Barriers to front propagation in laminar, three-dimensional fluid flows

We present experiments on one-way barriers that block reaction fronts in a fully three-dimensional (3D) fluid flow. Fluorescent Belousov-Zhabotinsky reaction fronts are imaged with laser-scanning in a laminar, overlapping vortex flow. The barriers are analyzed with a 3D extension to burning invariant manifold (BIM) theory that was previously applied to two-dimensional advection-reaction-diffusion processes. We discover tube and sheet barriers that guide the front evolution. The experimentally determined barriers are explained by BIMs calculated from a model of the flow.

Figure 1

Fluid cell, showing the magnetohydrodynamic forcing and the resulting flow formed from the superposition of a horizontal (red) and vertical (blue) chain of vortices. The fluid channel measures 0.80 $\times$ 8.0 cm horizontally with a height of 1.9 cm.

Figure 2

Sequences showing the evolution of reaction fronts for the middle two unit cells (4.0 cm) of the overlapping vortex flow. (a) $v_0=0.064$; the evolving front (viewed at an angle from above) is blocked by a quarter-tube, arch-like barrier that spans two vortices. The leading edge of this barrier is shown as a cyan curve in the 60 s image. (b) $v_0=0.16$; the evolving front (viewed at an angle from below) is blocked by a scroll-shaped, sheet barrier, the edge of which is shown in red in the 60 s image. Movies of these sequences can be found online in the Supplemental Material [29]. (c)-(d) Simulated burning invariant manifolds corresponding to the barriers seen in (a) and (b). The dots show the burning fixed points to which the BIMs are attached. The red (blue) arrows show the stable (unstable) directions of the fluid flow near the advective fixed point. A rotating animation of the theoretical BIM in (d) (in the Supplemental Material [29]) better illustrates its 3D nature.

Figure 3

Part of the quarter-tube, archlike barrier. (a) $v_0=0.16$; (b)-(c) $v_0=0.064$. The front left surface corresponds to the green grid in Fig. 2. In (a) and (b), the front enters the quarter-tube from the back and is blocked by the tube-barrier as it moves in the $-y$ direction near the top surface. In (c), a front moving in the $+y$ direction near the top penetrates into the quarter-tube. Panels (d) and (e) show numerically computed BIMs corresponding to the same viewing region and $v_0$ as columns a and b/c. An animation of the BIM in (d) is given in the Supplemental Materials [29].

Figure 4

Part of the sheet (scroll) barrier. (a) $v_0=0.064$; (b)-(c) $v_0=0.16$. The front right surface corresponds to the green grid in Fig. 2. In (a) and (b), a front above the barrier (partially drawn in cyan and red curves) is blocked from propagating downward by the barrier. In (c), a front propagating upward passes through the (one-way) barrier. Panels (d) and (e) show numerically computed BIMs corresponding to the same viewing region and $v_0$ as columns a and b/c. Compared to Fig. 2, part of a second BIM has also been plotted near $y=-0.5$.

Figure 5

(a) Illustration of a tubelike reaction barrier expected near an advective fixed point with SSU stability and equal stable flow rates. (b) Illustration of sheetlike reaction barriers expected near an SUU fixed point.

Figure 6

A front element with normal vector $\mathbf{n}$ and tangent vectors ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$.

Figure 7

(a) The surface (red) at which $v_0=\left|\mathbf{u}\right|$ for $v_0=0.16$ and $\mathbf{u}$ given by Eqs. (1) and (3). There are eight burning fixed points with two unstable BIM directions (blue circles) and eight burning fixed points with other stability types (green triangles). (b) The quarter-tube BIM. (c) The scroll BIM.