Schlick Conor P, Umbanhowar Paul B, Ottino Julio M, Lueptow Richard M
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Chaos. 2014 Mar;24(1):013109. doi: 10.1063/1.4862153.
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.
我们使用具有算子分裂的映射方法，通过计算研究了时间周期正弦流在自催化反应中的混沌平流和扩散。我们具体考虑了三种不同的自催化反应方案：单一自催化反应、竞争性自催化反应（其可用于深入了解手性对称性破缺和同手性问题）以及具有循环的竞争性自催化反应。在竞争性自催化反应中，物种B和C都与物种A发生自催化反应，使得[公式：见原文]和[公式：见原文]。少量初始时在空间上局部化的B和C以及大量空间均匀的A被速度场平流、扩散并反应，直到A被完全消耗，仅剩下B和C。我们发现局部有限时间李雅普诺夫指数（FTLEs）能够准确预测反应完成后B和C的最终平均浓度。起始于具有较大FTLE区域的物种，在反应结束时具有较高概率具有较大的平均浓度。如果B和C起始于具有相似FTLEs的区域，它们在反应结束时的平均浓度也将相似。当添加循环反应时，系统会朝着单一物种状态演化，FTLE通常有助于预测哪种物种会填满整个区域而哪种会耗尽。FTLE方法也在轴颈轴承流中的竞争性自催化反应中得到了验证，轴颈轴承流是一种可通过实验实现的产生混沌动力学的流动。
