1. Introduction 1
1.1. Fluid Mechanics 2
1.2. Units of Measurement 3
1.3. Solids, Liquids, and Gases 3
1.4. Continuum Hypothesis 5
1.5. Molecular Transport Phenomena 5
1.6. Surface Tension 8
1.7. Fluid Statics 9
1.8. Classical Thermodynamics 12
First Law of Thermodynamics 13
Equations of State 14
Specific Heats 14
Second Law of Thermodynamics 15
Property Relations 16
Speed of Sound 16
Thermal Expansion Coefficient 16
1.9. Perfect Gas 16
1.10. Stability of Stratified Fluid Media 18
Potential Temperature and Density 19
Scale Height of the Atmosphere 21
1.11. Dimensional Analysis 21
Step 1. Select Variables and Parameters 22
Step 2. Create the Dimensional Matrix 23
Step 3. Determine the Rank of the Dimensional Matrix 23
Step 4. Determine the Number of Dimensionless Groups 24
Step 5. Construct the Dimensionless Groups 24
Step 6. State the Dimensionless Relationship 26
Step 7. Use Physical Reasoning or Additional Knowledge to Simplify the Dimensionlcss Relationship 26
Exercises 30
Literature Cited 36
Supplemental Reading 37
2. Cartesian Tensors 39
2.1. Scalars, Vectors, Tensors, Notation 39
2.2. Rotation of Axes: Formal Definition of a Vector 42
2.3. Multiplication of Matrices 44
2.4. Second-Order Tensors 45
2.5. Contraction and Multiplication 47
2.6. Force on a Surface 48
2.7. Kronecker Delta and Alternating Tensor 50
2.8. Vector, Dot, and Cross Products 51
2.9. Gradient, Divergence, and Curl 52
2.10. Symmetric and Antisymmetric Tensors 55
2.11. Eigenvalues and Eigenvectors of a Symmetric Tensor 56
2.12. Gauss’ Theorem 58
2.13. Stokes’ Theorem 60
2.14. Comma Notation 62
Exercises 62
Literature Cited 64
Supplemental Reading 64
3. Kinematics 65
3.1. Introduction and Coordinate Systems 65
3.2. Particle and Field Descriptions of Fluid Motion 67
3.3. Flow Lines, Fluid Acceleration,and Galilean Transformation 71
3.4. Strain and Rotation Rates 76
Summary 81
3.5. Kinematics of Simple Plane Flows 82
3.6. Reynolds Transport Theorem 85
Exercises 89
Literature Cited 93
Supplemental Reading 93
4. Conservation Laws 95
4.1. Introduction 96
4.2. Conservation of Mass 96
4.3. Stream Functions 99
4.4. Conservation of Momentum 101
4.5. Constitutive Equation for a Newtonian Fluid 111
4.6. Navier-Stokes Momentum Equation 114
4.7. Noninertial Frame of Reference 116
4.8. Conservation of Energy 121
4.9. Special Forms of the Equations 125
Angular Momentum Principle for ary Stationa Control Volume 125
Bernoulli Equations 128
Neglect of Gravity in Constant Density Flows 134
The Boussinesq Approximation 135
Summary 137
4.10. Boundary Conditions 137
Moving and Deforming Boundaries 139
Surface Tension Revisited 139
4.11. Dimensionless Forms of the Equations and Dynamic Similarity 143
Exercises 151
Literature Cited 168
Supplemental Reading 168
5. Vorticity Dynamics 171
5.1. Introduction 171
5.2. Kelvin’s Circulation Theorem 176
5.3. Helmholtz’s Vortex Theorems 179
5.4. Vorticity Equation in a Nonrotating Frame 180
5.5. Velocity Induced by a Vortex Filament: Law of Biot and Savart 181
5.6. Vorticity Equation in a Rotating Frame 183
5.7. Interaction of Vortices 187
5.8. Vortex Sheet 191
Exercises 192
Literature Cited 195
Supplemental Reading 196
6. Ideal Flow 197
6.1. Relevance of Irrotational Constant-Density Flow Theory 198
6.2. Two-Dimensional Stream Function and Velocity Potential 200
6.3. Construction of Elementa Flows in Twory Dimensions 203
6.4. Complex Potential 216
6.5. Forces on a Two-Dimensional Body 219
Blasius Theorem 219
Kutta-Zhukhovsky Lift Theorem 221
6.6. Conformal Mapping 222
6.7. Numerical Solution Techniques in Two Dimensions 225
6.8. Axisymmetric Ideal Flow 231
6.9. Three-Dimensional Potential Flow and Apparent Mass 236
6.10. Concluding Remarks 240
Exercises 241
Literature Cited 251
Supplemental Reading 251
7. Gravity Waves 253
7.1. Introduction 254
7.2. Linear Liquid-Surface Gravity Waves 256
Approximations for Deep and Shallow Water 265
7.3. Influence of Surface Tension 269
7.4. Standing Waves 271
7.5. Group Velocity, Energy Flux, and Dispersion 273
7.6. Nonlinear Waves in Shallow and Deep Water 279
