fluid mechanics with multimedia dvd, fifth edition

NOTATION f ¼ principle-axis version of f, background or quiescent-fluid value of f, or average or ensemble average of f bf ¼ complex amplitude of f ~f ¼ full field value of f f0¼ derivat

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 ofFluid Mechanics, Vol 19 Copyright 1987 by Annual Reviews: www.AnnualReviews.org.Photograph of Geoffrey Ingram Taylor at age 69 in his laboratory reprinted withpermission from the AIP Emilio Segre` Visual Archieves Copyright, American Institute ofPhysics, 2000

FLUID MECHANICS

The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK

Ó 2012 Elsevier Inc All rights reserved.

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Notices

Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein 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

This revision to this textbook is dedicated to my wife and family who have patientlyhelped chip many sharp corners off my personality, and to the many fine instructors andstudents with whom I have interacted who have all in some way highlighted the allure ofthis 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 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 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 OceanographicCenter of Nova Southeastern University,where he remained until his death in 1994.During his career, Pijush contributed to

a number of sub-disciplines in physicaloceanography, most notably in the fields ofcoastal dynamics, mixed-layer physics,internal waves, and Indian-Ocean dynamics

He was a skilled data analyst, and, in thisregard, one of his accomplishments was tointroduce the “empirical orthogonal eigen-function” statistical technique to the oceano-graphic community

I arrived at Nova Southeastern Universityshortly after Pijush, and he and I workedclosely together thereafter I was immedi-ately impressed with the clarity of his scien-tific thinking and his thoroughness His mostimpressive and obvious quality, though, washis love of science, which pervaded all hisactivities Some time after we met, Pijushopened a drawer in a desk in his home office,showing me drafts of several chapters to

a book he had always wanted to write Adecade later, this manuscript became thefirst edition of Fluid Mechanics, the culmina-tion of his lifelong dream, which he dedi-cated to the memory of his mother, and tohis wife Shikha, daughter Tonushree, andson 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 continuum 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 static probes in a slightly ionized, collisiondominated gas” (1963; Physics of Fluids, 6,1492e1499), is to date the most highly citedpaper in the theory of electrostatic probesand plasma diagnostics

electro-During his doctoral work and for a fewyears beyond that, Ira collaborated with aworld-renowned mathematician/physicist,the late Dr Martin Kruskal (recipient ofNational Medal of Science, 1993) on the devel-opment of a monograph called “Asymptotol-ogy.” Professor Kruskal also collaboratedwith Professor Cohen on plasma physics.This was the basis for Ira’s strong foundation

in fluid dynamics that has been transmittedinto the prior editions of this textbook

In his forty-one years of service to theUniversity of Pennsylvania before his death

in December 2007, Professor Cohen guished himself with his integrity, his fiercedefense of high scholarly standards, andhis passionate commitment to teaching Hewill always be remembered for his candorand his sense of humor

distin-Professor Cohen’s dedication to ics was unrivalled In addition, his passionfor physical fitness was legendary Neitherrain nor sleet nor snow would deter himfrom his daily bicycle commute, which began

academ-at 5:00AM, from his home in Narberth to theUniversity of Pennsylvania His colleaguesgrew accustomed to seeing him drag hisforty-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 final 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 services there He

and his wife, Linda, were first marriednear Princeton, New Jersey, on February 13,

1960, when they eloped They were married

a second time four months later in a formalceremony He is survived by his wife, histwo children, Susan Cohen Bolstad andNancy Cohen Cavanaugh, and three grand-children, Melissa, Daniel, and Andrew

