Preface ........................................................ xi
Preface to the Third Edition ................................. xiii
Preface to the Second Edition .................................. xv
Preface to the First Edition ................................. xvii
1 Continuum Mechanics .......................................... 1
1.1 Continuum Assumption .................................... 3
1.2 Fundamental Concepts, Definitions, and Laws ............. 3
1.3 Space and Time .......................................... 5
1.4 Density, Velocity, and Internal Energy .................. 7
1.5 Interface between Phases ............................... 10
1.6 Conclusions ............................................ 12
Problems .................................................... 13
2 Thermodynamics .............................................. 15
2.1 Systems, Properties, and Processes ..................... 15
2.2 Independent Variables .................................. 16
2.3 Temperature and Entropy ................................ 16
2.4 Fundamental Equations of Thermodynamics ................ 18
2.5 Euler's Equation for Homogenous Functions .............. 19
2.6 Gibbs-Duhem Equation ................................... 20
2.7 Intensive Forms of Basic Equations ..................... 20
2.8 Dimensions of Temperature and Entropy .................. 21
2.9 Working Equations ...................................... 21
2.10 Ideal Gas .............................................. 22
2.11 Incompressible Substance ............................... 25
2.12 Compressible Liquids ................................... 26
2.13 Conclusions ............................................ 26
Problems .................................................... 26
3 Vector Calculus and Index Notation .......................... 28
3.1 Index Notation Rules and Coordinate Rotation ........... 29
3.2 Definition of Vectors and Tensors ...................... 32
3.3 Special Symbols and Isotropic Tensors .................. 33
3.4 Direction Cosines and the Laws of Cosines .............. 34
3.5 Algebra with Vectors ................................... 35
3.6 Symmetric and Antisymmetric Tensors .................... 37
3.7 Algebra with Tensors ................................... 38
3.8 Vector Cross-Product ................................... 41
3.9 Alternative Definitions of Vectors ..................... 42
3.10 Principal Axes and Values .............................. 44
3.11 Derivative Operations on Vector Fields ................. 45
3.12 Integral Formulas of Gauss and Stokes .................. 48
3.13 Leibnitz's Theorem ..................................... 51
3.14 Conclusions ............................................ 52
Problems .................................................... 53
4 Kinematics of Local Fluid Motion ............................ 54
4.1 Lagrangian Viewpoint ................................... 54
4.2 Eulerian Viewpoint ..................................... 57
4.3 Substantial Derivative ................................. 59
4.4 Decomposition of Motion ................................ 60
4.5 Elementary Motions in a Linear Shear Flow .............. 64
4.6 Proof of Vorticity Characteristics ..................... 66
4.7 Rate-of-Strain Characteristics ......................... 68
4.8 Rate of Expansion ...................................... 69
4.9 Streamline Coordinates ................................. 70
4.10 Conclusions ............................................ 72
5 Basic Laws .................................................. 74
5.1 Continuity Equation .................................... 74
5.2 Momentum Equation ...................................... 78
5.3 Surface Forces ......................................... 79
5.4 Stress Tensor Derivation ............................... 79
5.5 Interpretation of the Stress Tensor Components ......... 81
5.6 Pressure and Viscous Stress Tensor ..................... 83
5.7 Differential Momentum Equation ......................... 84
5.8 Moment of Momentum, Angular Momentum, and Symmetry of
Ту ..................................................... 89
5.9 Energy Equation ........................................ 90
5.10 Mechanical and Thermal Energy Equations ................ 92
5.11 Energy Equation with Temperature as the Dependent
Variable ............................................... 94
5.12 Second Law of Thermodynamics ........................... 94
5.13 Integral Form of the Continuity Equation ............... 95
5.14 Integral Form of the Momentum Equation ................. 97
5.15 Momentum Equation for a Deformable Particle of
Variable Mass ......................................... 100
5.16 Integral Form of the Energy Equation .................. 103
5.17 Integral Mechanical Energy Equation ................... 104
5.18 Jump Equations at Interfaces .......................... 106
5.19 Conclusions ........................................... 108
Problems ................................................... 108
