Interdisciplinary Applied Mathematics

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While these issues have been addressed to some extent in high-altitude rarefied flows, no significant progress has been made (Hash and Hassan, 1997). In addition, flows at microscales correspond to very viscous subsonic flows, whereas the previously studied flows corresponded to highspeed transonic and supersonic flows. A more appropriate approach is to develop interface conditions based on the overlap zone shown in Figure 15.4. The Navier-Stokes along with the slip conditions in the overlap zone will facilitate a smooth transition between the no-slip continuum flow and the large-slip rarefied flow. The size of the overlap is a parameter that will be selected in such a way that conservativity, solution continuity, and solution convergence are guaranteed. With regard to convergence in particular, the following acceleration scheme can be employed based on a relaxation procedure:

^MC1 = вфМс + (1 _ e)^NS,

where в is    the    acceleration    parameter    with    в G    (0,1).    Here the    subscript

(MC) denotes quantities on the DSMC side and (NS) on the Navier-Stokes side. Note that the microscopic quantities in the DSMC region will be computed based on the Chapman-Enskog distribution, which involves gradients of interpolated quantities, since Maxwellian distributions are certainly inappropriate for these nonequilibrium flows. The overlap region also provides a spatially homogeneous regime over which appropriate averages can be performed to reduce scatter from the DSMC solution. On the Navier-Stokes side, subsonic outflow conditions can be imposed following a characteristic decomposition that could involve fluxes (if the Marshak condition is enforced) or property interpolated quantities in analogy with the above interpolation procedure. In either case, appropriate formulations need to be developed in order to take into account the overlap zone.

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