A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions

TL;DR

This paper derives a generalized Reynolds equation for micropolar fluid flows over rough surfaces with nonzero boundary conditions, using homogenization and unfolding methods, and demonstrates its application in improving squeeze-film bearing performance.

Contribution

It introduces a new macroscopic model coupling roughness and boundary effects for micropolar fluids, extending classical Reynolds equations with novel boundary conditions and roughness considerations.

Findings

The derived models capture the influence of roughness and boundary conditions on flow behavior.

Numerical simulations show roughness can enhance bearing performance.

The critical case $rac{3}{2} ext{l} - rac{1}{2}$ is identified for model validity.

Abstract

Inspired by the lubrication framework, in this paper we consider a micropolar fluid flow through a rough thin domain, whose thickness is considered as the small parameter $\varepsilon$ while the roughness at the bottom is defined by a periodical function with period of order $\varepsilon^{\ell}$ and amplitude $\varepsilon^{\delta}$, with $\delta>\ell>1$. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case $\delta={3\over 2}\ell-{1\over 2}$ and obtain three macroscopic models coupling the effects of the rough bottom and the nonzero boundary conditions. In every case we provide the corresponding micropolar Reynolds equation. We apply these results to carry out a numerical study of a model of squeeze-film bearing lubricated with a micropolar fluid. Our simulations reveal the impact of the roughness coupled with the nonzero boundary conditions on the performance of the bearing, and suggest that the introduction of a rough geometry may contribute to enhancing the mechanical properties of the device.