MATH 3375 and 5375   Hydrodynamic Stability    

 

Second Semester 2009-10

Lectures and Examples Classes:

Monday

10am

Roger Stevens LT 02

 

Tuesday

11am

Roger Stevens LT 08

 

Wednesday

2pm

Roger Stevens LT 03

Assessment:

100% Final exam

 

 

 

 

Course Handouts

Examples sheets

Solutions

 

Course outline

Examples 1

Solutions 1

Vector calculus revision

Examples 2

Solutions 2

Fluid dynamics revision

Examples 3

Derivation of Navier-Stokes equation

Examples 4

Basic ideas of hydrodynamic stability

Examples for MATH 5375

Thermal Convection: Summary

Taylor Couette Experimental Results

Centrifugal Instability: Summary

Double Diffusive Convection

 

 

 

Course description

1. The governing equations of viscous fluid dynamics.

2. Introduction to the basic ideas of hydrodynamic stability (linear and nonlinear).

3. Linear theory of Rayleigh-Bénard convection. Derivation of governing equations in the Boussinesq approximation. Non-dimensionalisation and boundary conditions. Analysis of dispersion relation. Global bounds for stability.

4. Rotating flows. Linear theory of Taylor-Couette flow; Rayleigh's criterion.

5. The linear stability of parallel shear flows; Kelvin-Helmholtz instability, Squire's theorem, Rayleigh's inflexion point criterion,
Fjørtoft's criterion.

6. (Time permitting) Beyond linear stability analysis — an introduction to nonlinear theory.

7. For students taking the course as MATH 5375 there will be an additional topic covered by directed reading and some extra examples classes. This will be on a further aspect of convection (e.g. double-diffusive convection or convection in the presence of rotation).

Recommended reading


An excellent introduction to fluid dynamics is given by the folllowing two books:


D.J. Acheson

Elementary Fluid Dynamics, Clarendon Press, Oxford, 1990.

A.R. Paterson,

A First Course in Fluid Dynamics, Cambridge University Press,  1983.

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The following two books contain many excellent photographs illustrating important fluid dynamical phenomena:


M. Van Dyke

An Album of Fluid Motion, Parabolic Press, 1982.

M. Samimy, K.S. Breuer, L.G. Leal  and P.H. Steen

A Gallery of Fluid Motion, Cambridge University Press,  2004.


In a similar vein is the CD:


G.M. Homsy et al.

Multi-media Fluid Mechanics, Cambridge University Press, 2000.

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The following books all cover the ideas of hydrodynamic stability:

 

P.G. Drazin

Introduction to Hydrodynamic Stability, Cambridge University Press, 2002.

P.G. Drazin and W.H. Reid

Hydrodynamic Stability, Cambridge University Press, 1981.

S. Chandrasekar

Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford, 1961.