Tracking Particles in Flows near Invariant Manifolds via Balance Functions

TL;DR

This paper introduces balance functions based on Lyapunov exponents to determine particle entry and exit times near invariant manifolds in fluid flows, with applications to model flows and Navier-Stokes simulations.

Contribution

It develops a mathematical framework using balance functions and Lyapunov exponents to analyze particle trajectories near invariant manifolds in fluid dynamics.

Findings

Balance functions effectively predict particle entry and exit points.

Normal infinitesimal Lyapunov exponents provide an efficient balance function.

Framework applicable to both models and real data.

Abstract

Particles moving inside a fluid near, and interacting with, invariant manifolds is a common phenomenon in a wide variety of applications. One elementary question is whether we can determine once a particle has entered a neighbourhood of an invariant manifold, when it leaves again. Here we approach this problem mathematically by introducing balance functions, which relate the entry and exit points of a particle by an integral variational formula. We define, study, and compare different natural choices for balance functions and conclude that an efficient compromise is to employ normal infinitesimal Lyapunov exponents. We apply our results to two different model flows: a regularized solid-body rotational flow and the asymmetric Kuhlmann--Muldoon model developed in the context of liquid bridges. Furthermore, we employ full numerical simulations of the Navier-Stokes equations of a two-way coupled particle in a shear--stress-driven cavity to test balance functions for a particle moving near an invariant wall. In conclusion, our theoretically-developed framework seems to be applicable to models as well as data to understand particle motion near invariant manifolds.