| Issue |
ESAIM: M2AN
Volume 56, Number 4, July-August 2022
|
|
|---|---|---|
| Page(s) | 1255 - 1305 | |
| DOI | https://doi.org/10.1051/m2an/2022039 | |
| Published online | 27 June 2022 | |
A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions
1
Université de Paris and Sorbonne Université, CNRS, Laboratoire J-L. Lions/LJLL, F-75006 Paris, France
2
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
3
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/Tarfia s/n, 41012 Sevilla, Spain
* Corresponding author: mathieu.bonnivard@u-paris.fr
Received:
15
September
2021
Accepted:
20
April
2022
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 ε while the roughness at the bottom is defined by a periodical function with period of order εℓ and amplitude εδ, with δ> ℓ >1. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case δ = 3/2ℓ − 1/2 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.
Mathematics Subject Classification: 35B27 / 76D08
Key words: Thin-film flow / micropolar fluid / rough boundary / homogenization / unfolding method
© The authors. Published by EDP Sciences, SMAI 2022
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