Homogenization of a couple problem for sound propagation in porous media (Q1927703)

The authors describe an homogenization result for the sound propagation in porous media. They consider a 3D connected domain \(\Omega \) which is splitted in a solid part \(\Omega ^{s}\) and a fluid part \(\Omega ^{f}\). They write the displacement field \(U(x,t)=u_{f}(x)e^{i\omega t}\) in \(\Omega ^{f}\) and \(U(x,t)=u_{s}(x)e^{i\omega t}\) in \(\Omega ^{s}\) and the pressure \( P(x,t)=p(x)e^{i\omega t}\) in \(\Omega ^{s}\). In the fluid part, they write the incompressible Stokes equation \(-\rho _{f}\omega ^{2}u_{f}-i\omega \Delta u_{f}+\nabla p=f^{f}\) with \(\nabla \cdot u_{f}=0\). In the solid part, they consider a linear elastic medium of Lamé type \(-\rho _{s}\omega ^{2}u_{s}-(\lambda +\mu )\nabla \nabla \cdot u_{s}-\mu \Delta u_{s}=f^{s}\). Here \(f^{f}\) and \(f^{s}\) are applied forces which will finally be neglected. The authors impose mixed boundary conditions. They introduce a microstructure in \(\Omega \) depending on a small parameter \( \varepsilon \). They first prove uniform estimates on the solution of this problem. The main part of the paper describes the asymptotic behaviour of the solution, the authors using the two-scale convergence tools and the solutions of cell problems.