Effective dimensions and chemical reactions in fluid flows

We show that chemical activity in hydrodynamical flows can be understood as the outcome of three basic effects: the stirring protocol of the flow, the local properties of the reaction, and the global folding dynamics which also depends on the geometry of the container. The essence of each of these components can be described by simple functional relations. In an ordinary differential equation approach, they determine a new chemical rate equation for the concentration, which turns out to be coupled to the dynamics of an effective fractal dimension. The theory predicts an exponential convergence to the asymptotic chemical state. This holds even in flows characterized by a linear stirring protocol where transient fractal patterns are shown to exist despite the lack of any chaotic set of the advection dynamics. In the exponential case the theory applies to flows of chaotic time dependence (chaotic flows) as well.

Figure 1

(Color online) Concentration distribution $c\left(x\right)$ of material $C$ (dashed central peaks, red online) along the contracting direction $x$ of the flow and the definition of the bandwidth $\delta \left(t\right)$. (a) Autocatalytic reaction, material $A$ is in abundance outside of $C$. The transition region, the reaction front, is propagating with a fixed velocity $v$ relative to the fluid. (b) Acid-base reaction. The motion of the transition region, relative to the flow, is due to diffusion; there is no intrinsic front velocity $v$ in this case. If the transition region between material $C$ and the other components is narrow enough, the concentration of material $C$ can be well approximated by a function which is constant (of value $\overline{c}$) over an interval of length $\delta \left(t\right)$, and zero outside. Concentrations are considered here as number densities.

Figure 2

Distribution of the autocatalytic $C$ particles (black) in the random sinusoidal shear flow, with $A=0.7$ in Eq. (34). The simulations were performed on a grid of size $1024\times 1024$. Reactions occurred at each half period of the flow, when the direction of the flow velocity changed. The snapshots are shown at (a) 0, (b) 0.5, (c) 1, (d) 2, (e) 3, (f) 4, (g) 6, (h) 8 periods, right before the reaction events.

Figure 3

Concentration $c$ of reactant $C$ in the autocatalytic reaction of Fig. 2. Time is measured in periods of the flow, concentration is sampled just before the reaction events. The simulations were performed on a grid of size $8192\times 8192$. The average Lyapunov exponent of the flow is $\lambda =0.60\pm 0.02$. Crosses represent results obtained from the simulation, the straight line is predicted by Eq. (27). The chemical decay rate from the plot is $\sigma =0.625\pm 0.006$.

Figure 4

Time dependence of the effective dimension in the numerical experiment of Fig. 3. Crosses represent the measured $D_{\mathrm{eff}}$ values, while the straight line is predicted by Eq. (27). The chemical decay rate is $\sigma =0.639\pm 0.007$. An illustration of the box counting algorithm is shown in the inset, where the data correspond to time $t=5$, before the reaction event.

Figure 5

(Color online) Distribution of materials $A$ (medium gray, red online), $B$ (light gray, green online) and $C$ (dark gray, blue online) of the acid-base reaction $A+B\to 2C$ in the random sinusoidal shear flow with $A=0.7$ in Eq. (34). The simulations were performed on a grid of size $1024\times 1024$. The snapshots are shown at times (a) 0, (b) 0.5, (c) 1, (d) 2, (e) 3, (f) 4, (g) 6, (h) 8 periods, right after the reaction events.

Figure 6

Concentration $c$ of product $C$ in the acid-base reaction of Fig. 5. Time is measured in periods of the flow, concentration is sampled just after the reaction events. Simulations were performed on a grid of size $8192\times 8192$. The Lyapunov exponent of the flow is $\lambda =0.60\pm 0.03$. Crosses represent results obtained from the simulation, the straight line is predicted by Eq. (27). The relaxation exponent from the plot is $\sigma =0.160\pm 0.001$.

Figure 7

Distribution of the autocatalytic $C$ particles (black) in a frame comoving with the vortex pair (37). Simulations were performed on a grid of size $1024\times 1024$. Autocatalytic reactions occurred after time intervals of length 0.1. The snapshots are shown at times (a) 0, (b) 0.5, (c) 1, (d) 1.5, (e) 2, (f) 3, (g) 4, (h) 5 right before the reaction events. The locations of the vortex centers are denoted by small crosses.

Figure 8

Concentration $c$ of reactant $C$ in the autocatalytic reaction of Fig. 7. The simulations were performed on a grid of size $8192\times 8192$. Crosses represent results obtained from the simulation, the straight line is predicted by Eq. (27). The relaxation exponent from the plot is $\sigma =0.03272\pm 0.00004$.