Perturbation techniques in the theory of lubrication
Perturbation techniques are a very useful tool also in the theory of lubrication. Two typical situations are rigorously discussed in this paper: the bifurcation for the "shoe-bearing" and the asymptotic analysis for a finite gas slider bearing of narrow geometry. Furthermore the mathematical basis of a new application of the Reynolds equation is briefly treated. Introduction. This paper deals with various perturbation problems arising in the theory of lubrication. The main analytic tool of this branch of fluid mechanics is the Reynolds equation [1], [5] which reads ^-vp) = 6U(pH) x. (1.1) In (1.1), P( X), X = (x, y) is the pressure in a thin film of lubricating fluid of density p and viscosity fi. H{ X) is the film thickness and U the ^-component of the velocity of the moving surface. A bifurcation problem motivated by the study of the so-called "shoe-bearing" is examined in Section 2. For this kind of bearing the thickness of the film is not fixed because the upper surface (the shoe) is freely pivoted; thus the equilibrium positions under load need not be unique. The occurrence of bifurcation is proved with elementary analytic means. In Section 3 we are concerned with a singular perturbation problem for the nonlinear Reynolds equation of gas-lubrication. We take into account the width of the occurring boundary layer and prove a result of convergence previously stated without proof by Diprima and Shepherd (see [4]). * Received March 21, 1985. ^1986 Brown University 97