# Interdisciplinary Applied Mathematics

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ferent DSMC runs, 15 for nitrogen (diatomic molecules) and 5 for helium (monatomic molecules). The differences between the nitrogen and helium simulations are negligible, and thus this velocity scaling model is independent of the gas type. The linearized Boltzmann solution of (Ohwada et al., 1989a) for a monatomic gas is also shown in Figure 4.20 by triangles. This solution closely matches the DSMC predictions. Maxwell’s first-order boundary condition (b = 0) (shown by a solid line) erroneously predicts a uniform nondimensional velocity profile for large Knudsen number. The breakdown of slip flow theory based on the first-order slip boundary con-

dition is realized around Kn = 0.1 and Kn = 0.4 for the wall and the centerline velocity, respectively. This finding is consistent with the commonly accepted limits of the slip flow regime (Schaaf and Chambre, 1961). The prediction using b = —1 is shown by small dashed lines. The corresponding centerline velocity closely follows the DSMC results, while the slip velocity of the model with b = —1 deviates from DSMC in the intermediate range for 0.1 < Kn < 5. One possible reason for this is the effect of the Knudsen layer,    a    sublayer    that    is present    between    the    viscous    boundary    layer    and

the wall, with a thickness of approximately one mean free path. For small Kn flows    the    Knudsen    layer    is thin and    does    not    affect    the    velocity    slip

prediction too much. For very large Kn flows, the Knudsen layer covers the entire channel. However, for intermediate Kn values both the fully developed viscous flow (boundary layer) and the Knudsen layer exist in the channel. At this intermediate range, approximating the velocity profile to be parabolic neglects the Knudsen layers. For this reason, the model with b = —1 results in 10% error of the velocity slip at Kn = 1. However, the velocity distribution in the rest of the channel is described accurately for the entire flow regime.

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