Modeling non-Newtonian fluids in a thin domain perforated with cylinders of small diameter

Abstract

We consider the flow of a generalized Newtonian fluid through a thin porous medium of height $h_{\epsilon }$ perforated with $\epsilon$-periodically distributed solid cylinders of very small diameter $\epsilon {\delta }_{\epsilon }$, where the small parameters $\epsilon ,{\delta }_{\epsilon }$ and $h_{\epsilon }$ are devoted to tend to zero. We assume that the fluid is described by the 3D incompressible Stokes system with a non-linear power law viscosity of flow index $1<r<2$ (shear thinning). The particular case $h_{\epsilon }={\sigma }_{\epsilon }$, where ${\sigma }_{\epsilon }:=\epsilon /{\delta }_{\epsilon }^{\frac{2-r}{r}}\to 0$, was recently published in (Anguiano and Suárez-Grau, \emph{Mediterr. J. Math.} (2021) 18:175). In this paper, we generalize previous study for any $h_{\epsilon }$ and we provide a more complete description on the asymptotic behavior of non-Newtonian fluids in a thin porous medium composed by cylinders of small diameter. We prove that depending on the value of $\lambda :={\lim }_{\epsilon \to 0}{\sigma }_{\epsilon }/h_{\epsilon }\in [0,+\infty ]$, there exist three types of lower-dimensional asymptotic models: a non-linear Darcy law in the case $\lambda =0$, a non-linear Brinkman-type law in the case $\lambda \in (0,+\infty )$, and a non-linear Reynolds law in the case $\lambda =+\infty$.