Interdisciplinary Applied Mathematics

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FIGURE 4.26. Free-molecular scaling of Loyalka and Hamoodi’s linearized Boltzmann solutions (Loyalka and Hamoodi, 1990) and Tison’s experiments (Tison, 1993). Comparisons with the proposed model for both cases and Knudsen’s model are also presented.


In Figure 4.27 we present the mass flowrate variation (normalized with the corresponding no-slip value) as a function of Kn, up to Kn = 0.5. This covers the slip and the early transitional flow regimes. We see that Knudsen’s model is not accurate in this regime. Linearized Boltzmann solutions and experimental data both start with a slope of 4. Hence,


= 1 +4Kn+C>(Kn2);


Me


then the slope increases gradually with Kn. The model predicts this transition very accurately for the numerical and the experimental data. The increase in slope was observed by (Sreekanth, 1969) and explained as a change in the slip coefficient in Maxwell’s slip boundary conditions from 1.0 to 1.1466. If the change in the slope of the data is to be explained by an increase in the slip coefficient, the velocity scaling results shown in Figures 4.24 and 4.19 should be affected. However, it is clearly seen that such an effect is    not    present;    a    more    appropriate    explanation    of    the    slope change


is the change in the diffusion coefficient with Kn as presented in Section 4.2.2.


Duct Flow


We present    the    extensions    of the    new    model    for    duct    flows    in the    entire

FIGURE 4.27. Normalized flowrate variation in the slip and early transitional flow regimes for pipe flows. Symbols correspond to the linearized Boltzmann solutions of (Loyalka and Hamoodi, 1990) and experimental results of (Tison, 1993). Comparisons with the proposed model for both cases and Knudsen’s model are also presented.

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