fluid mechanics with multimedia dvd, fifth edition

FLUID MECHANICS
FIFTH 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 Segre
`
Visual Archieves. Copyright, American Institute of
Physics, 2000.
FLUID
MECHANICS
FIFTH EDITION
PIJUSH K. KUNDU
IRA M. COHEN
DAVID R. DOWLING
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
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Library of Congress Cataloging-in-Publication Data
Kundu, Pijush K.
Fluid mechanics / Pijush K. Kundu, Ira M. Cohen, David R. Dowling. – 5th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-382100-3 (alk. paper)
1. Fluid mechanics. I. Cohen, Ira M. II. Dowling, David R. III. Title.
QA901.K86 2012
620.1’06–dc22
2011014138
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
visit our website at www.elsevierdirect.com

Printed in the United States of America
11 12 13 14 10 9 8 7 6 5 4 3 2 1
Dedication
This revision to this textbook is dedicated to my wife and family who have patiently
helped chip many shar p corners off my personality, and to the many fine instructors and
students with whom I have interacted who have all in some way highlighted the allure of
this subject for me.
D.R.D.
In Memory of Pijush Kundu
Pijush Kanti Kundu was born in Calcutta,
India, on October 31, 1941. He received
a BS degree in Mechanical Engineering in
1963 from Shibpur Engineering College of
Calcutta University, earned an MS degree
in Engineering from Roo rkee 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 student at
Penn State University. With Dr. John L.
Lumley as his advisor, he studied instabil-
ities of viscoelastic fluids, receiving his
doctorate in 1972. He began his lifelong
interest in oceanography soon after his grad-
uation, 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 fiel ds 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 eigen-
function” statistical technique to the oceano-
graphic community.
I arrived at Nova Southeastern University
shortly after Pijush, and he and I worked
closely together thereafter. I was immedi-
ately 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. So me 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 culmina-
tion of his lifelong dream, which he dedi-
cated to the memory of his mother, and to
his wife Shikha, daughter Tonushree, and
son Joydip.
Julian P. McCreary, Jr.,
University of Hawaii

vi
In Memory of Ira Cohen
Ira M. Cohen earned his BS from Poly-
technic University in 1958 and his PhD from
Princeton University in 1963, both in aero-
nautical engineering. He taught at Brown
University for three years prior to joining
the University of Pennsylvania faculty as an
assistant professor in 1966. He served as chair
of the Department of Mechanical Engineering
and Applied Mechanics from 1992 to 1997.
Professor Cohen was a world-renowned
scholar in the areas of contin uum plasmas,
electrostatic probe theories and plasma
diagnostics, dynamics and heat transfer of
lightly ionized gases, low current arc
plasmas, laminar shear layer theory, and
matched asymptotics in fluid mechanics.
Most of his contributions appear in the
Physics of Fluids journal of the American
Institute of Physics. His seminal paper,
“Asymptotic theory of spherical electro-
static probes in a slightly ionized, collision
dominated gas” (1963; Physics of Fluids, 6,
1492e1499), is to date the most highly cited
paper in the theory of electrostatic probes
and plasma diagnostics.
During his doctoral work and for a few
years beyond that, Ira collaborated with a
world-renowned mathematician/physicist,

the late Dr. Martin Kruskal (recipient of
National Medal of Science, 1993) on the devel-
opment of a monograph called “Asymptotol-
ogy.” Professor Kruskal also collaborated
with Professor Cohen on plasma physics.
This was the basis for Ira’s strong foundation
in fluid dynamics that has been transmitted
into the prior editions of this textbook.
In his forty-one years of service to the
University of Pennsylvania before his death
in December 2007, Professor Cohen distin-
guished himself with his integrity, his fierce
defense of high scholarly standards, and
his passionate commitment to teaching. He
will always be remembered for his candor
and his sense of humor.
Professor Cohen’s dedication to academ-
ics was unrivalled. In addition, his passion
for physical fitness was legendary. Neither
rain nor sleet nor snow would det er him
from his daily bicycle commute, which began
at 5:00
AM, from his home in Narberth to the
University of Pennsylvania. His colleagues
grew accustomed to seeing him drag his
forty-year-old bicycle, with its original
vii
three-speed gearshift, up to his office. His
other great passion was the game of squash,
which he played with extraordinary skill

