Interdisciplinary Applied Mathematics

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• д

1-1-1-П 0.4

Model A

Model В


Maxwell    -| 0.3


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^ л д Д Д □ A

o □ * A A

О Da

° О о



0    0.1    0.2    0.3    0.4    0.5



FIGURE 4.17. Nondimensionalized velocity distribution in half of a microchannel (left). The linearized Boltzmann solution is from (Ohwada et al., 1989a). Error in the solution of the Navier-Stokes equations with various slip flow models at Kn = 0.6 (Right).

ing Schamberg’s boundary conditions. The models by Cercignani and by Deissler are almost identical for this case, and therefore only one is shown in the    figure.    The reason for    all    the    models    crossing at Y =    (y/h)    = 0.2    in

Figure 4.17 is an artifact of nondimensionalization of the velocity profiles with the    average    velocity    U. Due    to equation    (4.15),    we    have    U/U = 1    at

y/h = 0.2 for every slip model.

We now turn our attention to the flowrate in microducts. It is known from Knudsen’s and Gaede’s experiments in the transition flow regime that there is a minimum in the flowrate in pipe and channel flows at about Kn « 3 and Kn « 1, respectively. This behavior has been investigated by many researchers both theoretically (Cercignani and Daneri, 1963; Cercignani, 1963; Loyalka and Hamoodi, 1990; Ohwada et al., 1989a; Polard and Present, 1948; Kogan, 1969) and experimentally (Tison, 1993). It was first shown by Knudsen that in the free-molecular flow regime in pipes a diffusive transport process proportional to the pressure gradient but independent of

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