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Emergent reaction-diffusion phenomena in capillary tubes
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View: Figures


Image of FIG. 1.
FIG. 1.

A snapshot of autonomously formed wave patterns in capillary tubes (inner diameter ) containing soft-silica gel with ferroin catalyst. This image was captured after immersing the tubes into the reactant solution.

Image of FIG. 2.
FIG. 2.

Space-time representation of diffusion-driven wave patterns in a capillary tube: x axis, space; y axis, time; the time increases in the downwards direction. White and black indicate the oxidized (i.e., excited) and the reduced (i.e., unexcited) state of the ferroin catalyst, respectively. Each space-time frame consists of 256 horizontal data extracted along the center axis of the capillary at intervals (i.e., the span of each frame is 6 min) and each spatial datum is an average of two pixel rows. The tube length is .

Image of FIG. 3.
FIG. 3.

Top: Digitally enhanced gray-scale density plots from tomography layers (tube cross section) 5, 15, 25, 35, 45, 65, and 95. Bottom: Core radius as a function of distance from the open capillary tube end and its fitted curve approximation (dashed line). The total tube length is .

Image of FIG. 4.
FIG. 4.

Space-time plots from numerical 1D simulation exhibiting firing-type wave patterns (axes as in Fig. 2 ). Elevated levels of variable are shown as white and shades of gray. Equations (5)–(9) were integrated on a 1D grid using the explicit fourth-order Runge-Kutta method and utilizing three-point central-differences approximation for the 1D Laplacian operator (Ref. 36 ). Numerical parameters: grid length, 1024 points; grid spacing, ; time step (dimensionless), ; sampling interval, ; number of samples in each graph, 3333 (2083 time units). We considered a dimensionless space with immobilized catalyst and therefore set . Then, we fix the linear diffusion coefficients of and bromide to unity, , and use relative value (Table I in Ref. 37 ) for and bromomalonic acid. The initial conditions were uniformly throughout the medium and the values for other species set to zero except at the grid point on the left boundary . In order to simulate open tube end in contact with the BZR reactant solution, Dirichlet boundary conditions were applied to the left boundary with , , , and . The right boundary is zero flux (von Neumann conditions). The behavior remained unchanged when the time step was reduced to half.

Image of FIG. 5.
FIG. 5.

Normalized concentration profiles from the simulation shown in Fig. 4 at (a) , (b) , (c) , and (d) . Preceding wave leaves a slowly decaying region of [W] in its wake which causes annihilation of the subsequent wave.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Emergent reaction-diffusion phenomena in capillary tubes