Statistical topology of the streamlines of a two-dimensional flow

TL;DR

This paper develops a mathematical framework to analyze the topology of 2D flow streamlines, providing bounds and exact expressions for particle escape probabilities and trapped areas, supported by simulations.

Contribution

It introduces a novel approach for studying 2D flow topology, deriving bounds and formulas for particle trapping and escape probabilities, with analytical and numerical validation.

Findings

Established upper bounds on particle escape probability

Derived an exact power-series for trapped area

Validated results through numerical simulations

Abstract

Recent experiments on mucociliary clearance, an important defense against airborne pathogens, have raised questions about the topology of two-dimensional (2D) flows. We introduce a framework for studying ensembles of 2D time-invariant flow fields and estimating the probability for a particle to leave a finite area (to clear out). We establish two upper bounds on this probability by leveraging different insights about the distribution of flow velocities on the closed and open streamlines. We also deduce an exact power-series expression for the trapped area based on the asymptotic dynamics of flow-field trajectories and complement our analytical results with numerical simulations.