Homogenization of a micropolar fluid past a porous media with non-zero spin boundary condition

TL;DR

This paper derives a micropolar Darcy law for fluid flow in porous media with non-zero spin boundary conditions by analyzing the homogenization limit of a micropolar fluid with periodically distributed obstacles.

Contribution

It introduces a homogenization approach for micropolar fluids with specific boundary conditions, resulting in a novel Darcy law for such media.

Findings

Existence and uniqueness of solutions established.

Derived an effective Darcy law for micropolar fluids.

Analyzed the limit as obstacle size tends to zero.

Abstract

We consider a micropolar fluid flow in a media perforated by periodically distributed obstacles of size $\varepsilon$. A non-homogeneous boundary condition for microrotation is considered: the microrotation is assumed to be proportional to the rotation rate of the velocity on the boundary of the obstacles. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when $\varepsilon$ tends to zero, an analogue of the classical micropolar Darcy law in the theory of porous media is derived.