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We present a theoretical study on pattern formation occurring in misciblefluids reacting by a second-order reaction $A + B \to S$ in a verticalHele-Shaw cell under constant gravity. We have recently reported thatconcentration-dependent diffusion of species coupled with a frontalneutralization reaction can produce a multi-layer system where low densitydepleted zones could be embedded between the denser layers. This leads to theexcitation of chemoconvective modes spatially separated from each other by amotionless fluid. In this paper, we show that the layers can interact via adiffusion mechanism. Since diffusively-coupled instabilities initially havedifferent wavelengths, this causes a long-wave modulation of one pattern byanother. We have developed a mathematical model which includes a system ofreaction-diffusion-convection equations. The linear stability of a transientbase state is studied by calculating the growth rate of the Lyapunov exponentfor each unstable layer. Numerical simulations supported by the phase portraitreconstruction and Fourier spectra calculation have revealed that nonlineardynamics consistently passes through (i) a perfect spatially periodic system ofchemoconvective cells; (ii) a quasi-periodic system of the same cells; (iii) adisordered fingering structure. We show that in this system, the coordinateco-directed to the reaction front paradoxically plays the role of time, timeitself acts as a bifurcation parameter, and a complete spatial analog of theRuelle-Takens-Newhouse scenario of the chaos onset is observed.

A spatial analog of the Ruelle-Takens-Newhouse scenario developing in reactive miscible fluids