Senior FacultyDepartment of Mechanical Engineering

and Applied MechanicsUniversity of Pennsylvania

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 sincetaught and conducted research in fluidmechanics and acoustics He has authoredand co-authored more than 60 archival jour-nal articles and more than 100 conferencepresentations His published research in fluidmechanics includes papers on turbulent mix-ing, forced-convection heat transfer, cirrusclouds, molten plastic flow, interactions ofsurfactants with water waves, and hydrofoilperformance and turbulent boundary layercharacteristics at high Reynolds numbers.From January 2007 through June 2009, heserved as an Associate Chair and as theUndergraduate Program Director for theDepartment of Mechanical Engineering atthe University of Michigan He is a fellow

of the American Society of MechanicalEngineers and of the Acoustical Society ofAmerica He received the Student CouncilMentoring 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 Awardfrom the University of Michigan Chapter ofthe American Society for Engineering Educa-tion in 2009 Prof Dowling is an avidswimmer, is married, and has seven children

ix

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

Exercises 30Literature Cited 36Supplemental Reading 37

2 Cartesian Tensors 39

2.1 Scalars, Vectors, Tensors, Notation 392.2 Rotation of Axes: Formal Definition

of a Vector 422.3 Multiplication of Matrices 442.4 Second-Order Tensors 452.5 Contraction and Multiplication 472.6 Force on a Surface 48

2.7 Kronecker Delta and Alternating Tensor 502.8 Vector, Dot, and Cross Products 512.9 Gradient, Divergence, and Curl 522.10 Symmetric and Antisymmetric Tensors 552.11 Eigenvalues and Eigenvectors of

a Symmetric Tensor 562.12 Gauss’ Theorem 582.13 Stokes’ Theorem 602.14 Comma Notation 62Exercises 62

Literature Cited 64Supplemental Reading 64

3 Kinematics 65

3.1 Introduction and Coordinate Systems 653.2 Particle and Field Descriptions

of Fluid Motion 673.3 Flow Lines, Fluid Acceleration,and Galilean Transformation 713.4 Strain and Rotation Rates 76Summary 81

xi

3.5 Kinematics of Simple Plane Flows 82

3.6 Reynolds Transport Theorem 85

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

Moving and Deforming Boundaries 139

Surface Tension Revisited 139

4.11 Dimensionless Forms of the Equations and

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 1835.7 Interaction of Vortices 187

5.8 Vortex Sheet 191Exercises 192

Literature Cited 195Supplemental Reading 196

Kutta-Zhukhovsky Lift Theorem 2216.6 Conformal Mapping 222

6.7 Numerical Solution Techniques in TwoDimensions 225

6.8 Axisymmetric Ideal Flow 2316.9 Three-Dimensional Potential Flow andApparent Mass 236

6.10 Concluding Remarks 240Exercises 241

Literature Cited 251Supplemental Reading 251

7 Gravity Waves 253

7.1 Introduction 2547.2 Linear Liquid-Surface Gravity Waves 256Approximations for Deep and ShallowWater 265

7.3 Influence of Surface Tension 2697.4 Standing Waves 271

7.5 Group Velocity, Energy Flux, andDispersion 273

7.6 Nonlinear Waves in Shallow and DeepWater 279

7.7 Waves on a Density Interface 286

7.8 Internal Waves in a Continuously Stratified

Fluid 293

Internal Waves in a Stratified Fluid 296

Dispersion of Internal Waves in a Stratified

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

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 388Low Reynolds Numbers 389Moderate Reynolds Numbers 389High Reynolds Numbers 3929.9 Flow Past a Sphere and the Dynamics

of Sports Balls 395Cricket Ball Dynamics 396Tennis Ball Dynamics 398Baseball Dynamics 3999.10 Two-Dimensional Jets 3999.11 Secondary Flows 407Exercises 408

Literature Cited 418Supplemental Reading 419

10 Computational Fluid Dynamics 421

HOWARD H HU10.1 Introduction 42110.2 Finite-Difference Method 423Approximation to Derivatives 423Discretization and Its Accuracy 425Convergence, Consistency, andStability 426

10.3 Finite-Element Method 429Weak or Variational Form of PartialDifferential Equations 429Galerkin’s Approximation and Finite-Element Interpolations 430Matrix Equations, Comparison withFinite-Difference Method 431Element Point of View of the Finite-Element Method 434

10.4 Incompressible Viscous Fluid Flow 436Convection-Dominated Problems 437Incompressibility Condition 439Explicit MacCormack Scheme 440MAC Scheme 442

Q-Scheme 446Mixed Finite-Element Formulation 44710.5 Three Examples 449

Explicit MacCormack Scheme forDriven-Cavity Flow Problem 449Explicit MacCormack Scheme forFlow Over a Square Block 453

