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

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 $\varepsilon \ll 1$, perforated by periodically distributed solid cylinders of size $\varepsilon$. 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 $\varepsilon^{-γ}$, where $γ$ is a real constant, we prove the existence of a critical Reynolds number of order $1/\varepsilon$, 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 $1/\varepsilon$ (i.e. $γ= 1$). 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 $1/\varepsilon$. 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.