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# MMath Mathematics and Statistics / Course details

Year of entry: 2021

## Course unit details:Stability Theory

Unit code MATH45132 15 Level 4 Semester 2 Department of Mathematics No

### Overview

Many physical systems can become unstable in the sense that small disturbances superimposed on their basic state can amplify and significantly alter their initial state. In this course we introduce the basic theoretical and physical methodology required to understand and predict instability in a variety of situations with focus on hydrodynamic instabilities and on instabilities in reaction-diffusion systems.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Viscous Fluid Flow MATH35001 Pre-Requisite Compulsory
Partial Differential Equations and Vector Calculus A MATH20401 Pre-Requisite Compulsory
MATH45132 pre-requisites

Students are not permitted to take, for credit, MATH45132 in an undergraduate programme and then MATH65132 in a postgraduate programme at the University of Manchester, as the courses are identical.

This course is largely self-contained.

### Aims

The aim of this course unit is to introduce students to the basic concepts and techniques of modern stability theory, through case studies in fluid mechanics and transport phenomena.

### Learning outcomes

On successful completion of this 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 problem;
• Derive dispersion relations and use them to identify whether a basic state is stable or unstable for given values of the parameters;
• Perform a weakly non-linear stability analysis for simple systems near the instability threshold;
• Recognise the different physical mechanisms leading to instability for problems involving fluid flow or transport phenomena.

### Syllabus

Assuming general mechanics and fluid mechanics in particular (viscous/inviscid), as well as some aspects of dynamical systems as prerequisites for course.

1. Introduction to stability

Nonlinear dynamics. Linear instability versus nonlinear instability. Outline of the basic procedure involved in a linear stability analysis: dispersion relation, marginal stability curve. Role of weakly nonlinear theory, e.g. normal form for pitchfork bifurcation.

2. Linear stability analysis: a case study of Rayleigh-Benard convection

Introduction to physical system, Boussinesq equations, dimensional analysis, Basic state, linear theory, normal modes, marginal stability curve: Analytical approach for idealised boundary conditions.

3. Interfacial instabilities

Examples: Rayleigh-Taylor and capillary instabilities.

4. Shear flow instabilities

Inviscid/viscous, Squire's theorem. Rayleigh's equation, Rayleigh's inflexion point criterion, Howard's semi-circle theorem, Orr-Sommerfeld equation. Examples: plane Couette flow, plane Poiseuille flow, pipe flow, Taylor-Couette flow.

5. Stability in reaction diffusion systems.

Stability of propagating fronts.

6. Bifurcation theory

Local bifurcations, normal forms.

7. Nonlinear stability theory

Weakly nonlinear theory, derivation of Stuart-Landau equation, Ginzburg-Landau equation.

8. Introduction to pattern formation (if time allows)

Stripes, squares and hexagons, three-wave interactions, role of symmetry, long-wave instabilities of patterns: Eckhaus.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Mid-semester coursework: 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• P.G. Drazin, Introduction to hydrodynamic stability. Cambridge University Press (2002)
• F. Charru, Hydrodynamic Instabilities. Cambridge University Press (2011)
• P. Manneville, Instabilities, chaos and turbulence. Imperial College Press (2004)

### Study hours

Scheduled activity hours
Lectures 24
Tutorials 12
Independent study hours
Independent study 114

### Teaching staff

Staff member Role
Joel Daou Unit coordinator