Interdisciplinary Applied Mathematics

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Brownian motion.

1.2.2 Mixed Flow Regimes

In the examples of Section 1.1, the gas flow cannot be modeled based on the continuum hypothesis. The mean free path of air, which at standard atmospheric conditions is about 65 nm, is comparable to the characteristic geometric scale, and therefore microscopic effects are important. For example, in the case of computer hard drives, the load capacity predicted by the continuum Reynolds equations without slip is in error by more than 30% (Fukui and Kaneko, 1988; Alexander et al., 1994). This deviation of the state of    the    gas    from    continuum    is measured    by    the    Knudsen    number

Kn.    For    the    micromotor, using a length    scale    of L =    3pm    (the    gap    be

tween rotor and the base) and assuming that the operation conditions are atmospheric, we    obtain    the    value    Kn    =    0.022. For    the    magnetic    disk drive

(slider bearing) the Knudsen number is Kn = 1.3, and in ultralow clearances corresponding to increased recording capacity, the Knudsen number is well above unity. Also, in other capillary flows, such as in helium leak


FIGURE 1.12. Typical MEMS and nanotechnology applications in standard atmospheric conditions span the entire Knudsen regime (Continuum, slip, transition, and free-molecular flow). Here h denotes a characteristic length scale for the microflow.

detection microdevices and mass spectrometers, the Knudsen number may achieve values up to 200 (Tison, 1993).

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