Interdisciplinary Applied Mathematics

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Free-Molecular Regime

In this section, we follow (Park et al., 2004), to derive the velocity distribution and shear stress for oscillatory Couette flows in the free-molecular flow limit (Kn > 10). Our objectives are to provide a theoretical solution to compare and validate the DSMC results, and enhance our understanding of flow physics in this regime. As the Knudsen number is increased, inter-

FIGURE 3.12. Details of the flow dynamics for Kn = 1.0 and в = 2.5.

molecular collisions become negligible compared to the molecule/surface collisions. Therefore, the flow can be modeled using the collisionless Boltzmann equation

d f d f    .    .

aGui = 0‘    (328)

where f is the velocity distribution function and n is the cross-flow (y) component of the molecular velocity. Due to the simple geometry, f changes only in the cross-flow direction, and there are no external force fields. We assume that both top and bottom walls are fully diffusive, and a sinusoidal excitation is exerted on the top wall (y = L). The boundary conditions for equation (3.28) are


(a) Dynamic response of medium

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