Chaotic mixing induced transitions in reaction–diffusion systems

Abstract

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.

Homogenization induced by chaotic mixing and diffusion in an oscillatory chemical reaction

“…A similar result is obtained in Ref. [17] for an autocatalytic reaction. For larger values of Da, the concentrations remain oscillatory within the filaments that develop in the flow.…”

“…A one-dimensional Lagrangian model has previously proved to be useful in giving insight into the behavior of two-dimensional (2D) advection-reaction-diffusion problems [16][17][18][19][20] (for earlier work see Refs. [21][22][23][24][25][26]).…”

Time scales and species coexistence in chaotic flows

“…Species coexistence is however also found in stirred and seemingly homogeneous oceanic systems [3]. Horizontal mixing and turbulent flows have been offered as possible explanations, supported by theoretical models [4][5][6][7][8] and satellite observations of chlorophyll concentrations off the coast of South America [9,10]. Broadly speaking a flowing medium generates stable niches, spatially separated, and sufficiently long lived to let individual organisms thrive.…”

“…On the other hand stirring must not be too quick, as premature mixing would impede coexistence. Several studies suggest that patchiness in plankton populations is related to the Damköhler number characterising the ratio between the time scales associated with the flow and the biological processes respectively [5,6,11]. Theoretical models have been used to study the dynamics of species coexistence in flowing environments [6][7][8][12][13][14][15][16].…”

Chaotic mixing of a competitive–consecutive reaction

“…When the reactions take place in a liquid phase that is stirred continuously, their progress correspondingly depends strongly upon the details of the fluid mechanical mixing [1][2][3][4][5][6][7][8][9]. In this paper, we examine the influence of fluid mixing upon a competitiveconsecutive (or series-parallel) reaction, A + B → R, B + R → S [1], that takes place in a two-dimensional, laminar, chaotic fluid flow.…”