Chapter 1. The Physical Properties of Fluids

1.1 Solids， liquids and gases

1.2 The continuum hypothesis

1.3 Volume forces and surface forces acting on a fluid

Representation of surface forces by the stress tensor

The stress tensor in a fluid at rest

1.4 Mechanical equilibrium of a fluid

A body ‘floating’ in fluid at rest

Fluid at rest under gravity

1.5 Classical thermodynamics

1.6 Transport phenomena

The linear relation between flux and the gradient of a scalar intensity

The equations for diffusion and heat conduction in isotropic media at rest

Molecular transport of momentum in a fluid

1.7 The distinctive properties of gases

A perfect gas in equilibrium

Departures from the perfect-gas laws

Transport coefficients in a perfect gas

Other manifestations of departure from equilibrium of a perfect gas

1.8 The distinctive properties of liquids

Equilibrium properties

Transport coefficients

1.9 Conditions at a boundary between two media

Surface tension

Equilibrium shape of a boundary between two stationary fluids

Transition relations at a material boundary

Chapter 2. Kinematics of the Flow Field

2.1 Specification of the flow field

Differentiation following the motion of the fluid

2.2 Conservation of mass

Use of a stream function to satisfy the mass-conservation equation

2.3 Analysis of the relative motion near a point

Simple shearing motion

2.4 Expression for the velocity distribution with specified rate of expansion and vorticity

2.5 Singularities in the rate of expansion. Sources and sinks

2.6 The vorticity distribution

Line vortices

Sheet vortices

2.7 Velocity distributions with zero rate of expansion and zero vorticity

Conditions for ▽φ to be determined uniquely

Irrotational solenoidal flow near a stagnation point

The complex potential for irrotational solenoidal flow in two dimensions

2.8 Irrotational solenoidal flow in doubly-connected regions of space

Conditions for ▽φ to be determined uniquely

2.9 Three-dimensional flow fields extending to infinity

Asymptotic expressions for uoand uv

The behaviour of φ at large distances

Conditions for ▽φ to be determined uniquely

The expression of φ as a power series

Irrotational solenoidal flow due to a rigid body in translational motion

2.10 Two-dimensional flow fields extending to infinity

Irrotational solenoidal flow due to a rigid body in translational motion

Chapter 3. Equations Governing the Motion of a Fluid

3.1 Material integrals in a moving fluid

Rates of change of material integrals

Conservation laws for a fluid in motion

3.2 The equation of motion

Use of the momentum equation in integral form

Equation of motion relative to moving axes

3.3 The expression for the stress tensor

Mechanical definition of pressure in a moving fluid

The relation between deviatoric stress and rate-of-strain for a Newtonian fluid

The Navier-Stokes equation

Conditions on the velocity and stress at a material boundary

3.4 Changes in the internal energy of a fluid in motion

3.5 Bernoulli’s theorem for steady flow of a frictionless non-conducting fluid

Special forms of Bernoulli’s theorem

Constancy of H across a transition region in one-dimensional steady flow

3.6 The complete set of equations governing fluid flow

Isentropic flow

Conditions for the velocity distribution to be approximately solenoidal

3.7 Concluding remarks to chapters 1，2 and 3

Chapter 4. Flow of a Uniform Incompressible Viscous Fluid

4.1 Introduction

Modification of the pressure to allow for the effect of the body force

4.2 Steady unidirectional flow

Poiseuille flow

Tubes of non-circular cross-section

Two-dimensional flow

A model of a paint-brush

A remark on stability

4.3 Unsteady unidirectional flow

The smoothing-out of a discontinuity in velocity at a plane

Plane boundary moved suddenly in a fluid at rest

One rigid boundary moved suddenly and one held stationary

Flow due to an oscillating plane boundary

Starting flow in a pipe

4.4 The Ekman layer at a boundary in a rotating fluid

The layer at a free surface

The layer at a rigid plane boundary

4.5 Flow with circular streamlines

4.6 The steady jet from a point source of momentum

4.7 Dynamical similarity and the Reynolds number

Other dimensionless parameters having dynamical significance

4.8 Flow fields in which inertia forces are negligible

Flow in slowly-varying channels

Lubrication theory

The Hele-Shaw cell

Percolation through porous media

Two-dimensional flow in a corner

Uniqueness and minimum dissipation theorems

4.9 Flow due to a moving body at small Reynolds number

A rigid sphere

A spherical drop of a different fluid

A body of arbitrary shape

4.10 Oseen’s improvement of the equation for flow due to moving bodies at small Reynolds number

A rigid sphere

A rigid circular cylinder

4.11 The viscosity of a dilute suspension of small particles

The flow due to a sphere embedded in a pure straining motion

The increased rate of dissipation in an incompressible suspension

The effective expansion viscosity of a liquid containing gas bubbles

4.12 Changes in the flow due to moving bodies as R increases from I to about Ioo

Chapter 5. Flow at Large Reynolds Number：Effects of Viscosity

5.1 Introduction

5.2 Vorticity dynamics

The intensification of vorticity by extension of vortex-lines

5.3 Kelvin’s circulation theorem and vorticity laws for an inviscid fluid

The persistence of irrotationality

5.4 The source of vorticity in motions generated from rest

5.5 Steady flows in which vorticity generated at a solid surface is prevented by convection from diffusing far away from it

