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Index
Title Copyright Contents Preface Conventions and Notation 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 solenoidai flow near a stagnation point The complex potential for irrotational solenoidai flow in two dimensions 2.8 Irrotational solenoidai 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 ue and 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 Motionless 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 solenoidai 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 1 to about 100 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 Plates 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(z) 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 aerofoil and the basis for Joukowski’s hypothesis Aerofoils obtained by transformation of a circle Joukowski 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. Flaw 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 °C and 1 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
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