Front propagation and mode-locking in an advection-reaction-diffusion system

Mode-locking in advection-reaction-diffusion systems: An invariant manifold perspective

Fronts propagating in two-dimensional advection-reaction-diffusion systems exhibit a rich topological structure. When the underlying fluid flow is periodic in space and time, the reaction front can lock to the driving frequency. We explain this mode-locking phenomenon using the so-called burning invariant manifolds (BIMs). In fact, the mode-locked profile is delineated by a BIM attached to a relative periodic orbit (RPO) of the front element dynamics. Changes in the type (and loss) of mode-locking can be understood in terms of local and global bifurcations of the RPOs and their BIMs. We illustrate these concepts numerically using a chain of alternating vortices in a channel geometry.

Front propagation in a regular vortex lattice: Dependence on the vortex structure

We investigate the dependence on the vortex structure of the propagation of fronts in stirred flows. For this, we consider a regular set of vortices whose structure is changed by varying both their boundary conditions and their aspect ratios. These configurations are investigated experimentally in autocatalytic solutions stirred by electroconvective flows and numerically from kinematic simulations based on the determination of the dominant Fourier mode of the vortex stream function in each of them. For free lateral boundary conditions, i.e., in an extended vortex lattice, it is found that both the flow structure and the front propagation negligibly depend on vortex aspect ratios. For rigid lateral boundary conditions, i.e., in a vortex chain, vortices involve a slight dependence on their aspect ratios which surprisingly yields a noticeable decrease of the enhancement of front velocity by flow advection. These different behaviors reveal a sensitivity of the mean front velocity on the flow subscales. It emphasizes the intrinsic multiscale nature of front propagation in stirred flows and the need to take into account not only the intensity of vortex flows but also their inner structure to determine front propagation at a large scale. Differences between experiments and simulations suggest the occurrence of secondary flows in vortex chains at large velocity and large aspect ratios.

Avalanches dynamics in reaction fronts in disordered flows

We report on numerical studies of avalanches of an autocatalytic reaction front in a porous medium. The front propagation is controlled by an adverse flow resulting in upstream, static, or downstream regimes. In an earlier study focusing on front shape, we identified three different universality classes associated with this system by following the front dynamics experimentally and numerically. Here, using numerical simulations in the vicinity of the second-order transition, we identify an avalanche dynamics characterized by power-law distributions of avalanche sizes, durations, and lateral extensions. The related exponents agree well with the quenched-Kardar-Parisi-Zhang theory, which describes the front dynamics. However, the geometry of the propagating front differs slightly from that of the theoretical one. We show that this discrepancy can be understood in terms of the nonquasistatic correction induced by the finite front velocity.

Front propagation in a vortex lattice: dependence on boundary conditions and vortex depth

We experimentally address the propagation of reaction-diffusion fronts in vortex lattices by combining, in a Hele-Shaw cell and at low Reynolds number, forced electroconvective flows and an autocatalytic reaction in solution. We consider both vortex chains and vortex arrays, the former referring to mixed free/rigid boundary conditions for vortices and the latter to free boundary conditions. Varying the depth of the fluid layer, we observe no variation of the mean front velocities for vortex arrays and a noticeable variation for vortex chains. This questions the two-dimensional character of front propagation in low Reynolds number vortex lattices, as well as the mechanisms of this dependence.

Frozen reaction fronts in steady flows: A burning-invariant-manifold perspective

The dynamics of fronts, such as chemical reaction fronts, propagating in two-dimensional fluid flows can be remarkably rich and varied. For time-invariant flows, the front dynamics may simplify, settling in to a steady state in which the reacted domain is static, and the front appears "frozen." Our central result is that these frozen fronts in the two-dimensional fluid are composed of segments of burning invariant manifolds, invariant manifolds of front-element dynamics in xyθ space, where θ is the front orientation. Burning invariant manifolds (BIMs) have been identified previously as important local barriers to front propagation in fluid flows. The relevance of BIMs for frozen fronts rests in their ability, under appropriate conditions, to form global barriers, separating reacted domains from nonreacted domains for all time. The second main result of this paper is an understanding of bifurcations that lead from a nonfrozen state to a frozen state, as well as bifurcations that change the topological structure of the frozen front. Although the primary results of this study apply to general fluid flows, our analysis focuses on a chain of vortices in a channel flow with an imposed wind. For this system, we present both experimental and numerical studies that support the theoretical analysis developed here.

