# Interdisciplinary Applied Mathematics

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Normalized velocity amplitude

FIGURE 3.7. Velocity amplitudes for (a) quasi-steady and (b) low Kn cases.

that the DSMC results accurately capture the slip-flow limit even for large Stokes numbers. For high Stokes number flows there are deviations from the linear velocity profile, and the velocity amplitude distribution loses its symmetry beyond в =1-0 for Kn = 0.1 flow. High Stokes number cases (в > 5) result in bounded Stokes layers, where the flow is confined to a near wall region. Significant velocity slip can be observed with increased Stokes number beyond the quasi-steady flow limit, while the slip velocity for quasi-steady flows is independent of the Stokes number, as can be deduced by comparing the в < 1.0 cases with the в > 5 cases in Figure 3.7(b).

Figure 3.8 shows the dynamic response characteristics for Kn = 0.1 and в = 5.0 flow. Snapshots of velocity distribution at different times are shown in    Figure    3.8(a).    With    the    exception    of    the    velocity    slip, the    dynamics

are similar to those of no-slip continuum flows. The velocity distribution predicted by the extended slip model and the DSMC simulations are in good agreement, despite a slight phase difference between the DSMC and the model. In the context of this work,

• the phase angle is defined as the fraction of the time period by which the solution felt at any streamwise position lags or leads the reference velocity solution imposed on the oscillating wall.

A general representation of the velocity solution at an arbitrary location Y is given by

u(Y,t) = u0 sin(wt + Ф),    (3.21)

where Ф is the phase angle. Expanding and rearranging equation (3.21),

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