Homogenization of a stokes problem with non-homogeneous fourier boundary conditions in a thin perforated domain

In this paper, we consider a viscous fluid obeying the 3D Stokes system in a thin layer [Formula: see text] of thickness [Formula: see text], which is perforated by [Formula: see text]-periodically distributed cylinder shaped obstacles of size [Formula: see text]. On the boundary of the obstacles, we prescribe non-homogeneous Fourier boundary conditions with a parameter [Formula: see text]. Depending on the relation between [Formula: see text] and [Formula: see text], we prove the convergence of the homogenization process when [Formula: see text] goes to zero for different values of [Formula: see text]. As a result, we derive different 2D Darcy’s laws taking into account the microstructure of the domain by means of local problems. The dimensions of the local problems are 2D or 3D depending on the relation between [Formula: see text] and [Formula: see text], i.e. if [Formula: see text] or [Formula: see text], respectively. Moreover, depending on the relation between [Formula: see text] and [Formula: see text], the boundary conditions of the local problems on the reference obstacle are of three types: homogeneous Dirichlet if [Formula: see text], non-homogeneous Fourier if [Formula: see text], or homogeneous Fourier boundary conditions if [Formula: see text].