- UCAS course code
- FG3C
- UCAS institution code
- M20
Course unit details:
Stability Theory
Unit code | MATH45032 |
---|---|
Credit rating | 15 |
Unit level | Level 4 |
Teaching period(s) | Semester 2 |
Available as a free choice unit? | 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 |
---|---|---|---|
Mathematics of Waves and Fields | PHYS20171 | Pre-Requisite | Optional |
Partial Differential Equations & Vector Calculus | MATH24420 | Pre-Requisite | Optional |
Elasticity and Viscous Fluids | MATH35020 | Pre-Requisite | Compulsory |
Partial Differential Equations and Vector Calculus A | MATH20401 | Pre-Requisite | Compulsory |
Partial Differential Equations and Vector Calculus B | MATH20411 | Pre-Requisite | Compulsory |
MATH35002 | Pre-Requisite | Compulsory |
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.
Recommended reading
- 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 | 11 |
Tutorials | 11 |
Independent study hours | |
---|---|
Independent study | 128 |
Teaching staff
Staff member | Role |
---|---|
Alice Thompson | Unit coordinator |
Additional notes
This course unit detail provides the framework for delivery in 21/22 and may be subject to change due to any additional Covid-19 impact.
Please see Blackboard / course unit related emails for any further updates.