7.7. Waves on a Density Interface 286
7.8. Internal Waves in a Continuously Stratified Fluid 293
Internal Waves in a Stratified Fluid 296
Dispersion of Internal Waves in a Stratified Fluid 299
Energy Considerations for Internal Waves in a Stratified Fluid 302
Exercises 304
Literature Cited 307
8. Laminar Flow 309
8.1. Introduction 309
8.2. Exact Solutions for Steady Incompressible Viscous Flow 312
Steady Flow between Parallel Plates 312
Steady Flow in a Round Tube 315
Steady Flow between Concentric Rotating Cylinders 316
8.3. Elementary Lubrication Theory 318
8.4. Similarity Solutions for Unsteady Incompressible Viscous Flow 326
8.5. Flow Due to an Oscillating Plate 337
8.6. Low Reynolds Number Viscous Flow Past a Sphere 338
8.7. Final Remarks 347
Exercises 347
Literature Cited 359
Supplemental Reading 359
9. Boundary Layers and Related Topics 361
9.1. Introduction 362
9.2. Boundary-Layer Thickness Definitions 367
9.3. Boundary Layer on a Flat Plate:Blasius Solution 369
9.4. Falkner-Skan Similarity Solutions of the Laminar Boundary-Layer Equations 373
9.5. Von Karman Momentum Integral Equation 375
9.6. Thwaites’ Method 377
9.7. Transition, Pressure Gradients,and Boundary-Layer Separation 382
9.8. Flow Past a Circular Cylinder 388
Low Reynolds Numbers 389
Moderate Reynolds Numbers 389
High Reynolds Numbers 392
9.9. Flow Past a Sphere and the Dynamics of Sports Balls 395
Cricket Ball Dynamics 396
Tennis Ball Dynamics 398
Baseball Dynamics 399
9.10. Two-Dimensional Jets 399
9.11. Secondary Flows 407
Exercises 408
Literature Cited 418
Supplemental Reading 419
10. Computational Fluid Dynamics HOWARD H. HU 421
10.1. Introduction 421
10.2. Finite-Difference Method 423
Approximation to Derivatives 423
Discretization and Its Accuracy 425
Convergence, Consistency, and Stability 426
10.3. Finite-Element Method 429
Weak or Variational Form of Partial Differential Equations 429
Galerkin’s Approximation and Finite-Element Interpolations 430
Matrix Equations, Comparison with Finite-Difference Method 431
Element Point of View of the Finite-Element Method 434
10.4. Incompressible Viscous Fluid Flow 436
Convection-Dominated Problems 437
Incompressibility Condition 439
Explicit MacCormack Scheme 440
MAC Scheme 442
O-Scheme 446
Mixed Finite-Element Formulation 447
10.5. Three Examples 449
Explicit MacCormack Scheme for Driven-Cavity Flow Problem 449
Explicit MacCormack Scheme for Flow Over a Square Block 453
Finite-Element Formulation for Flow Over a Cylinder Confined in a Channel 459
10.6. Concluding Remarks 470
Exercises 470
Literature Cited 471
Supplemental Reading 472
11. Instability 473
11.1. Introduction 474
11.2. Method of Normal Modes 475
11.3. Kelvin-Helmholtz Instability 477
11.4. Thermal Instability: The Benard Problem 484
11.5. Double-Diffusive Instability 492
11.6. Centrifugal Instability: Taylor Problem 496
11.7. Instability of Continuously Stratified Parallel Flows 502
11.8. Squire’s Theorem and the Orr-Sommerfeld Equation 508
11.9. Inviscid Stability of Parallel Flows 511
11.10. Results for Parallel and Nearly Parallel Viscous Flows 515
Two-Stream Shear Layer 515
Plane Poiseuille Flow 516
Plane Couette Flow 517
Pipe Flow 517
Boundary Layers with Pressure Gradients 517
11.11. Experimental Verification of Boundary-Layer Instability 520
11.12. Comments on Nonlinear Effects 522
11.13. Transition 523
11.14. Deterministic Chaos 524
Closure 531
Exercises 532
Literature Cited 539
12. Turbulence 541
12.1. Introduction 542
12.2. Historical Notes 544
12.3. Nomenclature and Statistics for Turbulent Flow 545
12.4. Correlations and Spectra 549
12.5. Averaged Equations of Motion 554
12.6. Homogeneous Isotropic Turbulence 560
12.7. Turbulent Energy Cascade and 564
Spectrum 564
12.8. Free Turbulent Shear Flows 571
12.9. Wall-Bounded Turbulent Shear Flows 581
Inner Layer: Law of the Wall 584
Outer Layer: Velocity Defect Law 585
Overlap Layer: Logarithmic Law 585
Rough Surfaces 590
12.10. Turbulence Modeling 591
A Mixing Length Model 593
One-Equation Models 595
