# Interdisciplinary Applied Mathematics

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Dc = у (£).    (3.39)

Using this value as a reference, we define

D* = Dc/D(Kn,e),    (3.40)

where D(Kn,e)    is    the    energy    dissipation    per    cycle    obtained    using    the

DSMC results as a function of в and Kn. Figure 3.21 displays the effects of Kn and в on D*. Owing to the smaller viscous dissipation due to the rarefaction effects, D* increases as Kn is increased. We also observe that D* decreases with increasing в in the slip flow regime. This is due to the enhanced shear stress with increased в. However, such influence of the Stokes number is drastically reduced with increasing Kn, since both the gas velocity and the shear stress on the wall are reduced by increasing Kn (see Figures 3.19 and 3.20).

Assuming that the viscous dissipation for free molecular flows can be obtained using eqution (3.38), it is possible to predict the behavior of D* in

FIGURE 3.21. Rarefaction and Stokes number effects on the normalized dissipation parameter D*.

the free-molecular flow regime using the analytical solution of the linearized Boltzmann equation. Since, Тсо^/т^ = ^ Kn, and uWt9 = 0.5uq, we obtain

D*

5n

16

Kn,

as Kn

00.

(3.41)

Therefore, D* continuously increases with increasing Kn. For example, Kn = 10 flow in Figure 3.21 reaches D* = 39.27 regardless of the Stokes number. The figure shows that the Stokes number dependence of D* is important until Kn « 1. For higher Kn values, D* can be predicted using equation (3.41).

###### 3.3.3 Summary

In this section, time-periodic Couette flows are studied systematically in the entire Knudsen regime and a wide range of Stokes numbers using the unsteady DSMC method. Simulations show that the quasi-steady flow conditions, which result in linear velocity distribution with equal velocity slip on the oscillating and stationary surfaces, diminish beyond a certain Stokes number. Although this limit also depends on Kn, в < 0.25 can be taken as the limit for quasi-steady flows. The empirical models presented in Section 3.3.1 are valid in this regime for Kn < 12, and they can be easily substituted in place of the DSMC simulations. For moderate Stokes number flows, the aforementioned empirical model is valid only in the slip flow regime.

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