Homogenization of a non-stationary non-Newtonian flow in a porous medium containing a thin fissure

We consider a non-stationary incompressible non-Newtonian Stokes system in a porous medium with characteristic size of the pores g and containing a thin fissure of width The viscosity is supposed to obey the power law with flow index 5/3 <= q <= 2. The limit when size of the pores tends to zero gives the homogenized behaviour of the flow. We obtain three different models depending on the magnitude eta(epsilon) with respect to epsilon: if eta(epsilon) << epsilon(q/2q-1 )the homogenized fluid flow is governed by a time-dependent non-linear Darcy law, while if eta(epsilon) >> epsilon(q/2q-1) is governed by a time-dependent non-linear Reynolds problem. In the critical case, eta(epsilon) approximate to epsilon(q/2q-1), the flow is described by a time-dependent non-linear Darcy law coupled with a time-dependent non-linear Reynolds problem.