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

TL;DR

This paper characterizes the local directions of stable and unstable manifolds in nonautonomous flows, linking them to velocity shear, and proposes a method to control these directions in finite-time flows, with applications in microfluidics.

Contribution

It introduces a novel approach to identify and control time-varying stable and unstable manifold directions using velocity shear in finite-time flows.

Findings

Directions can be characterized through accumulated velocity shear.

Control methodology for prescribed time-varying directions is developed.

Finite-time Lyapunov exponent fields verify the control approach.

Abstract

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.