1. The Concept of the Continuum and Kinematics .................. 1
1.1. Properties of Fluids, Continuum Hypothesis .............. 1
1.2. Kinematics .............................................. 7
1.2.1. Material and Spatial Descriptions ................ 7
1.2.2. Pathlines, Streamlines, Streaklines ............. 10
1.2.3. Differentiation with Respect to Time ............ 14
1.2.4. State of Motion, Rate of Change of Line,
Surface and Volume Elements ..................... 17
1.2.5. Rate of Change of Material Integrals ............ 29
2. Fundamental Laws of Continuum Mechanics ..................... 35
2.1. Conservation of Mass, Equation of Continuity ........... 35
2.2. Balance of Momentum .................................... 37
2.3. Balance of Angular Momentum ............................ 44
2.4. Momentum and Angular Momentum in an Accelerating
Frame .................................................. 46
2.5. Applications to Turbomachines .......................... 54
2.6. Balance of Energy ...................................... 65
2.7. Balance of Entropy ..................................... 69
2.8. Thermodynamic Equations of State ....................... 71
3. Constitutive Relations for Fluids ........................... 75
4. Equations of Motion for Particular Fluids ................... 95
4.1. Newtonian Fluids ....................................... 95
4.1.1. The Navier-Stokes Equations ..................... 95
4.1.2. Vorticity Equation .............................. 98
4.1.3. Effect of Reynolds' Number ..................... 100
4.2. Inviscid Fluids ....................................... 106
4.2.1. Euler's Equations .............................. 106
4.2.2. Bernoulli's Equation ........................... 107
4.2.3. Vortex Theorems ................................ 112
4.2.4. Integration of the Energy Equation ............. 138
4.3. Initial and Boundary Conditions ....................... 141
4.4. Simplification of the Equations of Motion ............. 145
5. Hydrostatics ............................................... 151
5.1. Hydrostatic Pressure Distribution ..................... 151
5.2. Hydrostatic Lift, Force on Walls ...................... 156
5.3. Free Surfaces ......................................... 162
6. Laminar Unidirectional Flows ............................... 167
6.1. Steady Unidirectional Flow ............................ 168
6.1.1. Couette Flow ................................... 168
6.1.2. Couette-Poiseuille Flow ........................ 169
6.1.3. Flow Down an Inclined Plane .................... 171
6.1.4. Flow Between Rotating Concentric Cylinders ..... 174
6.1.5. Hagen-Poiseuille Flow .......................... 175
6.1.6. Flow Through Noncircular Conduits .............. 180
6.2. Unsteady Unidirectional Flows ......................... 183
6.2.1. Flow Due to a Wall Which Oscillates in its
Own Plane ...................................... 183
6.2.2. Flow Due to a Wall Which is Suddenly Set in
Motion ......................................... 186
6.3. Unidirectional Flows of Non-Newtonian Fluids .......... 188
6.3.1. Steady Flow Through a Circular Pipe ............ 188
6.3.2. Steady Flow Between a Rotating Disk and
a Fixed Wall ................................... 190
6.3.3. Unsteady Unidirectional Flows of a Second
Order Fluid .................................... 191
6.4. Unidirectional Flows of a Bingham Material ............ 197
6.4.1. Channel Flow of a Bingham Material ............. 197
6.4.2. Pipe Flow of a Bingham Material ................ 202
7. Fundamentals of Turbulent Flow ............................. 205
7.1. Stability and the Onset of Turbulence ................. 205
7.2. Reynolds' Equations ................................... 207
7.3. Turbulent Shear Flow Near a Wall ...................... 213
7.4. Turbulent Flow in Smooth Pipes and Channels ........... 223
7.5. Turbulent Flow in Rough Pipes ......................... 226
8. Hydrodynamic Lubrication ................................... 229
8.1. Reynolds' Equation of Lubrication Theory .............. 229
8.2. Statically Loaded Bearing ............................. 232
8.2.1. Infinitely Long Journal Bearing ................ 232
8.2.2. Infinitely Short Journal Bearing ............... 238
8.2.3. Journal Bearing of Finite Length ............... 239
8.3. Dynamically Loaded Bearings ........................... 240
8.3.1. Infinitely Long Journal Bearing ................ 240
8.3.2. Dynamically Loaded Slider Bearing .............. 241
