Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model

Pattern formation in biological, chemical and physical problems has received considerable attention, with much attention paid to dissipative systems. For example, the Ginzburg--Landau equation is a normal form that describes pattern formation due to the appearance of a single mode of instability in a wide variety of dissipative problems. In a similar vein, a certain "single-wave model" arises in many physical contexts that share common pattern forming behavior. These systems have Hamiltonian structure, and the single-wave model is a kind of Hamiltonian mean-field theory describing the patterns that form in phase space. The single-wave model was originally derived in the context of nonlinear plasma theory, where it describes the behavior near threshold and subsequent nonlinear evolution of unstable plasma waves. However, the single-wave model also arises in fluid mechanics, specifically shear-flow and vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter physics, and other Hamiltonian theories characterized by mean field interaction. We demonstrate, by a suitable asymptotic analysis, how the single-wave model emerges from a large class of nonlinear advection-transport theories. An essential ingredient for the reduction is that the Hamiltonian system has a continuous spectrum in the linear stability problem, arising not from an infinite spatial domain but from singular resonances along curves in phase space whereat wavespeeds match material speeds (wave-particle resonances in the plasma problem, or critical levels in fluid problems). The dynamics of the continuous spectrum is manifest as the phenomenon of Landau damping when the system is ... Such dynamical phenomena have been rediscovered in different contexts, which is unsurprising in view of the normal-form character of the single-wave model

Rotation shields chaotic mixing regions from no-slip walls

We report on the decay of a passive scalar in chaotic mixing protocols where the wall of the vessel is rotated, or a net drift of fluid elements near the wall is induced at each period. As a result the fluid domain is divided into a central isolated chaotic region and a peripheral regular region. Scalar patterns obtained in experiments and simulations converge to a strange eigenmode and follow an exponential decay. This contrasts with previous experiments [Gouillart et al., Phys. Rev. Lett. 99, 114501 (2007)] with a chaotic region spanning the whole domain, where fixed walls constrained mixing to follow a slower algebraic decay. Using a linear analysis of the flow close to the wall, as well as numerical simulations of Lagrangian trajectories, we study the influence of the rotation velocity of the wall on the size of the chaotic region, the approach to its bounding separatrix, and the decay rate of the scalar.Comment: 4 pages, 12 figures, RevTeX 4 styl

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

Moving walls accelerate mixing

Mixing in viscous fluids is challenging, but chaotic advection in principle allows efficient mixing. In the best possible scenario,the decay rate of the concentration profile of a passive scalar should be exponential in time. In practice, several authors have found that the no-slip boundary condition at the walls of a vessel can slow down mixing considerably, turning an exponential decay into a power law. This slowdown affects the whole mixing region, and not just the vicinity of the wall. The reason is that when the chaotic mixing region extends to the wall, a separatrix connects to it. The approach to the wall along that separatrix is polynomial in time and dominates the long-time decay. However, if the walls are moved or rotated, closed orbits appear, separated from the central mixing region by a hyperbolic fixed point with a homoclinic orbit. The long-time approach to the fixed point is exponential, so an overall exponential decay is recovered, albeit with a thin unmixed region near the wall.Comment: 17 pages, 13 figures. PDFLaTeX with RevTeX 4-1 styl

Topological entropy and secondary folding

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.Comment: 13 pages, 8 figures. pdfLaTeX with RevTeX4 macro

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

Can phoretic particles swim in two dimensions?

Artificial phoretic particles swim using self-generated gradients in chemical species (self-diffusiophoresis) or charges and currents (self-electrophoresis). These particles can be used to study the physics of collective motion in active matter and might have promising applications in bioengineering. In the case of self-diffusiophoresis, the classical physical model relies on a steady solution of the diffusion equation, from which chemical gradients, phoretic flows, and ultimately the swimming velocity may be derived. Motivated by disk-shaped particles in thin films and under confinement, we examine the extension to two dimensions. Because the two-dimensional diffusion equation lacks a steady state with the correct boundary conditions, Laplace transforms must be used to study the long-time behavior of the problem and determine the swimming velocity. For fixed chemical fluxes on the particle surface, we find that the swimming velocity ultimately always decays logarithmically in time. In the case of finite Péclet numbers, we solve the full advection-diffusion equation numerically and show that this decay can be avoided by the particle moving to regions of unconsumed reactant. Finite advection thus regularizes the two-dimensional phoretic problem.The research was supported by NSF Grants DMS-1109315 and DMS-1147523 (Madison) and by the European Union through a CIG grant (Cambridge)

Open-flow mixing: Experimental evidence for strange eigenmodes

We investigate experimentally the mixing dynamics in a channel flow with a finite stirring region undergoing chaotic advection. We study the homogenization of dye in two variants of an eggbeater stirring protocol that differ in the extent of their mixing region. In the first case, the mixing region is separated from the side walls of the channel, while in the second it extends to the walls. For the first case, we observe the onset of a permanent concentration pattern that repeats over time with decaying intensity. A quantitative analysis of the concentration field of dye confirms the convergence to a self-similar pattern, akin to the strange eigenmodes previously observed in closed flows. We model this phenomenon using an idealized map, where an analysis of the mixing dynamics explains the convergence to an eigenmode. In contrast, for the second case the presence of no-slip walls and separation points on the frontier of the mixing region leads to non-self-similar mixing dynamics.Comment: 12 pages, 8 figures

The influence of periodic islands in the flow on a scalar tracer in the presence of a steady source

Copyright © 2009 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Physics of Fluids 21 (2009) and may be found at http://link.aip.org/link/?PHFLE6/21/067103/1In this paper we examine the influence of periodic islands within a time periodic chaotic flow on the evolution of a scalar tracer. The passive scalar tracer is injected into the flow field by means of a steady source term. We examine the distribution of the tracer once a periodic state is reached, in which the rate of injected scalar balances advection and diffusion with the molecular diffusion К. We study the two-dimensional velocity field u(x,y,t)=2 cos2(ωt)(0,sin χ)+2 sin2(ωt)(sin y,0). As ω is reduced from an O(1) value the flow alternates through a sequence of states which are either globally chaotic, or contain islands embedded in a chaotic sea. The evolution of the scalar is examined numerically using a semi-Lagrangian advection scheme. By time-averaging diagnostics measured from the scalar field we find that the time-averaged lengths of the scalar contours in the chaotic region grow like К−1/2 for small К, for all values of ω, while the behavior of the time-averaged maximum scalar value, Cmax, for small К depends strongly on ω. In the presence of islands Cmax˜К−α for some α between 0 and 1 and with К small, and we demonstrate that there is a correlation between α and the area of the periodic islands, at least for large ω. The limit of small ω is studied by considering a flow field that switches from u=(0,2 sin x) to u=(2 sin y ,0) at periodic intervals. The small К limit for this flow is examined using the method of matched asymptotic expansions. Finally the role of islands in the flow is investigated by considering the time-averaged effective diffusion of the scalar field. This diagnostic can distinguish between regions where the scalar is well mixed and regions where the scalar builds up