Self-Similar Spatiotemporal Structure of Intermaterial Boundaries in Chaotic Flows

Álvarez, Mario Moisés; Muzzio, Fernando J.; Cerbelli, Stefano; Adrover, Alessandra; Giona, Massimiliano

doi:10.1103/physrevlett.81.3395

Cited by 113 publications

(90 citation statements)

References 17 publications

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“…Also the direction of the stretching fluctuates and foldings of the filament may lead to large curvatures whose effect is not captured by our one-dimensional description. Another effect is the non-uniform density of the unstable foliation pointed out by Alvarez et al [38]. Thus the advected filament can overlap with itself well before it fills the whole domain.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…This is because the instantaneous stretching rate fluctuates and the increase of the total length is dominated by the growth of a line elements that experience a faster than average stretching. In dynamical systems language the contour lengthening rate θ is given by the topological entropy [17,37,38] of the advection dynamics.…”

Section: The Lagrangian Filament Slice Modelmentioning

confidence: 99%

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Chaotic mixing induced transitions in reaction–diffusion systems

Neufeld

Haynes

Tél

2002

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study the evolution of a localized perturbation in a chemical system with multiple homogeneous steady states, in the presence of stirring by a fluid flow. Two distinct regimes are found as the rate of stirring is varied relative to the rate of the chemical reaction. When the stirring is fast localized perturbations decay towards a spatially homogeneous state. When the stirring is slow (or fast reaction) localized perturbations propagate by advection in form of a filament with a roughly constant width and exponentially increasing length. The width of the filament depends on the stirring rate and reaction rate but is independent of the initial perturbation. We investigate this problem numerically in both closed and open flow systems and explain the results using a one-dimensional "mean-strain" model for the transverse profile of the filament that captures the interplay between the propagation of the reaction-diffusion front and the stretching due to chaotic advection.The problem of chemical or biological activity in fluid flows has recently become an area of active research [1][2][3][4][5][6][7][8]. Apart from theoretical interest this problem has a number of industrial [9] and environmental [10,11] applications. One of the simplest manifestation of non-linear behavior in reaction-diffusion systems is the possibility of travelling front solutions. In this paper we study the effect of chaotic mixing, by an unsteady laminar flow, in reaction-diffusion systems supporting front propagation.

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Section: Summary and Discussionmentioning

confidence: 99%

Section: The Lagrangian Filament Slice Modelmentioning

confidence: 99%

Chaotic mixing induced transitions in reaction–diffusion systems

Neufeld

Haynes

Tél

2002

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The general mechanisms that make laminar mixing effective, independently of the process being either natural or artificial, are the repetitive stretching and folding of material elements leading to a reduction of the characteristic length scales, 6 and to an increase of the intermaterial contact area. 7 At smaller length scales the effect of molecular diffusion becomes more significant, under the influence of which homogenization of the mixture occurs.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

2012

Self Cite

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This paper extends the mapping matrix formalism to include the effects of molecular diffusion in the analysis of mixing processes in chaotic flows. The approach followed is Lagrangian, by considering the stochastic formulation of advection-diffusion processes via the Langevin equation for passive fluid particle motion. In addition, the inclusion of diffusional effects in the mapping matrix formalism permits to frame the spectral properties of mapping matrices in the purely convective limit in a quantitative way. Specifically, the effects of coarse graining can be quantified by means of an effective Péclet number that scales as the second power of the linear lattice size. This simple result is sufficient to predict the scaling exponents characterizing the behavior of the eigenvalue spectrum of the advection-diffusion operator in chaotic flows as a function of the Péclet number, exclusively from purely kinematic data, by varying the grid resolution. Simple but representative model systems and realistic physically realizable flows are considered under a wealth of different kinematic conditions-from the presence of large quasi-periodic islands intertwined by chaotic regions, to almost globally chaotic conditions, to flows possessing "sticky islands"-providing a fairly comprehensive characterization of the different numerical phenomenologies that may occur in the quantitative analysis of mapping matricesultimately of chaotic mixing processes. © 2012 American Institute of Physics

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“…First, there are two obvious candidates: an (averaged) Lyapunov exponent, which represents local stretching, and the topological entropy, which represents the growth rate of finite material lines, and which generally exceeds the Lyapunov exponent [43][44][45]. Second, the distribution of stretch rates along any material line is nonuniform [24], which causes difficulties in choosing a single value for µ.…”

Section: Single Planar Interfacementioning

confidence: 99%

“…In our simulations of this model, we shall choose to take λ to be the Lyapunov exponent of the sine flow, although it can be argued that it is more appropriate to take instead the (slightly larger) topological entropy [43,44]. Our justification for this choice is that clearly the separation of reaction and diffusion on the one hand, and stretching and folding on the other is itself a crude device, and so the impression of precision in the specification of τ = (log 2)/λ is illusory; and, of course, many features of the two-dimensional flow are not captured at all by any of our lamellar models (the nonuniformity of stretching in the sine flow; the presence of regular islands; the curvature of the striations, for instance).…”

Section: Lamellar Model With Discrete Stretching and 'Folding' (Bakermentioning

confidence: 99%

Chaotic mixing of a competitive–consecutive reaction

Cox

2004

Physica D: Nonlinear Phenomena

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The evolution of a competitive-consecutive chemical reaction is computed numerically in a two-dimensional chaotic fluid flow with initially segregated reactants. Results from numerical simulations are used to evaluate a variety of reduced models commonly adopted to model the full advection-reaction-diffusion problem. Particular emphasis is placed upon fast reactions, where the yield varies most significantly with Péclet number (the ratio of diffusive to advective time scales). When effects of the fluid mechanical mixing are strongest, we find that the yield of the reaction is underestimated by a one-dimensional lamellar model that ignores the effects of fluid mixing, but overestimated by two other lamellar models that include fluid mixing.

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Self-Similar Spatiotemporal Structure of Intermaterial Boundaries in Chaotic Flows

Cited by 113 publications

References 17 publications

Chaotic mixing induced transitions in reaction–diffusion systems

Chaotic mixing induced transitions in reaction–diffusion systems

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

Chaotic mixing of a competitive–consecutive reaction

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