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 ϵ and containing a thin fissure of width ηϵ. The viscosity is supposed to obey the power law with flow index$\frac{5}{3}\leq q\leq 2$. 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 ηϵwith respect to ϵ: if ηϵ≪$\varepsilon^{q\over 2q-1}$the homogenized fluid flow is governed by a time-dependent non-linear Darcy law, while if ηϵ≫$\varepsilon^{q\over 2q-1}$is governed by a time-dependent non-linear Reynolds problem. In the critical case, ηϵ≈$\varepsilon^{q\over 2q-1}$, the flow is described by a time-dependent non-linear Darcy law coupled with a time-dependent non-linear Reynolds problem.