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

Experimental evidence of chaotic advection in a convective flow

Lagrangian chaos is experimentally investigated in a convective flow by means of Particle Tracking Velocimetry. The Fnite Size Lyapunov Exponent analysis is applied to quantify dispersion properties at different scales. In the range of parameters of the experiment, Lagrangian motion is found to be chaotic. Moreover, the Lyapunov depends on the Rayleigh number as ${\cal R}a^{1/2}$. A simple dimensional argument for explaining the observed power law scaling is proposed.Comment: 7 pages, 3 figur

Walls Inhibit Chaotic Mixing

We report on experiments of chaotic mixing in a closed vessel, in which a highly viscous fluid is stirred by a moving rod. We analyze quantitatively how the concentration field of a low-diffusivity dye relaxes towards homogeneity, and we observe a slow algebraic decay of the inhomogeneity, at odds with the exponential decay predicted by most previous studies. Visual observations reveal the dominant role of the vessel wall, which strongly influences the concentration field in the entire domain and causes the anomalous scaling. A simplified 1D model supports our experimental results. Quantitative analysis of the concentration pattern leads to scalings for the distributions and the variance of the concentration field consistent with experimental and numerical results.Comment: 4 pages, 3 figure

Slow decay of concentration variance due to no-slip walls in chaotic mixing

Chaotic mixing in a closed vessel is studied experimentally and numerically in different 2-D flow configurations. For a purely hyperbolic phase space, it is well-known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincare section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1-D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.Comment: 17 pages, 12 figure

Transport and diffusion in the embedding map

We study the transport properties of passive inertial particles in a $2-d$ incompressible flows. Here the particle dynamics is represented by the $4-d$ dissipative embedding map of $2-d$ area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter $\alpha$, and the dissipation parameter $\gamma$. We obtain the statistical characterisers of transport for this system in these dynamical regimes. These are, the recurrence time statistics, the diffusion constant, and the distribution of jump lengths. The recurrence time distribution shows a power law tail in the dynamical regimes where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion constant shows behaviour of three types - normal, subdiffusive and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behaviour, as well as the behaviour of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes, and in the behaviour of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterisers of transport. We discuss the implications of our results for realistic systems.Comment: 28 pages, 14 figures, To Appear in Phys. Rev. E; Vol. 79 (2009

Targeted mixing in an array of alternating vortices

Transport and mixing properties of passive particles advected by an array of vortices are investigated. Starting from the integrable case, it is shown that a special class of perturbations allows one to preserve separatrices which act as effective transport barriers, while triggering chaotic advection. In this setting, mixing within the two dynamical barriers is enhanced while long range transport is prevented. A numerical analysis of mixing properties depending on parameter values is performed; regions for which optimal mixing is achieved are proposed. Robustness of the targeted mixing properties regarding errors in the applied perturbation are considered, as well as slip/no-slip boundary conditions for the flow

Advection of vector fields by chaotic flows

We have introduced a new transfer operator for chaotic flows whose leading eigenvalue yields the dynamo rate of the fast kinematic dynamo and applied cycle expansion of the Fredholm determinant of the new operator to evaluation of its spectrum. The theory hs been tested on a normal form model of the vector advecting dynamical flow. If the model is a simple map with constant time between two iterations, the dynamo rate is the same as the escape rate of scalar quantties. However, a spread in Poincar\'e section return times lifts the degeneracy of the vector and scalar advection rates, and leads to dynamo rates that dominate over the scalar advection rates. For sufficiently large time spreads we have even found repellers for which the magnetic field grows exponentially, even though the scalar densities are decaying exponentially.Comment: 12 pages, Latex. Ask for figures from [email protected]

Granular size segregation in underwater sand ripples

We report an experimental study of a binary sand bed under an oscillating water flow. The formation and evolution of ripples is observed. The appearance of a granular segregation is shown to strongly depend on the sand bed preparation. The initial wavelength of the mixture is measured. In the final steady state, a segregation in volume is observed instead of a segregation at the surface as reported before. The correlation between this phenomenon and the fluid flow is emphasised. Finally, different ``exotic'' patterns and their geophysical implications are presented.Comment: 8 page

Unstable periodic orbits in a chaotic meandering jet flow

We study the origin and bifurcations of typical classes of unstable periodic orbits in a jet flow that was introduced before as a kinematic model of chaotic advection, transport and mixing of passive scalars in meandering oceanic and atmospheric currents. A method to detect and locate the unstable periodic orbits and classify them by the origin and bifurcations is developed. We consider in detail period-1 and period-4 orbits playing an important role in chaotic advection. We introduce five classes of period-4 orbits: western and eastern ballistic ones, whose origin is associated with ballistic resonances of the fourth order, rotational ones, associated with rotational resonances of the second and fourth orders, and rotational-ballistic ones associated with a rotational-ballistic resonance. It is a new kind of nonlinear resonances that may occur in chaotic flow with jets and/or circulation cells. Varying the perturbation amplitude, we track out the origin and bifurcations of the orbits for each class

Intermittency of velocity time increments in turbulence

We analyze the statistics of turbulent velocity fluctuations in the time domain. Three cases are computed numerically and compared: (i) the time traces of Lagrangian fluid particles in a (3D) turbulent flow (referred to as the "dynamic" case); (ii) the time evolution of tracers advected by a frozen turbulent field (the "static" case), and (iii) the evolution in time of the velocity recorded at a fixed location in an evolving Eulerian velocity field, as it would be measured by a local probe (referred to as the "virtual probe" case). We observe that the static case and the virtual probe cases share many properties with Eulerian velocity statistics. The dynamic (Lagrangian) case is clearly different; it bears the signature of the global dynamics of the flow.Comment: 5 pages, 3 figures, to appear in PR