Interdisciplinary Applied Mathematics

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C. This velocity slip coefficient (Ci) is in essence a correction term applied to extend the validity of the original first-order slip boundary condition. In this model, the values of в were determined as во = 1.2977, в1 = 0.7185, в2 =    —1.1749,    and    вз    =    0.5864,    using a    least square    fit    to    the    linearized

Boltzmann solutions presented in (Sone et al., 1990).

Figure 3.3 also presents the predictions obtained by the new model (solid lines). Unlike the first-order model, the new model accurately matches the velocity profile in the bulk flow region for a wide range of Knudsen numbers. It must be noted that the new model fails to predict the velocity distribution in the Knudsen layer. This is expected, since the model is based on the Navier-Stokes equations, and the velocity profiles cannot be different from a linear    profile    for    Couette    flows.    For    Kn    <    0.1,    presence of the    Knudsen

layers can be neglected by extrapolating the bulk flow toward the wall. When Kn > 1, the Knudsen layer occupies the entire channel.

The results presented in Figure 3.3 are obtained for M = 0.05, and they exhibit rarefaction effects alone. Although the compressibility and viscous heating effects are insignificant for low-speed flows, they become important for nonisothermal as well as high-speed flows, where viscous heating creates temperature variations in the domain, as discussed in Section 3.1. Figure 3.4 shows the compressibility effects on Couette flow velocity distribution. The results are obtained using hard-sphere DSMC at various wall speeds. The new velocity model and the linearized Boltzmann solution in (Sone et al., 1990); exhibit significant deviations from the hard-sphere DSMC simulation results when M > 0.3. In order to extend the velocity model for M > 0.3, a slip boundary condition that couples the velocity and temperature fields is necessary, as presented in (Schamberg, 1947). For the range of simulations performed here, the maximum deviation of the new velocity model from the DSMC results is around 7% when M = 0.5, and the deviation is 14% when M =1.0.

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