# Interdisciplinary Applied Mathematics

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value proportional to loge(L/h) (see Figure 4.18). Therefore, for finite-length two-dimensional channel flows, the coefficient a should smoothly vary from zero in the slip flow regime to an appropriate constant value in the free-molecular flow regime. It is difficult to verify this variation using the DSMC simulations due to the statistical scatter of the DSMC method. However, progress can be made if we assume an approximate value of a, which we denote by a, and determine the value of it for a specific gas in a finite-length channel. Such an analysis has been performed for nitrogen flow in a channel of length to height ratio L/h = 20, resulting in a = 2.2.

Using this approximate value we compare the predictions of the model for mass flowrate versus DSMC results. By integrating equation (4.22) from the inlet to    the    outlet    of the channel,    we    derive    an expression    for    the    mass

flowrate per unit width:

Ml

h3Pa AP 2ffi0RT0 ~L~

(П + 1) + 2[61— cr] KnQ

J у

+ 12

2jv b + a av П — 1

##### Kn2 loge (

where the subscript o refers to the outlet conditions, П = P;/Po (inlet-to-outlet pressure ratio), and L is the channel length. The comparison of the corrected model with the DSMC data is given in Figure 4.22. The model predicts Knudsen’s minimum obtained by the DSMC calculations quite accurately at Kn « 1.0. Consistent with the DSMC solutions, the model predicts a flowrate independent of Kn for the free-molecular flow limit. However, this constant flowrate for larger Kn is slightly (13%) lower than the DSMC    predictions.    For    comparisons    with    the    model    and    the    DSMC

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