Finite-time barriers to front propagation in two-dimensional fluid flows

Recent theoretical and experimental investigations have demonstrated the role of certain invariant manifolds, termed burning invariant manifolds (BIMs), as one-way dynamical barriers to reaction fronts propagating within a flowing fluid. These barriers form one-dimensional curves in a two-dimensional fluid flow. In prior studies, the fluid velocity field was required to be either time-independent or time-periodic. In the present study, we develop an approach to identify prominent one-way barriers based only on fluid velocity data over a finite time interval, which may have arbitrary time-dependence. We call such a barrier a burning Lagrangian coherent structure (bLCS) in analogy to Lagrangian coherent structures (LCSs) commonly used in passive advection. Our approach is based on the variational formulation of LCSs using curves of stationary "Lagrangian shear," introduced by Farazmand et al. [Physica D 278-279, 44 (2014)] in the context of passive advection. We numerically validate our technique by demonstrating that the bLCS closely tracks the BIM for a time-independent, double-vortex channel flow with an opposing "wind."

Experimental studies of coherent structures in an advection-reaction-diffusion system

We present experimental studies of reaction front propagation in a single vortex flow with an imposed external wind. The fronts are produced by the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. The flow is generated using an electromagnetic forcing technique: an almost-radial electrical current interacts with a magnetic field from a magnet below the fluid layer to produce the vortex. The magnet is mounted on crossed translation stages allowing for movement of the vortex through the flow. Reaction fronts triggered in or in front of the moving vortex form persistent structures that are seen experimentally for time-independent (constant motion), time-periodic, and time-aperiodic flows. These results are examined with the use of burning invariant manifolds that act as one-way barriers to front motion in the flows. We also explore the usefulness of finite-time Lyapunov exponent fields as an instrument for analyzing front propagation behavior in a fluid flow.

Front propagation in cellular flows for fast reaction and small diffusivity

We investigate the influence of fluid flows on the propagation of chemical fronts arising in Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type models. We develop an asymptotic theory for the front speed in a cellular flow in the limit of small molecular diffusivity and fast reaction, i.e., large Péclet (Pe) and Damköhler (Da) numbers. The front speed is expressed in terms of a periodic path--an instanton--that minimizes a certain functional. This leads to an efficient procedure to calculate the front speed, and to closed-form expressions for (logPe)(-1) ≪ Da ≪ Pe and for Da ≫ Pe. Our theoretical predictions are compared with (i) numerical solutions of an eigenvalue problem and (ii) simulations of the advection-diffusion-reaction equation.

Chemical patterns in translating vortices: inter- and intra-cellular mixing effects

Stationary chemical patterns-flow distributed oscillations (FDOs)-are obtained when the Belousov-Zhabotinsky reaction is coupled with translating vortex flow in a Vortex Flow Reactor. For certain conditions, the FDOs are unstable with the observation of disappearing bands or complex patterns. The transitions between modes of pattern formation are reproduced in a modified Oregonator model consisting of two-zone cells connected in series. We show that increasing inter-cellular mixing of the outer zones results in a transition from FDO to absolute instabilities (AI) and increasing intra-cellular mixing between the core and outer zones can drive the reverse transition between modes (AI to FDO).

Attraction and repulsion of spiral waves by inhomogeneity of conduction anisotropy--a model of spiral wave interaction with electrical remodeling of heart tissue

Various forms of heart disease are associated with remodeling of the heart muscle, which results in a perturbation of cell-to-cell electrical coupling. These perturbations may alter the trajectory of spiral wave drift in the heart muscle. We investigate the effect of spatially extended inhomogeneity of transverse cell coupling on the spiral wave trajectory using a simple active media model. The spiral wave was either attracted or repelled from the center of inhomogeneity as a function of cell excitability and gradient of the cell coupling. High levels of excitability resulted in an attraction of the wave to the center of inhomogeneity, whereas low levels resulted in an escape and termination of the spiral wave. The spiral wave drift velocity was related to the gradient of the coupling and the initial position of the wave. In a diseased heart, a region of altered transverse coupling corresponds with local gap junction remodeling that may be responsible for stabilization-destabilization of spiral waves and hence reflect potentially important targets in the treatment of heart arrhythmias.

Invariant manifolds and the geometry of front propagation in fluid flows

Recent theoretical and experimental work has demonstrated the existence of one-sided, invariant barriers to the propagation of reaction-diffusion fronts in quasi-two-dimensional periodically driven fluid flows. These barriers were called burning invariant manifolds (BIMs). We provide a detailed theoretical analysis of BIMs, providing criteria for their existence, a classification of their stability, a formalization of their barrier property, and mechanisms by which the barriers can be circumvented. This analysis assumes the sharp front limit and negligible feedback of the front on the fluid velocity. A low-dimensional dynamical systems analysis provides the core of our results.

