# 2020/21 Taught Postgraduate Module Catalogue

### 20 creditsClass Size: 47

Email: A.J.Barker@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2020/21

### Pre-requisites

 MATH3620 Fluid Dynamics 2

### This module is mutually exclusive with

 MATH3375 Hydrodynamic Stability

Module replaces

MATH5375M

This module is not approved as an Elective

### Module summary

This module provides an introduction to the idea of the instability of fluid flows. This is a very important concept in hydrodynamics. For example, it is straightforward to derive an expression for a simple, laminar flow of fluid down a pipe. But will this simple flow always be realised in practice? This depends on whether the flow is stable or not. The ideas will be illustrated by looking in detail at three problems; the instability of fluids due to convection, the instability of rotating fluids, and the instability of shear flows. The bulk of the module will consider linear instability, i.e. instability to infinitesimal disturbances. The module will finish with an introduction to some of the important ideas of the nonlinear evolution of fluid instabilities.

### Objectives

On completion of the module, students should be able to:
- Understand the equations of viscous and inviscid fluid dynamics and the ideas of hydrodynamic stability theory.
- Apply the ideas of linear stability theory to the various problems of Rayleigh-Bénard convection, swirling flows and parallel shear flows.
- Understand the weakly nonlinear behaviour of certain fluid instabilities.

Learning outcomes
On completion of the module, students should be able to:
- Understand the equations of viscous and inviscid fluid dynamics and the ideas of hydrodynamic stability theory.
- Apply the ideas of linear stability theory to the various problems of Rayleigh-Bénard convection, swirling flows and parallel shear flows.
- Understand the weakly nonlinear behaviour of certain fluid instabilities.

### Syllabus

- Revision of the governing equations of inviscid and viscous fluid dynamics.
- Introduction to the ideas of hydrodynamic stability (linear and nonlinear).
- Linear theory of Rayleigh-Bénard convection. Derivation of governing equations in the Boussinesq approximation. Nondimensionalisation and boundary conditions. Analysis of dispersion relation. Global bounds for stability.
- Linear theory of swirling flows; Rayleigh's criterion. Application to Taylor-Couette flow.
- The linear stability of parallel shear flows; Squire's theorem, Rayleigh's inflexion point criterion, Fjørtoft's criterion. Kelvin-Helmholtz instability. Stability of piecewise linear flows. Effect of stratification: Richardson number criterion.
- Introduce some of the ideas of nonlinear behaviour through consideration of weakly nonlinear models of fluid instabilities.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Private study hours 178.00 Total Contact hours 22.00 Total hours (100hr per 10 credits) 200.00

Examples classes

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Open Book exam 3 hr 00 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated