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

Abstract

The form of the Reynolds-type equation which governs the macroscopic mechanics of hydrodynamic lubrication interfaces with a microscopic texture is well-accepted. The central role of the ratio of the mean film thickness to the texture period in determining the flow factor tensors that appear in this equation had been highlighted in a pioneering theoretical study through a rigorous two-scale derivation (Bayada and Chambat, 1988, “New Models in the Theory of the Hydrodynamic Lubrication of Rough Surfaces,” ASME J. Tribol., 110, pp. 402–407). However, the resulting homogenization theory still remains to be numerically investigated. For this purpose, after a comprehensive review of the literature, three microscopic regimes of lubrication will be outlined, and the transition between these three regimes for different texture types will be extensively demonstrated. In addition to conventional textures, representative re-entrant textures will also be addressed.

Microscopic design and optimization of hydrodynamically lubricated dissipative interfaces

“…Subsequently, denoting the out‐of‐plane position of the upper/lower physical surface by n ± , the film thickness at any point on may be expressed as where a dependence on time ( t ) is also included. The film thickness is not well‐defined in the presence of re‐entrant features on the surfaces, although a definition based on a simple modification of the surfaces may still enable one to proceed with the following analysis in the context of homogenization . The mechanics of lubrication is possibly active only over a subdomain of , but this will not be explicitly indicated.…”

“…These velocities do not depend on position or time in the present setting. The film thickness rate of change can now be locally evaluated via The sum and difference of the tangential velocities will be indicated by The relative motion of the surfaces, together with the boundary conditions on the interface geometry, generates a pressure p with gradient at the interface that is governed by the Reynolds equation where the fluid flux q has a combined Poiseuille‐Couette constitutive form that is obtained from , ie, Here, the following coefficients have been defined, together with e for future reference: Note that, despite the original assumption of a smoothly varying film thickness in the derivation of the Reynolds equation, the theory retains its predictive capability in the presence of sharp transitions as well with respect to a more general framework based on the Stokes equations in the context of homogenization . Here, a sharp transition refers to a steep slope in the film thickness within a localized region of the cell and, in the extreme case, could indicate a jump.…”

“…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 .…”

“…The film thickness is not well-defined in the presence of re-entrant features on the surfaces, although a definition based on a simple modification of the surfaces may still enable one to proceed with the following analysis in the context of homogenization. 28 The mechanics of lubrication is possibly active only over a subdomain of , but this will not be explicitly indicated. Within the Reynolds equation approximation, the tangential velocity variationŨ(n) of the fluid in the normal direction is quadratic and defines the fluid flux at any point on  via…”

A Novel Approach for Modeling Surface Effects in Hydrodynamic Lubrication

“…By using the Navier-Stokes equations, flow processes in the fluid can be depicted in greater detail. This is fundamental to being able to investigate the effects of structured and textured surfaces [33].…”