# Interdisciplinary Applied Mathematics

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sider highly rarefied gas flow with Kno = 4.167 and a high Л = 1264 value. Under these conditions, the minimum magnetic spacing is ho = 15 nm and the Mach number corresponding to the platter speeds is M = 0.5. The next-generation hard drives with ultrahigh storage densities would require slider bearings with a similar configuration as in case (c). Again, the solution of the generalized Reynolds equation is in good agreement with the DSMC data.

In summary, the generalized Reynolds equation for multidimensional problems is given by

(6.15)

Л • У {PH) + s-

V • [(QP(D0PH)VP — QT(D0PH)PVTW) PH3] d(PH)

dt8

where Л is the bearing number vector defined by the lateral and stream-wise components of the slider plate velocity, and t8 is the nondimensional time normalized with a characteristic frequency    in case of time-harmonic

excitation of the slider plate. The squeeze number S is given by

_ 12 iiuJqL? ph20

The generalized Reynolds equation is analogous to the Navier-Stokes based model given    in equation    (6.2),    and    we    presented    several    different    ways    to

model the Qp term in equation (6.15).

The Reynolds equation in the free-molecular flow regime employs the free-molecular flow solution between two parallel plates, which increases logarithmically as the inverse Knudsen number D apporaches 0 (Fukui and Kaneko, 1988):

Qp(D) —> ( —1/л/7г) log-D.

We have shown in Section 4.2 that the volumetric flowrate obtained by the linearized Boltzmann equations in microducts reaches finite asymptotic values in the free-molecular flow regime; see also (Sone and Hasegawa, 1987). The distinction between the two-dimensional channel and rectangular duct flows can be seen in Figure 4.30. This striking difference brings up the question of the finite dimension effects on the slider problem:

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