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Module Leader(s): Dr Magda Carr
Owning School: Mathematics, Statistics and Physics
Teaching Location: Newcastle City Campus
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 Activities
Lecture
20
1:00
20:00
Formal Lectures – Present in Person
Scheduled Learning And Teaching Activities
Lecture
2
1:00
2:00
Revision Lectures – Present in Person
Scheduled Learning And Teaching Activities
Lecture
5
1:00
5:00
Problem Classes – Synchronous On-Line
Guided Independent Study
Assessment preparation and completion
15
1:00
15:00
Completion of in course assessments
Guided Independent Study
Independent study
58
1:00
58:00
Preparation time for lectures, background reading, coursework review
Total
100:00
Jointly Taught With
Code
Title
MAS3808
Instabilities
MAS8808
Instabilities
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 Examination
120
1
A
80
N/A
Exam Pairings
Module Code
Module Title
Semester
Comment
MAS3808
Instabilities
1
N/A
MAS8808
Instabilities
2
N/A
Other Assessment
Description
Semester
When Set
Percentage
Comment
Prob solv exercises
1
M
10
Coursework assignment
Prob solv exercises
1
M
10
Coursework 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.
