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 $0<\varepsilon\ll 1$, the thickness of the thin porous medium is defined by a parameter $0<h_\varepsilon\ll 1$, and the thickness of the thin film is defined by a small parameter $0<\eta_\varepsilon\ll 1$, where $h_\varepsilon$ and $\eta_\varepsilon$ are devoted to tend to zero when $\varepsilon\to 0$. 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 $\varepsilon$, $\eta_\varepsilon$ and $h_\varepsilon$. More precisely, in this critical regime given by $h_\varepsilon\approx \eta_\varepsilon^{2r-1\over r-1}\varepsilon^{-{r\over r-1}}$, we prove that the effective flow when $\varepsilon\to 0$ is described by a 1D Darcy law coupled with a 1D Reynolds law.