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

TL;DR

This work develops a rigorous homogenization framework for micropolar fluids flowing in a thin, ribbed domain with nonzero boundary conditions on the rough bottom. By an unfolding-based asymptotic analysis, it identifies a critical scaling $δ = 3/2 ℓ - 1/2$ and derives three macroscopic models that couple riblet geometry with slip at the fluid-solid interface, culminating in generalized Reynolds equations for the pressure. The results include precise effective boundary conditions, the dependence on micropolar parameters through coefficients like $Θ_λ$ and $E_λ$, and a detailed treatment of sub- and super-critical regimes. The theory is then illustrated with a squeeze-film bearing, where numerical experiments show that introducing roughness can enhance mechanical performance under appropriate micropolar and geometric parameters.

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 $\varepsilon^δ$, with $δ>\ell>1$. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case $δ={3\over 2}\ell-{1\over 2}$ 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.

Figures (10)

• Figure 1: Representation of the geometry of riblets (upper) and of their cross section (lower). The riblets are periodic with period $\varepsilon^{\ell}$ in the $x_1$ direction, constant in the $x_2$ direction and oscillate with an amplitude of order $\varepsilon^{\delta}$ in the $x_3$ direction.
• Figure 2: Example of function $\Theta_{\lambda}$ and its asymptotic development $\Theta_0 - C_{j} E\lambda^2\Theta_1$ plotted against $\lambda$, for the set of parameters $N=0.3, R_c = 0.1, \delta=1, E=10$, and with $\bar{\nu}_b = 0.1$ (case $\alpha\neq 1$, left) and $\bar{\nu}_b = 1$ (case $\alpha=1$, right).
• Figure 3: $T_{\textrm{half}}$ plotted against $N$, for $\delta=1$, $E=0$ and different values of $R_c\in \{0.025, 0.05, 0.1, 0.2\}$, with $\bar{\nu}_b\in \{0.05,0.1,0.2,0.4\}$.
• Figure 4: $T_{\textrm{half}}$ plotted against $N$, for $\delta=1$, $E=10$ and different values of $R_c\in \{0.025, 0.05, 0.1, 0.2\}$, with $\bar{\nu}_b\in \{0.05,0.1,0.2,0.4\}$.
• Figure 5: $T_{\textrm{half}}$ plotted against $N$, for $\bar{\nu}_b=0.1$, $R_c\in \{0.025, 0.05, 0.1, 0.2\}$ and $\delta=1$. From left to right and top to bottom: $E=0$, $E=1$, $E=3$, $E=5$, $E=7$ and $E=10$.

Theorems & Definitions (20)

• Definition 1
• Remark 1
• Theorem 1
• Remark 2
• Theorem 2
• Remark 3
• Theorem 3
• Remark 4
• Theorem 4
• Proposition 1