# Homogenization of Bingham Flow in thin porous media

## TL;DR

This paper investigates the steady flow of Bingham fluids in thin porous media using homogenization, revealing how the flow's nonlinear behavior depends on the geometry and scale parameters of the medium.

## Contribution

It introduces a novel analysis of Bingham fluid flow in thin porous structures, accounting for the dependence of yield stress on geometric parameters, and derives three limit problems as the scale parameter tends to zero.

## Key findings

- Three distinct limit problems depending on geometric ratios.
- Flow retains nonlinear Bingham characteristics in all limit cases.
- Dependence of yield stress on the medium's small parameters.

## Abstract

By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter $\varepsilon$ tends to zero, following the ratio between the height $\varepsilon$ of the porous medium and the relative dimension $a_\varepsilon$ of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow.