Homogenization of ferrofluid flow models in porous media with Langevin magnetization law - Archive ouverte HAL Access content directly
Journal Articles Journal of Mathematical Analysis and Applications Year : 2023

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

Youcef Amirat
• Function : Author
• PersonId : 1133758
Kamel Hamdache
• Function : Author

#### 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.

#### Domains

Mathematics [math]

### Dates and versions

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

### Licence

Attribution - NonCommercial - NoDerivatives - CC BY 4.0

### Identifiers

• HAL Id : hal-03995934 , version 1
• DOI :

### Cite

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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