Fluid Mechanics (eBook)

Front Cover 1
Fluid Mechanics 4
Copyright Page 5
Dedication 6
About the Author 7
Table of Contents 8
Preface 18
Preface to Third Edition 20
Preface to Second Edition 22
Preface to First Edition 24
Author’s Notes 28
Chapter 1. Introduction 30
1. Fluid Mechanics 30
2. Units of Measurement 31
3. Solids, Liquids, and Gases 32
4. Continuum Hypothesis 33
5. Transport Phenomena 34
6. Surface Tension 37
7. Fluid Statics 38
8. Classical Thermodynamics 41
9. Perfect Gas 45
10. Static Equilibrium of a Compressible Medium 47
Exercises 51
Literature Cited 53
Supplemental Reading 53
Chapter 2. Cartesian Tensors 54
1. Scalars and Vectors 54
2. Rotation of Axes: Formal Definition of a Vector 55
3. Multiplication of Matrices 58
4. Second-Order Tensor 59
5. Contraction and Multiplication 61
6. Force on a Surface 62
7. Kronecker Delta and Alternating Tensor 65
8. Dot Product 66
9. Cross Product 67
10. Operator .: Gradient, Divergence, and Curl 67
11. Symmetric and Antisymmetric Tensors 69
12. Eigenvalues and Eigenvectors of a Symmetric Tensor 70
13. Gauss’ Theorem 73
14. Stokes’ Theorem 76
15. Comma Notation 78
16. Boldface vs Indicial Notation 78
Exercises 79
Literature Cited 80
Supplemental Reading 80
Chapter 3. Kinematics 82
1. Introduction 82
2. Lagrangian and Eulerian Specifications 83
3. Eulerian and Lagrangian Descriptions: The Particle Derivative 84
4. Streamline, Path Line, and Streak Line 86
5. Reference Frame and Streamline Pattern 88
6. Linear Strain Rate 89
7. Shear Strain Rate 90
8. Vorticity and Circulation 91
9. Relative Motion near a Point: Principal Axes 93
10. Kinematic Considerations of Parallel Shear Flows 96
11. Kinematic Considerations of Vortex Flows 97
12. One-, Two-, and Three-Dimensional Flows 100
13. The Streamfunction 102
14. Polar Coordinates 104
Exercises 106
Supplemental Reading 108
Chapter 4. Conservation Laws 110
1. Introduction 111
2. Time Derivatives of Volume Integrals 111
3. Conservation of Mass 113
4. Streamfunctions: Revisited and Generalized 116
5. Origin of Forces in Fluid 117
6. Stress at a Point 119
7. Conservation of Momentum 121
8. Momentum Principle for a Fixed Volume 122
9. Angular Momentum Principle for a Fixed Volume 127
10. Constitutive Equation for Newtonian Fluid 129
11. Navier–Stokes Equation 133
12. Rotating Frame 134
13. Mechanical Energy Equation 140
14. First Law of Thermodynamics: Thermal Energy Equation 144
15. Second Law of Thermodynamics: Entropy Production 145
16. Bernoulli Equation 147
17. Applications of Bernoulli’s Equation 151
18. Boussinesq Approximation 153
19. Boundary Conditions 158
Exercises 163
Literature Cited 165
Supplemental Reading 166
Chapter 5. Vorticity Dynamics 168
1. Introduction 168
2. Vortex Lines and Vortex Tubes 169
3. Role of Viscosity in Rotational and Irrotational Vortices 170
4. Kelvin’s Circulation Theorem 173
5. Vorticity Equation in a Nonrotating Frame 178
6. Velocity Induced by a Vortex Filament: Law of Biot and Savart 180
7. Vorticity Equation in a Rotating Frame 181
8. Interaction of Vortices 186
9. Vortex Sheet 190
Exercises 190
Literature Cited 192
Supplemental Reading 192
Chapter 6. Irrotational Flow 194
1. Relevance of Irrotational Flow Theory 194
2. Velocity Potential: Laplace Equation 196
3. Application of Complex Variables 198
4. Flow at a Wall Angle 200
5. Sources and Sinks 202
6. Irrotational Vortex 203
7. Doublet 203
8. Flow past a Half-Body 204
9. Flow past a Circular Cylinder without Circulation 207
10. Flow past a Circular Cylinder with Circulation 209
11. Forces on a Two-Dimensional Body 213
12. Source near a Wall: Method of Images 218
13. Conformal Mapping 219
14. Flow around an Elliptic Cylinder with Circulation 221
15. Uniqueness of Irrotational Flows 223
16. Numerical Solution of Plane Irrotational Flow 224
