## Announcements

There are no announcements

Forum

## 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

#### 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.

## Lecturers

Chris Jones (main contact)
 Email cajones@maths.leeds.ac.uk Phone (0113) 3435107

 Email dwh@maths.leeds.ac.uk Phone (0113) 3435105 Interests Astrophysical fluids

 Email A.M.Rucklidge@leeds.ac.uk Phone (0113) 3435161 Interests Pattern Formation, Nonlinear Dynamics, Astrophysical Fluid Dynamics

 Email smt@maths.leeds.ac.uk Phone (0113) 3435172

## Students

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

## 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

Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

## Assessment

No assessment information is available yet.

Example sheet 1

 Files: Exam paper Deadline: Friday 24 February 2012 (2038.7 days ago)

Example Sheet 2

 Files: Exam paper Deadline: Wednesday 7 March 2012 (2026.7 days ago)

Example Sheet 3

 Files: Exam paper Deadline: Wednesday 21 March 2012 (2012.7 days ago)

Example Sheet 4

 Files: Exam paper Deadline: Thursday 12 April 2012 (1990.8 days ago)

## Files

Files marked L are intended to be displayed on the main screen during lectures.

Week(s) File examples_1_2012.pdf examples_2_2012.pdf examples_3_2012.pdf examples_4_2012.pdf 1-2 hydro_01.pdf L 2-3 hydro_02.pdf L 3-4 hydro_03.pdf L 3-4 hydro_03_supp_integral_relations.pdf 3-4 hydro_03_supp_self_adjoint.pdf 5 hydro_04.pdf L 6-7 hydro_05.pdf L 6-7 sh2d_15_zz_anim.gif L 6-7 sh2d_17_stable_anim.gif L 6-7 sh2d_18_eckhaus_anim.gif L 6-7 sh2d_hex_anim.gif L 6-7 sh2d_linear_anim.gif L 6-7 sh2d_rolls_anim.gif L 7 hydro_05_extra_notes_120227.pdf 7 hydro_05_extra_notes_120229.pdf 8-10 hydro_07.pdf L 8 Couette flow analysis.pdf 8 hydro_06.pdf L