Finite-Element Formulation for

Flow Over a Cylinder Confined in

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

Spectrum 56412.8 Free Turbulent Shear Flows 57112.9 Wall-Bounded Turbulent Shear Flows 581Inner Layer: Law of the Wall 584Outer Layer: Velocity Defect Law 585Overlap Layer: Logarithmic Law 585Rough Surfaces 590

12.10 Turbulence Modeling 591

A Mixing Length Model 593One-Equation Models 595Two-Equation Models 59512.11 Turbulence in a Stratified Medium 596The Richardson Numbers 597Monin-Obukhov Length 598Spectrum of Temperature Fluctuations 60012.12 Taylor’s Theory of Turbulent Dispersion 601Rate of Dispersion of a Single Particle 602Random Walk 605

Behavior of a Smoke Plume in the Wind 606Turbulent Diffusivity 607

12.13 Concluding Remarks 607Exercises 608

Literature Cited 618Supplemental Reading 620

13 Geophysical Fluid Dynamics 621

13.1 Introduction 62213.2 Vertical Variation of Density in theAtmosphere and Ocean 62313.3 Equations of Motion 62513.4 Approximate Equations for a Thin Layer on

a Rotating Sphere 628f-Plane Model 630b-Plane Model 63013.5 Geostrophic Flow 630Thermal Wind 632Taylor-Proudman Theorem 632

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

14.4 Conformal Transformation for

Generating Airfoil Shapes 702

14.5 Lift of a Zhukhovsky Airfoil 70614.6 Elementary Lifting Line Theory forWings of Finite Span 708Lanchester Versus Prandtl 71614.7 Lift and Drag Characteristics ofAirfoils 717

14.8 Propulsive Mechanisms of Fishand Birds 719

14.9 Sailing against the Wind 721Exercises 722

Literature Cited 728Supplemental Reading 728

15 Compressible Flow 729

15.1 Introduction 730Perfect Gas Thermodynamic Relations 73115.2 Acoustics 732

15.3 Basic Equations for One-DimensionalFlow 736

15.4 Reference Properties in CompressibleFlow 738

15.5 Area-Velocity Relationship inOne-Dimensional Isentropic Flow 74015.6 Normal Shock Waves 748

Stationary Normal Shock Wave in aMoving Medium 748

Moving Normal Shock Wave in aStationary Medium 752Normal Shock Structure 75315.7 Operation of Nozzles at DifferentBack Pressures 755

Convergent Nozzle 755ConvergenteDivergent Nozzle 75715.8 Effects of Friction and Heating inConstant-Area Ducts 761Effect of Friction 763Effect of Heat Transfer 76415.9 Pressure Waves in Planar CompressibleFlow 765

15.10 Thin Airfoil Theory in Supersonic Flow 773Exercises 775

Literature Cited 778Supplemental Reading 778

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 841The Pressure Pulse Curve in the RightVentricle 842

Effect of Pulmonary Arterial Pressure onPulmonary Resistance 843

16.4 Introduction to the Fluid Mechanics

of Plants 844Exercises 849Acknowledgment 850Literature Cited 851Supplemental Reading 852

Appendix A 853 Appendix B 857 Appendix C 869 Appendix D 873 Index 875

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 reception of the first 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 andsimulations

• Dozens of new interactive demonstrationsand animations

Additional new features:

• Improved navigation via sidebars thatprovide rapid overviews of modules andguided browsing

• Media libraries for each chapter thatgive a snapshot of videos, each withdescriptive labels

• Ability to create movie playlists, whichare invaluable in teaching

• Higher-resolution graphics, with full orpart screen viewing options

• Operates on either a PC or a Mac OSX

xvii

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 ensuing

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

Editionof 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 thetext’s overall utility to students and instruc-tors by adding introductory material andreferences to the first few chapters, byincreasing the prominence of engineeringapplications of fluid mechanics, and byproviding a variety of new exercises (morethan 200) and figures (more than 100) Forthe chapters receiving the most attention(1e9, 11e12, and 14) this has meant approx-imately doubling, tripling, or quadruplingthe number of exercises Some of the newexercises have been built from derivationsthat previously had appeared in the body

of the text, and some involve simple kitchen

or bathroom experiments My hope for

a future edition is that there will be time tofurther expand the exercise offerings, espe-cially in Chapters 10, 13, 15, and 16

In preparing this 5th Edition, some ganization, addition, and deletion of mate-rial has also taken place Dimensionalanalysis has been moved to Chapter 1.The stream function’s introduction andthe dynamic-similarity topic have beenmoved to Chapter 4 Reynolds transporttheorem now occupies the final section ofChapter 3 The discussion of the wave equa-tion has been placed in the acoustics sec-tion of Chapter 15 Major topical additionsare: apparent mass (Chapter 6), elemen-tary lubrication theory (Chapter 8), andThwaites method (Chapter 9) The sectionscovering the laminar shear layer, andboundary-layer theory from a purely math-ematical perspective, and coherent struc-tures in wall-bounded turbulent flow have

reor-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 changes 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 isincomplete because of my imperfect mem-ory and record keeping Therefore, I standready to correctly attribute the origins ofany problem contained herein Furthermore,

I welcome the opportunity to correct anyerrors you find, to hear your opinion ofhow this book might be improved, and toinclude exercises you might suggest; justcontact me at drd@umich.edu

David R DowlingAnn Arbor, Michigan

April 2011

COMPANION WEBSITE

An updated errata sheet is available onthe book’s companion website To accessthe errata, visit www.elsevierdirect.com/

9780123821003 and click on the companionsite link Instructors teaching with this bookmay access the solutions manual and imagebank by visiting www.textbooks.elsevier.com and following the online instructions

to log on and register

The current version of this textbook

has benefited from the commentary and

suggestions provided by the reviewers of

the initial revision proposal and the

re-viewers of draft versions of several of the

chapters Chief among these reviewers is

Professor John Cimbala of the Pennsylvania

State University I would also like to

recog-nize and thank my technical mentors,

Professor Hans W Liepmann uate advisor), Professor Paul E Dimotakis(graduate advisor), and Professor Darrell

(undergrad-R Jackson (post-doctoral advisor); and myfriends and colleagues who have contrib-uted to the development of this text bydiscussing ideas and sharing their exper-tise, humor, and devotion to science andengineering

xxi

NOTATION

f ¼ principle-axis version of f, background or

quiescent-fluid value of f, or average or

ensemble average of f

bf ¼ complex amplitude of f

~f ¼ full field value of f

f0¼ derivative of f with respect to its

argu-ment, or perturbation of f from its

reference state

f¼ complex conjugate of f, dimensionless

version of f, or the value of f at the sonic

fN¼ reference value of f or value of f far

away from the point of interest

Df ¼ change in f

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

a¼ triangular area, cylinder radius,

sphere radius, amplitude

a0¼ initial tube radius

a¼ generic vector, Lagrangian acceleration(3.1)

A¼ generic second-order (or higher) tensor

A, A ¼ a constant, an amplitude, area,

surface, surface of a materialvolume, planform area of a wingA* ¼ control surface, sonic throat area

Ao¼ Avogadro’s number

A0¼ reference area

Aij¼ representative second-order tensor

b ¼ angle of rotation, coefficient of densitychange due to salinity or other constit-uent, variation of the Coriolis frequencywith latitude, camber parameter

b¼ generic vector, control surface velocity(3.35)

B, B ¼ a constant, Bernoulli function (4.70),log-law intercept parameter (12.88)

B, Bij¼ generic second-order (or higher)

tensor

Bo ¼ Bond number (4.118)

c¼ speed of sound (1.19, 15.6), phase speed(7.4), chord length (14.2), pressure pulsewave speed, concentration of solutes

cj¼ pressure pulse wave speed in tube j

c¼ phase velocity vector (7.8)

c , cg¼ group velocity magnitude (7.68)

Cp¼ specific heat capacity at constant

Cij¼ matrix of direction cosines between

original and rotated coordinate system

d ¼ Dirac delta function (B.4.1),

similarity-variable length scale (8.32),

boundary-layer thickness, generic length 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)

dij¼ Kronecker delta function (2.16)

d99¼ 99% layer thickness

D¼ distance, drag force, diffusion

coeffi-cient, Dean number (16.179)

