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

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 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.

Roughness‐induced effects on the thermomicropolar fluid flow through a thin domain

“…25, Corollary 3.4] (see also Refs. [9, 26). Proposition The following decomposition for $$p^\varepsilon \in L^2_0(\Omega ^\varepsilon )$$ holds $$$$\begin{equation} p^\varepsilon =p^\varepsilon _0+p^\varepsilon _1, \end{equation}$$$$where $$p^\varepsilon _0\in H^1(\omega )$$, which is independent of x 2 , and $$p^\varepsilon _1\in L^2(\Omega ^\varepsilon )$$.…”

“…Throughout the mathematical literature, one can find many papers on the rigorous derivation of the asymptotic models describing the isothermal flow of a micropolar fluid, see, for example, Refs. 7–13. Although there have been a number of recent papers concerning engineering applications of the thermomicropolar fluid model (see, e.g., Refs.…”