucefYoucefAmiratLMBP - Laboratoire de Mathématiques Blaise Pascal - CNRS - Centre National de la Recherche Scientifique - UCA - Université Clermont AuvergneHamdache, KamelKamelHamdachePULV - Pôle Universitaire Léonard de VinciHomogenization of ferrofluid flow models in porous media with Langevin magnetization lawHAL CCSD2023Ferrofluid flow in porous mediaStokes equationsLangevin magnetization lawhomogenizationtwo-scale convergenceDarcy law[MATH] Mathematics [math]Amirat, Youcef2023-02-18 22:51:142023-03-24 14:53:312023-02-22 13:28:20enJournal articleshttps://uca.hal.science/hal-03995934/document10.1016/j.jmaa.2023.127129application/pdf1The paper is concerned with the homogenization of the equations describing the flow of a ferrofluid through a heterogeneous porous medium $\Omega$ 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$M=\lambda_0 \, {1}_{\Omega_f} H\, \mbox{ in } \Omega,$where $\lambda_0$ is a positive constant and ${1}_{\Omega_f}$ is the characteristic function of $\Omega_f$ (the pore space). In the second model, $M$ and $H$ satisfy the Langevin magnetization law$M= M_s \frac{L(b_1\, |H|)}{|H|} {1}_{\Omega_f} \, H,$where ${L}$ is the Langevin function given by $L(x)= \frac{1}{\tanh x} - \frac{1}{ x}$,$M_s$ is the saturation magnetization and $b_1$ 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.