Interdisciplinary Applied Mathematics

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FIGURE 3.17. Details of the flow dynamics for Kn = 10 and в = 2.5.

Kn, which is constant irrespective of в. If we consider changing the characteristic length    scale    from    the    distance    between    the two    plates    (L)    to

the penetration    length    (S),    we    would    observe    that    the    effective    Knudsen

numbers for such cases are indeed in the free-molecular flow regime. For example, for the Kn = 2.5,в = 7.5 case, the penetration depth S is equal to 0.15L (see    Figure    3.18).    Hence,    the    effective    Knudsen    number    for    this

case is about Kn’ « 17, which corresponds to a free-molecular regime.

Slip Velocity Variation

Here we present the slip velocity variation as a function of the Knudsen and Stokes numbers. We simply present the slip amplitude variations, since the flow is time-periodic and there are no phase differences between the wall oscillation frequency and the slip velocity response. Figure 3.20 shows Kn and в effects on the slip amount on the moving wall. We had to determine the slip velocity by extrapolation of the DSMC cell-center velocities onto the oscillating wall. For low Knudsen flows, slip velocity increases with increasing в. This is due to the increased velocity gradients near the wall. For a    fixed    в,    the    slip    amount    increases    by    increasing Kn.    Noticeably,    all


FIGURE 3.18. Effects of Kn and в on penetration depth.


FIGURE 3.19. Effects of Kn and в on wall shear stress.

slip values reach the limit of 0.5w0 for high Kn values, as theoretically predicted in Section 3.3.2.

Energy Dissipation

Characterization of energy dissipation in laterally oscillating microstructures is important for designing several microsystem components, such as the microcomb drive mechanisms. The MEMS community utilizes the “quality (Q)    factor”    concept,    which is    the    ratio of the    total    energy    of    the

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