Modified Reynolds Equation for Ultra-Thin Film Gas Lubrication Using 1.5-Order Slip-Flow Model and Considering Surface Accommodation Coefficient

Abstract

A 1.5-order modified Reynolds equation for solving the ultra-thin film gas lubrication problem is derived by using an accurate higher-order slip-flow model. This model features two key differences from the current second-order slip-flow model. One is the involvement of an accommodation coefficient for momentum. The other is that the coefficient of the second-order slip-flow term is 4/9 times smaller than that for the current model. From the physical consideration of momentum transfer, the accommodation coefficient is found to have no affect on the second-order slip-flow term. Numerical calculations using the 1.5-order modified Reynolds equation are performed. The results are compared with those obtained using three kinds of currently employed modified Reynolds equations: those employing the first- and second-order slip-flow models and those utilizing the Boltzmann equation. These comparisons confirm that the present modified Reynolds equation provides intermediate characteristics between those derived from the first- and second-order slip-flow models, and produces an approximation closer to the exact solution resulting from the Boltzmann-Reynolds equation.

A slip model for rarefied gas flows at arbitrary Knudsen number

“…The mass flow rates for other models are available from the listed references. [2][3][4][5] The figure shows that the current slip model predicts a mass flow rate very close to the FK model and in this sense outperforms other slip models in the whole modified inversed Knudsen number range. Slight divergence between the current model and the FK model is observed when the modified inversed Knudsen number is reduced below 0.02.…”

“…The values of ␤ and ␥ for other slip models are available in literature. [2][3][4]6 To compare the performance of the newly derived slip model with other models, we first plot in Fig. 2 number, while is density, T 0 is temperature, and R is the gas constant.…”

“…The 1.5 order slip model has a factor of 2/9 instead of 1/2 in front of the second order derivative term. 4 The coefficients of correction terms on the RHS of Eq. ͑1͒ depend on the length scale used to do Taylor series expansion of u͑z͒ in the normal direction z.…”

“…Widely applied slip models are the 1st order slip model ͑Maxwell slip model͒, 1,2 2nd order slip model, 3 1.5 order slip model, 4 and Fukui-Kaneko ͑FK͒ model. 5 The 1st order, 2nd order, and 1.5 order slip models are derived from kinetic theory by considering the momentum transfer rate of gas molecules impinging on the wall.…”

Dynamics of a small gap gas lubricated bearing with Navier slip boundary conditions

“…A gas lubricated inclined plane slider bearing has been examined by Burgdofer (1959) in the slip flow regime using a first-order slip model with the boundary slip velocity given at a mean free path distance from the wall. Hsia & Domoto (1983) incorporated a second-order slip model and Mitsuya (1993) developed a modified second-order slip model through additional physical considerations, referred to as a 1.5-order slip model. Slip effects in a journal bearing were investigated using a first-order model for compressible flow by Malik (1984) and for incompressible flow by Maureau (1997).…”