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School of Mathematics

# MATH45132 - 2008/2009

General Information
• Title: Hydrodynamic Stability Theory
• Unit code: MATH45132
• Credits: 15
• Prerequisites: MATH35001 Viscous Fluid Flow, MATH45101 Perturbation Methods in Fluid Mechanics.
• Co-requisite units: None
• School responsible: Mathematics
• Members of staff responsible:
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## Specification

### Aims

The aim of this course unit is to look at various topics in hydrodynamic stability theory and introduce students to some of the classical as well as more modern ideas and techniques.

### Brief Description of the unit

Many fluid flows are unstable in the sense that small disturbances superimposed on the basic mean flow can amplify and significantly distort the basic state. In this course we investigate the hydrodynamic stability of a variety of flows ranging from thin layers heated from below, to the flow between rotating cylinders, shear and boundary layer flows.

### Learning Outcomes

On successful completion of the course unit students will be able to

• derive linearised stability equations for a given basic state;
• perform a normal-mode analysis leading to an eigenvalue problems;
• use the ideas of weakly non-linear stability theory in simple flows;
• appreciate the different physical mechanisms leading to instability in fluid flows.

None.

### Syllabus

1. Basic concepts of stability theory, stability, instability, normal modes, marginal stability, neutral curves, temporal/spatial growth.
2. Rayleigh-Benard instability. Navier-Stokes equations and formulation of the linearised stability problem. Cell patterns. experimental observations.
3. Shear Flow boundary layer instability. Stability of parallel flows. Inviscid stability theory and properties of Rayleigh equation. Inflexion point criteria, Fjortoft's theorem. Howard's semi-circle theorem. Viscous/Tollmien-Schlichting instability. Orr-Sommerfeld equation. Parallel flow approximation and application to boundary layers (if time permits). Gaster Transformation (if time permits).
4. Introduction to nonlinear stability theory. The Stuart-Landau equation. Local bifurcation theory: Saddle-node, Pitchfork, Hopf and transcritical bifurcations. Structural (topological) stability. The Ginsberg-Landau equation and modulation.
5. Benjamin-Feir instability.
6. Time-dependent flows. Mathieu's equation and the parametric pendulum (if time permits).

### Textbooks

• P.G. Drazin and W. Reid, Hydrodynamic Stability, C.U.P. 1982.
• J.T. Stuart (ed. L. Rosenhead), Laminar Boundary Layers, Dover paperback (Chapter IX) 1988.
• S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover paperback 1981.

### Teaching and learning methods

Three lectures and an examples class each week.

### Assessment

Mid-semester coursework: weighting 20%
End of semester examination: two and a half hours weighting 80%

## Arrangements

Online course materials are available for this unit.