Fluid mechanics third edition

Fluid Mechanics, Third Edition


Founders of Modern Fluid Dynamics

Ludwig Prandtl
(1875–1953)

G. I. Taylor
(1886–1975)

(Biographical sketches of Prandtl and Taylor are given in Appendix C.)

Photograph of Ludwig Prandtl is reprinted with permission from the Annual Review of Fluid
Mechanics, Vol. 19, Copyright 1987 by Annual Reviews www.AnnualReviews.org.
Photograph of Geoffrey Ingram Taylor at age 69 in his laboratory reprinted with permission
from the AIP Emilio Segr`e Visual Archieves. Copyright, American Institute of Physics, 2000.


Fluid Mechanics

Third Edition

Pijush K. Kundu
Oceanographic Center
Nova University
Dania, Florida

Ira M. Cohen
Department of Mechanical Engineering and

Applied Mechanics
University of Pennsylvania
Philadelphia, Pennsylvania

with a chapter on Computational Fluid Dynamics by Howard H. Hu

AMSTERDAM
BOSTON
HEIDELBERG
LONDON
NEW YORK
OXFORD
PARIS
SAN DIEGO
SAN FRANCISCO
SINGAPORE
SYDNEY
TOKYO


Elsevier Academic Press
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∞

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Printed in the United States of America
04 05 06 07 08 9 8 7 6 5 4 3 2 1


The third edition is dedicated to the memory of Pijush K. Kundu and also to my wife
Linda and daughters Susan and Nancy who have greatly enriched my life.

“Everything should be made as simple as possible,
but not simpler.”
—Albert Einstein
“If nature were not beautiful, it would not be worth studying it.
And life would not be worth living.”
—Henry Poincar´e


In memory of Pijush Kundu
Pijush Kanti Kundu was born in Calcutta,

India, on October 31, 1941. He received a
B.S. degree in Mechanical Engineering in
1963 from Shibpur Engineering College of
Calcutta University, earned an M.S. degree
in Engineering from Roorkee University in
1965, and was a lecturer in Mechanical Engineering at the Indian Institute of Technology
in Delhi from 1965 to 1968. Pijush came to
the United States in 1968, as a doctoral student at Penn State University. With Dr. John
L. Lumley as his advisor, he studied instabilities of viscoelastic fluids, receiving his doctorate in 1972. He began his lifelong interest in
oceanography soon after his graduation, working as Research Associate in Oceanography at Oregon State University from 1968 until 1972. After spending a year at the
University de Oriente in Venezuela, he joined the faculty of the Oceanographic Center
of Nova Southeastern University, where he remained until his death in 1994.
During his career, Pijush contributed to a number of sub-disciplines in physical
oceanography, most notably in the fields of coastal dynamics, mixed-layer physics,
internal waves, and Indian-Ocean dynamics. He was a skilled data analyst, and, in
this regard, one of his accomplishments was to introduce the “empirical orthogonal
eigenfunction” statistical technique to the oceanographic community.
I arrived at Nova Southeastern University shortly after Pijush, and he and I worked
closely together thereafter. I was immediately impressed with the clarity of his scientific thinking and his thoroughness. His most impressive and obvious quality, though,
was his love of science, which pervaded all his activities. Some time after we met,
Pijush opened a drawer in a desk in his home office, showing me drafts of several
chapters to a book he had always wanted to write. A decade later, this manuscript
became the first edition of “Fluid Mechanics,” the culmination of his lifelong dream;
which he dedicated to the memory of his mother, and to his wife Shikha, daughter
Tonushree, and son Joydip.
Julian P. McCreary, Jr.,
University of Hawaii


Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Preface to Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Preface to First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx
Author’s Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

Chapter 1

Introduction
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Units of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solids, Liquids, and Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fluid Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Classical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Static Equilibrium of a Compressible Medium . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
3
4
5
8
9
12
16
17
22
23
23

Chapter 2

Cartesian Tensors
1. Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Rotation of Axes: Formal Definition of a Vector . . . . . . . . . . . . . . . . . .

24
25
vii


viii

Contents

3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.

Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Second-Order Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contraction and Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Force on a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kronecker Delta and Alternating Tensor . . . . . . . . . . . . . . . . . . . . . . . . .
Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operator ∇: Gradient, Divergence, and Curl . . . . . . . . . . . . . . . . . . . . .
Symmetric and Antisymmetric Tensors . . . . . . . . . . . . . . . . . . . . . . . . .
Eigenvalues and Eigenvectors of a Symmetric Tensor . . . . . . . . . . . . .
Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stokes’ Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Comma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boldface vs Indicial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28
29
31
32
35
36
36
37
38
40
42
45
46
47
47
49
49

Chapter 3

Kinematics
1.
2.
3.

4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lagrangian and Eulerian Specifications . . . . . . . . . . . . . . . . . . . . . . . . .
Eulerian and Lagrangian Descriptions: The Particle Derivative . . . .
Streamline, Path Line, and Streak Line . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference Frame and Streamline Pattern . . . . . . . . . . . . . . . . . . . . . . . .
Linear Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shear Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vorticity and Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relative Motion near a Point: Principal Axes . . . . . . . . . . . . . . . . . . . .
Kinematic Considerations of Parallel Shear Flows . . . . . . . . . . . . . . . .
Kinematic Considerations of Vortex Flows . . . . . . . . . . . . . . . . . . . . . .
One-, Two-, and Three-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . .
The Streamfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

51
53
54
56
57
58
59
61
64
65
68
69
72
73
75


ix

Contents

Chapter 4

Conservation Laws
1.
2.
3.
4.
5.
6.

7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time Derivatives of Volume Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Streamfunctions: Revisited and Generalized . . . . . . . . . . . . . . . . . . . . .
Origin of Forces in Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stress at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Momentum Principle for a Fixed Volume . . . . . . . . . . . . . . . . . . . . . . . .
Angular Momentum Principle for a Fixed Volume . . . . . . . . . . . . . . . .
Constitutive Equation for Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . .
Navier–Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mechanical Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
First Law of Thermodynamics: Thermal Energy Equation . . . . . . . . .
Second Law of Thermodynamics: Entropy Production . . . . . . . . . . . .
Bernoulli Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Applications of Bernoulli’s Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77
77
79
81
82
84
86
88
92
94
97
99
104
108
109
110
114
117
121
126
128
128

Chapter 5

Vorticity Dynamics
1.
2.
3.
4.
5.
6.
7.
8.
9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vortex Lines and Vortex Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Role of Viscosity in Rotational and Irrotational Vortices . . . . . . . . . .
Kelvin’s Circulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vorticity Equation in a Nonrotating Frame. . . . . . . . . . . . . . . . . . . . . . .
Velocity Induced by a Vortex Filament: Law of Biot and Savart. . . .
Vorticity Equation in a Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . .
Interaction of Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vortex Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129
130
130
134
138
140

141
146
149
150


x

Contents

Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151
152

Chapter 6

Irrotational Flow
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.

12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.

Relevance of Irrotational Flow Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity Potential: Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow at a Wall Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Irrotational Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Doublet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow past a Half-Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow past a Circular Cylinder without Circulation . . . . . . . . . . . . . . . .
Flow past a Circular Cylinder with Circulation . . . . . . . . . . . . . . . . . . .
Forces on a Two-Dimensional Body . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Source near a Wall: Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . .
Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow around an Elliptic Cylinder with Circulation . . . . . . . . . . . . . . . .
Uniqueness of Irrotational Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Solution of Plane Irrotational Flow . . . . . . . . . . . . . . . . . . .
Axisymmetric Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Streamfunction and Velocity Potential for Axisymmetric Flow . . . . .

Simple Examples of Axisymmetric Flows . . . . . . . . . . . . . . . . . . . . . . .
Flow around a Streamlined Body of Revolution . . . . . . . . . . . . . . . . . .
Flow around an Arbitrary Body of Revolution . . . . . . . . . . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153
155
157
159
161
162
162
164
165
168
171
176
177
179
181
182
187
190
191
193
194
195

196
198
198

Chapter 7

Gravity Waves
1.
2.
3.
4.
5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wave Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Some Features of Surface Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . .

