Mixing (stretching and folding of fluid elements) in chaotic flows is an iterative process generating presumably self-similar distributions of stretching and striation thickness. This hypothesis is investigated using scaling and multifractal techniques for two prototypical time-periodic chaotic flows: one containing no detectable islands (egg-beater flow), the other involving sizable islands as well as no-slip boundaries (flow between eccentric cylinders). The simplest picture arises in the egg-beater flow. Stretching is well described by multifractal scaling if the very high tail of the distribution of stretchings is neglected. Different methods for obtaining the spectrum of fractal dimensions f(α) agree reasonably well, producing a time-independent self-similar distribution. On the other hand, in the flow between eccentric cylinders, the negative moments do not scale, and the spectrum f(α) is time-dependent (and therefore, it is not self-similar). Due to the extremely wide range of values of stretching, a very large number of points needs to be considered in order to characterize mixing in chaotic systems using a multifractal formalism; this suggests that more work is needed in order to understand finite-size effects and how asymptotic states are reached. However, for cases where multifractal scaling applies, it is possible to relate coarse-grained variables (e.g., intermaterial area density) to microscopic features of the flow (e.g., finite-time Lyapunov exponents).
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