Barriers to front propagation in ordered and disordered vortex flows

We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.

Lamination and mixing in three fundamental flow sequences driven by electromagnetic body forces

This article pursues the idea that the degree of striations, called lamination, could be engineered to complement stretching and to design new sequential mixers. It explores lamination and mixing in three new mixing sequences experimentally driven by electromagnetic body forces. To generate these three mixing sequences, Lorentz body forces are dynamically controlled to vary the flow geometry produced by a pair of local jets. The first two sequences are inspired from the "tendril and whorl" and "blinking vortex" flows. The third novel sequence is called the "cat's eyes flip." These three mixing sequences exponentially stretch and laminate material lines representing the interface between two domains to be mixed. Moreover, the mixing coefficient (defined as 1-σ(2)/σ(0)(2) where σ(2)/σ(0)(2) is the rescaled variance) and its rate grow exponentially before saturation. This saturation of the mixing process is related to the departure of the mixing rate from an exponential growth when the striations' thicknesses reach the diffusive length scale of the measurements or species and dyes. Incidentally, in our experiments, for the same energy or forcing input, the cat's eyes flip sequence has higher lamination, stretching, and mixing rates than the tendril and whorl and the blinking vortex sequences. These features show that bakerlike in situ mixers can be conceived by dynamically controlling a pair of local jets and by integrating lamination during stirring stages with persistent geometries. Combined with novel insights provided by the quantification of the lamination, this paper should offer perspectives for the development of new sequential mixers, possibly on all scales.

Propagation of a chemical wave front in a quasi-two-dimensional superdiffusive flow

Pattern formation in reaction-diffusion systems is an important self-organizing mechanism in nature. Dynamics of systems with normal diffusion do not always reflect the processes that take place in real systems when diffusion is enhanced by a fluid flow. In such reaction-diffusion-advection systems diffusion might be anomalous for certain time and length scales. We experimentally study the propagation of a chemical wave occurring in a Belousov-Zhabotinsky reaction subjected to a quasi-two-dimensional chaotic flow created by the Faraday experiment. We present a novel analysis technique for the local expansion of the active wave front and find evidence of its superdiffusivity. In agreement with these findings the variance σ(2)(t)∝t(γ) of the reactive wave grows supralinear in time with an exponent γ>2. We study the characteristics of the underlying flow with microparticles. By statistical analysis of particle trajectories we derive flight time and jump length distributions and find evidence that tracer-particles undergo complex trajectories related to Lévy statistics. The propagation of active and passive media in the flow is compared.

Inward propagating chemical waves in Taylor vortices

Advection-reaction-diffusion (ARD) waves in the Belousov-Zhabotinsky reaction in steady Taylor-Couette vortices have been visualized using magnetic-resonance imaging and simulated using an adapted Oregonator model. We show how propagating wave behavior depends on the ratio of advective, chemical and diffusive time scales. In simulations, inward propagating spiral flamelets are observed at high Damköhler number (Da). At low Da, the reaction distributes itself over several vortices and then propagates inwards as contracting ring pulses--also observed experimentally.

Chemical activity induces dynamical force with global structure in a reaction-diffusion-convection system

We found a rotating global structure induced by the dynamical force of local chemical activity in a thin solution layer of excitable Belousov-Zhabotinsky reaction coupled with diffusion. The surface flow and deformation associated with chemical spiral waves (wavelength about 1 mm) represents a global unidirectional structure and a global tilt in the entire Petri dish (100 mm in diameter), respectively. For these observations, we scanned the condition of hierarchal pattern selection. From this result, the bromomalonic acid has an important role to induce the rotating global structure. An interaction between a reaction-diffusion process and a surface-tension-driven effect leads to such hierarchal pattern with different scales.

Design and control of patterns in reaction-diffusion systems

We discuss the design of reaction-diffusion systems that display a variety of spatiotemporal patterns. We also consider how these patterns may be controlled by external perturbation, typically using photochemistry or temperature. Systems treated include the Belousov-Zhabotinsky (BZ) reaction, the chlorite-iodide-malonic acid and chlorine dioxide-malonic acid-iodine reactions, and the BZ-AOT system, i.e., the BZ reaction in a water-in-oil reverse microemulsion stabilized by the surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT).

