# Interdisciplinary Applied Mathematics

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The calculated average value is a = 0.99912 (with standard deviation of arms = 0.01603). Thus, we can use these DSMC results to examine the validity of the slip boundary conditions given in equation (2.28) with av = 1. To this end, we examined several forms where slip information CX away from the surface is utilized (see equation (2.26)). We have determined that the best agreement with the DSMC results is achieved by obtaining the

FIGURE 4.8. Comparisons of Navier-Stokes and DSMC predictions. Left: Density variation along the channel. Simulation conditions are for П = 2.28 and M0 = 0.19. Right: Pressure variation along the channel for pressure ratio П = 2.28. The solid line corresponding to pFlow Kn0 = 0.2 prediction is indistinguishable from the DSMC results.

slip information A away from the surface (C = 1).

The density distribution (normalized with the inlet density) along the channel for the case П = 2.28 predicted by DSMC and p,Flow is shown in Figure 4.8 (left). Good agreement between the microscopic and macroscopic simulations is achieved. The corresponding centerline pressure distribution along the channel is plotted in Figure 4.8 (right). The pressure distribution is nonlinear, as expected for a compressible channel flow. As has been shown earlier, rarefaction and compressibility are competing effects in determining the curvature of the pressure distribution. While compressibility makes the curvature more pronounced, rarefaction makes the variation more linear. This is also verified in Figure 4.8 (right) by comparison of slip-based Navier-Stokes results with the no-slip results.

Velocity profiles for П = 2.28, normalized with the reference inlet velocity at three different x/L locations, are shown in Figure 4.9. The DSMC and p,Flow results are in good agreement. The velocity profiles obtained by both methods are parabolic. The velocity slip variation along the channel wall is shown in Figure 4.10. Both the DSMC method and p,Flow predict similar magnitudes.

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