MATH 3375 and 5376   Hydrodynamic Stability    

 

First Semester 2011-12

Lectures and ClassesTuesday12 noonRoger Stevens LT 5
for MATH 3375 and 5376:Thursday11 amRoger Stevens LT 3
Friday10 amRoger Stevens LT 12
Additional lecture for MATH 5376:Friday12 noonRoger Stevens LT 6

 

 

 

Course Handouts

Examples sheets

Solutions

 

Course outline

Examples 1

Solutions 1

Vector calculus revision

Examples 2

Solutions 2

Fluid dynamics revision

Examples 3

Solutions 3

Derivation of Navier-Stokes equation

Examples 4

Solutions 4

Basic ideas of hydrodynamic stability

Examples 5

Solutions 5

Thermal convection: summary

Examples 1 for MATH 5376

Solutions 1 for MATH 5376

Double diffusive convection: summary (MATH5376)

Examples 2 for MATH 5376

Solutions 2 for MATH 5376

Taylor Couette experimental results

Centrifugal instability: summary

Shear flow instability: summary

Algebra for Example (iii), Section 5.8

Rotating Convection Stability Boundaries (MATH 5376)

 

 

 

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.

7. For students taking the course as MATH 5376 there will be an additional lecture/class each week (starting week 2), which will cover additional material relating to further aspects of convection, such as double-diffusive convection and convection in the presence of rotation.


Recommended reading


An excellent introduction to fluid dynamics is given by the following 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 (2nd ed. 2004).

S. Chandrasekar

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