Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions

We consider the Stokes system in a thin porous medium ${\mathrm{\Omega }}_{\epsilon }$ of thickness $\epsilon$ which is perforated by periodically distributed solid cylinders of size $\epsilon$. On the boundary of the cylinders we prescribe non-homogeneous slip boundary conditions depending on a parameter $\gamma$. The aim is to give the asymptotic behavior of the velocity and the pressure of the fluid as $\epsilon$ goes to zero. Using an adaptation of the unfolding method, we give, following the values of $\gamma$, different limit systems.