Influence of the Reynolds number on non-Newtonian flow in thin porous media

Abstract

We study the effect of the Reynolds number on the flow of a generalized Newtonian fluid through a thin porous medium in ${\mathbb{R}}^3$. This medium is a domain of thickness $\epsilon \ll 1$, perforated by periodically distributed solid cylinders of size $\epsilon$. We consider the nonlinear stationary Navier-Stokes system with viscosity following the Carreau law. Using tools from homogenization theory and assuming that the Reynolds number scales as ${\epsilon }^{-\gamma }$, where $\gamma$ is a real constant, we prove the existence of a critical Reynolds number of order $1/\epsilon$, in the sense that the inertial term in the Navier-Stokes system has no influence in the limit if the Reynolds number is of order smaller than or equal to $1/\epsilon$ (i.e. $\gamma =1$). In this case, we derive linear or nonlinear Darcy laws connecting velocity to pressure gradient. Conversely, we expect a contribution from the inertial term in the homogenized problem if the Reynolds number is greater than $1/\epsilon$. Finally, we propose a numerical method to solve nonlinear Darcy laws describing effective flow in the critical case and demonstrate its practical applicability on several examples.