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

Abstract

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 5/3 ≤ q ≤ 2. The limit when size of the pores tends to zero gives the homogenized behavior of the flow. We obtain three different models depending on the magnitude ηε with respect to ε: if ηε ≪ ε^{q/(2q−1)} the homogenized fluid flow is governed by a time-dependent nonlinear Darcy law, while if ηε ≫ ε^{q/(2q−1)} is governed by a time-dependent nonlinear Reynolds problem. In the critical case, ηε ≈ ε^{q/(2q−1)} , the flow is described by a time-dependent nonlinear Darcy law coupled with a time-dependent nonlinear Reynolds problem.