# Interdisciplinary Applied Mathematics

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(c)) (Boyd and Sun, 2001). (Shen et al., 2003) utilized the DSMC and IP-DSMC methods to simulate low-speed flows in long microchannels. The IP results agree with the experimental measurements of pressure distribution and mass flow through microchannels in (Arkilic et al., 1994; Liu et al., 1993; Pong    et    al.,    1994).    The    IP and DSMC    also    predict    the    Knudsen’s

x/chord

-0 2    0 0    0 2    0 4    0 6    0 8    1 0

x/chord

-0 2    0 0    0 2    0 4    0 6    0 8    1 0

-0 2    0 0    0 2    0 4    0 6    0 8    1 0

(c) Continuum approach with a slip boundary condition

FIGURE 15.3. Density contours of flow past an NACA 0012 airfoil, obtained by the DSMC (a), DSMC-IP (b), and Navie-Stokes with slip (c) algorithms. The free stream flow conditions are M = 0.1, Re = 1, Kn = 0.013. (Courtesy of I.D. Boyd.)

minimum equally well.

The DSMC-IP method is relatively new, and further developments for nonisothermal flow conditions, which require preservation of the internal energy, are necessary. However, the result presented in Figure 15.3 is a clear indication of significant advancement of the DSMC-IP over the classical DSMC method for nearly isothermal flow conditions. Another direction in which the IP method has been developed and utilized is as an interface in hybrid methods that combine particle-based DSMC with continuum-based discretizations; see (Sun et al., 2004), and also Section 15.2.

##### 15.2 DSM: Continuum Coupling

In this section we discuss possible procedures for coupling the DSMC method with the Navier-Stokes equations. This is important particularly for simulation of gas flows in MEMS components. If we consider the microcomb drive mechanism, the flow in most of the device can be simulated by    slip    continuum-based    solvers.    Only    when    the    gap    between    the

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