A Darcy law with memory by homogenisation for evolving microstructure

We consider the homogenisation of the instationary Stokes equations in a porous medium with an a priori given evolving microstructure. In order to pass to the homogenisation limit, we transform the Stokes equations to a domain with a fixed periodic microstructure. The homogenisation result is a Darcy-type equation with memory term and has the form of an integro–differential equation. The evolving microstructure leads to a time- and space-dependent permeability coefficient and the local change of the porosity causes an additional source term for the pressure.