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 $h_\varepsilon$ perforated with $\varepsilon$-periodically distributed solid cylinders of very small diameter $\varepsilon\delta_\varepsilon$, where the small parameters $\varepsilon, \delta_\varepsilon$ and $h_\varepsilon$ 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 $h_\varepsilon=\sigma_\varepsilon$, where $\sigma_\varepsilon:=\varepsilon/\delta_\varepsilon^{{2-r\over r}}\to 0$, was recently published in (Anguiano and Su\'arez-Grau, \emph{Mediterr. J. Math.} (2021) 18:175). In this paper, we generalize previous study for any $h_\varepsilon$ 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 $\lambda:=\lim_{\varepsilon\to 0}\sigma_\varepsilon/h_\varepsilon\in [0,+\infty]$, 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 $\lambda=+\infty$.