A Comparison of the Roughness Regimes in Hydrodynamic Lubrication

Abstract

This work relates to previous studies concerning the asymptotic behavior of Stokes flow in a narrow gap between two surfaces in relative motion. It is assumed that one of the surfaces is rough, with small roughness wavelength μ, so that the film thickness h becomes rapidly oscillating. Depending on the limit of the ratio h/μ, denoted as λ, three different lubrication regimes exist: Reynolds roughness (λ = 0), Stokes roughness (0 < λ < ∞), and high-frequency roughness (λ = ∞). In each regime, the pressure field is governed by a generalized Reynolds equation, whose coefficients (so-called flow factors) depend on λ. To investigate the accuracy and applicability of the limit regimes, we compute the Stokes flow factors for various roughness patterns by varying the parameter λ. The results show that there are realistic surface textures for which the Reynolds roughness is not accurate and the Stokes roughness must be used instead.

Homogenization in Hydrodynamic Lubrication: Microscopic Regimes and Re-Entrant Textures

“…Remark. It was brought to our attention that, while our manuscript was still under review, a study with a similar goal as the present study was very recently accepted for publication [42]. Overall, while the two studies share the same major goal, their scopes are not identical but rather complementary in nature, and future readers will benefit from both of them.…”

“…On the numerical side, the study in Ref. [42] employs a commercially available software package to carry out representative two-and threedimensional analyses in order to demonstrate the convergence of the Stokes regime predictions to the two limits. In the threedimensional case, only a single macroscopic quantity, namely A 11 with the present notation, was computed due to the high cost associated with multiple cell problems in the Stokes regime.…”

Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

“…By means of homogenization thecniques, it is showed in Bayada and Chambat [6,7] that depending in the critical size, η ε ≈ ε with η ε /ε → λ, 0 < λ < +∞, there exist three types of flow regimes. This result has been successfully generalized to the unstationary case (the rough surface is moving) in Fabricius et al [18,19]. Below, we describe the three characteristic regimes:…”

Microscopic design and optimization of hydrodynamically lubricated dissipative interfaces

“…This has first been highlighted in the work of Elrod and numerically first demonstrated in another work of Elrod and subsequently in the work of Mitsuya and Fukui . A rigorous mathematical basis for this influence was provided through homogenization theory in the work of Bayada and Chambat, which also clarified how to efficiently formulate the macroscopic response for all physical values of the period, which was numerically demonstrated only recently in the works of Yıldıran et al and Fabricius et al Based on these studies, one may state that a physical realization of the micro‐textures designed in the present study by invoking the Reynolds equation on the microscale must be such that the micro‐texture period is small enough for homogenization to be meaningful but not too small to avoid violating the assumptions of the Reynolds equation, ie, the interface geometry must fall in the Reynolds regime . In this context, it is clear that the macroscopic characterization of the dissipation will also depend on the physical value of the period if the starting point is the Stokes equations, as indeed demonstrated in the work of Mitsuya and Fukui .…”