An Analytical Solution for Thermal Behavior of the Step Thrust Bearing

An analytical solution to the energy equation is presented for the one-dimensional step thrust bearing. Constant lubricant properties are assumed which decouples the momentum and energy equations resulting in a known velocity distribution according to the usual lubrication theory. The energy equation is then linear and yields to a separation of variables solution which results in Sturm-Liouville problems for both domains of the step, which are married at the discontinuity. Conduction, convection and dissipation are all considered and convenient non-dimensional parameters are developed for their relative magnitudes. Expressions for temperature distribution, (along and across the film) heat transfer, and dissipation rates are developed. Results are presented for the Rayleigh step geometry although any other step configuration is easily handled. A wide range of thermal behavior is obtained (rising or falling temperatures, higher temperatures before or after the step, etc.) depending on the dimensionless parameters. The solution presented cannot substitute for the more detailed numerical studies but does offer advantages. The effects of various thermal modes of operation are easily studied and the method is relatively simple to use requiring only a small computer program to evaluate the eigenvalues and eigenfunctions. In addition, general conclusions drawn here can probably be applied with caution to more complex bearing geometries.