* Corresponding author. Tel.: 00-39-06-44585892; fax: 00-39-06- 44585339. E-mail address: centro@giona.ing.uniromal.it (M. Giona) Chemical Engineering Science 55 (2000) 381}389 The geometry of mixing in 2-d time-periodic chaotic #ows Massimiliano Giona*, Alessandra Adrover, Fernando J. Muzzio, Stefano Cerbelli Centro Interuniversitario sui Sistemi Disordinati e sui Frattali nell 'Ingegneria Chimica, Dipartimento di Ingegneria Chimica, Universita & di Roma **La Sapienza++, via Eudossiana 18, 00184 Roma, Italy Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, NJ 08855, USA Received 31 March 1999; accepted 1 April 1999 Abstract This paper demonstrates that the geometry and topology of material lines in time-periodic chaotic #ows is controlled by a global geometric property referred to as asymptotic directionality. This property implies the existence of local asymptotic orientations at each point within the chaotic region determined by the unstable eigendirections of the Jacobian matrix of the n-period Poincare H map associated with the #ow. Asymptotic directionality also determines the topology of the invariant unstable manifolds of the Poincare H map, which are everywhere tangent to the "eld of asymptotic eigendirections. This fact is used to derive simple non-perturbative methods for reconstructing the invariant unstable manifolds associated with a Poincare H section to any desired level of detail. Since material lines evolved by a chaotic #ow are asymptotically attracted to the geometric global unstable manifold of the #ow (this concept is introduced in this article), such reconstructions can be used to characterize the topological and statistical properties of partially mixed structures quantitatively. Asymptotic directionality provides evidence of a global self-organizing structure character- izing physically realizable chaotic mixing systems which is analogous to that of Anosov di!eomorphisms, which turns out to represent the basic prototype of a mixing system. In this framework we show how partially mixed structures can be quantitatively characterized by a non-uniform stationary measure (di!erent from the ergodic measure) associated with the dynamical system generated by the "eld of asymptotic unstable eigenvectors.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Incompressible laminar #uid mixing; Chaotic #ows 1. Introduction The mixing properties of laminar chaotic #ows with low dimensionality have enormous importance for both fundamental and practical reasons. Aref (1984) was the "rst to realize that 2-d laminar #ows provide a physically comprehensible analogy for the phase-space dynamics of conservative dynamical systems, with the stream function playing the role of the Hamiltonian and the #ow domain providing a directly accessible analogue of the phase space. Mixing processes in such #ows are thus directly related to the phenomena controlling the properties of trajectories in the phase space. Many studies on mixing in chaotic #ows have been carried out in the past few years using both experimental and computational approaches, and transplanting many methods and strategies typical of dynamic system theory (Guckenheimer & Holmes, 1983). The majority of chaotic systems investigated have been two-dimensional, time-dependent #ows such as the blinking vortex system (Aref, 1984; Khakhar, Rising & Ottino, 1986) the tendril-whorl #ow (Aref, 1984) and the two physically realizable #ows, the #ow between eccentric cylinders (Swanson & Ottino, 1990) and the driven cavity #ow (Liu, Peskin, Muzzio & Leong, 1994). In spite of previous e!orts, the process of formation and evolution of partially mixed structures (such as ma- terial lines and the boundary of material elements) in chaotic #ows has yet to be completely understood. Both experiments and simulations have demonstrated that chaotic #ows generate mixtures with complex structures, possessing characteristic invariant features and strong self-similar attributes as well as coexisting length scales spanning many order of magnitudes (Alvarez, Muzzio, Adroner & Cerbelli, 1997). The existence of invariant properties in the evolution of material lines has in fact been analyzed in detail in 0009-2509/00/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 3 3 - 4