The objective of this work is to develop techniques to predict the results of experiments involving chaotic dispersion of passive tracers in two-dimensional low Reynolds number flows. We present the design of a flow apparatus which allows the unobstructed observation of the entire flow region. Whenever possible we compare the experimental results with those of computations. Conventional tracking of the boundaries of the tracer is inefficient and works well only for low stretches (order 102 at most). However, most mixing experiments involve extremely large perturbations from steady flow since this is where the best mixing occurs. The best prediction of widespread mixing and large stretching (order 104–105) is provided by lineal stretching plots; surprisingly the technique also works for relatively low numbers of periods (as low as 2 or 3). The second best prediction is provided by a combination of low-order unstable manifolds – which indicate where the tracer goes, especially for short times – and the eigendirections of low-order hyperbolic periodic points – which indicate the alignment of striations in the flow. On the other hand, Poincaré sections provide only a gross picture of the spreading: they can be used primarily to detect what regions are inaccessible to the dye. Comparison of computations and experiments invariably reveals that bifurcations within islands have little impact in the mixing process.

The statistics of stretching and stirring in time-periodic chaotic flows is studied numerically by following the evolution of stretching of O(105) points. The ratio between stretchings accumulated by each point at successive periods is referred to as a multiplier, and the total stretching is the product of multipliers. As expected, the mean stretching of the population increases exponentially whereas the probability density function of multipliers converges—in just two periods or so—to a time-invariant distribution. There is, however, a considerable degree of order in the spatial distribution of stretching in spite of conditions of global chaos. The self-correlation of multipliers shows as well considerable structure and often there are segregated populations of points: the largest population consists of points that experience extensive stretching, efficient stirring, and have a distribution of stretching values that evolves asymptotically—in about ten periods—into a limiting time-invariant scaling distribution. The remaining points experience slow stretching and, although they also exhibit scaling behavior, are effectively segregated from the rest of the system in the time scale of our simulations.

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