Predicting the Progress of Diffusively Limited Chemical Reactions in the Presence of Chaotic Advection

The effects of chaotic advection and diffusion on fast chemical reactions in two-dimensional fluid flows are investigated using experimentally measured stretching fields and fluorescent monitoring of the local concentration. Flow symmetry, Reynolds number, and mean path length affect the spatial distribution and time dependence of the reaction product. A single parameter $\overline{\lambda }N$, where $\overline{\lambda }$ is the mean Lyapunov exponent and $N$ is the number of mixing cycles, can be used to predict the time-dependent total product for flows having different dynamical features.

Figure 1

Streamlines computed from measured velocity fields at $\mathrm{Re}=56$ and path length $p=2.5$ for (a) disordered and (b) ordered magnet array configurations. The ordered case shows clear lines of mirror symmetry. Corresponding stretching fields for (c) disordered and (d) ordered flows, showing the magnitude of stretching over an interval $\Delta t=1$ period. Regions of strong stretching are localized in both cases.

Figure 2

Normalized product concentration fields ($\tilde{P}$) vs number of periods ($N$) at $p=2.5$. (a) $\mathrm{Re}=37$ (disordered), $N=10$; (b) same as (a) for $N=30$; (c) $\mathrm{Re}=56$ (disordered), $N=10$; (d) same as (c) for $N=30$; (e) $\mathrm{Re}=56$ (ordered), $N=10$; (f) same as (e) for $N=30$. Fully reacted regions appear in red, as shown in the color bar. Unreacted regions are shown in black.

Figure 3

(a) Spatial averaged normalized product concentration $⟨\tilde{P}⟩$ as a function of time (or $N$) and Re for several different flows ($D=\mathrm{\text{disordered}}$, $O=\mathrm{\text{ordered}}$, all at $p=2.5$). (b) $⟨\tilde{P}⟩$ vs mean Lyapunov exponent ($\overline{\lambda }$) for various $N$ (disordered flows for various Re). Lines are added to guide the eye. Product growth is slow for small $\overline{\lambda }$ but accelerates for large $\overline{\lambda }$ at later times.

Figure 4

Dependence of $⟨\tilde{P}⟩$ on the mean Lyapunov exponent ($\overline{\lambda }$) and period $N$ for disordered flows at various Re and $p$. Insert: early behavior for both ordered and disordered flows (at various Re and $p$). The dashed and solid lines correspond to Eq. (1) ($a=0.00263$) and the empirical fit described in the text, respectively.