On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication
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abstract
Reynolds's lubrication approximation, one of the cornerstones of fluid 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 modified Reynolds equation. We then provide a comparison between the solutions to the classical equation of Reynolds's and our modified equation. The modified equation results in slightly higher pressures, but at significantly higher viscosities, than the classical Reynolds equation.
