Competitive autocatalytic reactions in chaotic flows with diffusion: Prediction using finite-time Lyapunov exponents

Abstract

We investigate chaotic advection and diffusion in autocatalytic reactions for time-periodic sine flow computationally using a mapping method with operator splitting. We specifically consider three different autocatalytic reaction schemes: a single autocatalytic reaction, competitive autocatalytic reactions, which can provide insight into problems of chiral symmetry breaking and homochirality, and competitive autocatalytic reactions with recycling. In competitive autocatalytic reactions, species B and C both undergo an autocatalytic reaction with species A such that [Formula: see text] and [Formula: see text]. Small amounts of initially spatially localized B and C and a large amount of spatially homogeneous A are advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that local finite-time Lyapunov exponents (FTLEs) can accurately predict the final average concentrations of B and C after the reaction completes. The species that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If B and C start in regions with similar FTLEs, their average concentrations at the end of the reaction will also be similar. When a recycling reaction is added, the system evolves towards a single species state, with the FTLE often being useful in predicting which species fills the entire domain and which is depleted. The FTLE approach is also demonstrated for competitive autocatalytic reactions in journal bearing flow, an experimentally realizable flow that generates chaotic dynamics.

Optimal Stretching in Advection-Reaction-Diffusion Systems

“…The correlation changes with advection speed. For small values of Re, reacted regions lie primarily along vortex edges and in narrow filaments where stretching is strongest, consistent with simulations showing that stretching enhances reaction [1,37,38] [31].…”

“…When two competing autocatalytic species were simulated in double-gyre flow, it was observed that the one initiated in a region of stronger stretching almost always dominated long term [38]. Similarly, in simulations of competing species in sine flow, another two-dimensional prescribed model, it was observed that local FTLEs accurately predict long-term concentrations [1]. Experiments have shown that the global averaged FTLE is correlated with global reaction rate [39].…”

“…Introduction.-In advection-reaction-diffusion (ARD) systems, some quantity grows locally, spreads by random motions, and is carried along by the surrounding fluid flow. Many natural phenomena and industrial technologies are accurately described as ARD systems, including bulk chemical reactors [1], wildfires [2], combustion engines [3,4], microfluidic reactors [5], cellular growth [6], reaction in porous media [7,8], and phytoplankton ecology [9][10][11]. For any of these applications, improved understanding of ARD dynamics would allow forecasts of local and global concentrations and reaction rates.…”

Optimal stretching in the reacting wake of a bluff body

“…43 Similarly, if two reactions compete, the one that is triggered where FTLEs are smaller is eventually overwhelmed. 2,44 Earlier experiments showed that the overall reaction rate increased when the average FTLE value increased. 45 However, recent evidence 1 suggests that strong stretching may inhibit the growth of reacted regions.…”

“…[7][8][9][10][11][12][13] Even without advection, reaction and diffusion give rise to nonlinear phenomena like traveling waves and spiral waves, [14][15][16] so including advection, which may be chaotic or turbulent itself, makes forecasting ARD systems a challenging task. But being able to forecast the spreading of the reacted region (where the concentration of the product species is high) and knowing how its edges will be constrained would allow better prediction of the concentration and uniformity of products in chemical reactors 2,17 and microfluidic devices, 18 more effective safety responses to forest fires, 19,20 and better phytoplankton models for understanding Earth's carbon budget. 12,21,22 In this paper, we present experimental measurements of an ARD system in the wake of a bluff body in a laminar channel flow.…”