Preface

**1 Continuum Mechanics**

1.1 Continuum Assumption

1.2 Fundamental Concepts, Definitions, and Laws

1.3 Space and Time

1.4 Density, Velocity, and Internal Energy

1.5 Interface Between Phases

1.6 Conclusions

Problems

**2 Thermodynamics**

2.1 Systems, Properties, and Processes

2.2 Independent Variables

2.3 Temperature and Entropy

2.4 Fundamental Equations of Thermodynamics

2.5 Euler’s Equation for Homogenous Functions

2.6 Gibbs-Duhem Equation

2.7 Intensive Forms of Basic Equations

2.8 Dimensions of Temperature and Entropy

2.9 Working Equations

2.10 Ideal Gas

2.11 Incompressible Substance

2.12 Compressible Liquids

2.13 Conclusions

Problems

**3 Vector Calculus and Index Notation**

3.1 Index Notation Rules & Coordinate Rotation

3.2 Definition of Vectors and Tensors

3.3 Special Symbols and Isotrpic Tensors

3.4 Direction Cosines and the Laws of Cosines

3.5 Algebra with Vectors

3.6 Symmetric and Antisymmetric Tensors

3.7 Algebra with Tensors

3.8 Vector Cross-Product

3.9 Alternative Definitions of Vectors

3.10 Principal Axes and Values

3.11 Derivative Operations on Vector Fields

3.12 Integral Formulas of Gauss and Stokes

3.13 Leibnitz’s Theorem

3.14 Conclusions

Problems

4 **Kinematics of Local Fluid Motion**

4.1 Lagrangian Viewpoint

4.2 Eulerian Viewpoint

4.3 Substantial Derivative

4.4 Decomposition of Motion

4.5 Elementary Motions in a Linear Shear Flow

4.6 Proof of Vorticity Characteristics

4.7 Rate of Strain Characteristics

4.8 Rate of Expansion

4.9 Streamline Coordinates

4.10 Conclusions

Problems

**5 Basic Laws**

5.1 Continuity Equation

5.2 Momentum Equation

5.3 Surface Forces

5.4 Stress Tensor Derivation

5.5 Interpretation of the Stress Tensor Components

5.6 Pressure and Viscous Stress Tensor

5.7 Differential Momentum Equation

5.8 Moment of Momentum, Angular Momentum, and Symmetry of *T*_{ij}

5.9 Energy Equation

5.10 Mechanical and Thermal Energy Equations

5.11 Energy Equation with Temperature as the Dependent Variable

5.12 Second Law of Thermodynamics

5.13 Integral Form of the Continuity Equation

5.14 Integral Form of the Momentum Equation

5.15 Momentum Equation for a Deformable Particle of Variable Mass

5.16 Integral Form of the Energy Equation

5.17 Integral Mechanical Energy Equation

5.18 Jump Equations at Interfaces

5.19 Conclusions

Problems

**6 Newtonian Fluids and the Navier–Stokes Equations**

6.1 Newton’s Viscosity Law

6.2 Molecular Model of Viscous Effects

6.3 Non-Newtonial Liquids

6.4 Wall Boundary Conditions, The No-Slip Condition

6.5 Fourier’s Heat Conduction Law

6.6 Navier–Stokes Equations

6.7 Conclusions

Problems

**7 Some Incompressible Flow Patterns**

7.1 Pressure-Driven Flow in a Slot

7.2 Mechanical Energy, Head Loss, and Bernoulli Equation

7.3 Plane Couette Flow

7.4 Presure-Driven Flow in a Slot with a Moving Wall

7.5 Double Falling Film on a Wall

7.6 Outer Solution for Rotary Viscous Coupling

7.7 The Rayleigh Problem

7.8 Conclusions

Problems

**8 Dimensional Analysis**

8.1 Measurement, Dimensions, and Scale Change Ratios

8.2 Physical Variables and Functions

8.3 P Theorem and Its Applications

8.4 Pump or Blower Analysis: Use of Extra Assumptions

8.5 Number of Primary Dimensions

