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 < +∞, and scaled by a factor 𝜖𝛾, where 𝛾 ∈ R. Generalizing (Anguiano M.: et al. Q. J. Mech. Math., 75(1), 1–27 (2022)), where the particular case 𝑟 < 2 and 𝛾 = 1 was addressed, we perform a new and complete study on the asymptotic behavior 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 behavior of the effective systems.