CONTENTS
Engineering Societies Monographs............... v
Preface............................ vii
Foreword by L. Prandtl.................... xv
Introduction......................... 1
PART ONE THE STATICS OF LIQUIDS AND GASES
CHAPTER I
Equilibrium and Stability................... 7
1. The Conditions under Which Liquids and Gases Can Be Treated as Continua...................... 7
2. The Concept of Fluid Pressure............... 11
3. Relation between Pressure Distribution and Volume Force..14
4. Stable, Unstable, and Neutral Equilibrium......... 17
5. Equation of Hydrostatic Pressure............. 18
6. Applications of the Pressure Equation; Communicating Vessels 20
7. Hydrostatic Pressure on Walls and Floors........... 22
8. Hydrostatic Lift and Stability............... 25
9. Calculation of the Metacentric Height........... 26
CHAPTER II
Application of the Pressure Equation to Permanent Gases. Stability of Air Masses................. 29
10. Equation of State for Permanent Gases............ 29
11. Uniform Atmosphere................... 30
12. Isothermal Atmosphere.................. 31
13. Polytropic Atmosphere.................. 32
14. Determination of the Exponent n of the Polytrope...... 34
15. Significance of the Temperature Gradient in Relation to the Stability of Air Masses................. 35
16. Influence of Humidity.................. 37
17. Concept of Potential Temperature............. 39
18. Origin of Clouds..................... 42
CHAPTER III
Static Lift on Gas-filled Aircraft............... 47
19. Pressure on the Balloon Wall............... 47
20. Lift of a Gas-filled Balloon................ 48
21. Effect of Temperature on Lift............... 49
22. Equilibrium of Forces on a Balloon............. 50
23. Stability of a Balloon in Taut State under Adiabatic Conditions 51
24. Stability of a Balloon in the Limp State under Adiabatic Conditions...................... 53
25. Effect of Temperature Changes at Constant Pressure on a Balloon in the Taut State................ 55
26. Effect of Temperature Changes at Constant Pressure on a Balloon in the Limp State................ 56
27. Causes of Heat Changes; Behavior of Balloon during Travel. 58
CHAPTER IV
Surface Tension....................... 60
28. Physical Basis....................... 60
29. Relation between Surface Tension and Pressure Difference across a Surface.................... 61
30. Surface Tension at Place of Contact between Several Media. 62
31. Surface Effects under the Action of Gravity......... 63
32. Capillarity....................... 64
PART TWO KINEMATICS OF LIQUIDS AND GASES
CHAPTER V
Methods of Description.................... 69
33. Lagrangian Method.................... 69
34. Eulerian Method and Its Connection with That of Lagrange. 71
35. Streamlines and Paths of Particles; Steady Flow....... 72
36. Streak Lines....................... 73
37. Significance of System of Reference in Interpretation of the Form of Motion.................... 74
38. Construction of Path and Streak Lines........... 75
39. Stream Tubes...................... 77
CHAPTER VI
Geometry of the Vector Field................. 78
40. Linear Vector Function of Position............. 78
41. Geometrical Significance of the Individual Quantities of a Matrix Characterizing a Velocity Field.......... 79
42. Shearing and Rotating Velocities.............. 81
43. The Concept of the Tensor................ 83
44. Splitting a Tensor into a Symmetrical and Antisymmetrical Part......................... 84
45. Stokes's Theorem.................... 86
46. Gauss's Theorem..................... 89
47. Introduction of the Operator V............... 91
CHAPTER VII
Acceleration of a Fluid Particle...............95
48. Velocity Change of a Fluid Particle as a Function of the Time and the Velocity Field.................95
49. The Substantial Differential Is the Sum of the Local and Convective Differentials..................95
50. Kinematic Boundary Conditions; Theorem of Lagrange.... 97
51. Liquids and Gases Are Not to Be Considered as Ideal Media but as Quasi-continua..................98
CHAPTER VIII
Equation of Continuity.................... 100
52. Incompressible Homogeneous Fluids............ 100
53. Eulerian Derivation of the Continuity Equation for Gases... 101
54. The General Lagrangian Equation of Continuity....... 103
PART THREE THE DYNAMICS OF NON-VISCOUS FLUIDS
CHAPTER IX
The Eulerian Equation and Its Integration along a Streamline 107
55. General Remarks on the Action of Fluid Viscosity......107
56. Euler's Equation.....................110
57. Integration of Euler's Equation along a Streamline......112
