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

TL;DR

This paper compares stretching and folding with cutting and shuffling as methods of mixing, analyzing their dynamical properties, efficiency, and implications for continuum deformations in fluids and granular matter.

Contribution

It provides a detailed comparison of two mixing mechanisms, highlighting their dynamical differences and introducing a computational method to detect stretching in data.

Findings

Stretching and folding lead to chaotic mixing with positive Lyapunov exponents.

Cutting and shuffling have zero Lyapunov exponents and linear mixing rates.

Mixing efficiency is exponential with stretching and folding, linear with cutting and shuffling.

Abstract

We compare and contrast two types of deformations inspired by mixing applications -- one from the mixing of fluids (stretching and folding), 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. On the other hand, 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 exponentially fast 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.