17. Axisymmetric Irrotational Flow 230
18. Streamfunction and Velocity Potential for Axisymmetric Flow 232
19. Simple Examples of Axisymmetric Flows 234
20. Flow around a Streamlined Body of Revolution 235
21. Flow around an Arbitrary Body of Revolution 237
22. Concluding Remarks 238
Exercises 238
Literature Cited 241
Supplemental Reading 241
Chapter 7. Gravity Waves 242
1. Introduction 243
2. The Wave Equation 243
3. Wave Parameters 245
4. Surface Gravity Waves 248
5. Some Features of Surface Gravity Waves 252
6. Approximations for Deep and Shallow Water 258
7. Influence of Surface Tension 263
8. Standing Waves 266
9. Group Velocity and Energy Flux 267
10. Group Velocity and Wave Dispersion 271
11. Nonlinear Steepening in a Nondispersive Medium 275
12. Hydraulic Jump 277
13. Finite Amplitude Waves of Unchanging Form in a Dispersive Medium 279
14. Stokes’ Drift 282
15. Waves at a Density Interface between Infinitely Deep Fluids 284
16. Waves in a Finite Layer Overlying an Infinitely Deep Fluid 288
17. Shallow Layer Overlying an Infinitely Deep Fluid 291
18. Equations of Motion for a Continuously Stratified Fluid 292
19. Internal Waves in a Continuously Stratified Fluid 296
20. Dispersion of Internal Waves in a Stratified Fluid 299
21. Energy Considerations of Internal Waves in a Stratified Fluid 301
Exercises 305
Literature Cited 306
Chapter 8. Dynamic Similarity 308
1. Introduction 308
2. Nondimensional Parameters Determined from Differential Equations 309
3. Dimensional Matrix 313
4. Buckingham’s Pi Theorem 314
5. Nondimensional Parameters and Dynamic Similarity 316
6. Comments on Model Testing 319
7. Significance of Common Nondimensional Parameters 321
Exercises 323
Literature Cited 323
Supplemental Reading 323
Chapter 9. Laminar Flow 324
1. Introduction 324
2. Analogy between Heat and Vorticity Diffusion 326
3. Pressure Change Due to Dynamic Effects 326
4. Steady Flow between Parallel Plates 327
5. Steady Flow in a Pipe 331
6. Steady Flow between Concentric Cylinders 332
7. Impulsively Started Plate: Similarity Solutions 335
8. Diffusion of a Vortex Sheet 342
9. Decay of a Line Vortex 344
10. Flow Due to an Oscillating Plate 346
11. High and Low Reynolds Number Flows 349
12. Creeping Flow around a Sphere 351
13. Nonuniformity of Stokes’ Solution and Oseen’s Improvement 356
14. Hele-Shaw Flow 361
15. Final Remarks 363
Exercises 364
Literature Cited 366
Supplemental Reading 366
Chapter 10. Boundary Layers and Related Topics 368
1. Introduction 369
2. Boundary Layer Approximation 369
3. Different Measures of Boundary Layer Thickness 375
4. Boundary Layer on a Flat Plate with a Sink at the Leading Edge: Closed Form Solution 377
5. Boundary Layer on a Flat Plate: Blasius Solution 381
6. von Karman Momentum Integral 391
7. Effect of Pressure Gradient 393
8. Separation 395
9. Description of Flow past a Circular Cylinder 397
10. Description of Flow past a Sphere 404
11. Dynamics of Sports Balls 405
12. Two-Dimensional Jets 410
13. Secondary Flows 417
14. Perturbation Techniques 418
15. An Example of a Regular Perturbation Problem 423
16. An Example of a Singular Perturbation Problem 425
17. Decay of a Laminar Shear Layer 430
Exercises 436
Literature Cited 438
Supplemental Reading 439
Chapter 11. Computational Fluid Dynamics 440
1. Introduction 440
2. Finite Difference Method 442
3. Finite Element Method 447
4. Incompressible Viscous Fluid Flow 455
5. Three Examples 469
6. Concluding Remarks 490
Exercises 492
Literature Cited 493
Chapter 12. Instability 496
1. Introduction 496
2. Method of Normal Modes 498
3. Thermal Instability: The Bénard Problem 499
4. Double-Diffusive Instability 511
5. Centrifugal Instability: Taylor Problem 515
6. Kelvin–Helmholtz Instability 522
7. Instability of Continuously Stratified Parallel Flows 529
