New First and Second Order Slip Models for the Compressible Reynolds Equation

In the original derivations of the first order and the second order slip models of the generalized Reynolds equation in the literature [3,4], a length scale equal to the mean free path of the gas molecules was used in a Taylor series expansion of the mean velocity field. The coefficients of the correction terms in the derived lubrication equation depend on that length scale. This choice of the length scale is arbitrary to some extent. In this paper, new first order and the second order slip models are derived using a somewhat more physical approach, in which the requirement that the expansion length scale be the mean free path is relaxed. In this approach the momentum transfer rate across each surface element is obtained by summing up the contributions from each group of molecules impinging on the surface at an angle θ to the surface normal within a solid angle dω. The new second order slip lubrication equation appears to be preferable to the original one when the inverse Knudsen number is small, and it is free of any contact pressure singularity, whereas the new first order slip model continues to contain the unacceptable pressure singularity in the limit as the spacing approaches zero, as does the original first order model.