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Barriers to front propagation in ordered and disordered vortex flows
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Image of FIG. 1.
FIG. 1.

Chain of alternating vortices.

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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, .

Image of FIG. 5.
FIG. 5.

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 .

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

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.

Image of FIG. 14.
FIG. 14.

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.

Image of FIG. 15.
FIG. 15.

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

Image of FIG. 16.
FIG. 16.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Barriers to front propagation in ordered and disordered vortex flows