Interdisciplinary Applied Mathematics

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The corresponding mass flowrate is computed from


M = pf U(y)dy,


J 0


where p = P/RT, assuming we can still treat the rarefied gas as an ideal gas. Expressing the Knudsen number at a location x as a function of the local pressure, i.e., Kn = KnoPo/P, where the subscript “o” refers to outlet conditions, we obtain


h6P 2    2 — n


M = 24^ВТЬ[{П21} + 12^^(КПо(П — !) — Kn>gen)], (4.6)


where we have defined П = P[/Po as the pressure ratio between inlet and outlet. The corresponding flowrate without rarefaction effects is given by


h6P 2


M“ = 2Уа#1(п211    (4J>


Therefore, the increase in mass flowrate due to rarefaction can be expressed as


M



Mn



1 + 12



2 — nv



nv



KnQ


П+1



12



2    nv



Kn2



v



i°gen П5 — 1′



(4.8)


It is seen from this formula that the effect of the second-order correction is to reduce the increase in mass flowrate due to first-order slip. This is in disagreement with the experimental data, since in the transition flow regime, the flow rate increases faster than predictions of the first-order slip theory. This inconsistency will be addressed in Sections 4.1.3 and 4.2.2.


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


where B is a constant such that P(0) = П. Here we have defined P = P/Po, i . e    the    pressure    at    a    station x    normalized    with    the    outlet pressure.    The

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