Homogenisation of the Stokes equations for evolving microstructure

We consider the homogenisation of the quasi-stationary Stokes equations in a porous medium that evolves over time. The evolution is a priori given. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit employing the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method. The homogenisation result is a new version of Darcy's law. It features a time- and space-dependent permeability tensor, which accounts for the local pore structure, and a macroscopic inhomogeneous divergence condition, which induces a new source term for the pressure. In the case of a no-slip boundary condition at the interface, this source term relates to the change of the local pore volume.