This work reviews the present position of and surveys future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows.

Mixing processes occur in many technological and natural applications, with length and time scales ranging from the very small to the very large. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification? The authors show how a variety of flows in very different settings possess the characteristic of streamline crossing. This notion can be placed on firm mathematical footing via Linked Twist Maps (LTMs), which is the central organizing principle of this book. The authors discuss the definition and construction of LTMs, provide examples of specific mixers that can be analyzed in the LTM framework and introduce a number of mathematical techniques which are then brought to bear on the problem of fluid mixing. In a final chapter, they present a number of open problems and new directions.

We study the mixing properties of two systems: (i) a half-filled quasi-two-dimensional circular drum whose rotation rate is switched between two values and which can be analysed in terms of the existing mathematical formalism of linked twist maps; and (ii) a half-filled three-dimensional spherical tumbler rotated about two orthogonal axes bisecting the equator and with a rotational protocol switching between two rates on each axis, a system which we call a three-dimensional linked twist map, and for which there is no existing mathematical formalism. The mathematics of the three-dimensional case is considerably more involved. Moreover, as opposed to the two-dimensional case where the mathematical foundations are firm, most of the necessary mathematical results for the case of three-dimensional linked twist maps remain to be developed though some analytical results, some expressible as theorems, are possible and are presented in this work. Companion experiments in two-dimensional and three-dimensional systems are presented to demonstrate the validity of the flow used to construct the maps. In the quasi-two-dimensional circular drum, bidisperse (size-varying or density-varying) mixtures segregate to form lobes of small or dense particles that coincide with the locations of islands in computational Poincaré sections generated from the flow model. In the 3d spherical tumbler, patterns formed by tracer particles reveal the dynamics predicted by the flow model.

In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents. In rare cases, for instance in uniquely ergodic systems, it is possible to derive uniform estimates from non-uniform hypotheses. This allowed one of us to show in a previous paper that a strange non-chaotic attractor for a quasiperiodically forced system could not be the graph of a continuous function. This had been a conjecture for some time. In this paper we generalize the uniform convergence of time averages for uniquely ergodic systems to a broader range of systems. In particular, we show how conditions on growth rates with respect to all the invariant measures of a system can be used to derive one-sided uniform convergence in both the Birkhoff and the sub-additive ergodic theorems. We apply the latter to show that any strange compact invariant set for a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Liapunov exponent; in other words, it must contain some 'non-attracting' orbits. This was already known for the few examples of strange non-chaotic attractors that have rigorously been proved to exist. Finally, we generalize our semi-uniform ergodic theorems to arbitrary skew product systems and discuss the application of such extensions to the existence of attracting invariant graphs.

Dynamical systems theory has proven to be a successful approach to understanding mixing, with stretching and folding being the hallmark of chaotic mixing. Here we consider the mixing of a granular material in the context of a different mixing mechanism -cutting and shuffling-as a complementary viewpoint to that of traditional chaotic dynamics. Cutting and shuffling has a theoretical foundation in a relatively new area of mathematics called piecewise isometries (PWIs) with properties that are fundamentally different from the stretching and folding mechanism of chaotic advection. To demonstrate the effect of the cutting and shuffling combined with stretching and folding, we consider the mixing of granular materials of two different colors in a half-filled spherical tumbler that is rotated alternately about orthogonal axes. Mixing experiments using 1 mm particles in a 14 cm diameter tumbler are compared to PWI maps. The experiments are readily related to the PWI theory using continuum model simulations. By comparing experimental, simulation, and theoretical results, we demonstrate that mixing in a three-dimensional granular system can be viewed as mixing by traditional chaotic dynamics (stretching and folding) built on an underlying framework, or skeleton, of mixing due to cutting and shuffling. We further demonstrate that pure cutting and shuffling can generate a well-mixed system, depending on the angles through which the tumbler is rotated. We also explore the generation of interfacial area between the two colors of material resulting from both stretching in the flowing layer and cutting due to switching the axis of rotation.