Transport in Transitory, Three-Dimensional, Liouville Flows

TL;DR

This paper introduces an action-flux formula for calculating transport volumes between invariant structures in transitory, three-dimensional Liouville flows, demonstrated through examples involving fluid flows and microdroplet movement.

Contribution

The authors develop a novel, efficient method for computing transport volumes in transitory Liouville flows using minimal Lagrangian information, applicable to complex 3D systems.

Findings

The method accurately computes transport volumes in 3D flows.

Comparison with Monte Carlo methods validates the approach.

Applicable to fluid dynamics and microfluidic systems.

Abstract

We derive an action-flux formula to compute the volumes of lobes quantifying transport between past- and future-invariant Lagrangian coherent structures of n-dimensional, transitory, globally Liouville flows. A transitory system is one that is nonautonomous only on a compact time interval. This method requires relatively little Lagrangian information about the codimension-one surfaces bounding the lobes, relying only on the generalized actions of loops on the lobe boundaries. These are easily computed since the vector fields are autonomous before and after the time-dependent transition. Two examples in three-dimensions are studied: a transitory ABC flow and a model of a microdroplet moving through a microfluidic channel mixer. In both cases the action-flux computations of transport are compared to those obtained using Monte Carlo methods.