Interdisciplinary Applied Mathematics

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For high Stokes number flows, we observed formation of “bounded Stokes layers,” where the stationary wall no longer interacts with the flow. In the slip flow regime, this results in the classical Stokes’s second problem with velocity slip. Once again, the empirical model in Section 3.3.1 is valid in this regime, where the wave speed is constant outside the Knudsen layer, and the velocity amplitude decays exponentially as a function of the distance from the oscillating surface. However, there are small deviations from this behavior within the Knudsen layer. Such deviations are captured by the DSMC, but they cannot be modeled using continuum-based approaches. In the transition and free-molecular flow regimes we observed that the solution decay is not exactly exponential and the wave speed is no longer constant. These are interesting deviations, which are also validated using the analytical solution of the linearized collisionless Boltzmann equation in the free-molecular flow limit. In all simulations, the results have consistently shown that the slip velocity and wave propagation speed increase with increased Kn and в.


An interesting behavior is observed when Kn is increased while the Stokes number is fixed. For such cases, the slip velocity increases, and a bounded layer with a finite penetration depth is formed after a certain value of Kn. This is named the “bounded rarefaction layer” (Park et al., 2004). The penetration depth for this layer is a function of both Kn and в, and it becomes a new length scale in the problem. For such cases, it is necessary to redefine the Knudsen number based on the penetration depth, rather than the separation distance between the two plates. However, without a priori knowledge of the penetration depth it is not possible to predetermine Kn in the simulations. In order to remain consistent, we kept the characteristic length scale of the problem as the plate separation distance. However, the reader can use Figure 3.18 to estimate the actual Knudsen number based on the penetration depth. Due to this switch in the length scales, we observed that the shear stress on the oscillating wall reaches the asymptotic free-molecular limit at earlier Kn values. Solution of the linearized collisionless Boltzmann equation in the free-molecular flow limit indicates that the shear stress and the slip velocity amplitude for oscillatory Couette flows are identical to those of the steady plane Couette flows. This interesting finding is also confirmed by DSMC results (Park et al., 2004).

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