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PHY3047 : Instabilities

Semesters
Semester 1 Credit Value: 10
ECTS Credits: 5.0

Aims

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
Scheduled Learning And Teaching ActivitiesLecture201:0020:00Formal Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture51:005:00Problem Classes – Synchronous On-Line
Guided Independent StudyAssessment preparation and completion151:0015:00Completion of in course assessments
Guided Independent StudyIndependent study581:0058:00Preparation time for lectures, background reading, coursework review
Total100:00
Jointly Taught With
Code Title
MAS3808Instabilities
MAS8808Instabilities
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.

Assessment Methods

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

Exams
Description Length Semester When Set Percentage Comment
Written Examination1201A80N/A
Exam Pairings
Module Code Module Title Semester Comment
MAS3808Instabilities1N/A
MAS8808Instabilities2N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Coursework assignment
Prob solv exercises1M10Coursework assignment
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments 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.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.

Reading Lists

Timetable