^{1,a)}and Tom Solomon

^{1,b)}

### Abstract

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 flowvelocities. For the disordered flow, the measured BIMs are compared to those predicted using a measuredvelocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.

^{1}plankton blooms,

^{2}epidemics,

^{3}phase transitions in matter,

^{4}and chemical reactions.

^{5,6}The behavior of reaction fronts has been well-studied in the

*reaction-diffusion*(RD) limit, i.e., in a stagnant system in which molecular diffusion is the dominant mixing mechanism. In many real systems, however, the propagation of fronts is strongly influenced by the presence of fluid flows. The behavior of these more general

*advection-reaction-diffusion*(ARD) systems has been studied extensively for turbulent flows, especially for engineering applications involving turbulent combustion. The propagation of fronts in laminar flows or in flows with large coherent structures, however, is still poorly understood. This is a problem with significant applications, e.g., in microfluidic chemical reactions,

^{7}cellular or embryonic scale biological systems,

^{8,9}and population dynamics in oceanic flows with large-scale coherent vortices.

^{10}In this paper, we present experiments on front propagation in a simple, laminar flow composed of a chain of alternating vortices and in a spatially disordered, vortex-dominated flow. We show that front propagation is dominated by local barriers in the flow called

*burning invariant manifolds*

^{11}(BIMs) which could form the basis for a general theory of front propagation in ARD systems. These BIMs are generalizations of invariant manifolds that act as barriers to transport of passive impurities. The BIMs are shown to be directional—they block fronts propagating in one direction but allow fronts propagating the opposite way to pass through. We demonstrate experimentally how BIMs can be measured in both time-independent and time-periodic fluid flows, and we show that the BIMs collapse onto the invariant manifolds for passive transport in the limit of large flowvelocities.

The authors would like to thank John Mahoney and Kevin Mitchell for their assistance in guiding both our theoretical and experimental inquiry in this work. These studies are supported by the U.S. National Science Foundation under Grants DMR-1004744 and PHY-0552790.

I. INTRODUCTION

II. BACKGROUND

A. Passive mixing and invariant manifolds

B. Front propagation

C. Burning invariant manifolds

III. EXPERIMENTAL TECHNIQUES

IV. RESULTS: VORTEX CHAIN FLOW

V. DISORDERED VORTEX ARRAY

VI. DISCUSSION

### Key Topics

- Rotating flows
- 60.0
- Manifolds
- 55.0
- Chemically reactive flows
- 38.0
- Laminar flows
- 11.0
- Chemical reactions
- 9.0

##### F23

## Figures

Chain of alternating vortices.

Chain of alternating vortices.

Invariant manifolds for passive transport in vortex chain flow. The vertical (black) lines show the boundaries between adjacent vortices; these lines are also the invariant manifolds for passive transport for a time-independent (stationary) flow. The other (red) curve is an invariant manifold for a time-periodic flow where the vortices oscillate laterally.

Invariant manifolds for passive transport in vortex chain flow. The vertical (black) lines show the boundaries between adjacent vortices; these lines are also the invariant manifolds for passive transport for a time-independent (stationary) flow. The other (red) curve is an invariant manifold for a time-periodic flow where the vortices oscillate laterally.

Schematics of the disordered vortex flow apparatus. Top: Side view of apparatus. A spatially disordered array of magnets is placed under a container of electrolytic solution. A DC electrical current is passed through the solution using two electrodes. Bottom: Top view of the apparatus. Magnetic field is shown entering or exiting the plane of the image.

Schematics of the disordered vortex flow apparatus. Top: Side view of apparatus. A spatially disordered array of magnets is placed under a container of electrolytic solution. A DC electrical current is passed through the solution using two electrodes. Bottom: Top view of the apparatus. Magnetic field is shown entering or exiting the plane of the image.

Sequence of fronts for time-independent flows, along with the experimentally extracted BIMs (in red). (a) Maximum flow speed *U* = 0.045 cm/s, . (b) *U* = 0.090 cm/s, .

Sequence of fronts for time-independent flows, along with the experimentally extracted BIMs (in red). (a) Maximum flow speed *U* = 0.045 cm/s, . (b) *U* = 0.090 cm/s, .

