Interdisciplinary Applied Mathematics

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data, we also plot the flowrate obtained by the continuum and the first-order slip models in Figure 4.22. The continuum model behaves like 1/Kn and gives the wrong variation, while the slip flow model yields flowrate values three times less than the DSMC calculations. Systematic investigations to test the accuracy of the mass flowrate formula for different values of the tangential momentum accommodation coefficient av indicate that the error in the prediction does not exceed 15%.

The corresponding free-molecular mass flowrate of the new model can be calculated using an asymptotic expansion of equation (4.24) in 1/ Kn as Kn ^ to. The result is independent of both the Knudsen number and the pressure ratio (since Kno ^ to, (П — 1)/(bKno) ^ 1), i.e.,


h2 / тг 12 2RT





Having obtained the mass flowrate, the corresponding pressure distribution along the channel can be obtained as


— 1 + 2(6 Ta)^-^Kno(P-l)

J у

+ 2(6 b + a) -—— Кпд loge

J v

P6Kn0 16KnQ

в (l — l) , (4.26)

where B is a constant such that P(0) = (Pj/Po) = П. Here we have defined

P(x) = P (x)/Po,

i.e., the pressure at a station x normalized with the exit pressure. The above equation provides an implicit relation for P. The pressure distribution for a first-order boundary condition is obtained explicitly by neglecting the second-order terms O(Kn2) in the above pressure equation.

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