6 Newtonian Fluids and the Navier-Stokes Equations ........... 111
6.1 Newton's Viscosity Law ................................ 111
6.2 Molecular Model of Viscous Effects .................... 114
6.3 Non-Newtonian Liquids ................................. 118
6.4 Wall Boundary Conditions; The No-Slip Condition ....... 120
6.5 Fourier's Heat Conduction Law ......................... 123
6.6 Navier-Stokes Equations ............................... 125
6.7 Conclusions ........................................... 125
Problems ................................................... 126
7 Some Incompressible Flow Patterns .......................... 127
7.1 Pressure-Driven Flow in a Slot ........................ 127
7.2 Mechanical Energy, Head Loss, and Bernoulli Equation .. 132
7.3 Plane Couette Flow .................................... 136
7.4 Pressure-Driven Flow in a Slot with a Moving Wall ..... 138
7.5 Double Falling Film on a Wall ......................... 139
7.6 Outer Solution for Rotary Viscous Coupling ............ 142
7.7 The Rayleigh Problem .................................. 143
8 Conclusions ................................................ 148
Problems ................................................... 148
8.7 Proof of the Pi Theorem ............................... 167
8.8 Dynamic Similarity and Scaling Laws ................... 170
8.9 Similarity with Geometric Distortion .................. 171
8.10 Nondimensional Formulation of Physical Problems ....... 174
8.11 Conclusions ........................................... 179
Problems ................................................... 180
8 Dimensional Analysis ....................................... 150
8.1 Measurement, Dimensions, and Scale Change Ratios ...... 150
8.2 Physical Variables and Functions ...................... 153
8.3 Pi Theorem and Its Applications ....................... 155
8.4 Pump or Blower Analysis: Use of Extra Assumptions ..... 159
8.5 Number of Primary Dimensions .......................... 163
8.6 Proof of Bridgman's Equation .......................... 165
9 Compressible Flow .......................................... 182
9.1 Compressible Couette Flow: Adiabatic Wall ............. 182
9.2 Flow with Power Law Transport Properties .............. 186
9.3 Inviscid Compressible Waves: Speed of Sound ........... 187
9.4 Steady Compressible Flow .............................. 194
9.5 Conclusions ........................................... 197
Problems ................................................... 197
10 Incompressible Flow ........................................ 198
10.1 Characterization ...................................... 198
10.2 Incompressible Flow as Low-Mach-Number Flow with
Adiabatic Walls ....................................... 199
10.3 Nondimensional Problem Statement ...................... 201
10.4 Characteristics of Incompressible Flow ................ 205
10.5 Splitting the Pressure into Kinetic and Hydrostatic
Parts ................................................. 207
10.6 Mathematical Aspects of the Limit Process M2 → 0 ...... 210
10.7 Invariance of Incompressible Flow Equations under
Unsteady Motion ....................................... 211
10.8 Low-Mach-Number Flows with Constant-Temperature
Walls ................................................. 213
10.9 Energy Equation Paradox ............................... 216
10.10 Conclusions .......................................... 218
Problems ................................................... 219
11 Some Solutions of the Navier-Stokes Equations .............. 220
11.1 Pressure-Driven Flow in Tubes of Various Cross
Sections: Elliptical Tube ............................. 221
11.2 Flow in a Rectangular Tube ............................ 224
11.3 Asymptotic Suction Flow ............................... 227
11.4 Stokes's Oscillating Plate ............................ 228
11.5 Wall under an Oscillating Free Stream ................. 231
11.6 Transient for a Stokes Oscillating Plate .............. 234
11.7 Row in a Slot with a Steady and Oscillating Pressure
Gradient .............................................. 236
11.8 Decay of an Ideal Line Vortex (Oseen Vortex) .......... 241
11.9 Plane Stagnation Point How (Hiemenz Row) .............. 245
11.10 Burgers Vortex ....................................... 251
11.11 Composite Solution for the Rotary Viscous Coupling ... 253
11.12 Von Kármán Viscous Pump .............................. 257
11.13 Conclusions .......................................... 262
12 Streamfunctions and the Velocity Potential ................. 266
12.1 Streamlines ........................................... 266
12.2 Streamfunction for Plane Flows ........................ 269
12.3 Row in a Slot with Porous Walls ....................... 272
12.4 Streamlines and Streamsurfaces for a Three-
Dimensional Row ....................................... 274
12.5 Vector Potential and the E2 Operator .................. 277