five days a week at the Ringe Squash Courts
at Penn, where he was a fierce but fair
competitor. During the fina l year of his life,
Professor Cohen remained true to his bicy-
cling and squash-playing schedule, refusing
to allow his illness get in the way of the things
he loved.
Professor Cohen was a member of Beth
Am Israel Synagogue, and would on occa-
sion lead Friday night servi ces there. He
and his wife, Linda, were first married
near Princeton, New Jersey, on February 13,
1960, when they eloped. They were married
a second time four months later in a formal
ceremony. He is survived by his wife, his
two children, Su san Cohen Bolstad and
Nancy Cohen Cavanaugh, and three grand-
children, Melissa, Daniel, and Andrew.
Senior Faculty
Department of Mechanical Engineering
and Applied Mechanics
University of Pennsylvania
IN MEMORY OF IRA COHENviii
About the Third Author
David R. Dowling was born in Mesa,
Arizona, in 1960 but grew up in southern
California where early practical exposure to
fluid mechanicsdswimming, surfing, sailing,
flying model aircraft, and trying to throw
a curve ballddominated his free time. He

attended the California Institute of Tech-
nology continuously for a decade starting in
1978, earning a BS degree in Applied Physics
in 1982, and MS and PhD degrees in Aeronau-
tics in 1983 and 1988, respectively. After grad-
uate school, he worked at Boeing Aerospace
and Electronics and then took a post-doctoral
scientist position at the Applied Physics
Laboratory of the University of Washington.
In 1992, he started a faculty career in the
Department of Mechanical Engineering at
the University of Michigan where he has since
taught and conducted research in fluid
mechanics and acoustics. He has authored
and co-authored more than 60 archival jour-
nal articles and more than 100 conference
presentations. His published research in fluid
mechanics includes papers on turbulent mix-
ing, forced-convection heat transfer, cirrus
clouds, molten plastic flow, interactions of
surfactants with water waves, and hydrofoil
performance and turbulent boundary layer
characteristics at high Reynolds numbers.
From January 2007 through June 2009, he
served as an Associate Chair and as the
Undergraduate Program Director for the
Department of Mechanical Engineering at
the University of Michigan. He is a fellow
of the American Society of Mechanical
Engineers and of the Acoustical Society of

America. He received the Student Council
Mentoring Award of the Acoustical Soci-
ety of America in 2007, the University
of Michigan College of Engineering John
R. Ullrich Education Excellence Award in
2009, and the Outstanding Professor Award
from the University of Michigan Chapter of
the American Society for Engineering Educa-
tion in 2009. Prof. Dowling is an avid
swimmer, is married, and has seven children.
ix
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Contents
About the DVD xvii
Preface xix
Companion Website xx
Acknowledgments xxi
Nomenclature xxii
1. Introduction 1
1.1. Fluid Mechanics 2
1.2. Units of Measurement 3
1.3. Solids, Liquids, and Gases 3
1.4. Continuum Hypothesis 5
1.5. Molecular Transport Phenomena 5
1.6. Surface Tension 8
1.7. Fluid Statics 9
1.8. Classical Thermodynamics 12
First Law of Thermodynamics 13
Equations of State 14
Specific Heats 14

Second Law of Thermodynamics 15
Property Relations 16
Speed of Sound 16
Thermal Expansion Coefficient 16
1.9. Perfect Gas 16
1.10. Stability of Stratified Fluid Media 18
Potential Temperature and Density 19
Scale Height of the Atmosphere 21
1.11. Dimensional Analysis 21
Step 1. Select Variables and Parameters 22
Step 2. Create the Dimensional Matrix 23
Step 3. Determine the Rank of the
Dimensional Matrix 23
Step 4. Determine the Number of
Dimensionless Groups 24
Step 5. Construct the Dimensionless
Groups 24
Step 6. State the Dimensionless
Relationship 26
Step 7. Use Physical Reasoning or Additional
Knowledge to Simplify the Dimensionless
Relationship 26
Exercises 30
Literature Cited 36
Supplemental Reading 37
2. Cartesian Tensors 39
2.1. Scalars, Vectors, Tensors, Notation 39
2.2. Rotation of Axes: Formal Definition
of a Vector 42
2.3. Multiplication of Matrices 44

2.4. Second-Order Tensors 45
2.5. Contraction and Multiplication 47
2.6. Force on a Surface 48
2.7. Kronecker Delta and Alternating Tensor 50
2.8. Vector, Dot, and Cross Products 51
2.9. Gradient, Divergence, and Curl 52
2.10. Symmetric and Antisymmetric Tensors 55
2.11. Eigenvalues and Eigenvectors of
a Symmetric Tensor 56
2.12. Gauss’ Theorem 58
2.13. Stokes’ Theorem 60
2.14. Comma Notation 62
Exercises 62
Literature Cited 64
Supplemental Reading 64
3. Kinematics 65
3.1. Introduction and Coordinate Systems 65
3.2. Particle and Field Descriptions
of Fluid Motion 67
3.3. Flow Lines, Fluid Acceleration,
and Galilean Transformation 71
3.4. Strain and Rotation Rates 76
Summary 81
xi
3.5. Kinematics of Simple Plane Flows 82
3.6. Reynolds Transport Theorem 85
Exercises 89
Literature Cited 93
Supplemental Reading 93
4. Conservation Laws 95

