The authors consider a simply connected and bounded domain \(\Omega \subset \mathbb{R}^{d}\), \(d\geq 2\), of class \(C^{1,1}\), and the unit cell \( Y=(0,1)^{d}\) in \(\mathbb{R}^{d}\), which is decomposed into \(Y=Y_{s}\cup Y_{f}\cup \Gamma \), where \(Y_{s}\) represents the magnetic inclusion and \( Y_{f}\) the fluid domain which are open sets in \(\mathbb{R}^{d}\), and \(\Gamma \) is the closed \(C^{1,1}\)-interface between them. They deduce a decomposition of \(\Omega \) into its fluid part \(\Omega_{f}^{\varepsilon}\), its solid part \(\Omega_{s}^{\varepsilon}\) and the interface \(\Gamma^{\varepsilon}\) through a periodicity process, with \(\varepsilon >0\).
The purpose of the paper is to describe the homogenization of the nonlinear system modeling a suspension of rigid inclusions in a nonconducting carrier fluid and written as: \[ \begin{aligned} \rho_{f}(\frac{\partial u^{\varepsilon}}{\partial t} +(u^{\varepsilon}\cdot \nabla )u^{\varepsilon})-\operatorname{div}\sigma^{\varepsilon}=\rho_{f}g,\ \operatorname{div}u^{\varepsilon}=0,\ \operatorname{curl}H^{\varepsilon}=0 \text{ in }\Omega_{f}^{\varepsilon},\\ \mathbb{D}(u^{\varepsilon})=0, \ \frac{\partial B^{\varepsilon}}{\partial t}+\frac{1}{\nu_{es}} \operatorname{curl\ curl} H^{\varepsilon}=\operatorname{curl}(u^{\varepsilon}\times B^{\varepsilon}) \text{ in }\Omega_{s}^{\varepsilon}, \end{aligned} \] where \(\rho_{f}\) is the (mass) fluid density, \(\rho_{s}\) the density of inclusions, \(\nu_{es}\) the electric conductivity of inclusions, \(g\) the external force field, \(u^{\varepsilon}\) the fluid velocity, \(\sigma \) the Cauchy stress tensor which depends on the velocity and on the fluid pressure \(p^{\varepsilon}\), \(B^{\varepsilon}\) the divergence-free magnetic field, and \(H^{\varepsilon}\) the magnetizing field which is linked to the magnetic field through \(B^{\varepsilon}=\mu H^{\varepsilon}\) in \(\Omega \), \(\mu \) being the magnetic permeability equal to some \(\mu_{f}\) in \(\Omega_{f}^{\varepsilon}\) and to some \(\mu_{s}\) in \(\Omega_{s}^{\varepsilon}\).
In the main part of the paper, the authors prove the existence of a unique solution \(u^{\varepsilon}\in H_{0}^{1}(\Omega;\mathbb{R}^{d})\), \(p^{\varepsilon}\in L_{0}^{2}(\Omega )\), \(B^{\varepsilon}\in H_{n}^{1}(\Omega;\mathbb{R}^{d})\) to a modified problem derived from the preceding one through a change of variables, under appropriate hypotheses on the data. They prove that \(u^{\varepsilon},B^{\varepsilon}\) two-scale converge to \(u^{0},B^{0}\) in \(H^{1}(\Omega; \mathbb{R}^{d})\) and that \(p^{\varepsilon}\) two-scale converges to \(p^{0}\) in \(L_{0}^{2}(\Omega )\), and they derive the limit problem which involves an elliptic fourth-rank tensor \(\mathcal{N}\) and two constant, symmetric and elliptic matrices \(\mathcal{M}\) and \(\mathcal{E}\). For the proof, the authors build a variational formulation of the problem, from which they derive uniform estimates on the solution. They finally use the properties of the two-scale convergence that they recall.