# Interdisciplinary Applied Mathematics

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In Figure 3.5, we compare the normalized shear stress nxy predicted by the new model with the hard-sphere DSMC results, linearized Boltzmann solution of    (Sone    et    al.,    1990), and    4-moment    solution    of    (Gross    and Zier-

ing, 1958). All solutions uniformly approach to the free-molecular flow shear stress limit as Kn ^ ж. However, the asymptotic solution of Sone (derived for Kn ^ 0) systematically deviates from the linearized Boltzmann and DSMC results for Kn > 1.0. Differences between the current hard-sphere DSMC results and the linearized Boltzmann solution are almost invisible in the plot, and any discrepancy in the numerical values can be attributed to the statistical nature of the DSMC. The new model has a maximum

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FIGURE 3.5.    Variation    of    normalized    shear    stress    with    the    rescaled    Knudsen

number k.

deviation of 0.3% from the linearized Boltzmann solution, while the disagreement is about 1% for the solution of (Cercignani, 1963) and 0.7% for the empirical model in (Veijola and Turowski, 2001).

Results presented in Figure 3.5 were obtained for a Mach number M = 0.05 flow.    The    shear    stress    variation    as a    function    of    the    wall    speed    using

hard-sphere DSMC simulations results in deviations from this model. The maximum increase observed in the magnitude of the shear stress is about 6% as the Mach number is increased from 0.05 to 1.0 (Bahukudumbi et al., 2003).

In another study, (Lockerby and Reese, 2003) presented numerical solutions of Burnett equations for linear Couette flows using the no-slip, first-order slip (2.19) and second-order slip (2.42) boundary conditions. Their shear stress results for high-speed Couette flows at M = 3 are shown in Figure 3.6, where the Burnett predictions using various boundary conditions are compared with the DSMC computations of (Nanbu, 1983). It is clearly seen that the first-order slip condition is valid up to Kn = 0.1, while the second-order slip condition (2.42) is ineffective for this flow (see Remark 3 at the end of Section 2.3.3).

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