## Viscous Fluid Flow, Fourth Edition

By Frank M. White and Joseph Majdalani

**Contents**

Preface xiii

List of Symbols xvii

1. Preliminary Concepts 1

1.1 Historical Outline 1

1.2 Some Examples of Viscous-Flow Phenomena 2

1.3 Properties of a Fluid 8

1.4 Boundary Conditions for Viscous-Flow Problems 27

Summary 31

Problems 31

2. Fundamental Equations of Compressible Viscous Flow 38

2.1 Introduction 38

2.2 Classification of The Fundamental Equations 38

2.3 Conservation of Mass: The Equation of Continuity 39

2.4 Conservation of Momentum: The Navier–Stokes Equations 40

2.5 The Energy Equation (First Law of Thermodynamics) 46

2.6 Boundary Conditions for Viscous Heat-Conducting Flow 49

2.7 **Orthogonal Coordinate Systems** 50

2.8 Mathematical Character of the Equations of Motion 52

2.9 Dimensionless Parameters in Viscous Flow 56

2.10 Vorticity Considerations in Incompressible Viscous Flow 59

2.11 Two-Dimensional Considerations: The Stream Function 60

2.12 Non-Inertial Coordinate Systems 63

2.13 Control-Volume Formulations 63

Summary 65

Problems 66

3. Solutions of the Newtonian Viscous-Flow Equations 76

3.1 Introduction And Classification of Solutions 76

3.2 Couette Flows Due to Moving Surfaces 77

3.3 Poiseuille Flow Through Ducts 82

3.4 Unsteady Duct Flows 95

3.5 Unsteady Flows with Moving Boundaries 103

3.6 Asymptotic Suction Flows 108

3.7 Wind-Driven Flows: The Ekman Drift 117

3.8 Similarity Solutions 119

3.9 Low Reynolds Number: Linearized Creeping Motion 133

Summary 144

Problems 145

4. Laminar Boundary Layers 153

4.1 Introduction 153

4.2 Laminar-Boundary-Layer Equations 164

4.3 Similarity Solutions for Steady Two-Dimensional Flow 167

4.4 Free-Shear Flows 185

4.5 Other Analytic Two-Dimensional Solutions 190

4.6 Approximate Integral Methods 192

4.7 Numerical Solutions 199

4.8 Thermal-Boundary-Layer Calculations 204

4.9 Developing Inlet Flow in Ducts 210

4.10 Rotationally Symmetric Boundary Layers 212

4.11 Three-Dimensional Laminar Boundary Layers 219

4.12 Unsteady Boundary Layers: Separation Anxiety 226

4.13 Free-Convection Boundary Layers 227

Summary 232

Problems 232

5. The **Stability of Laminar Flows** 249

5.1 Introduction: The Concept of Small-Disturbance Stability 249

5.2 Linearized Stability of Parallel Viscous Flows 253

5.3 Parametric Effects in the Linear Stability Theory 261

5.4 Transition to Turbulence 271

5.5 Engineering Prediction of Transition 276

5.6 Biglobal Stability of Incompressible Viscous Flow 287

5.7 Biglobal Stability of Compressible Viscous Flow 295

Summary 310

Problems 311

6. Incompressible Turbulent Mean Flow 323

6.1 Physical and Mathematical Description of Turbulence 323

6.2 The Reynolds Equations of Turbulent Motion 328

6.3 The Two-Dimensional Turbulent-Boundary-Layer Equations 332

6.4 Velocity Profiles: The Inner, Outer, and Overlap Layers 334

6.5 Turbulent Flow in Pipes and Channels 341

6.6 The Turbulent Boundary Layer on a Flat Plate 347

6.7 Turbulence Modeling 353

6.8 Analysis of Turbulent Boundary Layers with a Pressure Gradient 362

6.9 Free Turbulence: Jets, Wakes, and Mixing Layers 375

6.10 Turbulent Convective Heat Transfer 383

Summary 392

Problems 393

7. Compressible Boundary-Layer Flow 400

7.1 Introduction: The Compressible-Boundary-Layer Equations 400

7.2 Similarity Solutions for Compressible Laminar Flow 405

7.3 Solutions for Laminar Flat-Plate and Stagnation-Point Flow 407

7.4 Compressible Laminar Boundary Layers Under Arbitrary Conditions 415

7.5 Special Topics in Compressible Laminar Flow 424

7.5 The Compressible-Turbulent-Boundary-Layer Equations 430

7.6 Wall and Wake Laws for Turbulent Compressible Flow 432

7.7 Compressible Turbulent Flow Past a Flat Plate 436

7.8 Compressible-Turbulent-Boundary-Layer Calculation with a

7.9 Pressure Gradient 441

7.10 Compressible-Flow Approximations for Quasi-Viscous Flows 444

Summary 451

Problems 451

Appendices 457

A. Vector Identities 457

B1. 3d Kinematic Expressions in Cartesian Coordinates 458

B2. Reduced Kinematic Expressions in Steady 2D Cartesian Coordinates 459

B3. 3D Kinematic Expressions in Cylindrical Coordinates 460

B4. Reduced Kinematic Expressions in Steady 2D Cylindrical Coordinates 462

C. Transport Properties of Various Newtonian Fluids 463

D. Einstein’s Indicial Notation 472

E1. Advanced Energy Loss Evaluation 474

E2. Traditional/Iterative Energy Loss Evaluation 476

F. Bragg–Hawthorne Equation for Axisymmetric Flow in Spherical Coordinates 478

G. Equations of Motion of Incompressible Newtonian Fluids in Cylindrical and Spherical Polar Coordinates 480

H1. Determination of Dimensionless Parameters 483

H2. List of Dimensionless Parameters 485

I. Trigonometric Identities 490

Bibliography 492

Index 517