200
200
202
205
209


xi

Contents

6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.

Approximations for Deep and Shallow Water . . . . . . . . . . . . . . . . . . . .
Influence of Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group Velocity and Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group Velocity and Wave Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonlinear Steepening in a Nondispersive Medium . . . . . . . . . . . . . . .
Hydraulic Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite Amplitude Waves of Unchanging Form in
a Dispersive Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stokes’ Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Waves at a Density Interface between Infinitely Deep Fluids . . . . . .
Waves in a Finite Layer Overlying an Infinitely Deep Fluid . . . . . . .
Shallow Layer Overlying an Infinitely Deep Fluid . . . . . . . . . . . . . . . .

Equations of Motion for a Continuously Stratified Fluid . . . . . . . . . .
Internal Waves in a Continuously Stratified Fluid . . . . . . . . . . . . . . . .
Dispersion of Internal Waves in a Stratified Fluid . . . . . . . . . . . . . . . .
Energy Considerations of Internal Waves in a Stratified Fluid . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215
219
222
224
227
231
233
236
238
240
244
246
248
251
254
256
260
261

Chapter 8

Dynamic Similarity
1.

2.
3.
4.
5.
6.
7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nondimensional Parameters Determined from Differential Equations
Dimensional Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Buckingham’s Pi Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nondimensional Parameters and Dynamic Similarity . . . . . . . . . . . . .
Comments on Model Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Significance of Common Nondimensional Parameters . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262
263
267
268
270
272
274
276
276
276

Chapter 9

Laminar Flow
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Analogy between Heat and Vorticity Diffusion . . . . . . . . . . . . . . . . . . .
3. Pressure Change Due to Dynamic Effects . . . . . . . . . . . . . . . . . . . . . . .

277
279
279


xii

Contents

4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

Steady Flow between Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Steady Flow in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Steady Flow between Concentric Cylinders . . . . . . . . . . . . . . . . . . . . .
Impulsively Started Plate: Similarity Solutions . . . . . . . . . . . . . . . . . . .
Diffusion of a Vortex Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decay of a Line Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow Due to an Oscillating Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
High and Low Reynolds Number Flows . . . . . . . . . . . . . . . . . . . . . . . . .
Creeping Flow around a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonuniformity of Stokes’ Solution and Oseen’s Improvement . . . . .
Hele-Shaw Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Final Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

280
283
285
288
295
296
298
301
303
308
312
314
315
317
317

Chapter 10

Boundary Layers and Related Topics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary Layer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Different Measures of Boundary Layer Thickness . . . . . . . . . . . . . . . .
Boundary Layer on a Flat Plate with a Sink at the Leading
Edge: Closed Form Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary Layer on a Flat Plate: Blasius Solution . . . . . . . . . . . . . . . .
von Karman Momentum Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Effect of Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Description of Flow past a Circular Cylinder . . . . . . . . . . . . . . . . . . . .
Description of Flow past a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamics of Sports Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Dimensional Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Secondary Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perturbation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Example of a Regular Perturbation Problem . . . . . . . . . . . . . . . . . .
An Example of a Singular Perturbation Problem . . . . . . . . . . . . . . . . .
Decay of a Laminar Shear Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

318
319
324
327
330
339
342
343
346
353
354
357
365
366
370
373
378

382
384
385


xiii

Contents

Chapter 11

Computational Fluid Dynamics by Howard H. Hu
1.
2.
3.
4.
5.
6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incompressible Viscous Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Four Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

386
388

393
400
416
447
449
450

Chapter 12

Instability
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Method of Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thermal Instability: The B´enard Problem . . . . . . . . . . . . . . . . . . . . . . .
Double-Diffusive Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Centrifugal Instability: Taylor Problem. . . . . . . . . . . . . . . . . . . . . . . . . .