Convergence of reaction front to a BIM. The blue symbols show the bottom-most edge of the downward-propagating reaction above the highest point of the right BIM in Fig. 4(a). The straight line (red) shows the speed at which a front would propagate in the absence of any flow. The ARD front slows to a speed an order of magnitude smaller than .

Convergence of reaction front to a BIM. The blue symbols show the bottom-most edge of the downward-propagating reaction above the highest point of the right BIM in Fig. 4(a). The straight line (red) shows the speed at which a front would propagate in the absence of any flow. The ARD front slows to a speed an order of magnitude smaller than .

Sequence of fronts for time-periodic flow with cm/s (), oscillating with frequency rad/s and (non-dimensional) amplitude . Time after trigger for fronts (in multiples of the oscillation period T) are (a) 0.3, 0.9, and 1.0; (b) 1.1 and 1.4; (c) 1.5 and 1.8; and (d) 1.9 and 2.1. (Each image includes the previous fronts.) Edge enhancements of the fronts are shown, in order to make it possible to see multiple fronts in the same image. The red curves in (a)-(c) show the experimentally extracted BIMs for the first two periods of oscillation of the vortex chain, and the additional red curve in (d) shows a continuation of the right-part of the BIMs after an additional oscillation period.

Sequence of fronts for time-periodic flow with cm/s (), oscillating with frequency rad/s and (non-dimensional) amplitude . Time after trigger for fronts (in multiples of the oscillation period T) are (a) 0.3, 0.9, and 1.0; (b) 1.1 and 1.4; (c) 1.5 and 1.8; and (d) 1.9 and 2.1. (Each image includes the previous fronts.) Edge enhancements of the fronts are shown, in order to make it possible to see multiple fronts in the same image. The red curves in (a)-(c) show the experimentally extracted BIMs for the first two periods of oscillation of the vortex chain, and the additional red curve in (d) shows a continuation of the right-part of the BIMs after an additional oscillation period.

Sequence of images showing BIMs for a time-periodic flow with cm/s (), rad/s, and . Time after trigger for fronts (in multiples of the oscillation period T) is (a) 0.42, (b) 0.55, (c) 0.87, (d) 1.10, (e) 1.60, and (f) 2.10. The red curves show the BIMs surrounding the bottom center Eulerian fixed point. (g) shows a superposition of gradient-enhanced images; ridges in this image help in determination of the BIM structure.

Sequence of images showing BIMs for a time-periodic flow with cm/s (), rad/s, and . Time after trigger for fronts (in multiples of the oscillation period T) is (a) 0.42, (b) 0.55, (c) 0.87, (d) 1.10, (e) 1.60, and (f) 2.10. The red curves show the BIMs surrounding the bottom center Eulerian fixed point. (g) shows a superposition of gradient-enhanced images; ridges in this image help in determination of the BIM structure.

Fronts triggered (a) to the left and (b) to the right of a pair of BIMs in a time-periodic vortex chain flow with the same experimental parameters as in Fig. 7. In each case, the front passes through the first BIM that it encounters and is then stopped by the second.

Fronts triggered (a) to the left and (b) to the right of a pair of BIMs in a time-periodic vortex chain flow with the same experimental parameters as in Fig. 7. In each case, the front passes through the first BIM that it encounters and is then stopped by the second.

Sequence of images of a front triggered at the left and propagating to the right in a time-periodic vortex chain flow; same experimental parameters as in Fig. 7. The images shown are taken in intervals of one period of oscillation. The red curves show the pair of BIMs immediately in front of the leading edge of the front. The front in this case mode-locks to the BIM structure.

Sequence of images of a front triggered at the left and propagating to the right in a time-periodic vortex chain flow; same experimental parameters as in Fig. 7. The images shown are taken in intervals of one period of oscillation. The red curves show the pair of BIMs immediately in front of the leading edge of the front. The front in this case mode-locks to the BIM structure.

Velocity field for the spatially disordered flow. (Top left) Irregular and sparse measured velocity data. (Top right) Velocity data interpolated to a uniform grid. Both panels show velocity field overlaid on a vorticity field (shown in red and blue) calculated from the interpolated velocity field. (Bottom) Enlarged velocity field, boxed region at bottom of top right image.

Velocity field for the spatially disordered flow. (Top left) Irregular and sparse measured velocity data. (Top right) Velocity data interpolated to a uniform grid. Both panels show velocity field overlaid on a vorticity field (shown in red and blue) calculated from the interpolated velocity field. (Bottom) Enlarged velocity field, boxed region at bottom of top right image.

