Incompressible flow

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     Problems 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 Tij 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     7.8 Conclusions 148     Problems 148     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     *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     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 Flow 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 Flow (Hiemenz Flow) 245     11.10 Burgers Vortex 251     11.11 Composite Solution for the Rotary Viscous Coupling 253     11.12 Von Karman Viscous Pump 257     11.13 Conclusions 262     Problems 263     12 Streamfunctions and the Velocity Potential 266     12.1 Streamlines 266     12.2 Streamfunction for Plane Flows 269     12.3 Flow in a Slot with Porous Walls 272     *12.4 Streamlines and Streamsurfaces for a Three-Dimensional Flow 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 Flow 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 Flow Patterns 327     14.2 Entrance Flows 330     14.3 Entrance Flow 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 Flow into a Cascade of Plates: Results 342     14.6 Flow Around a Circular Cylinder 346     14.7 Jeffrey   Hamel Flow in a Wedge 362     14.8 Limiting Case for Re     0
 Stokes Flow 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 Flow 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 Flows: 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 Flow 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 Flow 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 Flows 502     19.2 Swirling Flow Turned into an Annulus 504     19.3 Flow 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 Flow from a Doublet 513     19.8 Flow 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 Flow over a Flat Plate 533     20.2 Displacement Thickness 538     20.3 Von K'arm'an Momentum Integral 540     20.4 Von K'arm'an   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 Flow 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 Flows, Moffatt Vortices 616     21.5 Plane Internal Flows 623     21.6 Flows between Rotating Cylinders 628     21.7 Flows 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 Flow 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 Flows: Bubble in an Infinite Stream 683     23.6 Constant-Tension Flows: Capillary Waves 686     23.7 Moving Contact Lines 688     23.8 Constant-Tension Flows: Coating Flows 691     23.9 Marangoni Flows 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 Flows in Tubes and Channels 728     24.10 Liquid Flows 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 Flows 744     25.4 Orr   Sommerfeld Equation 746     25.5 Invsicid Stability of Nearly Parallel Flows 747     25.6 Viscous Stability of Nearly Parallel Flows 749     25.7 Experiments on Blasius Boundary Layers 752     25.8 Transition, Secondary, Instability, and Bypass 756     25.9 Spatially Developing Open Flows 759     25.10 Transition in Free Shear Flows 759     25.11 Poiseuille and Plane Couette Flows 761     25.12 Inviscid Instability of Flows 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 Gortler 769     25.16 Conclusions 771     26 Turbulent Flows 772     26.1 Types of Turbulent Flows 772     26.2 Characteristics of Turbulent Flows 773     26.3 Reynolds Decomposition 776     26.4 Reynolds Stress 777     *26.5 Correlation of Fluctuations 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     B Differential Operations in Cylindrical and Spherical Coordinates 828     C Basic Equations in Rectangular, Cylindrical, and Spherical Coordinates 833     D Streamfunction Relations in Rectangular, Cylindrical, and Spherical Coordinates 838     E Matlab R Stagnation Point Solver 842     F Matlab R Program for Cascade Entrance 844     G Matlab R Boundary Layer Program 847     References 851     Index 869