Lobatto Point Quadrature for Thermal Lubrication Problems Involving Compressible Lubricants

Refined solutions of thermal lubrication problems generally require fine mesh and many iteration steps. To resolve these difficulties, Elrod and Brewe (1) proposed an efficient algorithm based on the use of Lobatto point quadrature. Within this approach, the unknown temperature across the film is written in a series of Legendre polynomials. This paper presents a Lobatto point quadrature algorithm which is applicable for thermal lubrication problems with compressible lubricants. In this case both density and viscosity of the lubricant are taken to be temperature and pressure dependent. The transverse velocity is obtained from the continuity equation. Use of the Labatto point calculation method has resulted in greater accuracy without the use of a large number of grid points.