8. Squire’s Theorem and Orr–Sommerfeld Equation 536
9. Inviscid Stability of Parallel Flows 539
10. Some Results of Parallel Viscous Flows 543
11. Experimental Verification of Boundary Layer Instability 549
12. Comments on Nonlinear Effects 551
13. Transition 552
14. Deterministic Chaos 554
Exercises 562
Literature Cited 564
Chapter 13. Turbulence 566
1. Introduction 566
2. Historical Notes 568
3. Averages 570
4. Correlations and Spectra 572
5. Averaged Equations of Motion 576
6. Kinetic Energy Budget of Mean Flow 583
7. Kinetic Energy Budget of Turbulent Flow 585
8. Turbulence Production and Cascade 588
9. Spectrum of Turbulence in Inertial Subrange 591
10. Wall-Free Shear Flow 593
11. Wall-Bounded Shear Flow 599
12. Eddy Viscosity and Mixing Length 609
13. Coherent Structures in a Wall Layer 613
14. Turbulence in a Stratified Medium 615
15. Taylor’s Theory of Turbulent Dispersion 620
16. Concluding Remarks 627
Exercises 627
Literature Cited 629
Supplemental Reading 630
Chapter 14. Geophysical Fluid Dynamics 632
1. Introduction 632
2. Vertical Variation of Density in Atmosphere and Ocean 634
3. Equations of Motion 636
4. Approximate Equations for a Thin Layer on a Rotating Sphere 639
5. Geostrophic Flow 642
6. Ekman Layer at a Free Surface 646
7. Ekman Layer on a Rigid Surface 651
8. Shallow-Water Equations 654
9. Normal Modes in a Continuously Stratified Layer 657
10. High- and Low-Frequency Regimes in Shallow-Water Equations 663
11. Gravity Waves with Rotation 665
12. Kelvin Wave 668
13. Potential Vorticity Conservation in Shallow-Water Theory 673
14. Internal Waves 676
15. Rossby Wave 686
16. Barotropic Instability 692
17. Baroclinic Instability 694
18. Geostrophic Turbulence 702
Exercises 705
Literature Cited 706
Chapter 15. Aerodynamics 708
1. Introduction 708
2. The Aircraft and Its Controls 709
3. Airfoil Geometry 712
4. Forces on an Airfoil 713
5. Kutta Condition 713
6. Generation of Circulation 716
7. Conformal Transformation for Generating Airfoil Shape 717
8. Lift of Zhukhovsky Airfoil 721
9. Wing of Finite Span 724
10. Lifting Line Theory of Prandtl and Lanchester 726
11. Results for Elliptic Circulation Distribution 730
12. Lift and Drag Characteristics of Airfoils 733
13. Propulsive Mechanisms of Fish and Birds 735
14. Sailing against the Wind 737
Exercises 738
Literature Cited 740
Supplemental Reading 740
Chapter 16. Compressible Flow 742
1. Introduction 742
2. Speed of Sound 746
3. Basic Equations for One-Dimensional Flow 750
4. Stagnation and Sonic Properties 753
5. Area–Velocity Relations in One-Dimensional Isentropic Flow 758
6. Normal Shock Wave 762
7. Operation of Nozzles at Different Back Pressures 770
8. Effects of Friction and Heating in Constant-Area Ducts 776
9. Mach Cone 779
10. Oblique Shock Wave 781
11. Expansion and Compression in Supersonic Flow 785
12. Thin Airfoil Theory in Supersonic Flow 787
Exercises 790
Literature Cited 792
Supplemental Reading 792
Chapter 17. Introduction to Biofluid Mechanics 794
1. Introduction 794
2. The Circulatory System in the Human Body 795
3. Modelling of Flow in Blood Vessels 811
4. Introduction to the Fluid Mechanics of Plants 860
Exercises 866
Acknowledgment 867
Literature Cited 867
Appendix A. Some Properties of Common Fluids 870
A1. Useful Conversion Factors 870
A2. Properties of Pure Water at Atmospheric Pressure 871
A3. Properties of Dry Air at Atmospheric Pressure 871
A4. Properties of Standard Atmosphere 872
Appendix B. Curvilinear Coordinates 874
B1. Cylindrical Polar Coordinates 874
B2. Plane Polar Coordinates 876
B3. Spherical Polar Coordinates 876
Appendix C. Founders of Modern Fluid Dynamics 880
Ludwig Prandtl (1875 1953) 880
Geoffrey Ingram Taylor (1886 1975) 881
Supplemental Reading 882
Appendix D. Visual Resources 884
Index 886