Interdisciplinary Applied Mathematics

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Kn = 2.5 cases, we find that the slip velocity increases with increasing Kn at    constant в.    For    a    fixed    Kn,    the    Stokes    layer    thickness decreases with


increasing в, as expected.


The effect    of Kn    on    the    velocity    amplitude    for    moderate    Stokes    number


conditions is shown in Figure 3.11. It can be seen that the slip velocity magnitude on the oscillating wall increases with increasing Kn for a fixed Stokes    number.    For    в =    1.0,    quasi-steady    flow behavior    is    observed    for


Kn = 0.1, since the velocity amplitude distribution is linear and passes through (y/L,u/uo) = (0.5,0.5). Hence, the quasi-steady flow approximation also depends on Kn, as can be deduced by comparison of Figures 3.10 and 3.11. The most interesting observation in Figure 3.11 is the emergence of a “bounded    rarefaction    layer”    with    increasing    Kn.    By    this    term    we    em


phasize that this behavior is due to the rarefaction effects alone, and not to the influence of the Stokes number, which is kept constant. Transition to this “bounded rarefaction layer” occurs even at moderate Stokes number flows (see Figure 3.11(a)) by increasing Kn. However, these effects are more pronounced when the Stokes number increases, as can be deduced by comparing Figures 3.11 (a) and (b).


Normalized velocity amplitude


0

y/L



E


cc


a-


FIGURE 3.11. Effect of Knudsen number Kn for moderate values of the Stokes number в.


Figures 3.12, 3.13, and 3.14 show the dynamic response of the system for moderate and high Stokes number flows in the transition flow regime (Kn = 1.0,в    = 2.5;    Kn    =    1.0,в =    5.0,    and    Kn    =    5.0,в =    2.5).    Here,


we will not present detailed discussions of the dynamic system response for the individual cases, since the behavior is qualitatively similar to that of Figure 3.8. Comparing Figures 3.12(a) and 3.13(a), we observe that a more pronounced Stokes layer forms by increasing the Stokes number. Alternatively, comparing Figures 3.12(a) and 3.14(a), we observe a more pronounced “bounded rarefaction layer” when Kn is increased. In all three cases, reduced velocity amplitudes and different phase angles are observed at different streamwise locations. Note that while the phase angle reaches 210° at y/L = 0 in Figure 3.12(c), the same value is reached at y/L « 0.75 in Figure 3.13(c), and at y/L « 0.8 in Figure 3.14(c). This indicates that the phase speed, defined in equation (3.25), increases with increasing в and Kn. It is also worthwhile to compare the level of statistical scatter between these three results. Statistical scatter in Figure 3.12(d) is insignificant, since the normalized velocity amplitude does not drop below 1% of the solution. However, with increasing в and Kn, the normalized velocity amplitude drops below 1% outside the “bounded layers,” and the statistical scatter becomes important, as can be observed in Figures 3.13(d) and 3.14(d).

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