Spatial analog of the two-frequency torus breakup in a nonlinear system of reactive miscible fluids

We present a theoretical study on pattern formation occurring in miscible fluids reacting by a second-order reaction $A+B\to C$ in a vertical Hele-Shaw cell under constant gravity. We have recently reported that the concentration-dependent diffusion of species coupled with a frontal neutralization reaction can produce a multilayer system where low-density depleted zones could be embedded between the denser layers. This leads to the excitation of chemoconvective modes spatially separated from each other by a motionless fluid. In this Rapid Communication, we show that the layers can interact via a diffusion mechanism. Since diffusively coupled instabilities initially have different wavelengths, this causes a long-wave modulation of one pattern by another. We have developed a mathematical model which includes a system of reaction-diffusion-convection equations. The linear stability of a transient base state is studied by calculating the growth rate of the Lyapunov exponent for each unstable layer. Numerical simulations supported by phase portrait reconstruction and Fourier spectra calculation have revealed that nonlinear dynamics consistently passes through (i) a perfect spatially periodic system of chemoconvective cells, (ii) a quasiperiodic system of the same cells, and (iii) a disordered fingering structure. We show that in this system, the coordinate codirected to the reaction front paradoxically plays the role of time, time itself acts as a bifurcation parameter, and a complete spatial analog of the two-frequency torus breakup is observed.

Figure 1

Schematic presentation of two systems of convective rolls independently arising in a depleted zone low in density.

Figure 2

Instantaneous base state profiles of the density $\rho$ are plotted against the vertical axis $z$ at times $t=0$, 3, 10.

Figure 3

(a) Neutral curves for two instabilities which arise in two zones low in density shown in Fig. 2. (b) Real part of the growth rate of modes 1 (lines) and 2 (line points) are illustrated at different times. (c) Real part of the growth rate of modes 1 (black line and line points) and 2 (red line points) demonstrates the coupling between modes. The line points indicated by open squares have been calculated for the case when only mode 1 develops in the system, while mode 2 has been artificially suppressed. The same growth rate curve calculated by taking into account the presence of both modes is indicated by the solid line.

Figure 4

Nonlinear evolution of the total density $\rho$ with time. The frames from up to down pertain to times $t=0.1$, 0.5, 5, 10, and 15, respectively. The domain of integration is $0⩽x⩽125, -20⩽z⩽20$.

Figure 5

The power spectra and phase portraits (in the inset) reconstructed from the averaged density $\widehat{\rho }(t,x)$ at three consecutive times: (a) $t=0.5$, (b) $t=5$, and (c) $t=15$.