Persistent localized states for a chaotically mixed bistable reaction

We describe the evolution of a bistable chemical reaction in a closed two-dimensional chaotic laminar flow, from a localized initial disturbance. When the fluid mixing is sufficiently slow, the disturbance may spread and eventually occupy the entire fluid domain. By contrast, rapid mixing tends to dilute the initial state and so extinguish the disturbance. Such a dichotomy is well known. However, we report here a hitherto apparently unremarked intermediate case, a persistent highly localized disturbance. Such a localized state arises when the Damköhler number is great enough to sustain a “hot spot,” but not so great as to lead to global spread. We show that such a disturbance is located in the neighborhood of an unstable periodic orbit of the flow, and we describe some limited aspects of its behavior using a reduced, lamellar model.

Figure 1

(a) Time evolution of the spatially averaged concentration $⟨A⟩$ for $\mathrm{Pe}={10}^4$, from the initial condition (3), with $Q=1$, for various Damköhler numbers, as indicated.

Figure 2

The circles are centered on a period-6 periodic orbit of the sine flow (2), plotted at times $T,2T,\dots ,6T$ (recall that the line $x=0$ is identified with the opposite edge $x=1$). Also shown are the contours $A(x,y,t)=0.5$ at corresponding times, for a hot spot found at $\mathrm{Da}=6$ and $\mathrm{Pe}={10}^5$, indicating that the hot spot is located in the vicinity of the period-6 orbit.

Figure 3

Long-time behavior of (1), subject to (2, 3) with $Q=1$, in $\mathrm{Da}$-$\mathrm{Pe}$ parameter space. The three long-time fates are $⟨A⟩\to 0$ $(*)$; a localized hot spot $(+)$; $⟨A⟩\to 1(\times )$.

Figure 4

Time evolution of $⟨A⟩$ for $\mathrm{Da}=8$ and $\mathrm{Pe}={10}^4$, from the initial condition (3), for various values of $Q$. (The value of $Q$ corresponding to each curve may be determined as 100 times the initial value of $⟨A⟩$.) For $Q=2,1.5,1.4$, $A\to 1$ everywhere at long time; for $Q=0.38,0.35,0.3$, $A\to 0$ everywhere at long time. Intermediate values of $Q$ lead to the (same) hot spot solution; such results are shown for $Q=0.39$ and for $Q$ from $0.4$ to $1.3$ in steps of $0.1$.

Figure 5

Time evolution of $⟨A⟩$ for $\mathrm{Da}=6$, from the initial condition (3), for various Péclet numbers, as indicated.

Figure 6

Cross-section through the hot spot at $\mathrm{Pe}={10}^5$ and $\mathrm{Da}=6$. Dashed line, from (1), with $\xi$ being a coordinate roughly normal to the hot spot. Solid line, corresponding prediction from (4).