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

Abstract: 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 $0<\ep\ll 1$, the thickness of the thin porous medium is defined by a parameter $0<h_\ep\ll 1$, and the thickness of the thin film is defined by a small parameter $0<η\ep\ll 1$, where $h\ep$ and $η\ep$ are devoted to tend to zero when $\ep\to 0$. 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 $r\in (1, +\infty)$, and we prove that there exists a critical regime, which depends on $r$, between $\ep$, $η\ep$ and $h_\ep$. More precisely, in this critical regime given by $h_\ep\approx η_\ep^{2r-1\over r-1}\ep^{-{r\over r-1}}$, we prove that the effective flow when $\ep\to 0$ is described by a 1D Darcy law coupled with a 1D Reynolds law.