Reynolds equation of lubrication coupled with the energy transport equation most often governs the flow of thin lubricant films in journal and thrust bearings. The numerical solution of these two-coupled equations is a solved problem since many decades. However, it requires a computational effort that can render transient analyses vert time consuming. Indeed, if film rupture and reformation (traditionally designed as “cavitation”) are absent and if the flow regime is laminar and isothermal, then Reynolds equation is linear. A direct solver can be used after its discretization in the thin film plane. For non-isothermal flow regime, Reynolds and energy equations are coupled. The solution becomes time-consuming because Reynolds and energy equation must be solved in a sequential way. Secondly, the energy equation must be discretized also across the thin fluid film. The number of discretization points in this direction must be large enough in order to capture the wall temperature gradients. For the turbulent flow regime, where these gradients are much steeper, the number of discretization points across the film is at least one order of magnitude larger than for laminar flow conditions.

If film rupture and reformation are present, then Reynolds equation must be solved with a special approach. Simplified approaches that discard the regions of sub-atmospheric pressures cannot be used because they do not preserve mass. More elaborate solutions of the problem use either a free boundary formulation of the incompressible Reynolds equation or an artificial compressibility approach. These approaches must be rapid and robust because they are repeatedly solved when coupled with the energy equation.

An efficient approach for solving the energy equation makes use of the Legendre polynomials for describing the temperature variation across the thin fluid film. The coefficients of these approximations are obtained by a collocation method applied at the roots of the highest used Legendre polynomial. These roots are the Lobatto points and the method is known as the “Lobatto points collocation method” (LPCM). The method is known since decades but was used only on an ad hoc basis.

The present work presents a systematic comparison between the natural discretization method (NDM) of the energy equation and its LPCM approximation. A one dimensional (1D) inclined slider is used for numerical tests.

The results for the 1D slider compare the number of points needed by the NDM and by the LPCM for obtaining grid independent results. Both situations, with imposed wall temperatures and wall temperature gradients are analyzed. When wall temperatures are imposed, then the wall temperature gradient is used for estimating the accuracy of the solution and reciprocal. The results show how the NDM is excessively time consuming when high accuracy is needed and the net economy of computational time brought by the LPCM approach.

The same analyses is then carried on for a circular journal bearing with a circumferential, central, feeding groove. For evaluation purposes, the film rupture/reformation is dealt with by using the two approaches previously mentioned. The a free boundary formulation of the incompressible Reynolds equation was solved by using an efficient solver based on Fischer-Burmeister form, Newton algorithm and Shur's complement. The artificial compressibility approach was combined with a regularization technique for avoiding discontinuities and the discretized equation were solved with a Newton algorithm.

The two approaches show comparable results in terms of robustness and computational effort. The LPCM approach was again largely superior to the NDM solution in terms of computational time.

These results show that efficient numerical solutions of the coupled Reynolds and energy equations are possible if a good approximation of the temperature variation across the film is used together with robust film rupture/reformation algorithms.