（a） Flow along plane and circular walls with suction through the wall

（b） Flow toward a ‘stagnation point’ at a rigid boundary

（c） Centrifugal flow due to a rotating disk

5.6 Steady two-dimensional flow in a converging or diverging channel

Purely convergent flow

Purely divergent flow

Solutions showing both outflow and inflow

5.7 Boundary layers

5.8 The boundary layer on a flat plate

5.9 The effects of acceleration and deceleration of the external stream

The similarity solution for an external stream velocity proportional to xm

Calculation of the steady boundary layer on a body moving through fluid

Growth of the boundary layer in initially irrotational flow

5.10 Separation of the boundary layer

5.11 The flow due to bodies moving steadily through fluid

Flow without separation

Flow with separation

5.12 Jets， free shear layers and wakes

Narrow jets

Free shear layers

Wakes

5.13 Oscillatory boundary layers

The damping force on an oscillating body

Steady streaming due to an oscillatory boundary layer

Applications of the theory of steady streaming

5.14 Flow systems with a free surface

The boundary layer at a free surface

The drag on a spherical gas bubble rising steadily through liquid

The attenuation of gravity waves

5.15 Examples of use of the momentum theorem

The force on a regular array of bodies in a stream

The effect of a sudden enlargement of a pipe

Chapter 6. Irrotational Flow Theory and its Applications

6.1 The role of the theory of flow of an inviscid fluid

6.2 General properties of irrotational flow

Integration of the equation of motion

Expressions for the kinetic energy in terms of surface integrals

Kelvin’s minimum energy theorem

Positions of a maximum of q and a minimum of P

Local variation of the velocity magnitude

6.3 Steady flow： some applications of Bernoulli’s theorem and the momentum theorem

Efflux from a circular orifice in an open vessel

Flow over a weir

Jet of liquid impinging on a plane wall

Irrotational flow which may be made steady by choice of rotating axes

6.4 General features of irrotational flow due to a moving rigid body

The velocity at large distances from the body

The kinetic energy of the fluid

The force on a body in translational motion

The acceleration reaction

The force on a body in accelerating fluid

6.5 Use of the complex potential for irrotational flow in two dimensions

Flow fields obtained by special choice of the function w（x）

Conformal transformation of the plane of flow

Transformation of a boundary into an infinite straight line

Transformation of a closed boundary into a circle

The circle theorem

6.6 Two-dimensional irrotational flow due to a moving cylinder with circulation

A circular cylinder

An elliptic cylinder in translational motion

The force and moment on a cylinder in steady translational motion

6.7 Two-dimensional aerofoils

The practical requirements of aerofoils

The generation of circulation round an serofoil and the basis for Joukowski’s hypothesis

Aerofoils obtained by transformation of a circle

Joukowaki aerofoils

6.8 Axisymmetric irrotational flow due to moving bodies

Generalities

A moving sphere

Ellipsoids of revolution

Body shapes obtained from source singularities on the axis of symmetry

Semi-infinite bodies

6.9 Approximate results for slender bodies

Slender bodies of revolution

Slender bodies in two dimensions

Thin aerofoils in two dimensions

6.10 Impulsive motion of a fluid

Impact of a body on a free surface of liquid

6.11 Large gas bubbles in liquid

A spherical-cap bubble rising through liquid under gravity

A bubble rising in a vertical tube

A spherical expanding bubble

6.12 Cavitation in a liquid

Examples of cavity formation in steady flow

Examples of cavity formation in unsteady flow

Collapse of a transient cavity

Steady-state cavities

6.13 Free-streamline theory， and steady jets and cavities

Jet emerging from an orifice in two dimensions

Two-dimensional flow past a flat plate with a cavity at ambient pressure

Steady-state cavities attached to bodies held in a stream of liquid

Chapter 7. Flow of Effectively Inviscid Fluid with Vorticity

7.1 Introduction

The self-induced movement of a line vortex

The instability of a sheet vortex

7.2 Flow in unbounded fluid at rest at infinity

The resultant force impulse required to generate the motion

The total kinetic energy of the fluid

Flow with circular vortex-lines

Vortex rings

7.3 Two-dimensional flow in unbounded fluid at rest at infinity

Integral invariants of the vorticity distribution

Motion of a group of point vortices

Steady motions

7.4 Steady two-dimensional flow with vorticity throughout the fluid

Uniform vorticity in a region bounded externally

Fluid in rigid rotation at infinity

Fluid in simple shearing motion at infinity

7.5 Steady axisymmetric flow with swirl

The effect of a change of cross-section of a tube on a stream of rotating fluid

The effect of a change of external velocity on an isolated vortex

7.6 Flow systems rotating as a whole

The restoring effect of Coriolis forces

Steady flow at small Rossby number

Propagation of waves in a rotating fluid

Flow due to a body moving along the axis of rotation

7.7 Motion in a thin layer on a rotating sphere

Geostrophic flow

Flow over uneven ground

Planetary waves

7.8 The vortex system of a wing

General features of the flow past lifting bodies in three dimensions

Wings of large aspect ratio， and ‘lifting-line’ theory

The trailing vortex system far downstream

Highly swept wings

Appendices

1 Measured values of some physical properties of common fluids

（a） Dry air at a pressure of one atmosphere

（b） The Standard Atmosphere

（c） Pure water

（d） Diffusivities for momentum and heat at 15 ℃ and I atm

（e） Surface tension between two fluids

2 Expressions for some common vector differential quantities in orthogonal curvilinear co-ordinate systems

Publications referred to in the text

Subject Index