In the Lagrangian representation, the problem of advection of a passive marker particle by a prescribed flow defines a dynamical system. For two-dimensional incompressible flow this system is Hamiltonian and has just one degree of freedom. For unsteady flow the system is non-autonomous and one must in general expect to observe chaotic particle motion. These ideas are developed and subsequently corroborated through the study of a very simple model which provides an idealization of a stirred tank. In the model the fluid is assumed incompressible and inviscid and its motion wholly two-dimensional. The agitator is modelled as a point vortex, which, together with its image(s) in the bounding contour, provides a source of unsteady potential flow. The motion of a particle in this model device is computed numerically. It is shown that the deciding factor for integrable or chaotic particle motion is the nature of the motion of the agitator. With the agitator held at a fixed position, integrable marker motion ensues, and the model device does not stir very efficiently. If, on the other hand, the agitator is moved in such a way that the potential flow is unsteady, chaotic marker motion can be produced. This leads to efficient stirring. A certain case of the general model, for which the differential equations can be integrated for a finite time to produce an explicitly given, invertible, area-preserving mapping, is used for the calculations. The paper contains discussion of several issues that put this regime of chaotic advection in perspective relative to both the subject of turbulent advection and to recent work on critical points in the advection patterns of steady laminar flows. Extensions of the model, and the notion of chaotic advection, to more realistic flow situations are commented upon.

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Divi sion of En gineering, Brown Un iver sity, Providence, Rh ode Is land 02912It is indeed rather astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. G. H. Hardy, A Mathematician's Apology INTRODUCTORY REMARKSVortex dynamic s would appear to be exempt from Hardy 's pe ssimi stic verdict. On one hand, the evolution of vorticity, and thu s the motion s of vortice s, are esse ntial ingredients of virtually any real flow. He nce vortex dynamic s is of profound practical importance. On the other hand, vortex motion ha s alway s con stituted a mathematically so phi sticated branch of fluid mechanics that continue s to invite the application of novel analyti cal technique s. Indeed it is ne it her dull nor co mmonplace .Thi s central role of vorticity in fluid mechanic s is not difficult to understand. As we know, any velocity field, v, can be sp lit into a su m of two field s, one with the sa me divergence as v, and no curl, and one with the sa me cu rl as v and vani shing divergence. Th is important re sult is due to Stoke s and to Helmholtz (1858; se e Sommerfeld 1964). In incompre ssi ble flow , as we deal with exclu si vely here, the fir st part is irrotational and diver gence-free and thu s lead s to the linear problem of po tential flow. , The se cond part, however , derive s directly from the vorticity of the field ' v. In the dynamic s of thi s part lie s the esse nce of the problem. 345 0066-4189/8 3/011 5-03 45$ 02.00 Annu. Rev. Fluid Mech. 1983.15:345-389. Downloaded from www.annualreviews.org by NORTH CAROLINA STATE UNIVERSITY on 10/20/12. For personal use only. Quick links to online content Further ANNUAL REVIEWS

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