Interdisciplinary Applied Mathematics

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up (y)



h2 dp 2po dx



(1 + aKn)



Kn


1 + Kn +



У


h



У


h



2i



(6.14)


This results in the desired volumetric flowrate in equation (6.13). Figure 6.4 shows the normalized velocity profiles (£/*) for к = л/7г/2 Kn = 0.1,1.0,


and 10.0 flows, where the velocity profiles are normalized by yy-yrgyj • This way of normalization preserves both the magnitude and shape of the velocity profile,    as    shown    in the    figure.    Consistent    with    the    approach    of


Section 4.2.2, the velocity profiles uniformly match the linearized Boltzmann solutions obtained in (Ohwada et al., 1989a). This empirical unified model also predicts the shear stress, as demonstrated in (Bahukudumbi and Beskok, 2003).


Figure 6.5 compares the Poiseuille flow-rate coefficient (Qp) calculated using the linearized Boltzmann equation, continuum, first- and second-order slip flow models. The model developed in (Bahukudumbi and Beskok, 2003) is shown by Qcurr. We must note that the free-molecular flow solution between the two parallel plates results in a logarithmic increase in the flowrate. Although this behavior is questionable, all the aforementioned models utilized the same flowrate database, which is the current industry standard for air-bearing design. In the following, we present the lubrication results for several different slider configurations.


In Figure 6.6 (a), we present the solution of the generalized Reynolds equation for a slider bearing with minimum magnetic spacing of 50 nm at ambient conditions, and compare the results with the DSMC simulations in (Alexander et al., 1994), which was obtained for a slider bearing of length L = 5    pm    and a    slider speed    U0    =    25    m/s,    resulting in bearing    number

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