Kelvin–Helmholtz Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instability of Continuously Stratified Parallel Flows . . . . . . . . . . . . . .
Squire’s Theorem and Orr–Sommerfeld Equation . . . . . . . . . . . . . . . .
Inviscid Stability of Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Some Results of Parallel Viscous Flows . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Verification of Boundary Layer Instability . . . . . . . . . .
Comments on Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deterministic Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

453
454
455
467
471
476
484
490
494
498
503
505
506
508
516
518

Chapter 13

Turbulence
1.
2.
3.
4.
5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Correlations and Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Averaged Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

519
521
522
525
529


xiv

Contents

6.
7.
8.
9.
10.

11.
12.
13.
14.
15.

Kinetic Energy Budget of Mean Flow . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kinetic Energy Budget of Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . .
Turbulence Production and Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectrum of Turbulence in Inertial Subrange . . . . . . . . . . . . . . . . . . . .
Wall-Free Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wall-Bounded Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Eddy Viscosity and Mixing Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coherent Structures in a Wall Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Turbulence in a Stratified Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Taylor’s Theory of Turbulent Dispersion . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

535
537
540
543
545
551
559
562
565
569

576
577
578

Chapter 14

Geophysical Fluid Dynamics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vertical Variation of Density in Atmosphere and Ocean . . . . . . . . . . .
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Approximate Equations for a Thin Layer on a Rotating Sphere . . . .

Geostrophic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ekman Layer at a Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ekman Layer on a Rigid Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shallow-Water Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal Modes in a Continuously Stratified Layer . . . . . . . . . . . . . . . .
High- and Low-Frequency Regimes in Shallow-Water Equations . .
Gravity Waves with Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kelvin Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Potential Vorticity Conservation in Shallow-Water Theory . . . . . . . .
Internal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rossby Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Barotropic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Baroclinic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geostrophic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

579
581
583
586
588
593
598
601
603
610
612
615
619

622
632
637
639
647
650
651


xv

Contents

Chapter 15

Aerodynamics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Aircraft and Its Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Airfoil Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Forces on an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kutta Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generation of Circulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conformal Transformation for Generating Airfoil Shape . . . . . . . . . .
Lift of Zhukhovsky Airfoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wing of Finite Span. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lifting Line Theory of Prandtl and Lanchester . . . . . . . . . . . . . . . . . . .
Results for Elliptic Circulation Distribution . . . . . . . . . . . . . . . . . . . . .
Lift and Drag Characteristics of Airfoils. . . . . . . . . . . . . . . . . . . . . . . . .
Propulsive Mechanisms of Fish and Birds . . . . . . . . . . . . . . . . . . . . . . .
Sailing against the Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

653
654
657
657
659
660
662
666
669
670
675

677
679
680
682
684
684

Chapter 16

Compressible Flow
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basic Equations for One-Dimensional Flow . . . . . . . . . . . . . . . . . . . . .
Stagnation and Sonic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Area–Velocity Relations in One-Dimensional Isentropic Flow . . . . .
Normal Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operation of Nozzles at Different Back Pressures . . . . . . . . . . . . . . . .

Effects of Friction and Heating in Constant-Area Ducts . . . . . . . . . . .
Mach Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Oblique Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Expansion and Compression in Supersonic Flow . . . . . . . . . . . . . . . . .
Thin Airfoil Theory in Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

685
689
692
696
701
705
711
717
720
722
726
728
731


xvi

Contents

Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

732

733

Appendix A

Some Properties of Common Fluids
A1.
A2.
A3.
A4.

Useful Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of Pure Water at Atmospheric Pressure . . . . . . . . . . . . . . .
Properties of Dry Air at Atmospheric Pressure . . . . . . . . . . . . . . . . . .
Properties of Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . .

734
735
735
736

Appendix B

Curvilinear Coordinates
B1.
B2.
B3.

Cylindrical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plane Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

737
739
739

Appendix C

Founders of Modern Fluid Dynamics
Ludwig Prandtl (1875–1953) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geoffrey Ingram Taylor (1886–1975) . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplemental Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index

742
743
744
745


Preface
This edition provided me with the opportunity to include (almost) all of the additional
material I had intended for the Second Edition but had to sacrifice because of the
crush of time. It also provided me with an opportunity to rewrite and improve the
presentation of material on jets in Chapter 10. In addition, Professor Howard Hu
greatly expanded his CFD chapter. The expansion of the treatment of surface tension
is due to the urging of Professor E. F. "Charlie" Hasselbrink of the University of
Michigan.
I am grateful to Mr. Karthik Mukundakrishnan for computations of boundary
layer problems, to Mr. Andrew Perrin for numerous suggestions for improvement

and some computations, and to Mr. Din-Chih Hwang for sharing his latest results
on the decay of a laminar shear layer. The expertise of Ms. Maryeileen Banford in
preparing new figures was invaluable and is especially appreciated.
The page proofs of the text were read between my second and third surgeries
for stage 3 bladder cancer. The book is scheduled to be released in the middle of my
regimen of chemotherapy. My family, especially my wife Linda and two daughters
(both of whom are cancer survivors), have been immensely supportive during this
very difficult time. I am also very grateful for the comfort provided by my many
colleagues and friends.
Ira M. Cohen

xvii


Preface to Second Edition
My involvement with Pijush Kundu’s Fluid Mechanics first began in April 1991 with
a letter from him asking me to consider his book for adoption in the first year graduate
course I had been teaching for 25 years. That started a correspondence and, in fact,
I did adopt the book for the following academic year. The correspondence related
to improving the book by enhancing or clarifying various points. I would not have
taken the time to do that if I hadn’t thought this was the best book at the first-year
graduate level. By the end of that year we were already discussing a second edition
and whether I would have a role in it. By early 1992, however, it was clear that I
had a crushing administrative burden at the University of Pennsylvania and could not
undertake any time-consuming projects for the next several years. My wife and I met
Pijush and Shikha for the first time in December 1992. They were a charming, erudite,
sophisticated couple with two brilliant children. We immediately felt a bond of warmth
and friendship with them. Shikha was a teacher like my wife so the four of us had a
great deal in common. A couple of years later we were shocked to hear that Pijush had
died suddenly and unexpectedly. It saddened me greatly because I had been looking

forward to working with Pijush on the second edition after my term as department
chairman ended in mid-1997. For the next year and a half, however, serious family
health problems detoured any plans. Discussions on this edition resumed in July of
1999 and were concluded in the Spring of 2000 when my work really started. This
book remains the principal work product of Pijush K. Kundu, especially the lengthy
chapters on Gravity Waves, Instability, and Geophysical Fluid Dynamics, his areas of
expertise. I have added new material to all of the other chapters, often providing an
alternative point of view. Specifically, vector field derivatives have been generalized,
as have been streamfunctions. Additional material has been added to the chapters on
laminar flows and boundary layers. The treatment of one-dimensional gasdynamics
has been extended. More problems have been added to most chapters. Professor
Howard H. Hu, a recognized expert in computational fluid dynamics, graciously
provided an entirely new chapter, Chapter 11, thereby providing the student with an
entree into this exploding new field. Both finite difference and finite element methods
are introduced and a detailed worked-out example of each is provided.
I have been a student of fluid mechanics since 1954 when I entered college to
study aeronautical engineering. I have been teaching fluid mechanics since 1963 when
I joined the Brown University faculty, and I have been teaching a course corresponding
to this book since moving to the University of Pennsylvania in 1966. I am most grateful
to two of my own teachers, Professor Wallace D. Hayes (1918–2001), who expressed
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fluid mechanics in the clearest way I have ever seen, and Professor Martin D. Kruskal,
whose use of mathematics to solve difficult physical problems was developed to a
high art form and reminds me of a Vivaldi trumpet concerto. His codification of rules

of applied limit processes into the principles of “Asymptotology” remains with me
today as a way to view problems. I am grateful also to countless students who asked
questions, forcing me to rethink many points.
The editors at Academic Press, Gregory Franklin and Marsha Filion (assistant)
have been very supportive of my efforts and have tried to light a fire under me. Since
this edition was completed, I found that there is even more new and original material I
would like to add. But, alas, that will have to wait for the next edition. The new figures
and modifications of old figures were done by Maryeileen Banford with occasional
assistance from the school’s software expert, Paul W. Shaffer. I greatly appreciate
their job well done.
Ira M. Cohen


Preface to First Edition
This book is a basic introduction to the subject of fluid mechanics and is intended for
undergraduate and beginning graduate students of science and engineering. There is
enough material in the book for at least two courses. No previous knowledge of the
subject is assumed, and much of the text is suitable in a first course on the subject. On
the other hand, a selection of the advanced topics could be used in a second course. I
have not tried to indicate which sections should be considered advanced; the choice
often depends on the teacher, the university, and the field of study. Particular effort
has been made to make the presentation clear and accurate and at the same time easy
enough for students. Mathematically rigorous approaches have been avoided in favor
of the physically revealing ones.
A survey of the available texts revealed the need for a book with a balanced
view, dealing with currently relevant topics, and at the same time easy enough for
students. The available texts can perhaps be divided into three broad groups. One
type, written primarily for applied mathematicians, deals mostly with classical topics such as irrotational and laminar flows, in which analytical solutions are possible. A second group of books emphasizes engineering applications, concentrating on
flows in such systems as ducts, open channels, and airfoils. A third type of text is
narrowly focused toward applications to large-scale geophysical systems, omitting

small-scale processes which are equally applicable to geophysical systems as well as
laboratory-scale phenomena. Several of these geophysical fluid dynamics texts are
also written primarily for researchers and are therefore rather difficult for students. I
have tried to adopt a balanced view and to deal in a simple way with the basic ideas
relevant to both engineering and geophysical fluid dynamics.
However, I have taken a rather cautious attitude toward mixing engineering and
geophysical fluid dynamics, generally separating them in different chapters. Although
the basic principles are the same, the large-scale geophysical flows are so dominated
by the effects of the Coriolis force that their characteristics can be quite different
from those of laboratory-scale flows. It is for this reason that most effects of planetary
rotation are discussed in a separate chapter, although the concept of the Coriolis force
is introduced earlier in the book. The effects of density stratification, on the other hand,
are discussed in several chapters, since they can be important in both geophysical and
laboratory-scale flows.
The choice of material is always a personal one. In my effort to select topics,
however, I have been careful not to be guided strongly by my own research interests.
The material selected is what I believe to be of the most interest in a book on general
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Preface to First Edition

fluid mechanics. It includes topics of special interest to geophysicists (for example,
the chapters on Gravity Waves and Geophysical Fluid Dynamics) and to engineers
(for example, the chapters on Aerodynamics and Compressible Flow). There are also
chapters of common interest, such as the first five chapters, and those on Boundary
Layers, Instability, and Turbulence. Some of the material is now available only in
specialized monographs; such material is presented here in simple form, perhaps
sacrificing some formal mathematical rigor.
Throughout the book the convenience of tensor algebra has been exploited freely.

My experience is that many students feel uncomfortable with tensor notation in the
beginning, especially with the permutation symbol εijk . After a while, however, they
like it. In any case, following an introductory chapter, the second chapter of the book
explains the fundamentals of Cartesian Tensors. The next three chapters deal with
standard and introductory material on Kinematics, Conservation Laws, and Vorticity
Dynamics. Most of the material here is suitable for presentation to geophysicists as
well as engineers.
In much of the rest of the book the teacher is expected to select topics that are
suitable for his or her particular audience. Chapter 6 discusses Irrotational Flow; this
material is rather classical but is still useful for two reasons. First, some of the results
are used in later chapters, especially the one on Aerodynamics. Second, most of the
ideas are applicable in the study of other potential fields, such as heat conduction
and electrostatics. Chapter 7 discusses Gravity Waves in homogeneous and stratified
fluids; the emphasis is on linear analysis, although brief discussions of nonlinear
effects such as hydraulic jump, Stokes’s drift, and soliton are given.
After a discussion of Dynamic Similarity in Chapter 8, the study of viscous flow
starts with Chapter 9, which discusses Laminar Flow. The material is standard, but
the concept and analysis of similarity solutions are explained in detail. In Chapter 10
on Boundary Layers, the central idea has been introduced intuitively at first. Only
after a thorough physical discussion has the boundary layer been explained as a singular perturbation problem. I ask the indulgence of my colleagues for including the
peripheral section on the dynamics of sports balls but promise that most students
will listen with interest and ask a lot of questions. Instability of flows is discussed at
some length in Chapter 12. The emphasis is on linear analysis, but some discussion
of “chaos” is given in order to point out how deterministic nonlinear systems can lead
to irregular solutions. Fully developed three-dimensional Turbulence is discussed in
Chapter 13. In addition to standard engineering topics such as wall-bounded shear
flows, the theory of turbulent dispersion of particles is discussed because of its geophysical importance. Some effects of stratification are also discussed here, but the
short section discussing the elementary ideas of two-dimensional geostrophic turbulence is deferred to Chapter 14. I believe that much of the material in Chapters 8–13
will be of general interest, but some selection of topics is necessary here for teaching
specialized groups of students.

The remaining three chapters deal with more specialized applications in geophysics and engineering. Chapter 14 on Geophysical Fluid Dynamics emphasizes
the linear analysis of certain geophysically important wave systems. However, elements of barotropic and baroclinic instabilities and geostrophic turbulence are also
included. Chapter 15 on Aerodynamics emphasizes the application of potential theory to flow around lift-generating profiles; an elementary discussion of finite-wing

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theory is also given. The material is standard, and I do not claim much originality or
innovation, although I think the reader may be especially interested in the discussions
of propulsive mechanisms of fish, birds, and sailboats and the material on the historic
controversy between Prandtl and Lanchester. Chapter 16 on Compressible Flow also
contains standard topics, available in most engineering texts. This chapter is included
with the belief that all fluid dynamicists should have some familiarity with such topics
as shock waves and expansion fans. Besides, very similar phenomena also occur in
other nondispersive systems such as gravity waves in shallow water.
The appendices contain conversion factors, properties of water and air, equations
in curvilinear coordinates, and short biographical sketches of Founders of Modern
Fluid Dynamics. In selecting the names in the list of founders, my aim was to come
up with a very short list of historic figures who made truly fundamental contributions.
It became clear that the choice of Prandtl and G. I. Taylor was the only one that would
avoid all controversy.
Some problems in the basic chapters are worked out in the text, in order to
illustrate the application of the basic principles. In a first course, undergraduate engineering students may need more practice and help than offered in the book; in that
case the teacher may have to select additional problems from other books. Difficult
problems have been deliberately omitted from the end-of-chapter exercises. It is my
experience that the more difficult exercises need a lot of clarification and hints (the

degree of which depends on the students’ background), and they are therefore better
designed by the teacher. In many cases answers or hints are provided for the exercises.

Acknowledgements
I would like to record here my gratitude to those who made the writing of this book
possible. My teachers Professor Shankar Lal and Professor John Lumley fostered my
interest in fluid mechanics and quietly inspired me with their brilliance; Professor
Lumley also reviewed Chapter 13. My colleague Julian McCreary provided support,
encouragement, and careful comments on Chapters 7, 12, and 14. Richard Thomson’s
cheerful voice over the telephone was a constant reassurance that professional science
can make some people happy, not simply competitive; I am also grateful to him for
reviewing Chapters 4 and 15. Joseph Pedlosky gave very valuable comments on
Chapter 14, in addition to warning me against too broad a presentation. John Allen
allowed me to use his lecture notes on perturbation techniques. Yasushi Fukamachi,
Hyong Lee, and Kevin Kohler commented on several chapters and constantly pointed
out things that may not have been clear to the students. Stan Middleman and Elizabeth
Mickaily were especially diligent in checking my solutions to the examples and
end-of-chapter problems. Terry Thompson constantly got me out of trouble with my
personal computer. Kathy Maxson drafted the figures. Chuck Arthur and Bill LaDue,
my editors at Academic Press, created a delightful atmosphere during the course of
writing and production of the book.
Lastly, I am grateful to Amjad Khan, the late Amir Khan, and the late Omkarnath
Thakur for their music, which made working after midnight no chore at all. I recommend listening to them if anybody wants to write a book!
Pijush K. Kundu


Author’s Notes
Both indicial and boldface notations are used to indicate vectors and tensors. The
comma notation to represent spatial derivatives (for example, A,i for ∂A/∂xi ) is used
in only two sections of the book (Sections 5.6 and 13.7), when the algebra became

cumbersome otherwise. Equal to by definition is denoted by ≡; for example, the
ratio of specific heats is introduced as γ ≡ Cp /Cv . Nearly equal to is written as ≃,
proportional to is written as ∝, and of the order is written as ∼.
Plane polar coordinates are denoted by (r, θ), cylindrical polar coordinates are
denoted by either (R, ϕ, x) or (r, θ, x), and spherical polar coordinates are denoted by
(r, θ, ϕ) (see Figure 3.1). The velocity components in the three Cartesian directions
(x, y, z) are indicated by (u, v, w). In geophysical situations the z-axis points upward.
In some cases equations are referred to by a descriptive name rather than a number
(for example, “the x-momentum equation shows that . . . ”). Those equations and/or
results deemed especially important have been indicated by a box.
A list of literature cited and supplemental reading is provided at the end of most
chapters. The list has been deliberately kept short and includes only those sources that
serve one of the following three purposes: (1) it is a reference the student is likely to
find useful, at a level not too different from that of this book; (2) it is a reference that
has influenced the author’s writing or from which a figure is reproduced; and (3) it
is an important work done after 1950. In currently active fields, reference has been
made to more recent review papers where the student can find additional references
to the important work in the field.
Fluid mechanics forces us fully to understand the underlying physics. This is
because the results we obtain often defy our intuition. The following examples support
these contentions:
1. Infinitesmally small causes can have large effects (d’Alembert’s paradox).
2. Symmetric problems may have nonsymmetric solutions (von Karman vortex
street).
3. Friction can make the flow go faster and cool the flow (subsonic adiabatic flow
in a constant area duct).
4. Roughening the surface of a body can decrease its drag (transition from laminar
to turbulent boundary layer separation).
5. Adding heat to a flow may lower its temperature. Removing heat from a flow
may raise its temperature (1-dimensional diabatic flow in a range of subsonic

Mach number).

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Author’s Notes

6. Friction can destabilize a previously stable flow (Orr-Sommerfeld stability
analysis for a boundary layer profile without inflection point).
7. Without friction, birds could not fly and fish could not swim (Kutta condition
requires viscosity).
8. The best and most accurate visualization of streamlines in an inviscid (infinite
Reynolds number) flow is in a Hele-Shaw apparatus for creeping highly viscous
flow (near zero Reynolds number).
Every one of these counterintuitive effects will be treated and discussed in
this text.
This second edition also contains additional material on streamfunctions, boundary conditions, viscous flows, boundary layers, jets, and compressible flows. Most
important, there is an entirely new chapter on computational fluid dynamics that introduces the student to the various techniques for numerically integrating the equations
governing fluid motions. Hopefully the introduction is sufficient that the reader can
follow up with specialized texts for a more comprehensive understanding.
An historical survey of fluid mechanics from the time of Archimedes (ca.
250 B.C.E.) to approximately 1900 is provided in the Eleventh Edition of
The Encyclopædia Britannica (1910) in Vol. XIV (under “Hydromechanics,”
pp. 115–135). I am grateful to Professor Herman Gluck (Professor of Mathematics at the University of Pennsylvania) for sending me this article. Hydrostatics and
classical (constant density) potential flows are reviewed in considerable depth. Great
detail is given in the solution of problems that are now considered obscure and arcane
with credit to authors long forgotten. The theory of slow viscous motion developed by
Stokes and others is not mentioned. The concept of the boundary layer for high-speed

motion of a viscous fluid was apparently too recent for its importance to have been
realized.
IMC