Interdisciplinary Applied Mathematics

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FIGURE 4.15. Streamwise (left) and cross-flow (right) velocity profiles at various streamwise locations    specified    by Z.    Kn    =    0.03    (solid    lines), and    no-slip    (dashed

lines) cases.

cross-flow component of the velocity decreased for the slip flows compared to the continuum flows. Finally, the pressure distribution along the centerline of the channel is plotted in Figure 4.16. The centerline pressure drop is parabolic very close to the channel inlet and tends to a linear form at about К = 2.5 (Z = 0.75) for both slip and no-slip flows. High-order slip effects in pressure distribution are negligible at the centerline of the channel. A comparison of the slopes of the no-slip and slip flow pressure distributions show that a reduction in the pressure drop is obtained in these simulations, where the mass flowrate at the channel inlet is specified.

4.2 Transition and Free-Molecular Regimes

In this section we concentrate on transition and free-molecular flow regimes with the objective of developing unified flow models for pipe and duct flows. These models are based on a velocity scaling law, which is valid for a wide range of Knudsen number. Simple no-slip and slip-based descriptions of flowrate in channels and pipes are corrected for effects in transition and free-molecular flow regimes with the introduction of a rarefaction coefficient. We follow here the analysis first presented in (Beskok and Karniadakis, 1999).

A number of investigators have considered semianalytical and numerical solutions of the linearized Boltzmann equations for rarefied flow between two parallel plates or a pipe in both transition and free-molecular regimes. Starting with the pioneering work of Cercignani and his associates, the Knudsen’s minimum has been rigorously explored (Cercignani and Daneri, 1963). In such studies, simplifications for the collision integral based on the

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