Modeling of a non-Newtonian thin film passing a thin porous medium

This theoretical study deals with asymptotic behavior of a coupling between a thin film of fluid and an adjacent thin porous medium. We assume that the size of the microstructure of the porous medium is given by a small parameter 0 < ε ≪ 1, the thickness of the thin porous medium is defined by a parameter 0 < hε ≪ 1, and the thickness of the thin film is defined by a small parameter 0 < ηε ≪ 1, where hε and ηε are devoted to tend to zero when ε → 0. In this paper, we consider the case of a non-Newtonian fluid governed by the incompressible Stokes equations with power law viscosity of flow index r ∈ (1,+∞), and we prove that there exists a critical regime, which depends on r, between ε, ηε and hε. More precisely, in this critical regime given by hε ≈ ηε2r−1/r−1ε−(r/r−1), we prove that the effective flow when ε → 0 is described by a 1D Darcy law coupled with a 1D Reynolds law.