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

¹ Departamento de Análisis Matemático. Facultad de Matemáticas. Universidad de Sevilla, 41012 - Sevilla, Spain
² Departamento de Ecuaciones Diferenciales y Análisis Numérico. Facultad de Matemáticas. Universidad de Sevilla, 41012 - Sevilla, Spain

^* Corresponding author: fjsgrau@us.es

Received: 14 December 2024
Accepted: 16 July 2025

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 < ε ≪ 1, the thickness of the thin porous medium is defined by a parameter 0 < h_ε ≪ 1, and the thickness of the thin film is defined by a small parameter 0 < η_ε ≪ 1, where h_ε and η_ε are devoted to tend to zero when ε → 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 ∈ (1,+∞), and we prove that there exists a critical regime, which depends on r, between ε, η_ε and h_ε. More precisely, in this critical regime given by h_ε ≈ ηε^2r−1/r−1ε^−(r/r−1), we prove that the effective flow when ε → 0 is described by a 1D Darcy law coupled with a 1D Reynolds law.