Interdisciplinary Applied Mathematics

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Inherent in these new technologies is the need to develop the fundamental science and engineering of small devices. Microdevices tend to behave differently from the objects we are used to handling in our daily life (Gad-el-Hak, 1999; Ho and Tai, 1998). The inertial forces, for example, tend to be quite small, and surface effects tend to dominate the behavior of these small systems. Friction, electrostatic forces, and viscous effects due to the surrounding air or liquid become increasingly important as the devices become smaller. In general, properties (p) that are a function of the area of interaction (A) decrease more slowly than properties that depend on the volume (V), as expressed by the “square-cube” law:


Pi(A)    L2    1

P2(V) * L’

where L is the characteristic dimension of the microdevice; a typical order of    magnitude    is    106    m2/m3.    Surface    tension    effects    are dominant    at

these scales, and micropumps and microvalves have been fabricated taking advantage of this principle (Evans et al., 1997).

Typical early applications can be found in the micro- and nanoscale design of computer components such as the Winchester-type hard disk drive mechanism, where the read/write head floats 50 nm above the surface of the spinning platter (Tagawa, 1993). The head and platter together with the air layer in between form a slider bearing. The typical operating conditions correspond to low values of both Reynolds and Mach number, e.g., less than 0.6 and 0.3, respectively. The corresponding Knudsen number, which expresses the relative size of the mean free path to the size of the microflow domain, is relatively large. It is expected to increase further for the next generation of devices, since the smaller the gap between the spinning platter and the read/write head, the greater the recording capacity.

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