Homogenization of a non-stationary non-Newtonian flow in a porous medium containing a thin fissure
María Anguiano · European Journal of Applied Mathematics · 2018
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.