4.1. Introduction 96
4.2. Conservation of Mass 96
4.3. Stream Functions 99
4.4. Conservation of Momentum 101
4.5. Constitutive Equation for a Newtonian
Fluid 111
4.6. Navier-Stokes Momentum Equation 114
4.7. Noninertial Frame of Reference 116
4.8. Conservation of Energy 121
4.9. Special Forms of the Equations 125
Angular Momentum Principle for a
Stationary Control Volume 125
Bernoulli Equations 128
Neglect of Gravity in Constant Density
Flows 134
The Boussinesq Approximation 135
Summary 137
4.10. Boundary Conditions 137
Moving and Deforming Boundaries 139
Surface Tension Revisited 139
4.11. Dimensionless Forms of the Equations and
Dynamic Similarity 143
Exercises 151
Literature Cited 168
Supplemental Reading 168
5. Vorticity Dynamics 171
5.1. Introduction 171
5.2. Kelvin’s Circulation Theorem 176
5.3. Helmholtz’s Vortex Theorems 179
5.4. Vorticity Equation in a Nonrotating

Frame 180
5.5. Velocity Induced by a Vortex Filament: Law
of Biot and Savart 181
5.6. Vorticity Equation in a Rotating Frame 183
5.7. Interaction of Vortices 187
5.8. Vortex Sheet 191
Exercises 192
Literature Cited 195
Supplemental Reading 196
6. Ideal Flow 197
6.1. Relevance of Irrotational Constant-Density
Flow Theory 198
6.2. Two-Dimensional Stream Function and
Velocity Potential 200
6.3. Construction of Elementary Flows in Two
Dimensions 203
6.4. Complex Potential 216
6.5. Forces on a Two-Dimensional Body 219
Blasius Theorem 219
Kutta-Zhukhovsky Lift Theorem 221
6.6. Conformal Mapping 222
6.7. Numerical Solution Techniques in Two
Dimensions 225
6.8. Axisymmetric Ideal Flow 231
6.9. Three-Dimensional Potential Flow and
Apparent Mass 236
6.10. Concluding Remarks 240
Exercises 241
Literature Cited 251
Supplemental Reading 251

7. Gravity Waves 253
7.1. Introduction 254
7.2. Linear Liquid-Surface Gravity Waves 256
Approximations for Deep and Shallow
Water 265
7.3. Influence of Surface Tension 269
7.4. Standing Waves 271
7.5. Group Velocity, Energy Flux, and
Dispersion 273
7.6. Nonlinear Waves in Shallow and Deep
Water 279
7.7. Waves on a Density Interface 286
CONTENTSxii
7.8. Internal Waves in a Continuously Stratified
Fluid 293
Internal Waves in a Stratified Fluid 296
Dispersion of Internal Waves in a Stratified
Fluid 299
Energy Considerations for Internal Waves in
a Stratified Fluid 302
Exercises 304
Literature Cited 307
8. Laminar Flow 309
8.1. Introduction 309
8.2. Exact Solutions for Steady Incompressible
Viscous Flow 312
Steady Flow between Parallel Plates 312
Steady Flow in a Round Tube 315
Steady Flow between Concentric Rotating
Cylinders 316

8.3. Elementary Lubrication Theory 318
8.4. Similarity Solutions for Unsteady
Incompressible Viscous Flow 326
8.5. Flow Due to an Oscillating Plate 337
8.6. Low Reynolds Number Viscous Flow Past
a Sphere 338
8.7. Final Remarks 347
Exercises 347
Literature Cited 359
Supplemental Reading 359
9. Boundary Layers and Related
Topics 361
9.1. Introduction 362
9.2. Boundary-Layer Thickness Definitions 367
9.3. Boundary Layer on a Flat Plate:
Blasius Solution 369
9.4. Falkner-Skan Similarity Solutions of
the Laminar Boundary-Layer Equations 373
9.5. Von Karman Momentum Integral
Equation 375
9.6. Thwaites’ Method 377
9.7. Transition, Pressure Gradients,
and Boundary-Layer Separation 382
9.8. Flow Past a Circular Cylinder 388
Low Reynolds Numbers 389
Moderate Reynolds Numbers 389
High Reynolds Numbers 392
9.9. Flow Past a Sphere and the Dynamics
of Sports Balls 395
Cricket Ball Dynamics 396

