# Asymptotic behavior of a Bingham Flow in thin domains with rough boundary

## TL;DR

This paper analyzes the asymptotic behavior of Bingham flows in thin, rough domains as the thickness tends to zero, deriving a homogenized nonlinear Darcy law that captures microstructure effects.

## Contribution

It introduces an adapted linear unfolding method to derive the homogenized limit of Bingham flows in thin rough domains, preserving nonlinearity.

## Key findings

- Derived the homogenized limit problem for Bingham flow in thin domains.
- Identified the effective nonlinear Darcy law incorporating microstructure effects.
- Analyzed the influence of boundary roughness on flow behavior.

## Abstract

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter $\epsilon$ denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when $\epsilon$ tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law.