Interdisciplinary Applied Mathematics

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The change from a microscopic to macroscopic (or vice versa) behavior is gradual, and therefore the question arises as to where the interface separating rarefied from nonrarefied behavior should be located. A possible criterion for determining this interface is the local Knudsen number defined


Kn; = (Xfp)Vp

or a continuum breakdown parameter suggested by (Bird, 1994):

P = M (X/p)Vp,

where M is the Mach number. Values of P = 0.1 and 0.01 to 0.02, respectively, have been suggested by Bird and other researchers, e.g., for rarefied

NS    NS (Slip)    DSMC

FIGURE 15.4. Domain for DSMC-continuum coupling. The interface is an overlap region where DSMC and modified Navier-Stokes with slip are both valid.

flows encountered in high-altitude applications. However, for internal flows at microscales the criterion is application-specific. For small Reynolds number viscous flows, deviation of velocity distribution from the Maxwellian may provide a more definitive metric. This interface from continuum to rarefied has been treated as a single surface in previous attempts to couple DSMC with Navier-Stokes equations starting with the work of (Wadsworth and Erwin, 1992) and in subsequent work, see (Hash and Hassan, 1997), and references therein. It is perhaps better to treat the interface as a finite zone giving an overlap between the region of validity of DSMC and the Navier-Stokes equations. A typical situation is sketched in Figure 15.4, corresponding to the aforementioned example of a long narrow channel. The proposed interface extends from Кщ to Kn2, inside which the modified Navier-Stokes with the high-order slip condition (see equation (2.28)) and DSMC are both valid.

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