Di¼ lift-induced drag (14.15)

D/Dt ¼ material derivative (3.4) or (3.5)

DT¼ turbulent diffusivity of particles

(12.127)

D ¼ generalized field derivative (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

3ijk¼ alternating tensor (2.18)

e¼ internal energy per unit mass (1.10)

ei¼ unit vector in the i-direction (2.1)

e ¼ average kinetic energy of turbulent

E ¼ kinetic energy of the average flow(12.46)

bE1¼ total energy dissipation in a bloodvessel

f¼ generic function, Helmholtz free energyper unit mass, longitudinal correlationcoefficient (12.38), Coriolis frequency(13.8), dimensionless friction parameter(15.45)

f ¼ velocity potential (6.10), an angle

f¼ surface force vector per unit area(2.15, 4.13)

F¼ force magnitude, generic flow fieldproperty, average energy flux per unitlength of wave crest (7.44), generic orprofile function

F¼ force vector, average wave energyflux vector

F ¼ body force potential (4.18), mined spectrum function (12.53)

_g ¼ shear rate

g¼ body force per unit mass (4.13)

g¼ acceleration of gravity, undeterminedfunction, transverse correlation coeffi-cient (12.38)

G¼ gravitational constant,

pressure-gradient pulse amplitude, profile

function

Gn¼ Fourier series coefficient

G ¼ center of mass, center of vorticity

h¼ 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

hT¼ Batchelor microscale (12.114)

H¼ atmospheric scale height, water depth,

shape factor (9.46), profile function,

Hematocrit

i¼ an index, imaginary root

I¼ incident light intensity, bending moment

of inertia

j¼ an index

J, Js¼ jet momentum flux per unit span

(9.61)

Ji¼ Bessel function of order i

Jm¼ diffusive mass flux vector (1.1)

4 ¼ a function, azimuthal angle in

cylin-drical and spherical coordinates

k¼ 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)

ks¼ diffusivity of salt

kT¼ turbulent thermal diffusivity (12.95)

km¼ mass diffusivity of a passive scalar in

Fick’s law (1.1)

kmT¼ turbulent mass diffusivity (12.96)

k ¼ Boltzmann’s constant (1.21)

Kn¼ Knudsen number

K¼ a generic constant, magnitude of the

wave number vector (7.6), lift curve

slope, Dean Number (16.178)

Kp¼ constant proportional to tube wall

bending stiffness

K ¼ compliance of a blood vessel, degrees

Kelvin (16.48)

K¼ wave number vector, stiffness matrix

l¼ molecular mean free path, spanwisedimension, generic length scale, wavenumber component (7.5, 7.6), shearcorrelation in Thwaites method (9.45),length scale in turbulent flow

lT¼ mixing length (12.98)

L, L ¼ generic length dimension, genericlength scale, lift force

LM¼ Monin-Obukhov length scale (12.110)

l ¼ wavelength (7.1, 7.7), laminar layer correlation parameter (9.44), flowresistance ratio

boundary-lm¼ wavelength of the minimum phasespeed

lt¼ temporal Taylor microscale (12.19)

lf, lg¼ longitudinal and lateral spatial

Taylor microscale (12.39)

L ¼ lubrication-flow bearing number (8.16),Rossby radius of deformation, wingaspect ratio

Lf, Lg¼ longitudinal and lateral integral

spatial scales (12.39)

Lt¼ integral time scale (12.18)

m ¼ dynamic or shear viscosity (1.3), Machangle (15.49)

my¼ bulk viscosity (4.37)

m¼ molecular mass (1.22), generic mass,

an index, two-dimensional sourcestrength, moment order (12.1), wavenumber component (7.5, 7.6)

M, M ¼ generic mass dimension, mass,

Mach number (4.111), apparent oradded mass (6.108)

NA¼ basis or interpolation functions

n ¼ kinematic viscosity (1.4), cyclic quency, Prandtl-Meyer function (15.56)

fre-nT¼ turbulent kinematic viscosity (12.94)

q¼ heat added to a system (1.10), volume

flux per unit span, dimensionless heat

addition parameter (15.45)

