MAS3808 : Instabilities
- Offered for Year: 2018/19
- Module Leader(s): Dr Thomas Billam
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
|Semester 2 Credit Value:||10|
To introduce linear stability theory, and demonstrate how it can be used it to understand the behaviour of mathematical models representing real-world systems, particularly in the field of fluid mechanics.
Why can you hang an umbrella from a hook, but not stand it on its point? Why do some fluid flows remain smooth while others become turbulent? Linear stability theory provides a mathematical framework to answer such questions.
The time-evolution of innumerable real-world systems can be described using mathematical models, but the resulting equations can be complicated and nonlinear. Often there are no general solutions. Nonetheless, linear stability theory provides a way to determine whether a particular steady state of the system is stable against small perturbations. The theory also provides insight into the nature of the systems of equations themselves, and highlights profound connections between the theory of differential equations and linear algebra.
Outline Of Syllabus
Developing mathematical models:
• Dimensionless variables and parameters
• Equations of motion: ODEs and PDEs
Introduction to linear stability theory:
• Linearization around steady state
• Normal modes
• Classification of solutions: bifurcations and stability criteria
• Kelvin-Helmholtz instability
• Rayleigh-Benard thermal convection
|Scheduled Learning And Teaching Activities||Lecture||25||1:00||25:00||Formal lectures|
|Guided Independent Study||Assessment preparation and completion||1||6:00||6:00||Revision for class test|
|Guided Independent Study||Assessment preparation and completion||1||13:00||13:00||Revision for unseen exam|
|Guided Independent Study||Assessment preparation and completion||1||2:00||2:00||Unseen exam|
|Scheduled Learning And Teaching Activities||Lecture||1||1:00||1:00||Class test|
|Scheduled Learning And Teaching Activities||Lecture||3||1:00||3:00||Problem classes|
|Scheduled Learning And Teaching Activities||Lecture||2||1:00||2:00||Revision lectures|
|Guided Independent Study||Independent study||1||27:00||27:00||Studying, practising, and gaining understanding of course material|
|Guided Independent Study||Independent study||3||3:00||9:00||Review of coursework assignments and course test|
|Guided Independent Study||Independent study||2||6:00||12:00||Preparation for coursework assignments|
Teaching Rationale And Relationship
Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) will provide an opportunity for more direct contact between individual students and the lecturer.
The format of resits will be determined by the Board of Examiners
|Written Examination||40||2||M||5||Class test|
|Prob solv exercises||2||M||5||Coursework assignments|
Assessment Rationale And Relationship
A substantial formal unseen examination is appropriate for the assessment of the material in this module. The written exercises are expected to consist of two assignments of equal weight: the exact nature of assessment will be explained at the start of the module. The coursework assignments and the (in class) coursework test allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these are thus formative as well as summative assessments.