Tennis Ball Dynamics 398
Baseball Dynamics 399
9.10. Two-Dimensional Jets 399
9.11. Secondary Flows 407
Exercises 408
Literature Cited 418
Supplemental Reading 419
10. Computational Fluid Dynamics 421
HOWARD H. HU
10.1. Introduction 421
10.2. Finite-Difference Method 423
Approximation to Derivatives 423
Discretization and Its Accuracy 425
Convergence, Consistency, and
Stability 426
10.3. Finite-Element Method 429
Weak or Variational Form of Partial
Differential Equations 429
Galerkin’s Approximation and Finite-
Element Interpolations 430
Matrix Equations, Comparison with
Finite-Difference Method 431
Element Point of View of the Finite-
Element Method 434
10.4. Incompressible Viscous Fluid Flow 436
Convection-Dominated Problems 437
Incompressibility Condition 439
Explicit MacCormack Scheme 440
MAC Scheme 442
Q-Scheme 446

Mixed Finite-Element Formulation 447
10.5. Three Examples 449
Explicit MacCormack Scheme for
Driven-Cavity Flow Problem 449
Explicit MacCormack Scheme for
Flow Over a Square Block 453
CONTENTS xiii
Finite-Element Formulation for
Flow Over a Cylinder Confined in
a Channel 459
10.6. Concluding Remarks 470
Exercises 470
Literature Cited 471
Supplemental Reading 472
11. Instability 473
11.1. Introduction 474
11.2. Method of Normal Modes 475
11.3. Kelvin-Helmholtz Instability 477
11.4. Thermal Instability: The Be
´
nard
Problem 484
11.5. Double-Diffusive Instability 492
11.6. Centrifugal Instability: Taylor Problem 496
11.7. Instability of Continuously Stratified Parallel
Flows 502
11.8. Squire’s Theorem and the Orr-Sommerfeld
Equation 508
11.9. Inviscid Stability of Parallel Flows 511
11.10. Results for Parallel and Nearly Parallel

Viscous Flows 515
Two-Stream Shear Layer 515
Plane Poiseuille Flow 516
Plane Couette Flow 517
Pipe Flow 517
Boundary Layers with Pressure
Gradients 517
11.11. Experimental Verification of Boundary-Layer
Instability 520
11.12. Comments on Nonlinear Effects 522
11.13. Transition 523
11.14. Deterministic Chaos 524
Closure 531
Exercises 532
Literature Cited 539
12. Turbulence 541
12.1. Introduction 542
12.2. Historical Notes 544
12.3. Nomenclature and Statistics for Turbulent
Flow 545
12.4. Correlations and Spectra 549
12.5. Averaged Equations of Motion 554
12.6. Homogeneous Isotropic Turbulence 560
12.7. Turbulent Energy Cascade and
Spectrum 564
12.8. Free Turbulent Shear Flows 571
12.9. Wall-Bounded Turbulent Shear Flows 581
Inner Layer: Law of the Wall 584
Outer Layer: Velocity Defect Law 585
Overlap Layer: Logarithmic Law 585

Rough Surfaces 590
12.10. Turbulence Modeling 591
A Mixing Length Model 593
One-Equation Models 595
Two-Equation Models 595
12.11. Turbulence in a Stratified Medium 596
The Richardson Numbers 597
Monin-Obukhov Length 598
Spectrum of Temperature Fluctuations 600
12.12. Taylor’s Theory of Turbulent Dispersion 601
Rate of Dispersion of a Single Particle 602
Random Walk 605
Behavior of a Smoke Plume in the Wind 606
Turbulent Diffusivity 607
12.13. Concluding Remarks 607
Exercises 608
Literature Cited 618
Supplemental Reading 620
13. Geophysical Fluid Dynamics 621
13.1. Introduction 622
13.2. Vertical Variation of Density in the
Atmosphere and Ocean 623
13.3. Equations of Motion 625
13.4. Approximate Equations for a Thin Layer on
a Rotating Sphere 628
f-Plane Model 630
b-Plane Model 630
13.5. Geostrophic Flow 630
Thermal Wind 632
Taylor-Proudman Theorem 632

CONTENTSxiv
13.6. Ekman Layer at a Free Surface 633
Explanation in Terms of Vortex Tilting 637
13.7. Ekman Layer on a Rigid Surface 639
13.8. Shallow-Water Equations 642
13.9. Normal Modes in a Continuously Stratified
Layer 644
Boundary Conditions on j
n
646
Vertical Mode Solution for Uniform N 646
Summary 649
13.10. High- and Low-Frequency Regimes
in Shallow-Water Equations 649
13.11. Gravity Waves with Rotation 651
Particle Orbit 652
Inertial Motion 653
13.12. Kelvin Wave 654
13.13. Potential Vorticity Conservation in
Shallow-Water Theory 658
13.14. Internal Waves 662
WKB Solution 664
Particle Orbit 666
Discussion of the Dispersion Relation 668
Lee Wave 670
13.15. Rossby Wave 671
Quasi-Geostrophic Vorticity Equation 671
Dispersion Relation 673
13.16. Barotropic Instability 676
13.17. Baroclinic Instability 678

