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Journal Articles Journal of Mathematical Analysis and Applications Year : 2023

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 $\Omega$ 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.
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Dates and versions

hal-03995934 , version 1 (18-02-2023)

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Attribution - NonCommercial - NoDerivatives

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Youcef Amirat, Kamel Hamdache. Homogenization of ferrofluid flow models in porous media with Langevin magnetization law. Journal of Mathematical Analysis and Applications, In press, ⟨10.1016/j.jmaa.2023.127129⟩. ⟨hal-03995934⟩
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