Front propagation in a laminar cellular flow: shapes, velocities, and least time criterion

We experimentally investigate the propagation of chemical fronts in steady laminar cellular flows at large Péclet numbers and large Damköhler numbers. Fronts are generated in an aqueous solution by an autocatalytic oxydoreduction reaction. They propagate in a channel in which a chain of counter-rotative parallel vortices is induced by electroconvection. We first accurately determine the form, the dynamics and the mean velocity of these fronts in the whole Hele-Shaw regime of the flow. We then address the modeling of the evolution of their mean velocity with the flow amplitude. The structure of the front wakes yields us to reject an effective reaction-diffusion wave as a relevant model for large-scale front propagation. On the other hand, analysis of the role of front heads brings us to introduce a kinematic model at the vortex scale for uncovering the front dynamics. This model addresses the propagation of the front leading point in a chain of vortices whose field is modeled by a two-dimensional solid rotation complemented by a boundary layer. Interestingly, it sensitively relies on the effective trajectory followed by the front leading point. To account for this, a competition is worked out among a one-parameter family of potential trajectories. The actual trajectory is then selected as the fastest one with quite a good agreement with measurements and observations. In particular, the measured effective front velocities are well recovered from the model, including their intrinsic dependence on the boundary layer width. Accordingly, effective front propagation in a laminar steadily stirred medium is thus understood from an optimization principle similar to the Fermat principle of ray propagation in heterogeneous media.

Chemical reaction fronts in ordered and disordered cellular flows with opposing winds

We present experiments on the motion of chemical fronts in ordered and disordered vortex flows with imposed uniform winds. Fronts in a chain of alternating vortices are found to freeze (pin to the separatrix) for a wide range of opposing winds that grows nonlinearly with the characteristic strength of the underlying vorticity. Experiments in spatially disordered flows demonstrate that freezing of fronts is common to cellular flows; furthermore, it is not dependent on boundary conditions. We therefore anticipate similar pinning in a wide range of 2D cellular flows and front-producing systems.

Effective dimensions and chemical reactions in fluid flows

We show that chemical activity in hydrodynamical flows can be understood as the outcome of three basic effects: the stirring protocol of the flow, the local properties of the reaction, and the global folding dynamics which also depends on the geometry of the container. The essence of each of these components can be described by simple functional relations. In an ordinary differential equation approach, they determine a new chemical rate equation for the concentration, which turns out to be coupled to the dynamics of an effective fractal dimension. The theory predicts an exponential convergence to the asymptotic chemical state. This holds even in flows characterized by a linear stirring protocol where transient fractal patterns are shown to exist despite the lack of any chaotic set of the advection dynamics. In the exponential case the theory applies to flows of chaotic time dependence (chaotic flows) as well.

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 and/or boundary conditions for the flow.

Active media under rotational forcing

The bubble-free Belousov-Zhabotinsky reaction has been used to study the effects of centrifugal forces on autowave propagation. The reaction parameters were chosen such that the system oscillates naturally creating target waves. In the present study, the system was forced to rotate with a constant velocity around a central axis. In studying the effects of such a forcing on the system, we focused on target dynamics. The system reacts to this forcing in different ways, the most spectacular being a dramatic increase in the period of the target, the effect growing stronger as we move away from the center of rotation. A numerical study was carried out using the two-variable Oregonator model, modified to include convective effects through the diffusion coefficient. The numerical results showed a good qualitative agreement with those of the experiments.

Effective front propagation in steady cellular flows: A least time criterion

We experimentally study the propagation of a reaction front in a chain of counter-rotating vortices. The front is induced by an autocatalytic chemical reaction in an aqueous solution stirred by electroconvective means. The front propagates by getting quickly engulfed in a vortex and by slowly crossing the separatrix to the next vortex. Its mean velocity along the chain increases with the flow intensity but shows both a bending and a dependence on the vortex aspect ratio. We recover these features within a kinematic model of front propagation by seeking the quickest front path through a vortex.

Chemical transients in closed chaotic flows: the role of effective dimensions

We investigate chemical activity in hydrodynamical flows in closed containers. In contrast to open flows, in closed flows the chemical field does not show a well-defined fractal property; nevertheless, there is a transient filamentary structure present. We show that the effect of the filamentary patterns on the chemical activity can be modeled by the use of time-dependent effective dimensions. We derive a new chemical rate equation, which turns out to be coupled to the dynamics of the effective dimension, and predicts an exponential convergence. Previous results concerning activity in open flows are special cases of this new rate equation.