Interdisciplinary Applied Mathematics

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we obtain


u(Y, t) = u0 [sin (ш t) cos Ф + cos (ш t) sin Ф]    (3.22)


= A (Y) sin (ш t) + B (Y) cos (ш t),


where


A (Y)= u0 sin Ф, B (Y) = u0 cos Ф.    (3.23)


The phase angle can then be determined from equation (3.23) as






where the coefficients A and B are determined from the DSMC results, using a    x-square fit.    In    addition,    a    theoretical    expression    for    the    phase


angle of the extended slip model of (Bahukudumbi and Beskok, 2003) is also presented in (Park et al., 2004).


Figure 3.8(b) shows the velocity time history at various streamwise locations (y/L) in the flow domain. The velocity solutions at different y/L locations exhibit reduced amplitudes and different phase angles. Note that the peak values of the velocity solution in Figure 3.8(b) correspond to the velocity amplitudes in Figure 3.7. In Figure 3.8(c), the phase angle predicted by the extended slip model and DSMC show similar trends. However, the initial deviation at y/L > 0.9, due to the Knudsen layer effects, offsets the DSMC results from the model solution. The phase angle variation is essentially linear in most of the domain, except within the Knudsen layers near the walls. The wave propagation speed C (phase speed) can be computed from the phase angle variation using the relation






Consequently, the phase speed, computed using the above definition, is constant in the region of linearly varying phase angle. The extended slip model predicts a wave propagation speed of C = 1.770, which is in good agreement with the corresponding DSMC prediction of C = 1.790. The classical Stokes second problem without the stationary wall also predicts a very similar wave propagation speed, C = 1.777. The phase speed is not constant near the walls due to the presence of Knudsen layers. In addition, the phase speed decays near the stationary wall, due to the interference between the incident and reflected solutions. The normalized velocity amplitude, plotted in log-scale in Figure 3.8(d), shows exponential decay in the amplitude with small alterations when y/L < 0.1, due to the presence of the stationary wall. It can be seen that the slip model results and DSMC solution are consistent.

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