# Interdisciplinary Applied Mathematics

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FIGURE 9.12. Mean finite time Lyapunov exponent (FTLE) for the twisted pipe: optimization study. (Courtesy of Y.K. Lee.)

to Xp    = 0.31,    while    for    T =    8.0    and T =    4.0,    Xp =    0.2    and Xp =    0.25,

respectively.

In Figure 9.14, we present the Poincare sections obtained for pattern B-C at periods T =1, 2,4, and 6. The Poincare sections are obtained by tracking the motion of 121 particles for 100 time periods. Islands ofbad mixing zones are observed for T = 1, 2 cases. The island boundaries, also known as the Kolmogorov-Arnold-Moser (KAM) boundaries, separate the chaotic and regular regions    of the    flow (Ottino,    1997).    In    the    figure,    we    also    present

the KAM boundaries, obtained by tracking 20 particles that were initially located on the KAM boundaries, for 300 periods. We observe reduction in the bad mixing zone with increasing T. For example, the islands disappear for T > 6. The Poincare section for T = 8 is qualitatively similar to that of the T = 6 case, and is not shown in the figure. Destruction of KAM boundaries is desired for enhanced mixing, but is not a sufficient condition for the best mixing case. For example, the FTLE for T = 6 is considerably larger than that of the T = 8 case, and it corresponds to the best mixing case among the flow patterns and frequency ranges studied in (Kim, 2004). We must indicate that the FTLE values presented in Figure 9.13 were obtained for particles that were outside the bad mixing zones.

Calculation of the FTLE is often computationally expensive, and it requires accurate    knowledge    of    the    flow    field.    Qian and    Bau    (2002)    have

shown that the flow solution often requires accuracy levels comparable to the computer evaluation of the analytical solution. The same is true for calculation of the Poincare sections, especially for determination of the

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