# Interdisciplinary Applied Mathematics

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a Kn +2b

2h aKn2 + cKn+b

(1 + 2C1 Kn)

(3.27)

cosh(VJ/3) + л/j/3CmKn sinh(V7/3)

(1 + je2Cm Kn2) sinh

Kn cosh

exp(jwt)

Figure 3.9 shows the shear stress time history at the oscillating wall, normalized with its continuum value. Predictions of the current model and DSMC results are also presented for Kn = 2.5 flow at в = 0-1 and 0.25 conditions. There is a difference of very small magnitude in the shear stress between the    в =    0.1    and 0.25    cases    at    Kn    =    2.5.    The    above model    accu

rately predicts the shear stress magnitude on the oscillating wall, and it also matches the DSMC results for a wide range of Kn values under quasisteady flow conditions (в < 0.25).

In this section we present time-periodic oscillatory Couette flows in the transition and free-molecular flow regimes at relatively large Stokes numbers. Therefore, the quasi-steady approximation is no longer valid, and “bounded Stokes layer”-type behaviors are observed. We also discuss the physical aspects, such as the penetration depth, wall shear stress, and slip

Normalized velocity amplitude

FIGURE 3.10. Effect of в in transition flow regime.

velocity variations, as well as the energy dissipation and damping characteristics of oscillatory Couette flows. Computational details about the unsteady DSMC method can be found in Section 15.1.2.

Figure 3.10 shows the effect of Stokes number on the velocity amplitude in the transition flow regime. At fixed Kn, the slip velocity increases with increasing в. For Kn = 1.0, it can be seen that beyond в = 0.25 the quasisteady approximation breaks down. We observe a “bounded Stokes layer” type of    behavior    for    в > 5 in    both    figures.    Comparing    the    Kn    =    1.0    and

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