Elementary Fluid Dynamics
by
D. J. Acheson


Contents

1. Introduction

1.1 An experiment
1.2 Some preliminary ideas
1.3 Equations of motion for an ideal fluid
1.4 Vorticity: irrotational flow
1.5 The vorticity equation
1.6 Steady flow past a fixed wing
1.7 Concluding remarks
Exercises

2. Elementary Viscous Flow

2.1 Introduction
2.2 The equations of viscous flow
2.3 Some simple viscous flows: the diffusion of vorticity
2.4 Flow with circular streamlines
2.5 The convection and diffusion of vorticity
Exercises

3. Waves

3.1 Introduction
3.2 Surface waves on deep water
3.3 Dispersion: group velocity
3.4 Surface tension effects: capillary waves
3.5 Effects of finite depth
3.6 Sound waves
3.7 Supersonic flow past a thin aerofoil
3.8 Internal gravity waves
3.9 Finite-amplitude waves in shallow water
3.10 Hydraulic jumps and shock waves
3.11 Viscous shocks and solitary waves
Exercises

4. Classical Aerofoil Theory

4.1 Introduction
4.2 Velocity potential and stream function
4.3 The complex potential
4.4 The method of images
4.5 Irrotational flow past a circular cylinder
4.6 Conformal mapping
4.7 Irrotational flow past an elliptical cylinder
4.8 Irrotational flow past a finite flat plate
4.9 Flow past a symmetric aerofoil
4.10 The forces involved: Blasius's theorem
4.11 The Kutta-Joukowski Lift Theorem
4.12 Lift: the deflection of the airstream
4.13 D'Alembert's paradox
Exercises

5. Vortex motion

5.1 Kelvin's circulation theorem
5.2 The persistence of irrotational flow
5.3 The Helmholtz vortex theorems
5.4 Vortex rings
5.5 Axisymmetric flow
5.6 Motion of a vortex pair
5.7 Vortices in flow past a circular cylinder
5.8 Instability of vortex patterns
5.9 A steady viscous vortex maintained by a secondary flow
5.10 Viscous vortices: the Prandtl-Batchelor theorem
Exercises

6. The Navier-Stokes Equations

6.1 Introduction
6.2 The stress tensor
6.3 Cauchy's equation of motion
6.4 A Newtonian viscous fluid: the Navier-Stokes equations
6.5 Viscous dissipation of energy
Exercises

7. Very Viscous Flow

7.1 Introduction
7.2 Low Reynolds number flow past a sphere
7.3 Corner eddies
7.4 Uniqueness and reversibility of slow flows
7.5 Swimming at low Reynolds number
7.6 Flow in a thin film
7.7 Flow in a Hele-Shaw cell
7.8 An adhesive problem
7.9 Thin-film flow down a slope
7.10 Lubrication theory
Exercises

8. Boundary Layers

8.1 Prandtl's paper
8.2 The steady 2-D boundary layer equations
8.3 The boundary layer on a flat plate
8.4 High Reynolds number flow in a converging channel
8.5 Rotating flows controlled by boundary layers
8.6 Boundary layer separation
Exercises

9. Instability

9.1 The Reynolds experiment
9.2 Kelvin-Helmholtz instability
9.3 Thermal convection
9.4 Centrifugal instability
9.5 Instability of parallel shear flow
9.6 A general theorem on the stability of viscous flow
9.7 Uniqueness and non-uniqueness of steady viscous flow
9.8 Instability, chaos and turbulence
9.9 Instability at very low Reynolds number
Exercises

Appendix

Hints and answers for exercises

Bibliography

Index


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