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 , perforated by periodically distributed solid cylinders of size . 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 , and scaled by a factor , where . Generalizing (Anguiano M.: et al. Q. J. Mech. Math., 75(1), 1–27 (2022)), where the particular case and was addressed, we perform a new and complete study on the asymptotic behavior of the fluid as goes to zero. Depending on and the flow index , 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 behavior of the effective systems.