# Interdisciplinary Applied Mathematics

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Turning now to micro-electro-mechanical systems (MEMS), one of the first microfabricated products is a polysilicon, surface micromachined side-driven motor; its fabrication, operation, and performance have been studied extensively in (Mehregany et al., 1990; Tai et al., 1989; Trimmer, 1997). A diagram of    such    a motor    is    shown    in the sketch    of    Figure    1.1    along    with

the characteristic dimensions. It creates a variable capacitance by means of salient poles distributed along the periphery; a typical design may employ 12 stator poles and eight rotor poles. During operation, the shield (lower surface), bearing, and rotor are electrically grounded, and only one stator phase is excited at a given time. The motive torque has been determined through step transient measurements and it is of order 10 PN • m. The axial

STATOR

ROTOR

60 |i m

13 ц m ->

1.5 ц m

0.3 ц m

0.

2 ц m

3.0 ц m

BEARING    BUSHING

FIGURE 1.1. Diagram of an electrostatically side-driven micromotor with typical dimensions. The motor consists of a set of stator poles and rotor poles which are excited sequentially, creating an electrostatic motive torque.

force is of order 10~7 N, which is much larger than the typical weight of the motor, of order 10~10 N. Typical operating conditions for an angular speed of ш = 5000 rad/sec show that the Reynolds number Re « 1 based on the gap between the base and the rotor (3 p,m) and that the Mach number is less than 0.1 based on the rotor tip speed (rotor radius is 60 pm). Under macroscale continuum conditions it is possible to have creeping flows that result in small Reynolds number Re and Mach number M. However, in the case of microflows, the Reynolds number Re is small due to the small length scales of the microdevice rather than very small velocities. Therefore, higher Mach numbers M could be achieved in microflows compared to the creeping (i.e., very slow) continuum flows. Results from a steady-state axisymmetric analysis reported in    (Omar    et    al.,    1992)    showed    that    75%    of the    viscous

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