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

Maria Anguiano; Matthieu Bonnivard; Francisco Javier Suarez-Grau

Paper

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 et al., Q. J. Mech. Math., 75(1), 2022, 1-27), where the particular case

and

was addressed, we perform a new and complete study on the asymptotic behaviour 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 behaviour of the effective systems.