MAGIC014: Hydrodynamic Stability Theory

Course details

A core MAGIC course


Spring 2014
Monday, January 20th to Friday, March 28th


Live lecture hours
Recorded lecture hours
Total advised study hours


13:05 - 13:55
12:05 - 12:55


This is offered as a core course for Applied.


No prerequisites information is available yet.


  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.


  • JH

    Prof Jonathan Healey

    Keele University


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The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. Questions may be of different lengths. The number marks for each question will be indicated. The pass mark will be 50%.

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