# Interdisciplinary Applied Mathematics

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FIGURE 15.7. The geometry of the microfilter device. Also shown in the figure are the Stokes (or the Navier-Stokes) and DSMC subdomains and the overlap between the two subdomains.

To test the accuracy of the coupled DSMC/Stokes method, the geometry shown in Figure 15.7 is simulated by DSMC only (which will serve as the exact solution) and by the coupled approach. The coupled simulations use an overlap of dov = 0.0 p,m. Figures 15.8 and 15.9 compare the pressure and x-velocity, respectively, obtained with the DSMC and the coupled simulations for hc = 0.2 p,m. We observe that the coupled simulations are in good agreement with the DSMC results. The good agreement between the two methods establishes the accuracy of the coupled method. From the results, we can also conclude that the multiscale approach achieves proper coupling between the DSMC and Stokes subdomains.

An important aspect of the multiscale approach is its convergence behavior. The dependence of the convergence on various parameters contained in the multiscale approach needs to be understood to guarantee the convergence of the multiscale approach. Here we investigate the convergence characteristics on the overlap size, dov, the DSMC particle weight, wp, and the number of DSMC time steps between coupling iterations, Nstep.

Overlap size: The overlap size in Figure 15.7, dov, describes the overlap between the Stokes and the DSMC subdomains, i.e., both DSMC and Stokes equations are solved in the overlap region. The dependence of the convergence on the overlap is investigated by considering various overlap sizes: dov = 0.6, 0.4, 0.2, and 0.0 p,m. The overlap is increased or decreased by adjusting the distance dext (see Figure 15.7). The implementation of the interface boundary conditions for the dov = 0.0 p,m case is not different from

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