On the Hyperbolic Behavior of Laminar Chaotic Flows

Abstract

The subject of laminar chaotic mixing, also referred to as Lagrangian chaos, is a relatively new field of investigation, started about two decades ago with the publication of the article “Stirring by chaotic advection” (Aref, 1984) by Hassan Aref published in the Journal of Fluid Mechanics in 1984, which showed how it is possible to generate unexpectedly complex mixing structures through the application of a simple stirring protocol (the case of two two-dimensional ideal vortexes blinking periodically was used as a case study in the article in question). The idea underlying Aref’s paper was a major breakthrough in the fluid dynamics community. Yet, a mathematician or a physicist involved in the study of Hamiltonian mechanics would have considered this idea not altogether new, since it was known since Poincaré’s contribution in celestial mechanics (Diacu and Holmes, 1996) that a conservative system whose dynamics is governed by nonlinear equations is apt to display complex dynamical behavior. In point of fact, an article discussing the existence of nonintegrable Euler flows was published by Arnold almost twenty years before Aref’s paper (Arnold, 1966). One may wonder why it took so many years to appreciate the analogy between a conservative mechanical system and the kinematics induced by an incompressible flow.

This formal analogy holds, for instance, between a one-degree of freedom periodically perturbed Hamiltonian system and a two-dimensional time-periodic incompressible flow, with the x and y coordinates of the flow domain playing the role of position and momentum of the phase space associated with the mechanical system.