# Interdisciplinary Applied Mathematics

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FIGURE 4.16. Pressure distribution along the centerline for slip flow, using first-order (b = 0), and high-order (b = b(x)) slip boundary conditions, and no-slip flow as a function of X = |).

BGK model (Bhatnagar et al., 1954) (see also Section 15.4) were extensively used. Other investigators have derived solutions based on the hard-sphere and Maxwellian models for the collision integral (Huang et al., 1997; Sone, 1989; Ohwada et al., 1989a), and have also obtained solutions in cylindrical geometry (Loyalka and Hamoodi, 1990) and ducts with various cross sections (Aoki, 1989). Similar approaches have also been used successfully in modeling gas film lubrication in the transition regime (see Section 6.1 and (Fukui and Kaneko, 1988; Fukui and Kaneko, 1990)).

Before we start developing velocity and flowrate scaling models, we examine the validity of continuum-based slip models in the transition flow regime. In Figure 4.17 we present the velocity profiles obtained by the DSMC and linearized Boltzmann solutions at Kn = 0.6. The model based on equation    (2.29)    is indicated    as “Model    A,”    and    the    one    based on    equa

tion (2.43) is indicated as “Model B.” We have also included models by Cercignani, Deissler, Maxwell, Schamberg, and Hsia and Domoto (see Table 2.2). Model B    does    very    well    in    the    center    of    the    channel    but    gives

slight deviations near the walls. The maximum deviation of the model is observed at the slip location, where Model A is somewhat better. However, Model A gives larger errors than Model B toward the centerline of the channel. The errors in Maxwell’s first-order boundary condition and the other second-order models are larger than the errors of either model A or B. The maximum error occurs near the wall with 0.32 units of overestimation us-

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