# Interdisciplinary Applied Mathematics

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1.

The above approach was used in (Niu et al., 2003), to analyze the twisted pipe of (Jones et al., 1989), that we presented earlier, see equations (9.3) and (9.4). In particular, FTLE contours in the (x,y) planes were computed and compared to Poincare sections for different values of the parameters and c. Figure 9.11 (left) shows the Poincare section for x = n/2 and c = 100. Clearly, mixing is poor in regions where islands appear. Figure 9.11 (right) shows the FTLE contour for the same parameters, which is topologically similar to the Poincare section. The mean FTLE was computed for many values of (x, c), shown in Figure 9.12. The highest value corresponds to best mixing, which is achieved at x slightly greater than n/2. However, for realistic values of the parameter c the twisted pipe turns out not to be a very good mixer.

We now study the electroosmotic stirrer of Qian and Bau (2002), described in the previous section, in order to quantify its mixing effectiveness. Using the    flow    patterns shown in    Figure    9.8    for    half    a period    (T/2)    each,

we obtain six different pattern combinations (A-B, A-C, A-D, B-C, B-D, and C-D). We computed the FTLE for all of these patterns for nondimensional periods of T = 4, 6, and 8, where T is normalized by the convective time scale    (based    on    the    half-channel    height    and    the    electroosmotic    slip

velocity from equation (7.24)). The variation of the FTLE, as a function of time for T = 4, 6, and 8 is shown in Figure 9.13. For all the cases, the initial particle location was at (x,y) = (0.5,0.1), and the virtual particle was initially offset by a distance of 10~5 (channel half-height). The results show that the pattern B-C at T = 6.0 has the largest FTLE, corresponding

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