8.6 Proof of Bridgman’s Equation

8.7 Proof of the Pi Theorem

8.8 Dynamic Similarity and Scaling Laws

8.9 Similarity with Geometric Distortion

8.10 Nondimensional Formulation of Physical Problems

8.11 Conclusions

Problems

**9 Compressible Flow**

9.1 Compressible Couette Flow Adiabatic Wall

9.2 Flow with Power Law Transport Properties

9.3 Inviscid Compressible Waves: Speed of Sound

9.4 Steady Compressible Flow

9.5 Conclusions

Problems

**10 Incompressible Flow**

10.1 Characterization

10.2 Incompressible Flow as Low-Mach Number Flow with Adiabatic Walls

10.3 Nondimensional Problem Statement

10.4 Characteristics of Incompressible Flow

10.5 Splitting the Pressure into Kinetic and Hydrostatic Parts

10.6 Mathematical Aspects of the Limit Process *M*^{2} → 0

10.7 Invariance of Incompressible Flow Equations Under UnsteadyMotion

10.8 Low-Mach Number Flows with Constant Temperature Walls

10.9 Energy Equation Paradox

10.10 Conclusions

Problems

**11 Some Solutions of the Navier–Stokes Equations**

11.1 Pressure-Driven Flow in Tubes of Various Cross Sections: Elliptical Tube

11.2 Flow in a Rectangular Tube

11.3 Asymptotic Suction Flow

11.4 Stokes’s Oscillating Plate

11.5 Wall Under and Oscillating Free Stream

11.6 Transient for a Stokes Oscillating Plate

11.7 Flow in a Slot with a steady and Oscillating Pressure Gradient

11.8 Decay of an Ideal Line Vortex (Oseen Vortex)

11.9 Plane Stagnation-Point Flow (Hiemenz Flow)

11.10 Burgers Vortex

11.11 Composite Solution for the Rotary Viscous Coupling

11.12 Von Karman Viscous Pump

11.13 Conclusions

Problems

12 **Streamfunctions and the Velocity Potential**

12.1 Streamlines

12.2 Streamfunction for Plane Flows

12.3 Flow in a Slot with Porous Walls

12.4 Streamlines and Streamsurfaces for a Three Dimensional Flow

12.5 Vector Potential and the E^{2} Operator

12.6 Stokes Stream Function for Axisymmetric Flow

12.7 Velocity Potential and the Unsteady Bernoulli Equation

12.8 Flow Caused by a Sphere with Variable Radius

12.9 Conclusions

Problems

**13 Vorticity Dynamics**

13.1 Vorticity

13.2 Kinematic Results Concerning Voriticity

13.3 Vorticity Equation

13.4 Vorticity Diffusion

13.5 Vorticity Intensification by Straining Vortex Lines

13.6 Production of Vorticity at Walls

13.7 Typical Vorticity Distributions

13.8 Development of Vorticity Distributions

13.9 Helmholtz’s Laws for Inviscid Flow

13.10 Kelvin’s Theorem

13.11 Vortex Definitions

13.12 Inviscid Motion of Point Vortices

13.13 Circular Line Vortex

13.14 Fraenkel-Norbury Vortex Rings

13.15 Hill’s Spherical Vortex

13.16 Breaking and Reconnection of Vortex Lines

13.17 Vortex Breakdown

13.18 Conclusions

Problems

**14 Flows at Moderate Reynolds Numbers**

14.1 Some Unusual Flow Patterns

14.2 Entrance Flows

14.3 Entrance Flow into a Cascade of Plates: Computer Solution by the Streamfunction-Vorticity Method

14.4 Entrance Flow into a Cascade of Plate: Pressure Solution

14.5 Entrance Flow into a Cascade of Plates: Results

14.6 Flow Around a Circular Cylinder

14.7 Jeffrey-Hamel Flow in a Wedge

14.8 Limiting Case for Re → 0; Stokes Flow

14.9 Limiting Case for Re → –¥

14.10 Conclusions

Problems

