The Incompressible Lubrication Problem

Abstract

It is evident to all those engaged in the analysis of realistic lubrication problems that none of the existing techniques of computation proves satisfactory for practical purposes. In particular the most extensively used method of finite difference approximations to the relevant Reynolds differential equation suffers from serious disadvantages. Thus, the contour of the lubrication film has to be described by constant co-ordinate lines in special reference systems, unless one is prepared to accept more or less serious inaccuracies. Also auxiliary conditions must be introduced when dealing with discontinuous changes of lubrication data, like film thickness. These and other reasons demand the development of a general method which can deal with all current practical complexities. Clearly, the finite element technique should prove an ideal tool allowing completely arbitrary variation of geometry, film properties, and boundary conditions. The present note intends to explore the theoretical basis of the application of finite elements, using a variational formulation of Reynold's equation. The various elemental matrices are formulated and detailed for triangular TRIC or TRIM-like elements.