Interdisciplinary Applied Mathematics

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FIGURE 4.23. Deviations from linear pressure drop for nitrogen channel flow (П = 2.28, Kn0 = 0.2). Comparison of pFlow with the DSMC predictions and the new slip model (b = —1, a = 2.2).

(Loyalka and Hamoodi, 1990; Sone and Hasegawa, 1987), for pipe and duct flows, respectively.

Pipe Flow

First, we derive a similar model for the pipe flow. Assuming a parabolic velocity profile with the slip amount given by (2.43), it is possible to obtain the following equation for the nondimensionalized velocity scaling in a pipe flow:

where a is the pipe radius, and the Knudsen number is defined as Kn = X/a. We compare the nondimensional velocity scaling with the linearized Boltzmann solution of (Loyalka and Hamoodi, 1990) in Figure 4.24. The general slip coefficient is found to be b = —1 as before, consistent with the velocity profiles given in (Loyalka and Hamoodi, 1990), for cases Kn = 0.1, Kn = 1.0, and Kn = 10. The velocity profiles predicted with the first-order slip model and the general slip model (2.43) are also shown in Figure 4.24. It is seen that the first-order model gives erroneous velocity distributions in the transitional and free-molecular flow regimes. For example, for the

Kn = 10 case an almost uniform velocity distribution is predicted. However, the model of equation (2.43) predicts accurately the velocity distribution in most of the pipe with a small error in the velocity slip.

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