The line contact problem of elastohydrodynamic lubrication - II. Numerical solutions of the integrodifferential equations in the transition and exit layers

Abstract

The solution of the line contact problem of elastohydrodynamic lubrication in the asymptotic régime developed by Bissett ( Proc . R . Soc . Lond . A 424, 393–407 (1989)) exhibits two regions of rapid change: a transition layer between the inlet and contact zones, and a downstream exit layer. In these regions the governing Reynolds equation of lubrication theory is essentially nonlinear, although pressure and surface displacement continue to be linearly related by the singular integral equation of plane elasticity. In combination, the system in each region reduces to a nonlinear singular integrodifferential equation with Cauchy kernel for the surface displacement, to be satisfied on either an infinite interval (transition layer) or a semi-infinite interval (exit layer). A method is developed along lines used by Spence & Sharp ( Proc . R . Soc . Lond . A400, 289 (1985), Proc . R . Soc . Lond . A422, 173 (1989)) and Spence et al . ( J . Fluid Mech . 174, 135 (1987)) for approximating the solution in either case by a finite number of trigonometric terms (up to 900). Rapid convergence is achieved by judicious allowance for end point behaviours as deduced by asymptotic analysis of the governing equations. The equations contain an eigenvalue, representing the scaled exit film thickness, which also characterizes the film thickness in the contact zone. This eigenvalue is found with high accuracy in the course of solving the transition layer problem. Close agreement with certain results of Hooke & O’Donoghue ( J . mech . Engng Sci . 14 (1), 34 (1972)) is exhibited.