Hassan Aref

Chaotic advection by laminar flow in a twisted pipe

The appearance of chaotic particle trajectories in steady, laminar, incompressible flow through a twisted pipe of circular cross-section is demonstrated using standard dynamical systems diagnostics and a model flow based on Deans perturbation solutions. A study is performed to determine the parameters that control fluid stirring in this mixing device that has no moving parts. Insight into the chaotic dynamics are provided by a simple one-dimensional map of the pipe boundary onto itself. The results of numerical experiments illustrating the stretching of material lines, stirring of blobs of material, and the three-dimensional trajectories of fluid particles are presented. Finally, enhanced longitudinal particle dispersal due to the coupling between chaos in the transverse direction and the non-uniform longitudinal transport of particles is shown.

Chaotic advection in a Stokes flow

Chaotic advection can be produced whenever the kinematic equations of motion for passively advected particles give rise to a nonintegrable dynamical system. Although this interpretation of the phenomenon immediately shows that it is possible for flows at any value of Reynolds number, the notion of stochastic particle motion within laminar flows runs counter to common intuition to such a degree that the range of applicability of early model results has been questioned. To dispel lingering doubts of this type a study of advection in a two‐dimensional Stokes flow slowly modulated in time is presented. Even for this very low Reynolds number, manifestly ‘‘laminar’’ flow chaotic particle motion is readily realizable. Standard diagnostics of chaos are computed for various methods of time modulation. Relations to the general ideas of parametric resonance and adiabatic invariance are pointed out.

Chaotic advection in pulsed source--sink systems

The onset of chaos in passive advection of particles by flow caused by a pulsed source–sink system is documented. This type of model is of interest in various applications. It is of fundamental interest as the first example of a flow without circulation about any contour at any instant displaying chaotic particle paths. Standard chaos diagnostics such as Poincare sections and Lyapunov exponents are studied as are more conventional flow visualization measures such as streaklines. Numerical stirring experiments for various collections of particles are performed and the properties of a certain one‐dimensional map induced by the two‐dimensional flow are examined.

Integrable and Chaotic Motions of Four Vortices II. Collision Dynamics of Vortex Pairs

The interaction of two vortex pairs is investigated analytically and by numerical experiments from the vantage point of dynamical-systems theory. Vortex pairs can escape to infinity, so the phase space of this system is unbounded in contrast to that of four identical vortices investigated previously. Chaotic motion is nevertheless possible both for ‘bound states’ of the system and for ‘scattering states’. For the bound states standard Poincare section techniques suffice. For scattering states chaos appears as complex structure in the numerically generated plot of scattering angle against impact parameter. Interpretations of physical space mechanisms leading to chaos are given. Analytical characterizations of the system include a formal reduction to two degrees of freedom by canonical transformations and an identification and discussion of integrable cases of which one is apparently new.

Enhanced separation of diffusing particles by chaotic advection

Combining the reversibility of advection by a Stokes flow with the irreversibility of diffusion leads to a separation strategy for diffusing substances. This basic idea goes back to Taylor and Heller. It is shown here that the sensitivity of the method can be greatly enhanced by making the advection chaotic. The separation is particularly efficient when the thinnest structures resulting from advection are made comparable in size to a diffusion length. Simple heuristic estimates based on an understanding of chaotic motion and diffusion lead to a certain scaling that is seen in numerical experiments on this separation method.

Vortices, kinematics and chaos

Abstract Chaotic fluid motion is explored from a Lagrangian point of view. A number of flow phenomena then arise, including in particular chaotic advection and sound radiation from non-integrable vortex motions. Examples are given and some theoretical classification is attempted. The role of vorticity and vortices in bringing about the chaos is continually highlighted. A modelling attempt of the turbulent-laminar interface is discussed from the vantage point of the ideas developed for simpler situations.

Stochastic particle motion in laminar flows

The practical actions referred to as stirring, mixing, blending, and so on rely on our ability to produce stochastic trajectories of particles within given flow fields. There are time‐honored ways of achieving stochastic particle motion within a fluid, in particular, by exploiting molecular diffusion and through advection by a turbulent flow. There are, however, also other methods, particularly useful for laminar flows, that rely on the notion of chaos in a low‐dimensional dynamical system for their explanation and elucidation. Indeed, the theory of dynamical systems is one of the few places where the term ‘‘mixing’’ has a precise technical meaning. The paper reviews several of these concepts, many of them taken from ergodic theory, and attempts to explain their application to the fluid mechanical problems. All these developments rely on a correspondence between the ‘‘phase space’’ of the general dynamical system, and the real‐space motion of the fluid in the stirring/mixing problem. Other aspects of theo...

Point vortex dynamics: recent results and open problems

Abstract he concept of point vortex motion, a classical model in the theory of two-dimensional, incompressible fluid mechanics, was introduced by Helmholtz in 1858. Exploration of the solutions to these equations has made fitful progress since that time as the point vortex model has been brought to bear on various physical situations: atomic structure, large scale weather patterns, “vortex street” wakes, vortex lattices in superfluids and superconductors, etc. The point vortex equations also provide an interesting example of transition to chaotic behavior. We give a brief historical introduction to these topics and develop two of them in particular to the point of current understanding: (i) Steadily moving configurations of point vortices; and (ii) Collision dynamics of vortex pairs.

Mixing during vortex ring collision

Using a vortex‐in‐cell method, the interactions of two identical rings moving on a collision course along parallel lines are studied. Evolution of a passive nondiffusing scalar carried initially with the vortices is visualized and analyzed. Diagnostics of mixing of the scalar particles demonstrate an improvement in mixing as the offset of the lines of propagation of the vortices is increased from zero to about a quarter of the ring diameter. With further increase of the offset the mixing is gradually reduced due apparently to a vortex reconnection process that limits the amount of stretching of the configuration.

The effect of strain upon the topology of a soap froth

Abstract The question of the effect of an imposed strain on the topology of a soap froth is addressed through numerical simulations. Both the standard model of a ‘dry’ twodimensional foam and the case of small liquid fraction are explored. The froths are subjected to both extensional deformation and simple shear. The counter-intuitive result that application of strain can induce ordering in a disordered froth is obtained, quantified and discussed. A qualitative dynamical explanation is proposed.