Homogenization and two-scale convergence for a Stokes or Navier-Stokes flow in an elastic thin porous medium

In the general framework of the homogenization method we study the behavior of a thin elastic periodic structure immersed in a viscous fluid. After the proof of the convergence of the homogenization process by using the two-scale convergence method it is possible to take the limit as delta --> 0, the small parameter which represents the thickness of the solid part. We obtain a viscoelastic medium with fading memory. We consider two cases: the one in which we have a Stokes flow in the fluid part and the other one when we have a Navier-Stokes flow in the fluid part.