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

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.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The following article appeared in the American Journal of Physics and may be found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright 2011 American Association of Physics Teachers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the AAP

Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations

We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of intermaterial interfaces) that can be computed over given (finite) numbers of iterations. We find that good mixing can be achieved after a finite number of iterations of a one-dimensional cutting and shuffling map, even though such a map cannot be considered chaotic in the usual sense and/or it may not fulfill the conditions of the ergodic theorems for interval exchange transformations. Specifically, good shuffling can occur with only six or seven intervals of roughly the same length, as long as the rearrangement order is an irreducible permutation. This study has implications for a number of mixing processes in which discontinuities arise either by construction or due to the underlying physics.Comment: 21 pages, 10 figures, ws-ijbc class; accepted for publication in International Journal of Bifurcation and Chao

Fluid Elasticity Can Enable Propulsion at Low Reynolds Number

Conventionally, a microscopic particle that performs a reciprocal stroke cannot move through its environment. This is because at small scales, the response of simple Newtonian fluids is purely viscous and flows are time-reversible. We show that by contrast, fluid elasticity enables propulsion by reciprocal forcing that is otherwise impossible. We present experiments on rigid objects actuated reciprocally in viscous fluids, demonstrating for the first time a purely elastic propulsion set by the object's shape and boundary conditions. We describe two different artificial "swimmers" that experimentally realize this principle.Comment: 5 pages, 4 figure

Complete Chaotic Mixing in an Electro-osmotic Flow by Destabilization of Key Periodic Pathlines

The ability to generate complete, or almost complete, chaotic mixing is of great interest in numerous applications, particularly for microfluidics. For this purpose, we propose a strategy that allows us to quickly target the parameter values at which complete mixing occurs. The technique is applied to a time periodic, two-dimensional electro-osmotic flow with spatially and temporally varying Helmoltz-Smoluchowski slip boundary conditions. The strategy consists of following the linear stability of some key periodic pathlines in parameter space (i.e., amplitude and frequency of the forcing), particularly through the bifurcation points at which such pathlines become unstable.Comment: 14 pages, 11 figure

Global Diffusion in a Realistic Three-Dimensional Time-Dependent Nonturbulent Fluid Flow

We introduce and study the first model of an experimentally realizable three-dimensional time-dependent nonturbulent fluid flow to display the phenomenon of global diffusion of passive-scalar particles at arbitrarily small values of the nonintegrable perturbation. This type of chaotic advection, termed {\it resonance-induced diffusion\/}, is generic for a large class of flows.Comment: 4 pages, uuencoded compressed postscript file, to appear in Phys. Rev. Lett. Also available on the WWW from http://formentor.uib.es/~julyan/, or on paper by reques

Quantification of the performance of chaotic micromixers on the basis of finite time Lyapunov exponents

Chaotic micromixers such as the staggered herringbone mixer developed by Stroock et al. allow efficient mixing of fluids even at low Reynolds number by repeated stretching and folding of the fluid interfaces. The ability of the fluid to mix well depends on the rate at which "chaotic advection" occurs in the mixer. An optimization of mixer geometries is a non trivial task which is often performed by time consuming and expensive trial and error experiments. In this paper an algorithm is presented that applies the concept of finite-time Lyapunov exponents to obtain a quantitative measure of the chaotic advection of the flow and hence the performance of micromixers. By performing lattice Boltzmann simulations of the flow inside a mixer geometry, introducing massless and non-interacting tracer particles and following their trajectories the finite time Lyapunov exponents can be calculated. The applicability of the method is demonstrated by a comparison of the improved geometrical structure of the staggered herringbone mixer with available literature data.Comment: 9 pages, 8 figure

Efficient topological chaos embedded in the blinking vortex system

Periodic orbits forhomeomorphisms on the plane give mathematical braids, which are topologically classified into three types by Thurston-Nielsen (T-N) theory; (1) periodic, (2) reducible, and (3) pseudo-Anosov (pA). If the braid is pA, then the homeomorphism must have an infinitely many number of pe-riodic orbits of distinct periods. This kind of complexity induced by the pA braid is called “topological chaos”, which was introduced by Boyland et. al [4] recently. We investigate numerically the topological chaos embedded in the particle mixing by the blinking vortex system introduced by Aref [1]. It has already been known that the system generates the chaotic advection due to the homoclinic chaos, but the chaotic mixing region is restricted locally in the vicinity of the vortex points. In the present study, we propose an in-genious operation of the blinking vortex system that defines a mathematical braid of pA type. The operation not onlygenerates the chaotic mixing region due to the topological chaos, but also ensures global particle mixing in the whole plane. We give a mathematical explanation for the phenomenon by the T-N theory and some numerical evidences to support the explanation. More-over, we makemention of the relation between the topological chaos and the homoclinic chaos in the blinking vortex system

Chaos induced coherence in two independent food chains

Coherence evolution of two food web models can be obtained under the stirring effect of chaotic advection. Each food web model sustains a three--level trophic system composed of interacting predators, consumers and vegetation. These populations compete for a common limiting resource in open flows with chaotic advection dynamics. Here we show that two species (the top--predators) of different colonies chaotically advected by a jet--like flow can synchronize their evolution even without migration interaction. The evolution is charaterized as a phase synchronization. The phase differences (determined through the Hilbert transform) of the variables representing those species show a coherent evolution.Comment: 5 pages, 5 eps figures. Accepted for publication in Phys. Rev.