12.6 Stokes's Streamfunction for Axisymmetric Flow ......... 282
12.7 Velocity Potential and the Unsteady Bernoulli
Equation .............................................. 283
12.8 Row Caused by a Sphere with Variable Radius ........... 284
12.9 Conclusions ........................................... 286
Problems ................................................... 287
13 Vorticity Dynamics ......................................... 289
13.1 Vorticity ............................................. 289
13.2 Kinematic Results Concerning Vorticity ................ 290
13.3 Vorticity Equation .................................... 292
13.4 Vorticity Diffusion ................................... 293
13.5 Vorticity Intensification by Straining Vortex Lines ... 295
13.6 Production of Vorticity at Walls ...................... 296
13.7 Typical Vorticity Distributions ....................... 300
13.8 Development of Vorticity Distributions ................ 300
13.9 Helmholtz's Laws for Inviscid Flow .................... 306
13.10 Kelvin's Theorem ..................................... 307
13.11 Vortex Definitions ................................... 308
13.12 Inviscid Motion of Point Vortices .................... 310
13.13 Circular Line Vortex ................................. 312
13.14 Fraenkel-Norbury Vortex Rings ........................ 314
13.15 Hill's Spherical Vortex .............................. 314
13.16 Breaking and Reconnection of Vortex Lines ............ 317
13.17 Vortex Breakdown ..................................... 317
13.18 Conclusions .......................................... 323
Problems ................................................... 324
14 Flows at Moderate Reynolds Numbers ......................... 326
14.1 Some Unusual Row Patterns ............................. 327
14.2 Entrance Rows ......................................... 330
14.3 Entrance Row into a Cascade of Plates: Computer
Solution by the Streamfunction-Vorticity Method ....... 331
14.4 Entrance Flow into a Cascade of Plates: Pressure
Solution .............................................. 341
14.5 Entrance Row into a Cascade of Plates: Results ........ 342
14.6 Row Around a Circular Cylinder ........................ 346
14.7 Jeffrey-Hamel Row in a Wedge .......................... 362
14.8 Limiting Case for Re → 0; Stokes Row .................. 367
14.9 Limiting Case for Re → -∞ ............................. 368
14.10 Conclusions .......................................... 372
Problems ................................................... 372
15 Asymptotic Analysis Methods ................................ 374
15.1 Oscillation of a Gas Bubble in a Liquid ............... 374
15.2 Order Symbols, Gauge Functions, and Asymptotic
Expansions ............................................ 377
15.3 Inviscid Row over a Wavy Wall ......................... 380
15.4 Nonuniform Expansions: Friedrich's Problem ............ 384
15.5 Matching Process: Van Dyke's Rule ..................... 386
15.6 Composite Expansions .................................. 391
15.7 Characteristics of Overlap Regions and Common Parts ... 393
15.8 Composite Expansions and Data Analysis ................ 399
15.9 Lagerstrom's Problems ................................. 403
15.10 Conclusions .......................................... 406
Problems ................................................... 407
16 Characteristics of High-Reynolds-Number Flows .............. 409
16.1 Physical Motivation ................................... 409
16.2 Inviscid Main Rows: Euler Equations ................... 411
16.3 Pressure Changes in Steady Flows: Bernoulli
Equations ............................................. 414
16.4 Boundary Layers ....................................... 418
16.5 Conclusions ........................................... 428
Problems ................................................... 428
17 Kinematic Decomposition of Flow Fields ..................... 429
17.1 General Approach ..................................... 429
17.2 Helmholtz's Decomposition; Biot-Savart Law ........... 430
17.3 Line Vortex and Vortex Sheet ......................... 431
17.4 Complex Lamellar Decomposition ....................... 434
17.5 Conclusions .......................................... 437
Problems ................................................... 437
18 Ideal Flows in a Plane ..................................... 438
18.1 Problem Formulation for Plane Ideal Flows ............. 439
18.2 Simple Plane Flows .................................... 442
18.3 Line Source and Line Vortex ........................... 445
18.4 Flow over a Nose or a Cliff ........................... 447
18.5 Doublets .............................................. 453
18.6 Cylinder in a Stream .................................. 456
18.7 Cylinder with Circulation in a Uniform Stream ......... 457
18.8 Lift and Drag on Two-Dimensional Shapes ............... 460
18.9 Magnus Effect ......................................... 462
