# Lagrangian transport and chaotic advection in three-dimensional laminar flows

## TL;DR

This paper reviews the role of Lagrangian transport and chaotic advection in three-dimensional laminar flows, emphasizing their importance for understanding and designing efficient transport in practical fluid systems.

## Contribution

It highlights the need to extend Lagrangian transport analysis from 2D to 3D flows and promotes integrating this approach into practical flow analysis and design.

## Key findings

- Lagrangian transport is key to understanding laminar flow mixing.
- 3D flows exhibit diverse Lagrangian transport behaviors.
- The review advocates for broader application of chaotic advection concepts.

## Abstract

Transport and mixing of scalar quantities in fluid flows is ubiquitous in industry and Nature. Turbulent flows promote efficient transport and mixing by their inherent randomness. Laminar flows lack such a natural mixing mechanism and efficient transport is far more challenging. However, laminar flow is essential to many problems and insight into its transport characteristics of great importance. Laminar transport, arguably, is best described by the Lagrangian fluid motion (`advection') and the geometry, topology and coherence of fluid trajectories. Efficient laminar transport being equivalent to `chaotic advection' is a key finding of this approach. The Lagrangian framework enables systematic analysis and design of laminar flows. However, the gap between scientific insights into Lagrangian transport and technological applications is formidable primarily for two reasons. First, many studies concern two-dimensional (2D) flows yet the real world is three dimensional (3D). Second, Lagrangian transport is typically investigated for idealised flows yet practical relevance requires studies on realistic 3D flows. The present review aims to stimulate further development and utilisation of know-how on 3D Lagrangian transport and its dissemination to practice. To this end 3D practical flows are categorised into canonical problems. First, to expose the diversity of Lagrangian transport and create awareness of its broad relevance. Second, to enable knowledge transfer both within and between scientific disciplines. Third, to reconcile practical flows with fundamentals on Lagrangian transport and chaotic advection. This may be a first incentive to structurally integrate the `Lagrangian mindset' into the analysis and design of 3D practical flows.