Rob Sturman

Frontiers of chaotic advection

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.

Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows

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 Poincare sections generated from the flow model. In the 3d spherical tumbler, patterns formed by tracer particles reveal the dynamics predicted by the flow model.

Mixing by cutting and shuffling

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.

Cycling Attractors of Coupled Cell Systems and Dynamics with Symmetry

Dynamical systems with symmetries show a number of atypical behaviours for generic dynamical systems. As coupled cell systems often possess symmetries, these behaviours are important for understanding dynamical effects in such systems. In particular the presence of symmetries gives invariant subspaces that interact with attractors to give new types of instability and intermittent attractor. In this paper we review and extend some recent work (Ashwin, Rucklidge and Sturman 2002) on robust non-ergodic attractors consists of cycles between invariant subspaces, called ‘cycling chaos’ by (1995).

Deceleration of one-dimensional mixing by discontinuous mappings

We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equally sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes nonmonotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map M in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counterintuitively predicts a deceleration in the asymptotic mixing rate with an increasing diffusivity rate. The implications of the results on finite time mixing are discussed.

Decelerating defects and non-ergodic critical behaviour in a unidirectionally coupled map lattice

Abstract We examine a coupled map lattice (CML) consisting of an infinite chain of logistic maps coupled in one direction by inhibitory coupling. We find that for sufficiently strong coupling strength there are dynamical states with ‘decelerating defects’, where defects between stable patterns (with chaotic temporal evolution and average spatial period two) slow down but never stop. These defects annihilate each other when they meet. We show for certain states that this leads to a lack of convergence (non-ergodicity) of averages taken from observables in the system and conjecture that this is typical for the system.

Lyapunov Exponents for the Random Product of Two Shears

We give lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. These types of random products arise in applications such as fluid stirring devices. The bounds, obtained by considering invariant cones in tangent space, give excellent accuracy compared to standard and general bounds, and are increasingly accurate with increasing shear. Bounds on generalised exponents are useful for testing numerical methods, since these exponents are difficult to compute in practice.

A Parametric Study of Mixing in a Granular Flow a Biaxial Spherical Tumbler

We report on a computational parameter space study of mixing protocols for a half-full biaxial spherical granular tumbler. The quality of mixing is quantified via the intensity of segregation (concentration variance) and computed as a function of three system parameters: angles of rotation about each tumbler axis and the flowing layer depth. Only the symmetric case is considered in which the flowing layer depth is the same for each rotation. We also consider the dependence on \(\bar{R}\), which parametrizes the concentric spheroids (“shells”) that comprise the volume of the tumbler. The intensity of segregation is computed over 100 periods of the mixing protocol for each choice of parameters. Each curve is classified via a time constant, \(\tau \), and an asymptotic mixing value, bias. We find that most choices of angles and most shells throughout the tumbler volume mix well, with mixing near the center of the tumbler being consistently faster (small \(\tau \)) and more complete (small bias). We conclude with examples and discussion of the pathological mixing behaviors of the outliers in the so-called \(\tau \)-bias scatterplots.

A dynamical systems approach to musical tuning

At the end of the 1990s Jaroslav Stark was supervising my PhD studies, on the subject of strange nonchaotic attractors in quasiperiodically forced systems [R.J. Sturman, Strange nonchaotic attractors in quasiperiodically forced systems, Ph.D. thesis, University College London, 2001]. Like several others at the time [A.S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors, Chaos Interdis. J. Nonlinear Sci. 5(1) (1995), p. 253; A. Witt, U. Feudel, and A. Pikovsky, Birth of strange nonchaotic attractors due to interior crisis, Physica D: Nonlinear Phenomena 109(1–2) (1997), pp. 180–190], we approximated irrational rotations on the circle by rational approximations of increasing accuracy [R. Sturman, Scaling of intermittent behaviour of a strange nonchaotic attractor, Phys. Lett. A 259(5) (1999), pp. 355–365]. During one conversation about circle maps, Jaroslav asked me about the connection between musical tuning and rational and irrational rotations; he had correctly recognized the link between this dichotomy and that which separates equal temperament from Pythagorean tuning. He was delighted when I described to him the natural extension to nonlinear circle maps which forms the basis of this article. Recently the ideas for such nonlinear tunings have been used as the basis for original compositions.