Interdisciplinary Applied Mathematics

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The Reynolds equation can be nondimensionalized by normalizing the pressure using the ambient pressure p0, and the length scales in the x and y directions with channel lengths L and h0, respectively. This results in (Alexander et al., 1994)


d


dX



H3P-



tdP


dX



Л



d


dX



(PH),



(6.7)


where X = x/L, H = h/h0, P = p/p0 and


_ 6pU0L ~ Pohl


is the bearing number.


The Reynolds equation gives good results for low-speed flows. For highspeed flows, viscous heating may affect the isothermal flow approximation, and thus some deviations from the predictions of the Reynolds equation are expected. Due to the small length scales of MEMS devices, typical surface speeds correspond to low subsonic flow conditions, and therefore viscous dissipation is negligible. However, very high angular speeds present in certain MEMS devices can lead to near-sonic conditions. We conclude then that    the    analysis presented    here    will    be invalid    for    cases    with    significant


heat transfer and viscous dissipation effects.


Reynolds Equation in the Slip Flow Regime


For gas microflows the rarefaction effects can be incorporated into the Reynolds equation using the first-order velocity slip condition. The following equation has been obtained (Burgdorfer, 1959):









where Kn = X/h is the local Knudsen number. This relation is valid strictly in the    slip    flow regime    (Kn    < 0.1).    Since    for    air,    Xa;r    ~ 65 nm    under


ambient conditions, a gap height of ho = 0.6 p,m is the smallest clearance for which equation (6.8) can be used. Although this is a reasonably small clearance for many MEMS applications, it does not cover all applications. For example, the distance between the read/write head and the media in current computer hard disk drives is of order 50 nm. The design goal of the next generation hard disk drives is to reach a read/write density of 100 Gbit/in2, which requires 5 to 10 nm separation distance between the head and the media. Hence, the current hard drives already operate in the transition flow regime, while the next-generation drives will push this limit toward the free-molecular flow regime. Therefore, successful design of such microfluidic systems requires development of lubrication equations valid in a wide range of Knudsen number.

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