Interdisciplinary Applied Mathematics

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Ж = 1 + 4Kn.


Mc


If we used a constant a in the entire flow regime, the model would have resulted in an incorrect form similar to Knudsen’s model in the slip flow regime. In order to obtain the variation of a as a function of Knudsen number for the transitional flow regime, we can solve for a from equation (4.33) to obtain


a



64



M


Mfm



3^(1 +



4Kn


1-bKn



1


In’


where M is the flowrate data obtained numerically or experimentally (and normalized with Mfm). The 1/ Kn behavior in this analytical expression makes it difficult to predict the value of a for small Kn. Therefore, we must rely on accurate numerical or experimental data. For this purpose we use linearized Boltzmann solutions of (Loyalka and Hamoodi, 1990) and experimental data of S. Tison (NIST, private communications). In Figure 4.25 we present the variation of a as a function of Kn (symbols). The value of a is initially small (close to zero), and it gradually increases with Kn, reaching a constant value in the free-molecular flow regime. The physical meaning of this behavior is that the dynamic viscosity remains the standard diffusion coefficient in the early slip flow regime. The value of increases slowly with Kn in the slip flow regime, and therefore the effect of change of the diffusion coefficient is second-order in Kn. For this reason the experimental slip flow results are accurately predicted by the slip flow theory, which does not require change of the diffusion coefficient length scale from A to channel height h. Variation of a as a function of Kn found by the numerical and experimental data can be represented accurately with the following relation:

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