Competitive autocatalytic reactions in chaotic flows with diffusion: prediction using finite-time Lyapunov exponents.
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.
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Conor P
CP
Schlick
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Paul B
PB
Umbanhowar
Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Julio M
JM
Ottino
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Richard M
RM
Lueptow
D002384:Catalysis; D008956:Models, Chemical; D017711:Nonlinear Dynamics