Interdisciplinary Applied Mathematics

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Basic Characteristics

In the following we present results for the penetration depth, shear stress, slip velocity, and viscous dissipation as functions of the Knudsen and Stokes numbers.

Penetration Depth Variation

It is important to note that the bounded Stokes and rarefaction layers observed in    the    results create    a    new    length    scale    in the problem.    This    new

length scale is related to the thickness of the Stokes/rarefaction layers, and becomes particularly important for high values of Kn or /3. The Stokes layer thickness (6 « Jv/ui), also referred to as the “penetration depth,” is defined as the distance from the moving wall where the velocity amplitude decays to 1% of its excitation value = 0.01). Most of the flow is confined within this layer, and the moving wall no longer interacts with the stationary wall. For these cases, the characteristic length scale of the problem should be based on the penetration depth S, rather than the separation distance between the two plates. This would require redefinition of the nondimensional parameters Kn and в, based on the penetration depth

(Kn’ = ^,/3′ = J^p-)- However, there are no relations for variation of 6 as a function of Kn and в. Hence, a priori estimation of the penetration

FIGURE 3.15. Velocity amplitudes for the free-molecular flow regime.

depth is not possible. For the sake of consistency, Kn and в are defined using the plate separation distance throughout this work. Hence, no switch is made in the characteristic length scale. However, change in the characteristic length scale has physical implications. For example, the actual Knudsen number for these cases can be found by Kn’ = Kn L/S. Figure 3.18 shows variation of the normalized penetration depth (S/L) with Kn and в. For the cases not shown in this figure, the solution does not attenuate enough to observe a “bounded layer.” The penetration depth decreases with increasing в, as expected. The penetration depth approaches different values in the free-molecular limit for different Stokes numbers. For fixed в, the penetration depth decreases by increasing Kn, reflecting the “bounded rarefaction layer” concept introduced in (Park et al., 2004). It can be seen from Figure 3.18 that j- oc ^ for a given /3. This figure also clarifies the need for    redefinition    of    the    characteristic    length    scale    for    high в and    Kn

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