Transport between two fluids across their mutual flow interface: the streakline approach

Mixing between two different miscible fluids with a mutual interface must be initiated by fluid transporting across this fluid interface, caused for example by applying an unsteady velocity agitation. In general, there is no necessity for this physical flow barrier between the fluids to be associated with extremal or exponential attraction as might be revealed by applying Lagrangian coherent structures, finite-time Lyapunov exponents or other methods on the fluid velocity. It is shown that streaklines are key to understanding the breaking of the interface under velocity agitations, and a theory for locating the relevant streaklines is presented. Simulations of streaklines in a cross-channel mixer and a perturbed Kirchhoff's elliptic vortex are quantitatively compared to the theoretical results. A methodology for quantifying the unsteady advective transport between the two fluids using streaklines is presented

Local stable and unstable manifolds and their control in nonautonomous finite-time flows

It is well-known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite and finite time-intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.Comment: Under consideration for publication in the Journal of Nonlinear Science

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory

Nonautonomous control of stable and unstable manifolds in two-dimensional flows

We outline a method for controlling the location of stable and unstable manifolds in the following sense. From a known location of the stable and unstable manifolds in a steady two-dimensional flow, the primary segments of the manifolds are to be moved to a user-specified time-varying location which is near the steady location. We determine the nonautonomous perturbation to the vector field required to achieve this control, and give a theoretical bound for the error in the manifolds resulting from applying this control. The efficacy of the control strategy is illustrated via a numerical example

Unifying Lyapunov exponents with probabilistic uncertainty quantification

The Lyapunov exponent is well-known in deterministic dynamical systems as a measure for quantifying chaos and detecting coherent regions in physically evolving systems. In this Letter, we show how the Lyapunov exponent can be unified with stochastic sensitivity (which quantifies the uncertainty of an evolving uncertain system whose initial condition is certain) within a finite time uncertainty quantification framework in which both the dynamics and the initial condition of a continuously evolving $n$-dimensional state variable are uncertain

Absolute flux optimising curves of flows on a surface

Given a flow on a surface, we consider the problem of connecting two distinct trajectories by a curve of extremal (absolute) instantaneous flux. We develop a complete classification of flux optimal curves, accounting for the possibility of the flux having spatially and temporally varying weight. This weight enables modelling the flux of non-equilibrium distributions of tracer particles, pollution concentrations, or active scalar fields such as vorticity. Our results are applicable to all smooth autonomous flows, area preserving or not

Optimal frequency for microfluidic mixing across a fluid interface

A new analytical tool for determining the optimum frequency for a micromixing strategy to mix two fluids across their interface is presented. The frequency dependence of the flux is characterized in terms of a Fourier transform related to the apparatus geometry. Illustrative microfluidic mixing examples based on electromagnetic forcing and fluid pumping strategies are presented.Sanjeeva Balasuriy

Energy constrained transport maximization across a fluid interface

With enhancing mixing in micro- or nanofluidic applications in mind, the problem of maximizing fluid transport across a fluid interface subject to an available energy budget is examined. The optimum cross-interface perturbing velocity is obtained explicitly in the time-periodic instance using an Euler-Lagrange constrained optimization approach. Numerical investigations which calculate transferred lobe areas and cross-interface flux are used to verify that the predicted strategy achieves optimum transport. Explicit active protocols for achieving this optimal transport are suggested.Sanjeeva Balasuriya, Matthew D. Fin

Quantifying transport within a two-cell microdroplet induced by circular and sharp channel bends

Abstract not availableSanjeeva Balasuriya