# MAGIC014: Hydrodynamic Stability Theory

## Course details

A core MAGIC course

### Semester

Spring 2014
Monday, January 20th to Friday, March 28th

### Hours

Live lecture hours
20
Recorded lecture hours
0
0

Mondays
13:05 - 13:55
Fridays
12:05 - 12:55

## Description

This is offered as a core course for Applied.

### Prerequisites

No prerequisites information is available yet.

### Syllabus

1. Introduction (2 lectures)
• Derivation of the Navier-Stokes equations
• Boundary conditions
• Non-dimensionalisation
• Additional forces and equations: Coriolis force, buoyancy
• Boussinesq approximation
2. Basics of stability theory (2 lectures)
• Swift-Hohenberg equation as a model
• Linear stability. Dispersion relation.
• Marginal stability curve.
• Weakly nonlinear theory.
• Normal form for pitchfork bifurcation
• Global stability
3. Rayleigh-Benard convection (4 lectures)
• Basic state. Linear theory. Normal modes.
• Marginal stability curve.
• Weakly nonlinear theory. Modified perturbation theory.
• Global stability for two-dimensional solutions
• Truncation: the Lorenz equations
4. Double-diffusive convection (2 lectures)
• Thermosolutal convection. Salt fingers.
• Linear theory: real and complex eigenvalues.
• Rotating convection, plane layer and spherical geometry
• Taylor-Proudman theorem.
5. The Taylor-Couette problem (1 lecture)
6. Instabilities of parallel flows (6 lectures)
• Instabilities of invicid shear flows. Linear theory.
• Squire's theorem. Rayleigh's equation.
• Plane Couette flow.
• Rayleigh's inflexion point criterion.
• Howard's semi-circle theorem.
• Examples: Kelvin-Helmholtz, bounded shear layer.
• Role of stratification. Role of viscosity, global stability.
• Shear flow instabilities of viscous fluids.
• Orr-Sommerfeld equation.
• Examples: plane Couette flow, plane Poiseuille flow, pipe flow, Taylor-Couette flow.
• Problems with normal mode analysis.
• Pseudo-spectrum and non-normality.
• Absolute and convective instabilities.
• Finite domain effects.
7. Introduction to pattern formation (3 lectures)
• Stripes, squares and hexagons. Weakly nonlinear theory.
• Three-wave interactions.
• The role of symmetry.
• Long-wave instabilities of patterns: Eckhaus.

## Lecturer

• JH

### Prof Jonathan Healey

University
Keele University

## Bibliography

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## Assessment

### Description

The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. Questions may be of different lengths. The number marks for each question will be indicated. The pass mark will be 50%.

### Assessment not available

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