**15 Asymptotic Analysis Methods**

15.1 Oscillation of Gas Bubble in a Liquid

15.2 Order Symbols, Gauge Functions, and Asymptotic Expansions

15.3 Inviscid Flow Over a Wavy Wall

15.4 Nonuniform Expansions: Friedrich’s Problem

15.5 Matching Process: Van Dyke’s Rule

15.6 Composite Expansions

15.7 Characteristics of Overlap Regions and Common Parts

15.8 Composite Expansions And Data Analysis

15.9 Lagerstrom’s Problems

15.10 Conclusions

Problems

**16 Characteristics of High-Reynolds-Number Flows**

16.1 Physical Motivation

16.2 Inviscid Main Flows: Euler Equations

16.3 Pressure Changes in Steady Flows: Bernoulli Equations

16.4 Boundary Layers

16.5 Conclusions

Problems

**17 Kinematic Decomposition of Flow Fields**

17.1 General Approach

17.2 Helmholtz’s Decomposition; Biot-Savart Law

17.3 Line Vortex and Vortex Sheet

17.4 Complex Lamellar Decomposition

17.5 Conclusions

Problems

**18 Ideal Flows in a Plane**

18.1 Problem Formulation for Plane Ideal Flows

18.2 Simple Plane Flows

18.3 Line Source and Line Vortex

18.4 Flow Over a Nose or a Cliff

18.5 Doublets

18.6 Cylinder in a Stream

18.7 Cylinder with Circulation in a Uniform Stream

18.8 Lift and Drag on Two-Dimensional Shapes

18.9 Magnus Effect

18.10 Conformal Transformations

18.11 Joukowski Transformation: Airfoil Geometry

18.12 Kutta Condition

18.13 Flow Over a Joukowski Airfoil: Airfoil Lift

18.14 Numerical Method for Airfoils

18.15 Actual Airfoils

18.16 Schwarz-Christoffel Transformation

18.17 Diffuser or Contraction Flow

18.18 Gravity Waves in Liquids

18.19 Conclusions

Problems

**19 Three-Dimensional Ideal Flows**

19.1 General Equations and Characteristics of Three Dimensional Ideal Flows

19.2 Swirling Flow Turned into and Annulus

19.3 Flow Over a Weir

19.4 Point Source

19.5 Rankine Nose Shape

19.6 Experiments of the Nose Drag of Slender Shapes

19.7 Flow from a Doublet

19.8 Flow Over a Sphere

19.9 Work to Move a Body in a Still Fluid

19.10 Wake Drag of Bodies

19.11 Induced Drag Due to Lift

19.12 Lifting Line Theory

19.13 Winglets

19.14 Added Mass of Accelerating Bodies

19.15 Conclusions

Problems

**20 Boundary Layers**

20.1 Blasius Flow Over a Flat Plate

20.2 Displacement Thickness

20.3 Von Karman Momentum Integral

20.4 Von Karman Pohlhausen Approximate: Method

20.5 Falkner–Skan Similarity Solutions

20.6 Arbitrary Two Dimensinoal Layers: Crank-Nicolson Difference Method

20.7 Vertical Velocity

20.8 Joukowski Airfoil Boundary Layer

20.9 Boundary Layer on a Bridge Piling

20.10 Boundary Layers Beginning at Infinity

20.11 Plane Boundary Layer Separation

20.12 Axisymmteric Boundary Layers

20.13 Jets

20.14 Far Wake of Nonlifting Bodies

20.15 Free Shear Layers

20.16 Unsteady and Erupting Boundary Layers

20.17 Entrance Flow into a Cascade, Parabolized Navier-Stokes Equations

20.18 Three-Dimensional Boundary Layers

20.19 Boundary Layer with a Constant Transverse Pressure Gradient

20.20 Howarth’s Stagnation Point

20.21 Three-Dimensional Separation Patterns

20.22 Conclusions

Problems

**21 Flow at Low Reynolds Numbers**

21.1 General Relations for Re → 0: Stokes’s Equations

21.2 Global Equations for Stokes Flow

21.3 Streamfunction for Plane and Axisymmetric Flows

21.4 Local Flows, Moffatt Vortices

