The exact analytical solution for the gas lubricated bearing in the slip and continuum flow regime

Abstract

The exact analytical solution for the compressible two-dimensional gas flow in the microbearing is presented. The general slip-corrected Reynolds lubrication equation is derived and it is shown that it possesses an exact analytical solution. It is obtained by a suitable transformation of the independent variable, and it provides the pressure distribution in the bearing as well as the mass flow rate through it. By neglecting the rarefaction effect, this solution is also applicable to the continuum gas flow in the bearing, which also does not exist in the open literature. The obtained analytical solution can be usefully applied for testing the other, experimental or numerical results.

An exact analytical solution for the second order slip-corrected Reynolds lubrication equation

“…2, more extensive validation of this analytical solution was already presented in [8] by comparison of the analytically obtained results with available numerical results. Namely, the slip flow results for a wide range of Knudsen number and the continuum flow conditions, provided by the general analytical solution from this paper and [8], are in excellent agreement with Fukui and Kaneko [15] numerical solution of the Boltzmann equation.…”

“…The obtained differential equation written in the form in which variables H and Π are separated (eq. 16) can be transformed in the differential equation with separated variables presented by equation (2.14) in [8]. This indicates that despite the fact that final analytical solutions obtained by transformations of the dependent and independent variable don't have the same form, the both solutions give the same result.…”

“…This equation is also nonlinear. To the best of our knowledge only one exact analytical solution of this equation exists and it is presented in [8]. It was found by suitably transforming the independent variable in the slip-corrected Reynolds equation.…”

Gap Flow Simulation Methods in High Pressure Variable Displacement Axial Piston Pumps

“…Stevanovic et. al [ 19 ] derived general slip-corrected Reynolds lubrication equation and showed that it possesses an exact analytical solution. Wieczorek and Ivantysynova [ 20 ] developed a method to calculate non-isothermal gap flow in the connected gaps of swash plate type axial piston machines by combining Matlab and C language routines that analyzed the motion equation of all moveable parts of the rotating group and performed gap simulation of all concerned gaps.…”

First integrals and exact solutions of the generalized models of magnetic insulation

“…The knowledge of explicit exact solutions of the nonlinear PDE is very useful for the qualitative analysis [14]. The obtained analytical solution can be usefully applied for testing the other, experimental or numerical results for adequate modeling of modern technologies [15].…”