Perturbation Vorticity Equation 679
Wave Solution 681
Instability Criterion 682
Energetics 684
13.18. Geostrophic Turbulence 685
Exercises 688
Literature Cited 690
Supplemental Reading 690
14. Aerodynamics 691
14.1. Introduction 692
14.2. Aircraft Terminology 692
Control Surfaces 693
14.3. Characteristics of Airfoil Sections 696
Historical Notes 701
14.4. Conformal Transformation for
Generating Airfoil Shapes 702
14.5. Lift of a Zhukhovsky Airfoil 706
14.6. Elementary Lifting Line Theory for
Wings of Finite Span 708
Lanchester Versus Prandtl 716
14.7. Lift and Drag Characteristics of
Airfoils 717
14.8. Propulsive Mechanisms of Fish
and Birds 719
14.9. Sailing against the Wind 721
Exercises 722
Literature Cited 728
Supplemental Reading 728
15. Compressible Flow 729
15.1. Introduction 730

Perfect Gas Thermodynamic Relations 731
15.2. Acoustics 732
15.3. Basic Equations for One-Dimensional
Flow 736
15.4. Reference Properties in Compressible
Flow 738
15.5. Area-Velocity Relationship in
One-Dimensional Isentropic Flow 740
15.6. Normal Shock Waves 748
Stationary Normal Shock Wave in a
Moving Medium 748
Moving Normal Shock Wave in a
Stationary Medium 752
Normal Shock Structure 753
15.7. Operation of Nozzles at Different
Back Pressures 755
Convergent Nozzle 755
ConvergenteDivergent Nozzle 757
15.8. Effects of Friction and Heating in
Constant-Area Ducts 761
Effect of Friction 763
Effect of Heat Transfer 764
15.9. Pressure Waves in Planar Compressible
Flow 765
15.10. Thin Airfoil Theory in Supersonic Flow 773
Exercises 775
Literature Cited 778
Supplemental Reading 778
CONTENTS xv
16. Introduction to Biofluid

Mechanics 779
PORTONOVO S. AYYASWAMY
16.1. Introduction 779
16.2. The Circulatory System in the Human
Body 780
The Heart as a Pump 785
Nature of Blood 788
Nature of Blood Vessels 793
16.3. Modeling of Flow in Blood Vessels 796
Steady Blood Flow Theory 797
Pulsatile Blood Flow Theory 805
Blood Vessel Bifurcation: An Application of
Poiseuille’s Formula and Murray’s Law 820
Flow in a Rigid-Walled Curved Tube 825
Flow in Collapsible Tubes 831
Laminar Flow of a Casson Fluid in a Rigid-
Walled Tube 839
Pulmonary Circulation 841
The Pressure Pulse Curve in the Right
Ventricle 842
Effect of Pulmonary Arterial Pressure on
Pulmonary Resistance 843
16.4. Introduction to the Fluid Mechanics
of Plants 844
Exercises 849
Acknowledgment 850
Literature Cited 851
Supplemental Reading 852
Appendix A 853
Appendix B 857

Appendix C 869
Appendix D 873
Index 875
CONTENTSxvi
About the DVD
We are pleased to include a free copy of
the DVD Multimedia Fluid Mechanics, 2/e,
with this copy of Fluid Mechanics, Fifth
Edition. You will find it in a plastic sleeve
on the inside back cover of the book. If you
are purchasing a used copy, be aware that
the DVD might have been removed by
a previous owner.
Inspired by the receptionof thefirst edition,
the objectives in Multimedia Fluid Mechanics,
2/e, remain to exploit the moving image and
interactivity of multimedia to improve the
teaching and learning of fluid mechanics in
all disciplines by illustrating fundamental
phenomena and conveying fascinating fluid
flows for generations to come.
The completely new edition on the DVD
includes the following:
• Twice the coverage with new modules on
turbulence, control volumes, interfacial
phenomena, and similarity and scaling
• Four times the number of fluid videos,
now more than 800
• Now more than 20 virtual labs and
simulations

• Dozens of new interactive demonstrations
and animations
Additional new features:
• Improved navigation via sidebars that
provide rapid overviews of modules and
guided browsing
• Media libraries for each chapter that
give a snapshot of videos, each with
descriptive labels
• Ability to create movie playlists, whic h
are invaluable in teaching
• Higher-resolution graphics, with full or
part screen viewing options
• Operates on either a PC or a Mac OSX
xvii
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Preface
In the fall of 2009, Elsevier approached
me about possibly taking over as the lead
author of this textbook. After some consider-
ation and receipt of encouragement from
faculty colleagues here at the University of
Michigan and beyond, I agreed. The ensu ing
revision effort then tenaciously pulled all the
slack out of my life for the next 18 months.
Unfortunately, I did not have the honor or
pleasure of meeting or knowing either prior
author, and have therefore missed the
opportunity to receive their advice and guid-
ance. Thus, the revisions made for this 5th

