Modeling of a non-Newtonian thin film passing a thin porous medium

TL;DR

This theoretical study analyzes the asymptotic behavior of a non-Newtonian thin film flowing over a thin porous medium, revealing a critical regime where the flow is described by coupled 1D Darcy and Reynolds laws.

Contribution

It introduces a new asymptotic analysis of non-Newtonian fluid flow in coupled thin film and porous media, identifying a critical regime depending on microstructure and thickness parameters.

Findings

Existence of a critical regime depending on parameters

Effective flow described by coupled 1D Darcy and Reynolds laws

Asymptotic behavior characterized as parameters tend to zero

Abstract

This theoretical study deals with asymptotic behavior of a coupling between a thin film of fluid and an adjacent thin porous medium. We assume that the size of the microstructure of the porous medium is given by a small parameter $0<\varepsilon\ll 1$, the thickness of the thin porous medium is defined by a parameter $0<h_\varepsilon\ll 1$, and the thickness of the thin film is defined by a small parameter $0<\eta_\varepsilon\ll 1$, where $h_\varepsilon$ and $\eta_\varepsilon$ are devoted to tend to zero when $\varepsilon\to 0$. In this paper, we consider the case of a non-Newtonian fluid governed by the incompressible Stokes equations with power law viscosity of flow index $r\in (1, +\infty)$, and we prove that there exists a critical regime, which depends on $r$, between $\varepsilon$, $\eta_\varepsilon$ and $h_\varepsilon$. More precisely, in this critical regime given by $h_\varepsilon\approx \eta_\varepsilon^{2r-1\over r-1}\varepsilon^{-{r\over r-1}}$, we prove that the effective flow when $\varepsilon\to 0$ is described by a 1D Darcy law coupled with a 1D Reynolds law.