18.10 Conformal Transformations ............................ 464
18.11 Joukowski Transformation: Airfoil Geometry ........... 468
18.12 Kutta Condition ...................................... 473
18.13 Row over a Joukowski Airfoil: Airfoil Lift ........... 475
18.14 Numerical Method for Airfoils ........................ 482
18.15 Actual Airfoils ...................................... 484
18.16 Schwarz-Christoffel Transformation ................... 487
18.17 Diffuser or Contraction Row .......................... 489
18.18 Gravity Waves in Liquids ............................. 494
18.19 Conclusions .......................................... 499
Problems ................................................... 499
19 Three-Dimensional Ideal Flows .............................. 502
19.1 General Equations and Characteristics of Three-
Dimensional Ideal Rows ................................ 502
19.2 Swirling Row Turned into an Annulus ................... 504
19.3 Row over a Weir ....................................... 505
19.4 Point Source .......................................... 507
19.5 Rankine Nose Shape .................................... 508
19.6 Experiments on the Nose Drag of Slender Shapes ........ 510
19.7 Row from a Doublet .................................... 513
19.8 Row over a Sphere ..................................... 515
19.9 Work to Move a Body in a Still Fluid .................. 516
19.10 Wake Drag of Bodies .................................. 518
19.11 Induced Drag: Drag due to Lift ....................... 519
19.12 Lifting Line Theory .................................. 524
19.13 Winglets ............................................. 525
19.14 Added Mass of Accelerating Bodies .................... 526
19.15 Conclusions .......................................... 531
Problems ................................................... 531
20 Boundary Layers ............................................ 533
20.1 Blasius Row over a Rat Plate .......................... 533
20.2 Displacement Thickness ................................ 538
20.3 Von Kármán Momentum Integral .......................... 540
20.4 Von Kármán-Pohlhausen Approximate Method .............. 541
20.5 Falkner-Skan Similarity Solutions ..................... 543
20.6 Arbitrary Two-Dimensinoal Layers: Crank-Nicolson
Difference Method ..................................... 547
20.7 Vertical Velocity ..................................... 556
20.8 Joukowski Airfoil Boundary Layer ...................... 558
20.9 Boundary Layer on a Bridge Piling ..................... 563
20.10 Boundary Layers Beginning at Infinity ................ 564
20.11 Plane Boundary Layer Separation ...................... 570
20.12 Axisymmteric Boundary Layers ......................... 573
20.13 Jets ................................................. 576
20.14 Far Wake of Nonlifting Bodies ........................ 579
20.15 Free Shear Layers .................................... 582
20.16 Unsteady and Erupting Boundary Layers ................ 584
20.17 Entrance Row into a Cascade, Parabolized Navier-
Stokes Equations ..................................... 587
20.18 Three-Dimensional Boundary Layers .................... 589
20.19 Boundary Layer with a Constant Transverse Pressure
Gradient ............................................. 593
20.20 Howarth's Stagnation Point ........................... 598
20.21 Three-Dimensional Separation Patterns ................ 600
20.22 Conclusions .......................................... 603
Problems ................................................... 605
21 Flow at Low Reynolds Numbers ............................... 607
21.1 General Relations for Re → 0: Stokes' s
Equations ............................................ 607
21.2 Global Equations for Stokes Flow ..................... 611
21.3 Streamfunction for Plane and Axisymmetric Flows ....... 613
21.4 Local Rows, Moffatt Vortices .......................... 616
21.5 Plane Internal Rows ................................... 623
21.6 Rows between Rotating Cylinders ....................... 628
21.7 Rows in Tubes, Nozzles, Orifices, and Cones ........... 631
21.8 Sphere in a Uniform Stream ............................ 636
21.9 Composite Expansion for Flow over a Sphere ............ 641
21.10 Stokes Row near a Circular Cylinder .................. 642
21.11 Axisymmetric Particles ............................... 644
21.12 Oseen's Equations .................................... 646
21.13 Interference Effects ................................. 647
21.14 Conclusions 648 Problems ............................. 649
22 Lubrication Approximation ............................... 650
22.1 Basic Characteristics: Channel Flow ................... 650
22.2 Flow in a Channel with a Porous Wall .................. 653
22.3 Reynolds Equation for Bearing Theory .................. 655
22.4 Slipper Pad Bearing ................................... 657
22.5 Squeeze-Film Lubrication: Viscous Adhesion ............ 659
