Interdisciplinary Applied Mathematics

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Generalized Reynolds Equation


A generalized gas film lubrication analysis can be obtained using the Boltzmann equation, which is valid in the entire Knudsen regime. Based on this idea, Fukui and Kaneko analyzed gas film lubrication using the linearized Boltzmann equations (Fukui and Kaneko, 1988; Fukui and Kaneko, 1990), and presented their results as a function of the local inverse Knudsen number





where D0 = p0h0/(рл/PRTf) is the characteristic inverse Knudsen number defined using the minimum film thickness (ho) and the ambient pressure (po). The primary assumptions in linearizing the Boltzmann equations are (see also Section 15.4):


1. Small flow velocities and thermal fluctuations; hence, local isotropic equilibrium conditions are assumed.


2.    Small lubrication film thickness and small cross-flow velocity components.


Based on these two assumptions, the following nondimensional Reynolds equation is obtained (Fukui and Kaneko, 1988):




where Qp(D0PH) and Qp(D0PH) show the relative volumetric flowrate of pressure-driven and thermal creep channel flows, normalized by the no-slip Poiseuille flowrate. The last term in equation (6.10) corresponds to the linear Couette flow volumetric flowrate, which is independent of the Knudsen number. The thermal creep flow Qp(D0PH) is present under tangential temperature gradients, as shown in Section 5.1. The term in equation (6.10) is the nondimensional temperature variation along the surface. Since the thermal creep effects are present only under surface temperature gradients, the term Qp(D0PH) is zero for isothermal surfaces.


DSMC studies of slider bearing flows were presented in (Alexander et al., 1994). For low subsonic slider motion, good agreement between the DSMC and the predictions of equation (6.10) is shown. One peculiar difference between the results of the Reynolds equation and DSMC is in the loadcarrying capacity of the slider bearing for high-speed flows. For near-sonic conditions, calculation of pressure on the slider surface using the ideal gas law agreed well with the Reynolds-equation-based calculations. However, the load capacity calculated from the time-averaged change in the momentum of particles striking the wall predicted 20% lower load capacity. This difference was attributed to the increased nonequilibrium effects for high-speed rarefied gas flows (Alexander et al., 1994).

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