Description

How do you best describe the flow of a complex fluid like a liquid or gas? This project will study the so-called Lagrangian picture of fluid mechanics. This is actually the obvious approach, where one follows each fluid particle around according to Newton's second law. But mathematically, this turns out to be trickier than the common Eulerian picture, in which one abandons trying to follow individual particles, instead measuring the fluid properties at fixed points in space (leading to the famous Navier-Stokes equations).

For understanding the complex flows of realistic fluids, however, the Lagrangian approach has undergone something of a resurgence, with the availability of modern computing power. The idea is to characterise the behaviour of fluid particle trajectories in a particular flow, using ideas from dynamical systems. Particular importance is attached to Lagrangian coherent structures, which are strongly attracting/repelling surfaces that act as a kind of underlying skeleton for the fluid flow. This skeleton organises transport within the flow, and leads to many of the observed structures that you see in real fluids. Oil spills, clouds of volcanic ash, and chlorophyll patterns in the ocean are just a few (large-scale) examples.

This project will consider the mathematical definition of Lagrangian coherent structures (still a matter for debate in the literature), and their identification. There is ample scope for practical work for the numerically inclined (for example, with the Python programming language).

Lagrangian coherent structures (in red and blue) in the flow around a swimming jellyfish (Sledge et al.).

Prerequisites

Analysis in Many Variables II is essential, and I would strongly recommend Dynamical Systems III and/or Continuum Mechanics III as prerequisites. If you are interested in numerical work then you should take Numerical Differential Equations IV alongside the project.

Resources

A good place to start is the Physics Today article by Peacock and Haller, or this article from The Economist. Useful resources include this recent review article by Haller, or this book chapter by Shadden, who also has an online tutorial. You might like to try this interactive FTLE demonstration from Douglas Lipinski.