MAS3808 : Instabilities

Semester 2 Credit Value: 10
ECTS Credits: 5.0


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.

Module summary
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
Advanced examples:
•       Kelvin-Helmholtz instability
•       Rayleigh-Benard thermal convection

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion16:006:00Revision for class test
Guided Independent StudyAssessment preparation and completion113:0013:00Revision for unseen exam
Guided Independent StudyAssessment preparation and completion12:002:00Unseen exam
Scheduled Learning And Teaching ActivitiesLecture11:001:00Class test
Scheduled Learning And Teaching ActivitiesLecture31:003:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture251:0025:00Formal lectures
Scheduled Learning And Teaching ActivitiesDrop-in/surgery120:102:00Office hours
Guided Independent StudyIndependent study33:009:00Review of coursework assignments and course test
Guided Independent StudyIndependent study26:0012:00Preparation for coursework assignments
Guided Independent StudyIndependent study125:0025:00Studying, practising, and gaining understanding of course material
Jointly Taught With
Code Title
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: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Description Length Semester When Set Percentage Comment
Written Examination1202A90N/A
Written Examination402M5Class test
Exam Pairings
Module Code Module Title Semester Comment
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M5Coursework 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 assessments have a secondary formative purpose as well as their primary summative purpose.

Reading Lists