Graebel W.P. Advanced Fluid Mechanics
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Advanced Fluid Mechanics
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Advanced Fluid
Mechanics
W. P. Graebel
Professor Emeritus, The University of Michigan
AMSTERDAM
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BOSTON
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HEIDELBERG
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LONDON
NEW YORK
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OXFORD
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PARIS
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SAN DIEGO
SAN FRANCISCO
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SINGAPORE
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SYDNEY
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TOKYO
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Contents
Preface .........................................................xiv
Chapter 1
Fundamentals
1.1 Introduction............................................................ 1
1.2 Velocity, Acceleration, and the Material Derivative ....................... 4
1.3 The Local Continuity Equation .......................................... 5
1.4 Path Lines, Streamlines, and Stream Functions............................ 7
1.4.1 Lagrange’s Stream Function for Two-Dimensional Flows ........... 7
1.4.2 Stream Functions for Three-Dimensional Flows, Including
Stokes Stream Function .......................................... 11
1.5 Newton’s Momentum Equation .......................................... 13
1.6 Stress ................................................................. 14
1.7 Rates of Deformation ................................................... 21
1.8 Constitutive Relations .................................................. 24
1.9 Equations for Newtonian Fluids ......................................... 27
1.10 Boundary Conditions ................................................... 28
1.11 Vorticity and Circulation ................................................ 29
1.12 The Vorticity Equation.................................................. 34
1.13 The Work-Energy Equation ............................................. 36
1.14 The First Law of Thermodynamics....................................... 37
1.15 Dimensionless Parameters............................................... 39
1.16 Non-Newtonian Fluids .................................................. 40
1.17 Moving Coordinate Systems............................................. 41
Problems .................................................................... 43
vii
viii Contents
Chapter 2
Inviscid Irrotational Flows
2.1 Inviscid Flows .......................................................... 46
2.2 Irrotational Flows and the Velocity Potential .............................. 47
2.2.1 Intersection of Velocity Potential Lines and Streamlines
in Two Dimensions ............................................... 49
2.2.2 Basic Two-Dimensional Irrotational Flows ......................... 51
2.2.3 Hele-Shaw Flows................................................. 57
2.2.4 Basic Three-Dimensional Irrotational Flows ........................ 58
2.2.5 Superposition and the Method of Images ........................... 59
2.2.6 Vortices Near Walls .............................................. 61
2.2.7 Rankine Half-Body ............................................... 65
2.2.8 Rankine Oval .................................................... 67
2.2.9 Circular Cylinder or Sphere in a Uniform Stream ................... 68
2.3 Singularity Distribution Methods ......................................... 69
2.3.1 Two- and Three-Dimensional Slender Body Theory ................. 69
2.3.2 Panel Methods ................................................... 71
2.4 Forces Acting on a Translating Sphere .................................... 77
2.5 Added Mass and the Lagally Theorem .................................... 79
2.6 Theorems for Irrotational Flow ........................................... 81
2.6.1 Mean Value and Maximum Modulus Theorems ..................... 81
2.6.2 Maximum-Minimum Potential Theorem ............................ 81
2.6.3 Maximum-Minimum Speed Theorem .............................. 82
2.6.4 Kelvin’s Minimum Kinetic Energy Theorem ........................ 82
2.6.5 Maximum Kinetic Energy Theorem ................................ 83
2.6.6 Uniqueness Theorem ............................................. 84
2.6.7 Kelvin’s Persistence of Circulation Theorem ........................ 84
2.6.8 Weiss and Butler Sphere Theorems ................................ 84
Problems .................................................................... 85
Chapter 3
Irrotational Two-Dimensional Flows
3.1 Complex Variable Theory Applied to Two-Dimensional
Irrotational Flow ........................................................ 87
3.2 Flow Past a Circular Cylinder with Circulation ............................ 91
3.3 Flow Past an Elliptical Cylinder with Circulation .......................... 93
3.4 The Joukowski Airfoil ................................................... 95
3.5 Kármán-Trefftz and Jones-McWilliams Airfoils ............................ 98
3.6 NACA Airfoils.......................................................... 99
3.7 Lifting Line Theory .....................................................101
Contents ix
3.8 Kármán Vortex Street..................................................103
3.9 Conformal Mapping and the Schwarz-Christoffel Transformation ..........108
3.10 Cavity Flows..........................................................110
3.11 Added Mass and Forces and Moments for
Two-Dimensional Bodies .............................................. 112
Problems....................................................................114
Chapter 4
Surface and Interfacial Waves
4.1 Linearized Free Surface Wave Theory...................................118
4.1.1 Infinitely Long Channel .........................................118
4.1.2 Waves in a Container of Finite Size ..............................122
4.2 Group Velocity........................................................123
4.3 Waves at the Interface of Two Dissimilar Fluids ......................... 125
4.4 Waves in an Accelerating Container .................................... 127
4.5 Stability of a Round Jet ................................................128
4.6 Local Surface Disturbance on a Large Body of
Fluid—Kelvin’s Ship Wave ............................................ 130
4.7 Shallow-Depth Free Surface Waves—Cnoidal
and Solitary Waves ....................................................132
4.8 Ray Theory of Gravity Waves for Nonuniform Depths ................... 136
Problems...................................................................139
Chapter 5
Exact Solutions of the Navier-Stokes Equations
5.1 Solutions to the Steady-State Navier-Stokes Equations
When Convective Acceleration Is Absent................................140
5.1.1 Two-Dimensional Flow Between Parallel Plates ...................141
5.1.2 Poiseuille Flow in a Rectangular Conduit .........................142
5.1.3 Poiseuille Flow in a Round Conduit or Annulus ...................144
5.1.4 Poiseuille Flow in Conduits of Arbitrarily
Shaped Cross-Section ...........................................145
5.1.5 Couette Flow Between Concentric Circular Cylinders..............147
5.2 Unsteady Flows When Convective Acceleration Is Absent ................ 147
5.2.1 Impulsive Motion of a Plate—Stokes’s First Problem ..............147
5.2.2 Oscillation of a Plate—Stokes’s Second Problem ..................149
5.3 Other Unsteady Flows When Convective Acceleration
Is Absent ............................................................. 152
5.3.1 Impulsive Plane Poiseuille and Couette Flows.....................152
5.3.2 Impulsive Circular Couette Flow .................................153