Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua

Ivan C. Christov*, Richard M. Lueptow, Julio M. Ottino

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

We compare and contrast two types of deformations inspired by mixing applications-one from the mixing of fluids (stretching and folding) and the other from the mixing of granular matter (cutting and shuffling). The connection between mechanics and dynamical systems is discussed in the context of the kinematics of deformation, emphasizing the equivalence between stretches and Lyapunov exponents. The stretching and folding motion exemplified by the baker's map is shown to give rise to a dynamical system with a positive Lyapunov exponent, the hallmark of chaotic mixing. In contrast, cutting and shuffling does not stretch. When an interval exchange transformation is used as the basis for cutting and shuffling, we establish that all of the map's Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per unit volume, is shown to be exponential when there is stretching and folding but linear when there is only cutting and shuffling. We also discuss how a simple computational approach can discern stretching in discrete data.

Original languageEnglish (US)
Pages (from-to)359-367
Number of pages9
JournalAmerican Journal of Physics
Volume79
Issue number4
DOIs
StatePublished - Mar 14 2011

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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