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

TL;DR

This paper analyzes the asymptotic behavior of non-Newtonian shear-thinning fluids flowing through a thin, perforated domain with small cylinders, deriving different effective models depending on geometric parameters.

Contribution

It generalizes previous work by characterizing the asymptotic limits for various domain thicknesses and cylinder sizes, leading to three distinct reduced models.

Findings

Derivation of a non-linear Darcy law for certain geometries.

Identification of a non-linear Brinkman law in intermediate regimes.

Establishment of a non-linear Reynolds law when the domain thickness dominates.

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 $\lambda=+\infty$.