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General


This course is part of the MAGIC core.

Description

This is offered as a core course for Applied.
Semester

Spring 2012 (Monday, January 16 to Friday, March 23)

Timetable
  • Mon 12:05 - 12:55
  • Wed 10:05 - 10:55

Lecturers


Chris Jones (main contact)
Email cajones@maths.leeds.ac.uk
Phone (0113) 3435107
vcard
David Hughes
Email dwh@maths.leeds.ac.uk
Phone (0113) 3435105
Interests Astrophysical fluids
vcard
Photo of David Hughes
Alastair Rucklidge
Email A.M.Rucklidge@leeds.ac.uk
Phone (0113) 3435161
Interests Pattern Formation, Nonlinear Dynamics, Astrophysical Fluid Dynamics
vcard
Photo of Alastair Rucklidge
Steven Tobias
Email smt@maths.leeds.ac.uk
Phone (0113) 3435172
vcard


Students


Photo of Ali A
Ali A
(Loughborough)
Photo of Azwani  Alias
Azwani Alias
(Loughborough)
Photo of Chris Bennett
Chris Bennett
(Birmingham)
Photo of Matthew Buckley
Matthew Buckley
(Newcastle)
Photo of Richard Clift
Richard Clift
(Loughborough)
Photo of Laura Cole
Laura Cole
(Newcastle)
Photo of Sarah Dawson
Sarah Dawson
(Leeds)
Photo of Neil Deacon
Neil Deacon
(East Anglia)
Photo of Aiman Elragig
Aiman Elragig
(Exeter)
Photo of Meurig Gallagher
Meurig Gallagher
(Birmingham)
Photo of Mariano Galvagno
Mariano Galvagno
(Loughborough)
Photo of Sam Jones
Sam Jones
(Exeter)
Photo of Thorpe Maria
Thorpe Maria
(Manchester)
Photo of Pearce Philip
Pearce Philip
(Manchester)
Photo of Harvind Rai
Harvind Rai
(Birmingham)
Photo of Chris Rowlatt
Chris Rowlatt
(Cardiff)
Photo of Lucy Sherwin
Lucy Sherwin
(Newcastle)
Photo of Jonathan Stone
Jonathan Stone
(Southampton)
Photo of Julian Thompson
Julian Thompson
(East Anglia)
Photo of Martin Walters
Martin Walters
(East Anglia)
Photo of Michael Walters
Michael Walters
(Cardiff)
Photo of James Wright
James Wright
(Surrey)


Prerequisites


No prerequisites information is available yet.

Syllabus


  1. Introduction (2 lectures)
    • Derivation of the Navier-Stokes equations
    • Boundary conditions
    • Non-dimensionalisation
    • Additional forces and equations: Coriolis force, buoyancy
    • Boussinesq approximation
  2. Basics of stability theory (2 lectures)
    • Swift-Hohenberg equation as a model
    • Linear stability. Dispersion relation.
    • Marginal stability curve.
    • Weakly nonlinear theory.
    • Normal form for pitchfork bifurcation
    • Global stability
  3. Rayleigh-Benard convection (4 lectures)
    • Basic state. Linear theory. Normal modes.
    • Marginal stability curve.
    • Weakly nonlinear theory. Modified perturbation theory.
    • Global stability for two-dimensional solutions
    • Truncation: the Lorenz equations
  4. Double-diffusive convection (2 lectures)
    • Thermosolutal convection. Salt fingers.
    • Linear theory: real and complex eigenvalues.
    • Rotating convection, plane layer and spherical geometry
    • Taylor-Proudman theorem.
  5. The Taylor-Couette problem (1 lecture)
  6. Instabilities of parallel flows (6 lectures)
    • Instabilities of invicid shear flows. Linear theory.
    • Squire's theorem. Rayleigh's equation.
    • Plane Couette flow.
    • Rayleigh's inflexion point criterion.
    • Howard's semi-circle theorem.
    • Examples: Kelvin-Helmholtz, bounded shear layer.
    • Role of stratification. Role of viscosity, global stability.
    • Shear flow instabilities of viscous fluids.
    • Orr-Sommerfeld equation.
    • Examples: plane Couette flow, plane Poiseuille flow, pipe flow, Taylor-Couette flow.
    • Problems with normal mode analysis.
    • Pseudo-spectrum and non-normality.
    • Absolute and convective instabilities.
    • Finite domain effects.
  7. Introduction to pattern formation (3 lectures)
    • Stripes, squares and hexagons. Weakly nonlinear theory.
    • Three-wave interactions.
    • The role of symmetry.
    • Long-wave instabilities of patterns: Eckhaus.

Bibliography


Hydrodynamic stability Drazin and Reid
Introduction to hydrodynamic stability Drazin
The theory of hydrodynamic stability Lin
Hydrodynamic and hydromagnetic stability Chandrasekhar
An introduction to fluid dynamics Batchelor
Elementary fluid dynamics Acheson
Benard cells and Taylor vortices Koschmieder
Pattern formation: an introduction to methods Hoyle
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Assessment


No assessment information is available yet.

Assignments


Example sheet 1

Files:Exam paper
Deadline: Friday 24 February 2012 (882.0 days ago)


Example Sheet 2

Files:Exam paper
Deadline: Wednesday 7 March 2012 (870.0 days ago)


Example Sheet 3

Files:Exam paper
Deadline: Wednesday 21 March 2012 (856.0 days ago)


Example Sheet 4

Files:Exam paper
Deadline: Thursday 12 April 2012 (834.1 days ago)