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