Interdisciplinary Applied Mathematics

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Kn = 10. We also included the corresponding linearized Boltzmann solutions obtained in (Ohwada et al., 1989a). It is seen that the DSMC velocity distribution and the linearized Boltzmann solutions agree quite well. We can now use equation (4.18) and compare with the DSMC data by varying the parameter b, which for b = 0 corresponds to Maxwell’s first-order and for b = —1 to the second-order boundary condition in the slip regime only. Here we find that for b = —1 equation (4.18) results in an accurate model

FIGURE 4.20. Velocity scaling at wall and centerline of the channels for slip and transition flows. The linearized Boltzmann solution of (Ohwada et al., 1989a) is shown by triangles, and the DSMC simulations are shown by points. Theoretical predictions of    velocity    scaling    for    different    values    of    b,    and Hsia    and Domoto’s

second-order slip (large dashed line) boundary condition are also shown (Hsia and Domoto, 1983).

of the velocity distribution for a wide range of Knudsen number. From the figure, it    is    clear that    the    velocity    slip    is    slightly    overestimated    with    the

proposed model for the Kn = 1 case. To obtain a better velocity slip, we varied the value of the parameter b by imposing, for example, b = —1.8 for the Kn = 1 case. Although a better agreement is achieved for the velocity slip, the accuracy of the model in the rest of the channel is destroyed.

In Figure 4.20 we show the nondimensionalized velocity distribution along the centerline and along the wall of the channels for the entire Knudsen number regime considered here, i.e., 0.01 < Kn < 30. We included in the    plot    data    for    the    velocity    slip    and centerline    velocity    from    20    dif

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