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

We consider the flow of a generalized Newtonian fluid through a thin porous medium of thickness $\epsilon$, perforated by periodically distributed solid cylinders of size $\epsilon$. 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 $\epsilon^{\gamma}$, where $\gamma\in \mathbb{R}$. Generalizing (Anguiano et al., Q. J. Mech. Math., 75(1), 2022, 1-27), where the particular case $r<2$ and $\gamma=1$ was addressed, we perform a new and complete study on the asymptotic behaviour of the fluid as $\epsilon$ goes to zero. Depending on $\gamma$ 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.