8.3.3. Squeeze Flow of a Bingham Material ............. 246
8.4. Thin-Film Flow on a Semi-Infinite Wall ................ 249
8.5. Flow Through Particle Filters ......................... 252
8.6. Flow Through a Porous Medium .......................... 254
8.7. Hele-Shaw Flows ....................................... 258
9. Stream Filament Theory ..................................... 261
9.1. Incompressible Flow ................................... 261
9.1.1. Continuity Equation ............................ 262
9.1.2. Inviscid Flow .................................. 263
9.1.3. Viscous Flow ................................... 266
9.1.4. Application to Flows with Variable Cross-
Section ........................................ 271
9.1.5. Viscous Jet .................................... 276
9.2. Steady Compressible Flow .............................. 279
9.2.1. Flow Through Pipes and Ducts with Varying
Cross-Section .................................. 279
9.2.2. Constant Area Flow ............................. 290
9.2.3. The Normal Shock Wave Relations ................ 294
9.3. Unsteady Compressible Flow ............................ 299
10.Potential Flows ............................................ 315
10.1. One-Dimensional Propagation of Sound ................. 316
10.2. Steady Compressible Potential Flow ................... 323
10.3.Incompressible Potential Flow ......................... 324
10.3.1.Simple Examples of Potential Flows ............. 326
10.3.2.Virtual Masses ................................. 348
10.4.Plane Potential Flow .................................. 354
10.4.1.Examples of Incompressible, Plane Potential
Flows .......................................... 354
10.4.2.Complex Potential for Plane Flows .............. 358
10.4.3.Blasius' Theorem ............................... 367
10.4.4.Kutta-Joukowski Theorem ........................ 370
10.4.5.Conformal Mapping .............................. 372
10.4.6.Schwarz-Christoffel Transformation ............. 374
10.4.7.Free Jets ...................................... 376
10.4.8.Flow Around Airfoils ........................... 382
10.4.9.Approximate Solution for Slender Airfoils
in Incompressible Flow ......................... 388
10.4.10.Slender Airfoils in Compressible Flow ......... 395
11.Supersonic Flow ............................................ 399
11.1.Oblique Shock Wave .................................... 400
11.2.Detached Shock Wave ................................... 402
11.3.Reflection of Oblique Shock Waves ..................... 403
11.4.Supersonic Potential Flow Past Slender Airfoils ....... 405
11.5.Prandtl-Meyer Flow .................................... 408
11.6.Shock Expansion Theory ................................ 414
12.Boundary Layer Theory ...................................... 417
12.1.Solutions of the Boundary Layer Equations ............. 421
12.1.1.Flat Plate ..................................... 422
12.1.2.Wedge Flows .................................... 426
12.1.3.Unsteady Stagnation Point Flow ................. 428
12.1.4.Flow Past a Body ............................... 429
12.2.Temperature Boundary Layer in Forced Convection ....... 431
12.3.Temperature Boundary Layer in Natural Convection ...... 437
12.4.Integral Methods of Boundary Layer Theory ............. 440
12.5.Turbulent Boundary Layers ............................. 443
13.Creeping Flows ............................................. 451
13.1.Plane and Axially-Symmetric Flows ..................... 451
13.1.1.Examples of Plane Flows ........................ 453
13.1.2.Plane Creeping Flow Round a Body (Stokes's
Paradox) ....................................... 465
13.1.3.Creeping Flow Round a Sphere ................... 465
A. Introduction to Cartesian Tensors .......................... 471
A.l. Summation Convention .................................. 471
A.2. Cartesian Tensors ..................................... 472
В. Curvilinear Coordinates .................................... 481
B.l. Cartesian Coordinates ................................. 488
B.2. Cylindrical Coordinates ............................... 490
B.3. Spherical Coordinates ................................. 493
С. Tables and Diagrams for Compressible Flow .................. 497
D. Physical Properties of Air and Water ....................... 515
References .................................................... 519
Index ......................................................... 521
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