Edition of Fluid Mechanics have been driven
primarily by my experience teaching and
interacting with undergraduate and grad-
uate students during the last two decades.
Overall, the structure, topics, and tech-
nical level of the 4th Edition have been
largely retained, so instructors who have
made prior use of this text should recognize
much in the 5th Edition. This textbook should
still be suitable for advanced-undergraduate
or beginning-graduate courses in fluid
mechanics. However, I have tried to make
the subject of fluid mechanics more acces-
sible to students who may have only studied
the subject during one prior semester, or
who may need fluid mechanics knowledge
to pursue research in a related field.
Given the long history of this important
subject, this textbook (at best) reflects one
evolving instructional approach. In my
experience as a student, teacher, and faculty
member, a textbook is most effective when
used as a supporting pedagogical tool for
an effective lecturer. Thus my primary
revision objective has been to improve the
text’s overall utility to students and instruc-
tors by adding introductory material and
references to the first few chapters, by
increasing the prominence of engineering
applications of fluid mechanics, and by

providing a variety of new exercises (more
than 200) and figures (more than 100). For
the chapters receiving the most attention
(1e9, 11e12, and 14) this has meant approx-
imately doubling, tripling, or quadrupling
the number of exercises. Some of the new
exercises have been built from derivations
that previously had appeared in the body
of the text, and some involve sim ple kitchen
or bathroom experiments. My hope for
a future edition is that there will be time to
further expand the exercise offerings, espe-
cially in Chapters 10, 13, 15, and 16.
In preparing this 5th Edition, some reor-
ganization, addition, and deletion of mate-
rial has also taken place. Dimensional
analysis has been moved to Chapter 1 .
The stream function’s introduction and
the dynamic-similarity topic have been
moved to Chapter 4. Reynolds transport
theorem now occupies the final section of
Chapter 3. The discussion of the wave equa-
tion has been placed in the acoustics sec-
tion of Chapter 15. Major topical additions
are: apparent mass (Chapter 6), elemen-
tary lubrication theory (Chapter 8), and
Thwaites method (Chapter 9). The sections
covering the laminar shear layer, and
boundary-layer theory from a purely m ath-
ematical perspective, and coherent struc-

tures in wall-bounded turbulent flow have
xix
been removed. The specialty chapters (10, 13,
and 16) have been left largely untouched
except for a few language changes and
appropriate renumbering of equations. In
addition, some sections have been combined
to save space, but this has been offset by an
expansion of nearly every figure caption and
the introduction of a nomenclature section
with more than 200 entries.
Only a few notation chang es have been
made. Index and vector notation predomi-
nate throughout the text. The comma nota-
tion for derivatives now only appears in
Section 5.6. The notation for unit vectors
has been changed from bold i to bold e to
conform to other texts in physics and engi-
neering. In addition, a serious effort was
made to denote two- and three-dimensional
coordinate systems in a consistent manner
from chapter to chapter. However, the
completion of this task, which involves
retyping literally hundreds of equations,
was not possible in the time available.
Thus, cylindrical coordinates (R, 4, z) pre-
dominate, but (r, q, x) still appear in Table
12.1, Chapter 16, and a few other places.
And, as a final note, the origins of many
of the new exercises are referenced to

individuals and other sources via footnotes.
However, I am sure that such referencing is
incomplete because of my imperfect mem-
ory and record keeping. Therefore, I stand
ready to correctly attribute the origins of
any problem contained herein. Furthermore,
I welcome the opportunity to correct any
errors you find, to hear your opinion of
how this book might be improved, and to
include exercises you might suggest; just
contact me at
David R. Dowling
Ann Arbor, Michigan
April 2011
COMPANION WEBSI TE
An updated errata sheet is available on
the book’s companion website. To access
the errata, visit www.elsevierdirect.com/
9780123821003 and click on the companion
site link. Instructors teaching with this book
may access the solutions manual and image
bank by visiting www.textbooks.elsevier
.com and following the online instructions
to log on and register.
PREFACExx
Acknowledgments
The current version of this textbook
has benefited from the commen tary and
suggestion s provided by the revie wers of
the initial revision proposal and the re-

viewers of draft versions of several of the
chapters. Chief among these reviewers is
Profess or John Cimbala of the Pennsylvania
State University. I would also like to recog-
nize and thank my technical mentors,
Profess or Hans W. Liepmann (undergrad-
uate a dvisor), Professor Paul E. Dimotakis
(graduate advisor), and Professor Darrell
R. Jackson (post-doctoral advisor); and my
friends and colleagues who have contrib-
uted to the development of this text by
discussing ideas and sharing their exper-
tise, humor, and devotion to science and
engineering.
xxi
Nomenclature
NOTATION
f ¼ principle-axis version of f, background or
quiescent-fluid val ue of f, or average or
ensemble average of f
b
f ¼ complex amplitude of f
~
f ¼ full field value of f
f
0
¼ derivative of f with respect to its argu-
ment, or perturbation of f from its
reference state
f

