# Interdisciplinary Applied Mathematics

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We also examine the differences in pressure between DSMC and p,Flow predictions by plotting the curvature in the pressure distribution, i.e., the deviation from the corresponding linear pressure drop (P — Pic)/(Po)) in Figure 4.23, where PC denotes the pressure of corresponding incompressible flow. The p,Flow simulation using the slip boundary condition (2.26) predicts larger curvature in pressure distribution than the DSMC results. The pressure distribution obtained by the first-order boundary condition is shown by dashed lines and lies between the p,Flow and DSMC results. Our second-order slip model without the correction of the rarefaction coefficient (Cr = 1 + aKn) gives identical results to p,Flow predictions. The corresponding continuum (no-slip) pressure distribution is also given in the figure. The reduction in the curvature of the pressure distribution with rarefaction is clearly demonstrated. Finally, the model, including the rarefaction coefficient Cr (Kn) shown by solid lines, gives results closest to the DSMC solution. This demonstrates the ability of the new model in predicting the pressure distribution for channel flows. At higher Knudsen number, the curvature in the pressure distribution is much smaller, with linear pressure drop observed as Kn ^ to.

The asymptotic value of flowrate for pipe and duct flows at high Knudsen number is constant, and this offers the possibility of obtaining a model for the rarefaction coefficient Cr (Kn) and in particular the coefficient a. The objective is to construct a unified expression for a(Kn) that represents the transition of a from zero in the slip flow regime to its asymptotic constant value    in    the    free-molecular    flow regime.    We    will    verify    the    velocity    and

new flowrate models using available experimental data for pipes (S. Tison, NIST, private communications) as well as numerical results obtained in

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