# Interdisciplinary Applied Mathematics

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FIGURE 4.18.    Variation    of    normalized    flowrate    in a    channel    as    a    function    of

Knudsen number. Comparisons are made between DSMC results and various solutions of the linearized Boltzmann equation.

density is observed. Accordingly, the free-molecular mass flowrate in pipes of diameter 2a and length L is (Kennard, 1938)

MlFM

4 3 ДР

-a

3 L

ГъМ

V RT’

(4.16)

Deviation from this behavior is expected for finite-length pipes (i.e., a ^ L ^ X) by a factor (1 — Ca/L) (up to first order in a/L) due to end effects (Polard and Present, 1948), where C is a constant. However, for free-molecular flow in two-dimensional very long channels where h ^ X ^ L, the flowrate increases asymptotically to a value proportional to

(l/7r)hoge(Kn)

in the limit Kn ^ to (Cercignani, 1963; Huang and Stoy, 1966). This logarithmic behavior is attributed to the degenerate geometry of the twodimensional channel (Kogan, 1969). For finite-length two-dimensional channels, the flowrate tends toward a finite limit (see the discussion of D.R. Willis at the end of (Cercignani, 1963)). For ducts, the flowrate tends toward a finite limit, resembling the pipe flow behavior. This has been documented in the experiments of (Gaede, 1913) and verified by linearized Boltzmann solutions (Sone and Hasegawa, 1987).

The variation of flowrate in a channel as obtained by DSMC simulations in the transition and early free-molecular flow regimes is shown in Figure

4.18. The volumetric flowrate data are presented at the average Knudsen number in the channel (corresponding to the mean pressure P between the inlet and outlet), and it is nondimensionalized in the form

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