BIFURCATIONS AND MIXING WINDOWS IN SINUSOIDAL CAVITY FLOW MIXING SYSTEMS

Abstract

Bifurcation analysis is applied to low periodic orbits of co- and counter-rotating sinusoidal cavity flow mixing systems, giving a quantitative account of the behavior observed and presenting an efficient way of finding nearly globally chaotic mixing systems, that can lead to uniform mixing. The corresponding parameter regions are called mixing windows. Symmetries of the Poincare maps deduced from those of continuous trajectories are used to help find periodic points. Both co- and counter-rotating flows have an integrable limit as T (the period of the sinusoidal motion of the walls) → 0. As T → ∞, most orbits in the co-rotating flows have an adiabatic invariant and are, therefore, regular, while the orbits in the counter-rotating flows do not, due to the existence of separatices. This qualitatively explains the better mixing performance of the counter-rotating flows compared with the co-rotating flows. The stream lines of the systems with small phase shift between the motion of two walls are nearly unchanged with time, and therefore, no good mixing can be expected. The bifurcation analysis reveals that a phase shift of half a period also results in bad mixing and that the best results are obtained with the phase shift of about a quarter period. The methodology presented in this paper saves computation time greatly compared with the traditional Poincare section method. Our results are compared with the existing experiments.