Interdisciplinary Applied Mathematics

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4, 6, and 8.

T =1 KAM Boundaries

KAM boundaries. However, we note that the Poincare sections can be obtained experimentally by long-time tracking of noninertial particles. Given the computational difficulties of obtaining FTLE and Poincare sections, it is desirable to develop other computation- and experiment-friendly methods to quantify the mixing quality. Dispersion of particles that are initially confined to a zone can be utilized to characterize mixing. In particular, uniform dispersion of particles to the entire mixing region can be considered as a homogeneously mixed state. However, such observations need to be quantified to obtain a reasonable measure of the mixing quality.

To this end, we employ the box counting method to quantify the rate at which particles are dispersed by the flow into small uniform boxes (Liu et al., 1994). In this method, selection of an appropriate box size is important, and it is related to the number of tracked particles. Jones recognized that a perfectly randomized population of particles has a Poisson bin-occupancy distribution, and used this information to determine the box size such that on average there was one particle per cell (Jones, 1991). If the box size is chosen such that for a perfectly random distribution of particles, 98% of the boxes contain at least one particle, then the box size s for a unit-square domain (of length 1 x 1) is approximately given by

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