Two-Equation Models 595
12.11. Turbulence in a Stratified Medium 596
The Richardson Numbers 597
Monin-Obukhov Length 598
Spectrum of Temperature Fluctuations 600
12.12. Taylor’s Theory of Turbulent Dispersion 601
Rate of Dispersion of a Single Particle 602
Random Walk 605
Behavior of a Smoke Plume in the Wind 606
Turbulent Diffusivity 607
12.13. Concluding Remarks 607
Exercises 608
Literature Cited 618
Supplemental Reading 620
13. Geophysical Fluid Dynamics 621
13.1. Introduction 622
13.2. Vertical Variation of Density in the Atmosphere and Ocean 623
13.3. Equations of Motion 625
13.4. Approximate Equations for a Thin Layer on a Rotating Sphere 628
f-Plane Model 630
β-Plane Model 630
13.5. Geostrophic Flow 630
Thermal Wind 632
Taylor-Proudman Theorem 632
13.6. Ekman Layer at a Free Surface 633
Explanation in Terms of Vortex Tilting 637
13.7. Ekman Layer on a Rigid Surface 639
13.8. Shallow-Water Equations 642
13.9. Normal Modes in a Continuously Stratified Layer 644
Boundary Conditions on ？n 646
Vertical Mode Solution for Uniform N 646
Summary 649
13.10. High- and Low-Frequency Regimes in Shallow-Water Equations 649
13.11. Gravity Waves with Rotation 651
Particle Orbit 652
Inertial Motion 653
13.12. Kelvin Wave 654
13.13. Potential Vorticity Conservation in Shallow-Water Theory 658
13.14. Internal Waves 662
WKB Solution 664
Particle Orbit 666
Discussion of the Dispersion Relation 668
Lee Wave 670
13.15. Rossby Wave 671
Quasi-Geostrophic Vorticity Equation 671
Dispersion Relation 673
13.16. Barotropic Instability 676
13.17. Baroclinic Instability 678
Perturbation Vorticity Equation 679
Wave Solution 681
Instability Criterion 682
Energetics 684
13.18. Geostrophic Turbulence 685
Exercises 688
Literature Cited 690
Supplemental Reading 690
14. Aerodynamics 691
14.1. Introduction 692
14.2. Aircraft Terminology 692
Control Surfaces 693
14.3. Characteristics of Airfoil Sections 696
Historical Notes 701
14.4. Conformal Transformation for Generating Airfoil Shapes 702
14.5. Lift of a Zhukhovsky Airfoil 706
14.6. Elementary Lifting Line Theory for Wings of Finite Span 708
Lanchester Versus Prandtl 716
14.7. Lift and Drag Characteristics of Airfoils 717
14.8. Propulsive Mechanisms of Fish and Birds 719
14.9. Sailing against the Wind 721
Exercises 722
Literature Cited 728
Supplemental Reading 728
15. Compressible Flow 729
15.1. Introduction 730
Perfect Gas Thermodynamic Relations 731
15.2. Acoustics 732
15.3. Basic Equations for One-Dimensional Flow 736
15.4. Reference Properties in Compressible Flow 738
15.5. Area-Velocity Relationship in One-Dimensional Isentropic Flow 740
15.6. Normal Shock Waves 748
Stationary Normal Shock Wave in a Moving Medium 748
Moving Normal Shock Wave in a Stationary Medium 752
Normal Shock Structure 753
15.7. Operation of Nozzles at Different Back Pressures 755
Convergent Nozzle 755
Convergent-Divergent Nozzle 757
15.8. Effects of Friction and Heating in Constant-Area Ducts 761
Effect of Friction 763
Effect of Heat Transfer 764
15.9. Pressure Waves in Planar Compressible Flow 765
15.10. Thin Airfoil Theory in Supersonic Flow 773
Exercises 775
Literature Cited 778
Supplemental Reading 778
16. Introduction to Biofluid Mechanics PORTONOVO S. AYYASWAMY 779
16.1. Introduction 779
16.2. The Circulatory System in the Human Body 780
The Heart as a Pump 785
Nature of Blood 788
Nature of Blood Vessels 793
16.3. Modeling of Flow in Blood Vessels 796
Steady Blood Flow Theory 797
Pulsatile Blood Flow Theory 805
Blood Vessel Bifurcation: An Application of Poiseuille’s Formula and Murray’s Law 820
Flow in a Rigid-Walled Curved Tube 825
Flow in Collapsible Tubes 831
Laminar Flow of a Casson Fluid in a Rigid-Walled Tube 839
Pulmonary Circulation 841
The Pressure Pulse Curve in the Right Ventricle 842
Effect of Pulmonary Arterial Pressure on Pulmonary Resistance 843
16.4. Introduction to the Fluid Mechanics of Plants 844
Exercises 849
Acknowledgment 850
Literature Cited 851
Supplemental Reading 852
Appendix A 853
Appendix B 857
Appendix C 869
Appendix D 873
Index 875