21.5 Plane Internal Flows

21.6 Flows Between Rotating Cylinders

21.7 Flows in Tubes, Nozzles, Orifices, and Cones

21.8 Sphere in a Uniform Stream

21.9 Composite Expansion for Flow Over a Sphere

21.10 Stokes Flow Near a Circular Cylinder

21.11 Axisymmetric Particles

21.12 Oseen’s Equations

21.13 Interference Effects

21.14 Conclusions

Problems

**22 Lubrication Approximation**

22.1 Basic Characteristics: Channel Flow

22.2 Flow in a Channel with a Porous Wall

22.3 Reynolds Equation for Bearing Theory

22.4 Slipper Pad Bearing

22.5 Squeeze-Film Lubrication: Viscous Adhesion

22.6 Journal Bearing

22.7 Hele-Shaw Flow

22.8 Conclusions

Problems

**23 Surface Tension Effects**

23.1 Interface Concepts and Laws

23.2 Statics: Plane Interfaces

23.3 Statics: Cylindrical Interfaces

23.4Statics: Attached Bubbles and Drops

23.5 Constant-Tension Flows: Bubble in an Infinite Stream

23.6 Constant-Tension Flows: Capillary Waves

23.7 Moving Contact Lines

23.8 Constant-Tension Flows: Coating Flows

23.9 Marangoni Flows

23.10 Conclusions

Problems

**24 Introduction to Microflows**

24.1 Molecules

24.2 Contiuum Description

24.3 Compressible Flow in Long Channels

24.4 Simple Solutions with Slip

24.5 Gases

24.6 Couette Flow in Gases

24.7 Poiseuille Flow in Gases

24.8 Gas Flow Over a Sphere

24.9 Liquid Flows in Tues and Channels

24.10 Liquid Flows Near Walls

24.11 Conclusions

**25 Stability and Transition**

25.1 Linear Stability and Normal Modes as Perturbations

25.2 Kelvin-Helmholtz Inviscid Shear Layer Instability

25.3 Stability Problems for Nearly Parallel Viscous Flows

25.4 Orr-Sommerfeld Equation

25.5 Invsicid Stability of Nearly Parallel Flows

25.6 Viscous Stability of Nearly Parallel Flows

25.7 Experiments on Blasius Boundary Layers

25.8 Transition, Secondary, Instability, and Bypass

25.9 Spatially Developing Open Flows

25.10 Transition in Free Shear Flows

25.11 Poiseuille and Plane Couette Flows

25.12 Inviscid Instability of Flows with Curved Streamlines

25.13 Taylor Instability of Couette Flow

25.14 Stability of Regions of Concentrated Vorticity

25.15 Other Instabilities: Taylor, Curved, Pipe, Capillary Jets, and Görtler

25.16 Conclusions

**26 Turbulent Flows**

26.1 Types of Turbulent Flows

26.2 Characteristics of Turbulent Flows

26.3 Reynolds Decomposition

26.4 Reynolds Stress

26.5 Correlation of Fluctuations

26.6 Mean and Turbulent Kinetic Energy

26.7 Energy Cascade: Kolmogorov Scales and Taylor Microscale

26.8 Wall Turbulence: Channel Flow Analysis

26.9 Channel and Pipe Flow Experiments

26.10 Boundary Layers

26.11 Wall Turbulence Fluctuations

26.12 Turbulent Structures

26.13 Free Turbulence: Plane Shear Layers

26.14 Free Turbulence: Turbulent Jet

26.15 Bifurcating and Blooming Jets

26.16 Conclusions

Appendix A Properties of Fluids

Appendix B Differential Operations in Cylindrical and Spherical Coordinates

Appendix C Basic Equations in Rectangular, Cylindrical, and Spherical Coordinates

Appendix D Streamfunction Relations in Rectangular, Cylindrical, and Spherical Coordinates

Appendix E Matlab Stagnation Point Solver

Appendix F Matlab program for Cascade Entrance

Appendix G Matlab Boundary Layer Program

References