Stephen Wiggins

An analytical study of transport, mixing and chaos in an unsteady vortical flow

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikovs technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

Introduction: mixing in microfluidics

In this paper we briefly review the main issues associated with mixing at the microscale and introduce the papers comprising the Theme Issue.

Foundations of chaotic mixing

The simplest mixing problem corresponds to the mixing of a fluid with itself; this case provides a foundation on which the subject rests. The objective here is to study mixing independently of the mechanisms used to create the motion and review elements of theory focusing mostly on mathematical foundations and minimal models. The flows under consideration will be of two types: two–dimensional (2D) ‘blinking flows’, or three–dimensional (3D) duct flows. Given that mixing in continuous 3D duct flows depends critically on cross–sectional mixing, and that many microfluidic applications involve continuous flows, we focus on the essential aspects of mixing in 2D flows, as they provide a foundation from which to base our understanding of more complex cases. The bakerstransformation is taken as the centrepiece for describing the dynamical systems framework. In particular, a hierarchy of characterizations of mixing exist, Bernoulli→mixing→ergodic, ordered according to the quality of mixing (the strongest first). Most importantly for the design process, we show how the so–called linked twist maps function as a minimal picture of mixing, provide a mathematical structure for understanding the type of 2D flows that arise in many micromixers already built, and give conditions guaranteeing the best quality mixing. Extensions of these concepts lead to first–principle–based designs without resorting to lengthy computations.

The geometry of reaction dynamics

The geometrical structures which regulate transformations in dynamical systems with three or more degrees of freedom (DOFs) form the subject of this paper. Our treatment focuses on the (2n − 3)-dimensional normally hyperbolic invariant manifold (NHIM) in nDOF systems associated with a centre × · ·· ×centre × saddle in the phase space flow in the (2n − 1)dimensional energy surface. The NHIM bounds a (2n − 2)-dimensional surface, called a ‘transition state’ (TS) in chemical reaction dynamics, which partitions the energy surface into volumes characterized as ‘before’ and ‘after’ the transformation. This surface is the long-sought momentum-dependent TS beyond two DOFs. The (2n − 2)-dimensional stable and unstable manifolds associated with the (2n − 3)-dimensional NHIM are impenetrable barriers with the topology of multidimensional spherical cylinders. The phase flow interior to these spherical cylinders passes through the TS as the system undergoes its transformation. The phase flow exterior to these spherical cylinders is directed away from the TS and, consequently, will never undergo the transition. The explicit forms of these phase space barriers can be evaluated using normal form theory. Our treatment has the advantage of supplying a practical algorithm, and we demonstrate its use on a strongly coupled nonlinear Hamiltonian, the hydrogen atom in crossed electric and magnetic fields.

On the integrability and perturbation of three-dimensional fluid flows with symmetry

SummaryThe purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikovs method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.

Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics

We study the flow obtained from a three-layer, eddy-resolving quasigeostrophic ocean circulation model subject to an applied wind stress curl. For this model we will consider transport between the northern and southern gyres separated by an eastward jet. We will focus on the use of techniques from dynamical systems theory, particularly lobe dynamics, in the forming of geometric structures that govern transport. By “govern”, we mean they can be used to compute Lagrangian transport quantities, such as the flux across the jet. We will consider periodic, quasiperiodic, and chaotic velocity fields, and thus assess the effectiveness of dynamical systems techniques in flows with progressively more spatiotemporal complexity. The numerical methods necessary to implement the dynamical systems techniques and the significance of lobe dynamics as a signature of specific “events”, such as rings pinching off from a meandering jet, are also discussed.

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al. [J. Stat. Phys. 50, 529 (1988)]. (c) 1999 American Institute of Physics.

Orbits homoclinic to resonances: the Hamiltonian case

Abstract In this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits, hyperbolic fixed points or combinations of hyperbolic fixed points and/or periodic orbits in a class of two-degree-of-freedom, integrable Hamiltonian systems subject to arbitrary Hamiltonian perturbations. Our methods differ from previous methods in that the invariant sets (periodic orbits, fixed points) are created, and become hyperbolic, as a result of the interaction of the perturbation with a resonance in the unperturbed system. This results in a very degenerate situation that requires a combination of geometric singular perturbation theory, higher-dimensional Melnikov-type methods, and transversality theory. We establish a simple energy-phase criterion which gives a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonance. We apply our methods to a two-mode truncation of the driven nonlinear Schrodinger equation first studied by Bishop et al. In this example we show that as the energy is increased at resonance, orbits homoclinic to hyperbolic periodic orbits are created in pairs in a global bifurcation that is best described as a saddle-node bifurcation of homoclinic orbits.

Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

Abstract. We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.

TIME-FREQUENCY ANALYSIS OF CLASSICAL TRAJECTORIES OF POLYATOMIC MOLECULES

We present a new method of frequency analysis for Hamiltonian Systems of 3 degrees of freedom and more. The method is based on the concept of instantaneous frequency extracted numerically from the continuous wavelet transform of the trajectories. Knowing the time-evolution of the frequencies of a given trajectory, we can define a frequency map, resonances, and diffusion in frequency space as an indication of chaos. The time-frequency analysis method is applied to the Baggott Hamiltonian to characterize the global dynamics and the structure of the phase space in terms of resonance channels. This 3-degree-of-freedom system results from the classical version of the quantum Hamiltonian for the water molecule given by Baggott [1988]. Since another first integral of the motion exists, the so-called Polyad number, the system can be reduced to 2 degrees of freedom. The dynamics is therefore simplified and we give a complete characterization of the phase space, and at the same time we could validate the results of the time-frequency analysis.