Ã
¼ complex conjugate of f , dimensionless
version of f, or the value of f at the sonic
condition
f
+
¼ the dimensionless, law-of-the-wall
value of f
f
cr
¼ critical value of f
f
CL
¼ centerline value of f
f
0
¼ reference, surface, or stagnation value
of f
f
N
¼ reference value of f or value of f far
away from the point of interest
Df ¼ change in f
SYMBOLS
)
a ¼ contact angle, thermal expansion coef-
ficient (1.20), angle of rotation, angle of
attack, Womersley number (16.12),
angle in a toroidal coordinate system,
area ratio

a ¼ triangular area, cylinder radius,
sphere radius, amplitude
a
0
¼ initial tube radius
a ¼ generic vector, Lagrangian acceleration
(3.1)
A ¼ generic second-order (or higher) tensor
A, A ¼ a constant, an amplitude, area,
surface, surface of a material
volume, planform area of a wing
A* ¼ control surface, sonic throat area
A
o
¼ Avogadro’s number
A
0
¼ reference area
A
ij
¼ representative second-order tensor
b ¼ angle of rotation, coefficient of density
change due to salinity or other constit-
uent, variation of the Coriol is frequency
with latitude, camber parameter
b ¼ generic vector, co ntrol surface velocity
(3.35)
B, B ¼ a constant, Bernoulli function (4.70),
log-law intercept parameter (12.88)
B, B

ij
¼ generic second-order (or higher)
tensor
Bo ¼ Bond number (4.118)
c ¼ speed of sound (1.19, 15.6), phase speed
(7.4), chord length (14.2), pressure pulse
wave speed, concentration of solutes
c
j
¼ pressure pulse wave speed in tube j
c ¼ phase velocity vector (7.8)
c
g
, c
g
¼ group velocity magnitude (7.68)
and vector (7.144)
c ¼ scalar stream function

C ¼ degrees centigrade
C ¼ a generic constant, hypotenuse length,
closed contour
Ca ¼ Capillary number (4.119)
C
f
¼ skin friction coefficient (9.32)
C
p
¼ coefficient of pressure (4.106, 6.32)
)

Relevant equation numbers appear in
parentheses
xxii
C
p
¼ specific heat capacity at constant pres-
sure (1.14)
C
D
¼ coefficient of drag (4.107 , 9.33)
C
L
¼ coefficient of lift (4.108)
C
v
¼ specific heat capacity at constant
volume (1.15)
C
ij
¼ matrix of direction cosines bet ween
original and rotated coordinate system
axes (2.5)
d ¼ diameter, distance, fluid layer
depth
d ¼ dipole strength vector (6.29), displace-
ment vector
d ¼ Dirac delta function (B.4.1), similarity-
variable length scale (8.32), boundary-
layer thickness, generic len gth scale,
small increment, flow deflection angle

(15.53), tube radius divided by tube
radius of curvature
d ¼ average boundary-layer thickness
d* ¼ boundary-layer displacement thickness
(9.16)
d
ij
¼ Kronecker delta function (2.16)
d
99
¼ 99% layer thickness
D ¼ distance, drag force, diffusion coeffi-
cient, Dean number (16.179)
D
i
¼ lift-induced drag (14.15 )
D/Dt ¼ material derivative (3.4) or (3.5)
D
T
¼ turbulent diffusivity of particles
(12.127)
D
¼ generalized field der ivative (2.31)
3 ¼ roughness height, kinetic energy dissi-
pation rate (4.58), a small distance, fine-
ness ratio h/L (8.14), downwash angle
(14.14)
3 ¼ average dissipation rate of the turbulent
kinetic energy (12.47)
3

T
¼ average dissipation rate of the variance
of temperature fluctuations (12.112)
3
ijk
¼ alternating tensor (2.18)
e ¼ internal energy per unit mass (1.10)
e
i
¼ unit vector in the i-direction (2.1)
e ¼ average kinetic energy of turbulent
fluctuations (12.47, 12.49)
Ec ¼ Eckert number (4.115)
E
k
¼ kinetic energy per unit horizontal area
(7.39)
E
p
¼ potential energy per unit horizontal
area (7.41)
E ¼ average energy per unit horizontal area
(7.43), Ekman number (13.18), Young’s
modulus
E ¼ kinetic energy of the average flow
(12.46)
b
E
1
¼ total energy dissipation in a blood

