Sanjeeva Balasuriya

Melnikov theory for finite-time vector fields

Melnikov theory provides a powerful tool for analysing time-dependent perturbations of autonomous vector fields that exhibit heteroclinic orbits. The standard theory requires that the perturbed vector field be defined, and bounded, for all times. In this paper, Melnikov theory is adapted so that it is applicable to vector fields that are defined over sufficiently large, but finite, time intervals. Such an extension is desirable when investigating Lagrangian trajectories in fluid flows under the effect of viscous perturbations; the resulting velocity field can only be guaranteed to be close to the unperturbed velocity field, corresponding to the inviscid limit, for finite times. Applications to transport in the viscous barotropic vorticity equation are given.

Viscous perturbations of vorticity-conserving flows and separatrix splitting

We examine the effect of the breaking of vorticity conservation by viscous dissipation on transport in the underlying fluid flow. The transport of interest is between regimes of different characteristic motion and is afforded by the splitting of separatrices. A base flow that is vorticity conserving is therefore assumed to have a separatrix that is either a homoclinic or heteroclinic orbit. The corresponding vorticity dissipating flow, with small time-dependent forcing and viscous parameter , maintains an closeness to the inviscid flow in a weak sense. An appropriate Melnikov theory that allows for such weak perturbations is then developed. A surprisingly simple expression for the leading-order distance between perturbed invariant (stable and unstable) manifolds is derived which depends only on the inviscid flow. Finally, the implications for transport in barotropic jets are discussed.

HIGH LEWIS NUMBER COMBUSTION WAVEFRONTS: A PERTURBATIVE MELNIKOV ANALYSIS ∗

The wavefronts associated with a one‐dimensional combustion model with Arrhenius kinetics and no heat loss are analyzed within the high Lewis number perturbative limit. This situation, in which fuel diffusivity is small in comparison to that of heat, is appropriate for highly dense fluids. A formula for the wavespeed is established by a nonstandard application of Melnikov’s method and slow manifold theory from dynamical systems, and compared to numerical results. A simple characterization of the wavespeed correction is obtained: it is proportional to the ratio between the exothermicity parameter and the Lewis number. The perturbation method developed herein is also applicable to more general coupled reaction‐diffusion equations with strongly differing diffusivities. The stability of the wavefronts is also tested using a numerical Evans function method.

Explicit invariant manifolds and specialised trajectories in a class of unsteady flows

A class of unsteady two- and three-dimensional velocity fields for which the associated stable and unstable manifolds of the Lagrangian trajectories are explicitly known is introduced. These invariant manifolds form the important time-varying flow barriers which demarcate coherent fluids structures, and are associated with hyperbolic trajectories. Explicit expressions are provided for time-evolving hyperbolic trajectories (the unsteady analogue of saddle stagnation points), which are proven to be hyperbolic in the sense of exponential dichotomies. Elliptic trajectories (the unsteady analogue of stagnation points around which there is rotation, i.e., the “centre of a vortex”) are similarly explicitly expressed. While this class of models possesses integrable Lagrangian motion since formed by applying time-dependent spatially invertible transformations to steady flows, their hyperbolic/elliptic trajectories can be made to follow any user-specified path. The models are exemplified through two classical flows...

A Tangential Displacement Theory for Locating Perturbed Saddles and Their Manifolds

The stable and unstable manifolds associated with a saddle point in two-dimensional non-area- preserving flows under general time-aperiodic perturbations are examined. An improvement to existing geometric Melnikov theory on the normal displacement of these manifolds is presented. A new theory on the previously neglected tangential displacement is developed. Together, these enable locating the perturbed invariant manifolds to leading order. An easily usable Laplace transform expression for the location of the perturbed time-dependent saddle is also obtained. The theory is illustrated with an application to the Duffing equation. While Melnikov methods can be used to determine how invariant manifolds move normal to the original manifolds, there has been no method in the literature in which the tangen- tial movement is characterized. This study addresses this issue, arriving at a Melnikov-like function for the tangential displacement, under general time-dependent perturbations. The original two-dimensional flow is assumed to contain a saddle structure but need not be area- preserving. The displacement is expressed as a function of the original position p on the manifold and the time-slice t. Along the way, a similar quantification for the normal dis- placement is obtained, in which potential divergence issues in the Melnikov function and the legitimacy of ignoring higher-order terms are explicitly addressed. The normal and tangential results together permit the locating of the perturbed stable and unstable manifolds of the

Cross-separatrix flux in time-aperiodic and time-impulsive flows

A theory for the fluid flux generated across heteroclinic separatrices under the influence of time-aperiodic perturbations is presented. The flux is explicitly defined as the amount of fluid transferred per unit time, and its detailed time-dependence monitored. The perturbations are allowed to be significantly discontinuous in time, including for example impulsive (Dirac delta type) discontinuities. The flux is characterized in terms of time-varying separatrices, with easily computable formulae (directly related to Melnikov functions) provided.

Dynamical systems techniques for enhancing microfluidic mixing

Achieving rapid mixing is often desirable in microfluidic devices, for example in improving reation rates in biotechnological assays. Enhancing mixing within a particular context is often achieved by introducing problem-specific strategies such as grooved or twisted channels, ac electromagnetic fields or oscillatory microsyringe flows. Evaluating the efficiency of these methods is challenging since either experimental fabrication and sensing, or computationally expensive direct numerical simulations with complicated boundary conditions, are required. A review of how mixing can be quantified when velocity fields have been obtained from such situations is presented. A less-known alternative to these methods is offered by dynamical systems, which characterizes the motion of collective fluid parcel trajectories by studying crucial interior flow barriers which move unsteadily, but nevertheless strongly govern mixing possibilities. The methodology behind defining these barriers and quantifying the fluid transport influenced by them is explained. Their application towards several microfluidic situations (e.g. best cross-flow positioning in cross-channel micromixers, usage of channel curvature to enhance mixing within microdroplets traveling in a channel, optimum frequencies of velocity agitations to use) is discussed.

Approach for maximizing chaotic mixing in microfluidic devices

This paper uses recent theoretical work to determine the best configurations for cross-channel micromixers in optimizing mixing between two fluids. Insight into the positioning, widths, and flow protocols within the lateral channels is provided.

Weak finite-time Melnikov theory and 3D viscous perturbations of Euler flows

Abstract The ordinary differential equations related to fluid particle trajectories are examined through a 3D Melnikov approach. This theory assesses the destruction of 2D heteroclinic manifolds (such as that present in Hill’s spherical vortex) under a perturbation which is neither differentiable in the perturbation parameter e, nor defined for all times. The rationale for this theory is to analyse viscous flows that are close to steady Euler flows; such closeness in e can only reasonably be expected in a weak sense for finite times. An expression characterising the splitting of the two-dimensional separating manifold is derived.

CONTROLLING THE UNSTEADY ANALOGUE OF SADDLE STAGNATION POINTS

It is well known that saddle stagnation points are crucial flow organizers in steady (autonomous) flows due to their accompanying stable and unstable manifolds. These have been extensively investigated experimentally, numerically, and theoretically in situations related to macro- and micromixers in order to either restrict or enhance mixing. Saddle points are also important players in the dynamics of mechanical oscillators, in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior. The entity analogous to a saddle point in an unsteady (nonautonomous) flow is a time-varying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time. Within the context of nearly steady flows, the unsteady velocity perturbation required to ensure that such a hyperbolic (saddle) trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained...