Cohen I.M., Kundu P.K. Fluid Mechanics
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Fluid Mechanics, Fourth Edition
Founders of Modern Fluid Dynamics
Ludwig Prandtl G. I. Taylor
(1875–1953) (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
Fourth Edition
Pijush K. Kundu
Oceanographic Center
Nova Southeastern University
Dania, Florida
Ira M. Cohen
Department of Mechanical Engineering and
Applied Mechanics
University of Pennsylvania
Philadelphia, Pennsylvania
with contributions by P. S. Ayyaswamy and H. H. Hu
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Library of Congress Cataloging-in-Publication Data
Kundu, Pijush K.
Fluid mechanics / Pijush K. Kundu, Ira M. Cohen. – 4th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-373735-9 (alk. paper) /
1. Fluid mechanics. I. Cohen, Ira M. II. Title.
QA901.K86 2008
620.1’06–dc22 2007042765
Reprinted January 2010 – ISBN: 978-0-12-381399-2
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
For information on all Academic Press publications
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Printed in the United States
10 11 12 13 14 10 9 8 7 6 5 4 3 2 1
The fourth 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 Engi-
neering at the Indian Institute of Technology
in Delhi from 1965 to 1968. Pijush came to
the United States in 1968, as a doctoral stu-
dent at Penn State University. With Dr. John
L. Lumley as his advisor, he studied instabili-
ties of viscoelastic fluids, receiving his doctor-
ate in 1972. He began his lifelong interest in
oceanography soon after his graduation, working as Research Associate in Oceano-
graphy 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 scien-
tific 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 Third Edition.................................... xix
Preface to Second Edition .................................. xxi
Preface to First Edition .................................... xxiii
Author’s Notes ............................................ xxvii
Chapter 1
Introduction
1. Fluid Mechanics ................................................. 1
2. Units of Measurement ............................................ 2
3. Solids, Liquids, and Gases ........................................ 3
4. Continuum Hypothesis ........................................... 4
5. Transport Phenomena ............................................ 5
6. Surface Tension .................................................. 8
7. Fluid Statics ..................................................... 9
8. Classical Thermodynamics ........................................ 12
9. Perfect Gas ...................................................... 16
10. Static Equilibrium of a Compressible Medium ...................... 18
Exercises ........................................................ 22
Literature Cited .................................................. 24
Supplemental Reading ............................................ 24
Chapter 2
Cartesian Tensors
1. Scalars and Vectors ............................................... 25
2. Rotation of Axes: Formal Definition of a Vector ..................... 26
3. Multiplication of Matrices ........................................ 29
4. Second-Order Tensor ............................................. 30
5. Contraction and Multiplication .................................... 32
vii
New in This Book
xvi
.........................................
viii Contents
6. Force on a Surface ............................................... 33
7. Kronecker Delta and Alternating Tensor ............................ 36
8. Dot Product ..................................................... 37
9. Cross Product.................................................... 38
10. Operator ∇: Gradient, Divergence, and Curl ........................ 38
11. Symmetric and Antisymmetric Tensors ............................. 40
12. Eigenvalues and Eigenvectors of a Symmetric Tensor ................ 41
13. Gauss’ Theorem ................................................. 44
14. Stokes’ Theorem ................................................. 47
15. Comma Notation ................................................. 49
16. Boldface vs Indicial Notation...................................... 49
Exercises ........................................................ 50
Literature Cited .................................................. 51
Supplemental Reading ............................................ 51
Chapter 3
Kinematics
1. Introduction ..................................................... 53
2. Lagrangian and Eulerian Specifications ............................. 54
3. Eulerian and Lagrangian Descriptions: The Particle Derivative ........ 55
4. Streamline, Path Line, and Streak Line ............................. 57
5. Reference Frame and Streamline Pattern ............................ 59
6. Linear Strain Rate ................................................ 60
7. Shear Strain Rate ................................................ 61
8. Vorticity and Circulation .......................................... 62
9. Relative Motion near a Point: Principal Axes ........................ 64
10. Kinematic Considerations of Parallel Shear Flows ................... 67
11. Kinematic Considerations of Vortex Flows .......................... 68
12. One-, Two-, and Three-Dimensional Flows ......................... 71
13. The Streamfunction .............................................. 73
14. Polar Coordinates ................................................ 75
Exercises ........................................................ 77
Supplemental Reading ............................................ 79
Chapter 4
Conservation Laws
1. Introduction ..................................................... 82
2. Time Derivatives of Volume Integrals .............................. 82
3. Conservation of Mass ............................................ 84
4. Streamfunctions: Revisited and Generalized ........................ 87
5. Origin of Forces in Fluid .......................................... 88
6. Stress at a Point .................................................. 90
Contents ix
7. Conservation of Momentum ....................................... 92
8. Momentum Principle for a Fixed Volume ........................... 93
9. Angular Momentum Principle for a Fixed Volume ................... 98
10. Constitutive Equation for Newtonian Fluid.......................... 100
11. Navier–Stokes Equation .......................................... 104
12. Rotating Frame .................................................. 105
13. Mechanical Energy Equation ...................................... 111
14. First Law of Thermodynamics: Thermal Energy Equation ............ 115
15. Second Law of Thermodynamics: Entropy Production ............... 116
16. Bernoulli Equation ............................................... 118
17. Applications of Bernoulli’s Equation ............................... 122
18. Boussinesq Approximation ........................................ 124
19. Boundary Conditions ............................................. 129
Exercises ........................................................ 134
Literature Cited .................................................. 136
Supplemental Reading ............................................ 137
Chapter 5
Vorticity Dynamics
1. Introduction ..................................................... 139
2. Vortex Lines and Vortex Tubes .................................... 140
3. Role of Viscosity in Rotational and Irrotational Vortices .............. 141
4. Kelvin’s Circulation Theorem ..................................... 144
5. Vorticity Equation in a Nonrotating Frame .......................... 149
6. Velocity Induced by a Vortex Filament: Law of Biot and Savart ....... 151
7. Vorticity Equation in a Rotating Frame ............................. 152
8. Interaction of Vortices ............................................ 157
9. Vortex Sheet ..................................................... 161
Exercises ........................................................ 161
Literature Cited .................................................. 163
Supplemental Reading ............................................ 163
Chapter 6
Irrotational Flow
1. Relevance of Irrotational Flow Theory.............................. 165
2. Velocity Potential: Laplace Equation ............................... 167
3. Application of Complex Variables ................................. 169
4. Flow at a Wall Angle ............................................. 171
5. Sources and Sinks ................................................ 173
6. Irrotational Vortex................................................ 174
7. Doublet ......................................................... 174
8. Flow past a Half-Body ............................................ 175