vessel
f ¼ generic function, Helmholtz free energy
per unit mass, longitudinal correlation
coefficient (12.38), Coriolis frequency
(13.8), dimensionless friction parameter
(15.45)
f ¼ velocity potential (6.10), an angle
f ¼ surface force vector per unit area
(2.15, 4.13)
F ¼ force magnitude, generic flow field
property, average energy flux per unit
length of wave crest (7.44), generic or
profile function
F ¼ force vector, average wave energy
flux vector
F ¼ body force potential (4.18), undeter-
mined spectrum function (12.53)
F
D
¼ drag force
F
L
¼ lift force
Fr ¼ Froude number (4.104)
g ¼ ratio of specific heats (1.24), velocity
gradient, vortex sheet strength, generic
dependent-field variable
_
g ¼ shear rate
g ¼ body force per unit mass (4.13)

g ¼ acceleration of gravity, undetermined
function, transverse correlation coeffi-
cient (12.38)
g
0
¼ reduced gravity (7.188)
G ¼ vertical temperature gradient or lapse
rate, circulation (3.18)
G
a
¼ adiabatic vertical temperature gradient
(1.30)
G
a
¼ circulation due to the absolute vorticity
(5.33)
NOMENCLATURE xxiii
G ¼ gravitational constan t, pressure-
gradient pulse amplitude, profile
function
G
n
¼ Fourier series coefficient
G ¼ center of mass, center of vorticity
h ¼ enthalpy per unit mass (1.13), height,
gap height, viscous layer thickness, grid
size, tube wall thickness
h ¼ free surface shape, waveform, similarity
variable (8.25, 8.32), Kolmogorov
microscale (12.50), radial tube-wall

displacement
h
T
¼ Batchelor microscale (12.114)
H ¼ atmospheric scale height, water depth,
shape factor (9.46), profile function,
Hematocrit
i ¼ an index, imaginary root
I ¼ incident light intensity, bending moment
of inertia
j ¼ an index
J, J
s
¼ jet momentum flux per unit span
(9.61)
J
i
¼ Bessel function of order i
J
m
¼ diffusive mass flux vector (1.1)
4 ¼ a function, azimuthal angle in cylin-
drical and spherical coordinates
k ¼ thermal conductivity (1.2), an index,
wave number (7.2), wave number
component
k ¼ thermal diffusivity, von Karman
constant (12.88), Dean number (16.171)
k
s

¼ diffusivity of salt
k
T
¼ turbulent thermal diffusivity (12.95)
k
m
¼ mass diffusivity of a passive scalar in
Fick’s law (1.1)
k
mT
¼ turbulent mass diffusivity (12.96)
k
B
¼ Boltzmann’s constant (1.21)
Kn ¼ Knudsen number
K ¼ a generic constant, magnitude of the
wave number vector (7.6), lift curve
slope, Dean Number (16.178)
K
p
¼ constant proportional to tube wall
bending stiffness
K ¼ compliance of a blood vessel, degrees
Kelvin (16.48)
K ¼ wave number vector, stiffness matrix
l ¼ molecular mean free path, spanwise
dimension, generic length scale, wave
number component (7.5, 7.6), shear
correlation in Thwaites method (9.45),
length scale in turbulent flow

l
T
¼ mixing length (12.98)
L, L ¼ generic length dimension, generic
length scale, lift force
L
M
¼ Monin-Obukhov length scale (12.110)
l ¼ wavelength (7.1, 7.7), laminar boundary-
layer correlation parameter (9.44), flow
resistance ratio
l
m
¼ wavelength of the minimum phase
speed
l
t
¼ temporal Taylor microscale (12.19)
l
f
, l
g
¼ longitudinal and lateral spatial
Taylor microscale (12.39)
L ¼ lubrication-flow bearing number (8.16),
Rossby radius of deformation, wing
aspect ratio
L
f
, L

g
¼ longitudinal and lateral integral
spatial scales (12.39)
L
t
¼ integral time scale (12.18)
m ¼ dynamic or shear viscosity (1.3), Mach
angle (15.49)
m
y
¼ bulk viscosity (4.37)
m ¼ molecular mass (1.22), generic mass,
an index, two-dimensional source
strength, moment order (12.1), wave
number component (7.5, 7.6)
M, M ¼ generic mass dimensi on, mass,
Mach number (4.111), apparent or
added mass (6.108)
M
w
¼ molecular weight
n ¼ number of molecules (1.21), an index,
generic integer number
n ¼ normal unit vector
n
s
¼ index of refraction
N ¼ Brunt-Va
¨
isa

¨
la
¨
or buoyancy frequency
(1.29, 7.128), number, number of pores
in a sieve plate
N
A
¼ basis or interpolation functions
n ¼ kinematic viscosity (1.4), cyclic fre-
quency, Prandtl-Meyer function (15.56)
NOMENCLATURExxiv