Interdisciplinary Applied Mathematics

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and less than 10~3. A similar study using the (entropic) lattice Boltzmann method has been performed by (Ansumali, 2004); see also Section 15.5.


In Figure 3.22 we plot the streamlines for a small Knudsen number Kn = 0.00485 and a larger Knudsen number Kn = 0.388. We see that the effect of rarefaction is to push the center of the vortex towards the moving wall. This was studied by (Nie et al., 1998) more systematically for a range of values of


Kn=0.000485    Kn=0.388



0


10 20 30


0


10 20 30 40


x


x

FIGURE 3.24. Pressure contours for the microcavity flow in the slip flow regime (left) and transition regime (right). (Courtesy of X. Nie, G.D. Doolen, and S. Chen.)


30


>* 20

0


Knudsen number, and their results are shown in Figure 3.23. The distance of the vortex center from the lower wall along with the mass flux between the lower wall and the vortex center are plotted. We see that in the slip flow regime the    vortex    center    moves    upward rapidly,    but    then    it    stabilizes    for


Kn > 0.3. The corresponding mass flux has a nonmonotonic trend. Initially, it increases due to larger area, but then rarefaction effects are strong and dominate, so the circulation motion is not as intense and the mass flux drops significantly. It is also interesting to examine the pressure contours in the microcavity. This is shown in Figure 3.24, obtained by Nie et al. In the slip flow regime the pressure contours are circular arcs centered on the corners of the moving lid. However, in the transition regime the pressure contours become almost vertical lines. The symmetry around the x = 20 plane is due to the low Reynolds number. Other work reported in (Hou et al., 1995) shows that for low Knudsen numbers, as the Reynolds number increases the vortex actually moves toward the lower wall, in obvious contrast with the low Reynolds number limit.

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