Applied Partial Differential Equations in Fluid Dynamics

Sponsored by

Lecturer

Dr Michael Dallaston, Queensland University of Technology

Synopsis

This course will cover foundational applied partial differential equation (PDE) methods that frequently arise in the study of fluid dynamics.  Particular focus will be placed on analysis and simulation of equations that arise from liquid films evolving due to interfacial phenomena such as surface tension.  We will cover standard equations of fluid mechanics, nondimensionlisation, numerical solution methods, linear stability theory, and self similarity.  While fluid mechanics provides the context for this course, the methods covered are fundamentally useful across all applications that use PDEs.

Course Overview

Week 1: Navier-Stokes equations, interfacial conditions, and lubrication theory

  • Introduction to the PDEs of fluid mechanics and nondimensionalisation
  • Approximations that reduce model complexity
  • Thin films that evolve due to surface tension and gravity.

Week 2: A crash course on numerical methods for solving transport equations

  • Conservative evolution equations
  • The finite volume method in one dimension, method of lines
  • Approximations of higher derivatives for high order PDEs

Week 3: Linear Stability Theory

  • Linearisation of Partial Differential Equations near steady states
  • Solution of linearised PDEs and interpretation
  • Visual representations: growth curves and neutral curves
  • Orr-Sommerfeld-type analysis vs stability of lubrication models

Week 4: Similarity solutions

  • Formulation and calculation of similarity solutions that explain the nonlinear behaviour of a selection of models, for example:
    • spreading under gravity
    • levelling due to surface tension

Prerequisites

  • Some knowledge of solution methods for linear PDEs (e.g. separation of variables, integral transforms)
  • Some coding experience in a high-level language (such as MATLAB, Python, Julia) very advantageous. The coding part of this course will be taught using Julia, which is freely available.

Assessment

  • TBA

Attendance requirements

  • TBA

Resources/pre-reading

The coding part of the unit will be taught in the open source programming environment Julia. To prepare for this material students should install Julia (https://julialang.org/install/), on a machine they will bring to the Summer School, with an IDE – I suggest VSCode. See the full installation instructions for here: https://code.visualstudio.com/docs/languages/julia

It will also be advantageous for students to experiment with solving ODE systems in Julia before this course – see https://docs.sciml.ai/DiffEqDocs/stable/

Note that if you are more familiar with MATLAB, and have a MATLAB licence from your university, everything done in this unit can be readily done in MATLAB also; however, the notes/lectures will be presented in Julia.

Not sure if you should sign up for this course?

Take this pre-enrolment QUIZ to self evaluate and get a measure of the key foundational knowledge required.

dallaston-web

Dr Michael Dallaston, Queensland University of Technology

I am a Senior Lecturer in Applied and Computational Mathematics at the Queensland University of Technology (QUT). I obtained my PhD from QUT in 2013, followed by postdoctoral research positions at the University of Oxford and Imperial College London, and a lecturing position at Coventry University in the UK, before returning to QUT in 2019.

My research interests are in theoretical fluid dynamics, and the applied mathematical techniques that arise in the study of PDE models in fluid dynamics, typically numerical methods, analysis of invariant solutions (such as travelling waves and similarity solutions), and perturbation methods. I also collaborate with researchers in other fields (for example mathematical biology) where applied PDE methods are also very useful.