Effective models for generalized Newtonian fluids through a thin porous medium following the Carreau law

Abstract

We consider the flow of a generalized Newtonian fluid through a thin porous medium of thickness $ε$, perforated by periodically distributed solid cylinders of size $ε$. We assume that the fluid is described by the 3D incompressible Stokes system, with a non-linear viscosity following the Carreau law of flow index $1<r<+\infty$, and scaled by a factor $ε^γ$, where $γ\in \mathbb{R}$. Generalizing (Anguiano et al., Q. J. Mech. Math., 75(1), 2022, 1-27), where the particular case $r<2$ and $γ=1$ was addressed, we perform a new and complete study on the asymptotic behaviour of the fluid as $ε$ goes to zero. Depending on $γ$ and the flow index $r$, using homogenization techniques, we derive and rigorously justify different effective linear and non-linear lower-dimensional Darcy's laws. Finally, using a finite element method, we study numerically the influence of the rheological parameters of the fluid and of the shape of the solid obstacles on the behaviour of the effective systems.