Propagation of a BZ reaction front in the spatially disordered flow. The first three panels show the front at times , and , and 840 s after the trigger. The fourth panel shows the extracted leading edges of these three fronts.

Propagation of a BZ reaction front in the spatially disordered flow. The first three panels show the front at times , and , and 840 s after the trigger. The fourth panel shows the extracted leading edges of these three fronts.

Superposition of reaction fronts as they evolve in time. (a) Fronts extracted from a reaction triggered at a hyperbolic fixed point in the flow. An individual snapshot of the reaction is in white. Time-advanced reaction fronts appear in green. Experimentally extracted BIMs are in red. Each front shown is advanced in time by 10 s. (b) and (c) Propagation of fronts triggered away from the Eulerian fixed points. In these two images, the same BIM structures are shown (red). Arrows are added to indicate the direction of reaction propagation. Despite the fronts being triggered far away from the Eulerian fixed point, they still converge on the BIMs. Additionally one observes the left BIM blocking a left-propagating front and vice versa, while the other BIM is burned through (video online).;. [URL: http://dx.doi.org/10.1063/1.4746764.1] [URL: http://dx.doi.org/10.1063/1.4746764.2]10.1063/1.4746764.110.1063/1.4746764.2

Superposition of reaction fronts as they evolve in time. (a) Fronts extracted from a reaction triggered at a hyperbolic fixed point in the flow. An individual snapshot of the reaction is in white. Time-advanced reaction fronts appear in green. Experimentally extracted BIMs are in red. Each front shown is advanced in time by 10 s. (b) and (c) Propagation of fronts triggered away from the Eulerian fixed points. In these two images, the same BIM structures are shown (red). Arrows are added to indicate the direction of reaction propagation. Despite the fronts being triggered far away from the Eulerian fixed point, they still converge on the BIMs. Additionally one observes the left BIM blocking a left-propagating front and vice versa, while the other BIM is burned through (video online).;. [URL: http://dx.doi.org/10.1063/1.4746764.1] [URL: http://dx.doi.org/10.1063/1.4746764.2]10.1063/1.4746764.110.1063/1.4746764.2

BIMs measured with three values of the ratio ; , 0.11, and 0.18 for the blue, green, and red curves, respectively. The case is measured by detecting transport barriers for passive mixing.

BIMs measured with three values of the ratio ; , 0.11, and 0.18 for the blue, green, and red curves, respectively. The case is measured by detecting transport barriers for passive mixing.

Theoretical calculation (in blue) of the pair of BIMs, along with experimental BIMs (in red). While not all branches of these BIMs could be calculated, the branches shown are in good agreement with experiment.

Theoretical calculation (in blue) of the pair of BIMs, along with experimental BIMs (in red). While not all branches of these BIMs could be calculated, the branches shown are in good agreement with experiment.

A sample of experimentally measured BIMs for the disordered vortex array. (a) Green and red BIMs block fronts moving in opposite directions. Adjacent green and red BIMs are pairs arising from the splitting of the invariant manifold for passive mixing in that region. (b) Same experimental BIMs with theoretical calculations overlaid. Blue calculated BIMs match red experimental BIMs and magenta calculations match green (video online). [URL: http://dx.doi.org/10.1063/1.4746764.3]10.1063/1.4746764.3

A sample of experimentally measured BIMs for the disordered vortex array. (a) Green and red BIMs block fronts moving in opposite directions. Adjacent green and red BIMs are pairs arising from the splitting of the invariant manifold for passive mixing in that region. (b) Same experimental BIMs with theoretical calculations overlaid. Blue calculated BIMs match red experimental BIMs and magenta calculations match green (video online). [URL: http://dx.doi.org/10.1063/1.4746764.3]10.1063/1.4746764.3

These panels show a front pinwheeling around the projection kink of a theoretically calculated BIM (in red). Each panel is separated by 30 s (video online). [URL: http://dx.doi.org/10.1063/1.4746764.4]10.1063/1.4746764.4

These panels show a front pinwheeling around the projection kink of a theoretically calculated BIM (in red). Each panel is separated by 30 s (video online). [URL: http://dx.doi.org/10.1063/1.4746764.4]10.1063/1.4746764.4

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