Interdisciplinary Applied Mathematics

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Since the Burnett equations are obtained by a second-order Chapman-Enskog expansion in Kn, they require second-order slip boundary conditions. Such boundary conditions were derived by (Schamberg, 1947); however, numerical experiments with aerodynamic rarefied flows (Zhong, 1993) showed that Schamberg’s boundary conditions are inaccurate for Kn > 0.2. Similar second-order slip boundary conditions have also been proposed in (Deissler, 1964) and (Sreekanth, 1969). Detailed discussions of performance of these second-order slip models will be presented in Sections 4.1.3 and 4.2, with comparisons of the DSMC and the linearized Boltzmann results against the analytical predictions for the velocity profile.

2.3.1 Derivation of High-Order Slip Models

Maxwell’s derivation of equation (2.19) is based on kinetic theory. A similar boundary condition can be derived by an approximate analysis of the motion of gas in isothermal conditions. We write the tangential momentum flux on a surface s located near the wall (see Figure 2.5) as



where ns is the number density of the molecules crossing surface s, m is the molecular mass, V is the mean thermal speed defined as

v — (8/nRT )05,

and us is the tangential (slip) velocity of the gas on this surface. If we assume that approximately half of the molecules passing through s are coming from a layer of gas at a distance proportional to one mean free path (A = p(RTn/2) 5 /p]) away from the surface, the tangential momentum flux of these incoming molecules is written as



where the subscript A indicates quantities evaluated one mean free path away from    the    surface.    Since    we    have    assumed    that half    of    the    molecules

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