Nonperfect mixing affects synchronization on a large number of chemical oscillators immersed in a chemically active time-dependent chaotic flow

The problem of synchronization of finite-size chemical oscillators described by active inertial particles is addressed for situations in which they are immersed in a reacting nonstationary chaotic flow. Active substances in the fluid will be modeled by Lagrangian particles closely following the fluid streamlines. Their interaction with the active inertial particles as well as the properties of the fluid dynamics will result in modifying the synchronization state of the chemical oscillators. This behavior is studied in terms of the exchange rate between the Lagrangian and inertial particles, and the finite-time Lyapunov exponents characterizing the flow. The coherence of the population of oscillators is determined by means of the order parameter introduced by Kuramoto. The different dynamics observed for the inertial particles (chemical oscillators) and Lagrangian particles (describing chemicals in the flow) lead to nonlinear interactions and patches of synchronized regions within the domain.

Figure 1

Scatterplots of beads and Lagrangian particles for the double-gyre flow (left and right panels,respectively, in the top row). Finite-time Lyapunov exponents $\sigma$ corresponding to the beads and tracers transport (left and right panels, respectively, of the bottom row). Darker (red) colors indicate larger values. Local maxima in the plot are repelling coherent structures, responsible for the flow stretching in their vicinity. Integration time $\tau =10$ for the scatterplots and $\tau =20$ for the FTLE. Parameters for the inertial particles are $\beta =1.1$ and ${\tau }_s=1$.

Figure 2

Binary images of Lagrangian particles phase ${\varphi }_j$ (top) and percentage of beads contributing to the exchange term Eq. (A4) in the tracers equations (A3) (bottom) for three different values of $\beta$. Parameters: ${\tau }_s=1, N_B/N_I=0.25$, and $\tau =20$.

Figure 3

Kuramoto order parameter $R$ as a function of (a) the Stokes time ${\tau }_s$, (b) the $\beta$ parameter (density fraction), and (c) the ratio $N_B/N_I$ (ratio of beads and tracers in the flow). Parameters: (a) $\beta =1.1, N_B/N_I=0.25$; (b) ${\tau }_s=1, N_B/N_I=0.25$; (c) $\beta =1.1, {\tau }_s=1$. The rest of the parameters are as in Fig. 1.

Figure 4

Oscillation periods for the Lagrangian particles at different integration times $\tau$. From top to bottom and from left to right, $\tau =5, \tau =15, \tau =25$, and $\tau =35$, respectively. Parameters: $\beta =1.1, {\tau }_s=1$, and $N_B/N_I=0.25$.

Figure 5

Time series of (a) $v^I$ and (b) the percentage of beads contributing to the exchange term, Eq. (A4), in the tracers equations (A3) for two Lagrangian particles (continuous and dashed lines, respectively) in the domain. Parameters: $\beta =1.1, {\tau }_s=1$, and $N_B/N_I=0.25$.