22.6 Journal Bearing ....................................... 660
22.7 Hele-Shaw Flow ........................................ 664
22.8 Conclusions ........................................... 667
Problems ................................................... 668
23 Surface Tension Effects .................................... 669
23.1 Interface Concepts and Laws ........................... 669
23.2 Statics: Plane Interfaces ............................. 676
23.3 Statics: Cylindrical Interfaces ....................... 679
23.4 Statics: Attached Bubbles and Drops ................... 681
23.5 Constant-Tension Rows: Bubble in an Infinite Stream ... 683
23.6 Constant-Tension Rows: Capillary Waves ................ 686
23.7 Moving Contact Lines .................................. 688
23.8 Constant-Tension Rows: Coating Rows ................... 691
23.9 Marangoni Rows ........................................ 695
23.10 Conclusions .......................................... 703
Problems ............................................. 705
24 Introduction to Microflows ................................. 706
24.1 Molecules ............................................. 706
24.2 Continuum Description ................................. 708
24.3 Compressible Flow in Long Channels .................... 709
24.4 Simple Solutions with Slip ............................ 712
24.5 Gases ................................................. 715
24.6 Couette Flow in Gases ................................. 719
24.7 Poiseuille Flow in Gases .............................. 722
24.8 Gas Flow over a Sphere ................................ 726
24.9 Liquid Rows in Tubes and Channels ..................... 728
24.10 Liquid Rows near Walls; Slip Boundaries .............. 730
24.11 Conclusions .......................................... 735
25 Stability and Transition ................................... 737
25.1 Linear Stability and Normal Modes as Perturbations .... 738
25.2 Kelvin-Helmholtz Inviscid Shear Layer Instability ..... 739
25.3 Stability Problems for Nearly Parallel Viscous Rows ... 744
25.4 Orr-Sommerfeld Equation ............................... 746
25.5 Invsicid Stability of Nearly Parallel Rows ............ 747
25.6 Viscous Stability of Nearly Parallel Rows ............. 749
25.7 Experiments on Blasius Boundary Layers ................ 752
25.8 Transition, Secondary, Instability, and Bypass ........ 756
25.9 Spatially Developing Open Rows ........................ 759
25.10 Transition in Free Shear Rows ........................ 759
25.11 Poiseuille and Plane Couette Rows .................... 761
25.12 Inviscid Instability of Rows with Curved
Streamlines .......................................... 763
25.13 Taylor Instability of Couette Flow ................... 765
25.14 Stability of Regions of Concentrated Vorticity ....... 767
25.15 Other Instabilities: Taylor, Curved, Pipe,
Capillary Jets, and Gцrtier .......................... 769
25.16 Conclusions .......................................... 771
26 Turbulent Flows ............................................ 772
26.1 Types of Turbulent Rows ............................... 772
26.2 Characteristics of Turbulent Rows ..................... 773
26.3 Reynolds Decomposition ................................ 776
26.4 Reynolds Stress ....................................... 777
26.5 Correlation of Ructuations ............................ 780
26.6 Mean and Turbulent Kinetic Energy ..................... 782
26.7 Energy Cascade: Kolmogorov Scales and Taylor
Microscale ............................................ 784
26.8 Wall Turbulence: Channel Flow Analysis ................ 789
26.9 Channel and Pipe Flow Experiments ..................... 797
26.10 Boundary Layers ...................................... 800
26.11 Wall Turbulence: Fluctuations ........................ 804
26.12 Turbulent Structures ................................. 811
26.13 Free Turbulence: Plane Shear Layers .................. 817
26.14 Free Turbulence: Turbulent Jet ....................... 822
26.15 Bifurcating and Blooming Jets ........................ 824
26.16 Conclusions .......................................... 825
A Properties of Fluids ....................................... 827
В Differential Operations in Cylindrical and Spherical
Coordinates ................................................ 828
С Basic Equations in Rectangular, Cylindrical, and
Spherical Coordinates ...................................... 833
D Streamfunction Relations in Rectangular, Cylindrical, and
Spherical Coordinates ...................................... 838
E Matlab® Stagnation Point Solver ............................ 842
F Matlab® Program for Cascade Entrance ....................... 844
G Matlab® Boundary Layer Program ............................. 847
References .................................................... 851
Index ......................................................... 869
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