58. Bernoulli's Equation...................114
59. Applications of the Bernoulli Equation...........116
CHAPTER X
Potential Motion.......................122
60. Simplification of Euler's Equation and Integration on Assuming a Velocity Potential..................122
61. Connection between the Integral of Euler's Equation for
Potential Motion and the Corresponding Integral along a Streamline.......................126
62. Equations Defining Potential and Pressure Functions.....128
63. The Potential Function for Incompressible Fluids......129
64. The Potential Function When the Velocity w Is Very Small.. 130
65. The Potential Function for Steady Motion.........132
66. The Potential Function for the One-dimensional Problem... 138
67. Simple Examples of Potential Motion for Incompressible Fluids 139
68. The Source and Sink Potential...............144
69. Description of Motion about a Body of Revolution by the Method of Sources and Sinks.............. 146
70. The Motion about a Sphere; Doublets........... 149
71. The Potential of a Rectilinear Vortex............ 153
72. Difference between Potential Motion with Circulation and a Motion with Rotation.................154
73. The Interpretation of Potential as Impulsive Pressure....155
CHAPTER XI
Two-dimensional Potential Motion.............. 157
74. The Real and Imaginary Parts of an Analytic Function of
Complex Argument Are Solutions of Laplace's Differential Equation....................... 157
75. The Cauchy-Riemann Differential Equations and Their Physical Interpretation................... 158
76. The Stream Function................... 161
77. Examples of the Application of the Stream Function F(z) to Simple Problems of Motion in Two Dimensions...... 162
78. The Motion Round a Straight Circular Cylinder....... 166
79. The Fundamentals of Conformal Transformation....... 169
80. Applications of Conformal Transformation......... 172
81. The Hodograph Method................. 178
82. Discontinuous Fluid Motions............... 183
CHAPTER XII
Vortex Motion........................189
83. The Kinematics of Vortex Motion.............189
84. Thomson's Theorem on the Permanence of Circulation.... 191
85. Extension of Thomson's Theorem to the Case of Non-homogeneous Fluids by V. Bjerkness........194
86. The Dynamics of Vortex Motion.............. 196
87. The Vortex Theorems of Helmholtz............. 197
88. The Velocity Field in the Neighborhood of an Isolated Vortex; the Law of Biot and Savart...............201
89. Simplified Construction of a Vortex Line by Assuming a Core of Constant Rotation..................207
90. The Motion and Mutual Influence of Single Vortices.....208
91. Pressure Distribution in the Neighborhood of a Rectilinear Vortex.........................213
92. The Relation between Vortex Motion and the Surface of Discontinuity or Separation................214
93. The Formation of Surfaces of Discontinuity.........216
94. Instability of the Surface of Discontinuity..........221
CHAPTER XIII
The Influence of Conpressibility...............224
95. General Remarks about the Justification for Treating Gases as Incompressible Fluids................224
96. Compressibility in Bernoulli's Equation...........225
97. The Effect of Compressibility on the Formula for Stagnation Pressure........................227
98. Compressibility in the Equation of Continuity........229
99. The Effect of Compressibility on the Streamlines When the Velocity Is Less than That of Sound...........231
CHAPTER XIV
Theorems of Energy and Momentum..............233
100. The Momentum Theorem for Steady Motion........233
101. Extension of the Momentum Theorem to Fluid Motion with a Steady Mean Flow...................239
102. Applications of the Theorem of Momentum.........240
103. The Energy Theorem for Non-steady Motion of Incompressible Fluids........................246
CHAPTER XV
The Equation of Navier-Stokes for Viscous Fluids.......251
104. The Fundamental Equation of Fluid Mechanics.......251
105. Decomposition of the Total Surface Force into the Elements of a Stress Tensor...................252
106. Relation of the Elements of the Stress Tensor to the Corresponding Rates of Change of Deformation......... 253
107. Relation between the Stress Tensor and the Velocity Tensor. 257
108. The Equation of Navier-Stokes.............. 258
109. Discussion of the Navier-Stokes Equation.......... 260
110. The Differential Equation of Creeping Motion........ 261
111. Oseen's Improvement of the Theory............ 264
Index.............................267
(Fundamentals of Hydro- and Aeromechanics (First Edition Second Impression) by O.G. Tietjens, Ph.D. 1934)