High-Speed Subsonic Compressible Lubrication

The present study extends the scope of compressible lubrication theory (CLT) by considering a more complete formulation of compressible flow in a thin film. A one-dimensional (1D) approximation is obtained, which is common in basic studies of compressible flow. A dimensionless formulation of the thin film compressible flow equations (continuity, momentum, energy, and perfect gas) is derived. There are three dimensionless governing parameters, the Mach number M, the compressibility or bearing number Λ, and a heat transfer number H (a sort of inverse Péclet number). The classical theory assumes isothermal conditions (a consequence of a large heat transfer number) and implicitly assumes low Mach number conditions. It turns out that neither of these conditions are met in high-speed applications such as foil bearings. Results are calculated by varying M and H in a parametric fashion. We find that the influence of Mach number is small (at least up to M = 0.5) but the influence of heat transfer is large: the classical predicted results are in error by a factor of four or so. The improved theory predicts much greater load than the traditional. This means that high-speed air bearing design based on CLT would function satisfactorily, as born out by their successful application; however, such bearings would be significantly over-designed.