Interdisciplinary Applied Mathematics

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with large b (b =    —► oo, since U’0 —► 0). This may result in a velocity


slip at the separation point based on some numerical truncation error in the calculations.


In this section we have developed various second- and higher-order slip conditions for gas microflows. We note that the Navier-Stokes equations require only the first-order slip conditions, and the second-order slip models should be used strictly for the second-order equations, such as the Burnett or Woods equations. Throughout this book we will utilize the second-order slip conditions routinely for the Navier-Stokes equations. This can be justified by the following arguments:


TABLE 2.2. Coefficients for first- and second-order slip models.


Author


Ci


c2


Cercignani (Cercignani and Daneri, 1963)


1.1466


0.9756


Cercignani (Hadjiconstantinou, 2003a)


1.1466


0.647


Deissler (Deissler, 1964)


1.0


9/8


Schamberg (Schamberg, 1947)


1.0


5tt/12


Hsia and Domoto (Hsia and Domoto, 1983)


1.0


0.5


Maxwell (Kennard, 1938)


1.0


0.0


Equation (2.29)


1.0


-0.5

In the small Reynolds number limit, i.e., Re ^ Kn ^ 1, asymptotic analysis of the Boltzmann equation shows that a consistent set of governing equations and boundary conditions up to O(Kn2) is the Stokes system with second-order slip boundary conditions; see Section 15.4.2 and for details (Sone, 2002; Aoki, 2001).


Rarefaction effects both in the aforementioned limit as well as in the limit of Re ~ O(1) ^ M ~ O(Kn) come in only through the boundary condition. This has been proven rigorously using the Boltzmann equation in (Sone, 2002).

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