MAGIC014: Hydrodynamic Stability Theory

Course details

A core MAGIC course

Semester

Spring 2025
Monday, January 27th to Friday, April 4th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Tuesdays
13:05 - 13:55 (UK)

Course forum

Visit the https://maths-magic.ac.uk/forums/magic014-hydrodynamic-stability-theory

Description

This is offered as a core course for Applied.

Prerequisites

It will be assumed that students are familiar with the Navier-Stokes equations.

Any previous experience of perturbation methods would be an advantage, but is not essential, as the main ideas will be introduced as needed. 

Syllabus

0. Some pictures of unstable flows (motivation)

1. Introduction The idea of instability(Approximately) parallel shear flows - e.g. pipe flow, boundary layers, channel flows, jets, wakes, mixing layersShear layer stability equations - reduction to linear ODEs

2. Inviscid stability theory Stability theorems - inflexion points, etc.Piecewise-linear profilesCritical points - Tollmien's solutionsEmergence of layered structures in the long-wave limitMatched asymptotic expansionsSecond order long-wave theory capturing critical layers

3. Viscous stability theory Thin viscous layers within inviscid flowDestabilizing effects of viscosityAn interpretation of the viscous instability mechanism

4. Weakly nonlinear theory Solvability conditions - when do solutions to forced equations exist?Higher order expansions in the amplitude parameter.Multiple-scales theory.Amplitude equations - supercritical/subcritical bifurcations.Wave interactions - resonant and nonresonant cases.

5. Absolute and convective instabilities Upstream and downstream propagation.Initial value problems.Saddle point methods.

Lecturer

  • JH

    Professor Jonathan Healey

    University
    Keele University

Bibliography

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Tuesday 22nd April 2025 at 00:00 and is due in before Monday 5th May 2025 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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