On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication

Abstract

Reynolds's lubrication approximation, one of the cornerstones of°uid mechanics, is constructed on the assumption that the viscosity is independent of the pressure. This assumption is reasonable at low pressures and is appropriate for a large class of applications. However, in an important instance that appeals to the approximation (elastohydrodynamic lubrication (EHL)), the liquid lubricant is subjected to extremely high pressures and the assumption that the viscosity is independent of the pressure no longer holds. On the contrary, pressure dependence of viscosity is severe and the viscosity can increase by several orders of magnitude due to pressure increase. Nevertheless, in the current literature the pressure dependence of viscosity in the derivation of the governing equations for EHL is only recognized a posteriori, that is, after the Reynolds equation has been stated under the assumption of constant viscosity.A consistent derivation of the equations of EHL that takes into account the pressure dependence of viscosity right from the outset leads to additional and hitherto neglected terms in the governing equations. Consequently, construction of a single pressure equation, analogous to the Reynolds equation, is no longer possible without additional, drastic, assumptions.In this study, we provide a consistent derivation of the equations of motion for EHL and, with additional, simplifying assumptions, derive a modi¯ed Reynolds equation. We then provide a comparison between the solutions to the classical equation of Reynolds's and our modi¯ed equation. The modi¯ed equation results in slightly higher pressures, but at signi¯cantly higher viscosities, than the classical Reynolds equation.

Pressure-dependent surface viscosity and its surprising consequences in interfacial lubrication flows

“…(22) and (23). The dependence of η s results in a nonlinear ODE, but one which may be separated [as in Eq.…”

“…3D studies of such "piezoviscous" liquids have used exponential and power-law η(p) relations, particularly in the contexts of polymer melt processing [20,21] and lubricating oils under exceedingly high pressures [22,23]. Such flows may depart from the classical Newtonian solutions [22,24], e.g., the extrusion flux of polymer melts may slow dramatically under high pressures [21].…”

A Novel Approach to the Description of Constitutive Relations

“…For example, the field of elastohydrodynamics is built around the notion that the material properties of the lubricant are a function of the mean normal stress of the fluid. When one recognizes that most lubricants have also shear rate dependent properties, we are naturally led to models that are implicit [see Rajagopal and Szeri (2003)]. When one deals with a generalization of the Navier-Stokes fluid with a pressure dependent viscosity (usually referred to in the literature as a piezoviscous fluid), one cannot express the stress explicitly in terms of the velocity gradient, but one can express the velocity gradient as a non-linear function of the stress.…”

Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane

“…Rajagopal and co-workers have studied issues concerning existence and uniqueness of flows as well as special flows of fluids with pressure-dependent viscosity. Of specific relevance is the paper by Rajagopal & Szeri (2003), who obtain the appropriate lubrication approximation, in the spirit of the early works by Reynolds, for the flows of such fluids, previous approximations being incorrect.…”

“…In this paper we consider a non-Newtonian fluid whose viscosity depends on both the mean normal stress (in the case of the model being considered, the pressure) and the shear rate (the fluid can shear-thin or shear-thicken). In a recent paper, Saccomandi & Vergori (2010), using the lubrication approximation appropriate to such fluids as developed by Rajagopal & Szeri (2003), studied in great detail the flow down an inclined plane of an incompressible fluid whose viscosity depends on pressure. They considered various flow regimes, namely flows wherein viscous effects, surface tension effects, etc, are predominant.…”