Q¼ thermodynamic heat per unit mass,

volume flux in two or three dimensions

q ¼ potential temperature (1.31), unit of

temperature, angle in polar coordinates,

momentum thickness (9.17), local phase,

an angle, angle in a toroidal coordinate

system

r ¼ mass density (1.1)

rm¼ mass density of a mixture

r ¼ average or quiescent density in a

strati-fied fluid

rq¼ potential density (1.33)

r¼ matrix rank, distance from the origin,

distance from the axis

r¼ particle trajectory (3.1, 3.8)

R¼ distance from the cylindrical axis, radius

of curvature, gas constant (1.23), generic

nonlinearity parameter, total peripheral

resistance (16.9), tube radius of

curvature

R ¼ viscous resistance per unit length,

reflection coefficient (16.204), (16.153)

Ru¼ universal gas constant (1.22)

Ri¼ radius of curvature in direction i (1.5)

R, Rij¼ rotation tensor (3.13), correlation

s¼ entropy (1.16), arc length, salinity,wingspan (14.1), dimensionless arclength

sij¼ viscous stress tensor (4.27)

S¼ salinity, scattered light intensity, an area,dimensionless speed index, entropy

Se¼ one-dimensional temporal longitudinalenergy spectrum (12.20)

S11¼ one-dimensional spatial longitudinalenergy spectrum (12.45)

ST¼ one-dimensional temperature tion spectrum (12.113, 12.114)

fluctua-S, Sij¼ strain rate tensor (3.12), symmetric

Ta ¼ Taylor number (11.52)

To¼ free stream temperature

Tw¼ wall temperature

Ti¼ tension in the i-direction

s ¼ shear stress (1.3), time lag

ui¼ fluid velocity components, fluctuating

velocity components

u ¼ friction velocity (12.81)

U¼ generic uniform velocity vector

Ui¼ ensemble average velocity components

U¼ generic velocity, average stream-wise

velocity

DU ¼ characteristic velocity difference

Ue¼ local free-stream flow speed above

a boundary layer (9.11), flow speed at

the effective angle of attack

UCL¼ centerline velocity (12.56)

UN¼ flow speed far upstream or far away

v¼ component of fluid velocity along the y

axis

v¼ generic vector

V¼ volume, material volume, average

stream-normal velocity, average

velocity, variational space, complex

velocity

V)¼ control volume

w¼ complex potential (6.42), vertical

component of fluid velocity, function in

the variational space, downwash

U ¼ angular velocity of a rotating frame ofreference

x¼ first Cartesian coordinate

x¼ position vector (2.1)

xi¼ components of the position vector (2.1)

x ¼ generic spatial coordinate, integrationvariable, similarity variable (12.84), axialtube wall displacement

y¼ second Cartesian coordinate

Y¼ mass fraction (1.1)

YCL¼ centerline mass fraction (12.69)

Yi¼ Bessel function of order i, admittance

j ¼ stream function (6.3, 6.75), waterpotential

J ¼ Reynolds stress scaling function (12.57),generic functional solution

J ¼ vector potential, three-dimensionalstream function (4.12)

z¼ third Cartesian coordinate, complexvariable (6.43)

z ¼ interface displacement, angular wall displacement, relative vorticity

tube-Z¼ impedance (16.151)

1 Introduction

Literature Cited 36Supplemental Reading 37

CHAPTER OBJECTIVES

• To properly introduce the subject of fluid

mechanics and its importance

• To state the assumptions upon which the

subject is based

• To review the basic background science of

liquids and gases

• To present the relevant features of fluidstatics

• To establish dimensional analysis as anintellectual tool for use in the remainder ofthe text

1

Fluid Mechanics, Fifth Edition DOI: 10.1016/B978-0-12-382100-3.10001-0 Ó 2012 Elsevier Inc All rights reserved.

1.1 FLUID MECHANICS

Fluid mechanics is the branch of science concerned with moving and stationary fluids.Given that the vast majority of the observable mass in the universe exists in a fluid state,that life as we know it is not possible without fluids, and that the atmosphere and oceanscovering this planet are fluids, fluid mechanics has unquestioned scientific and practicalimportance Its allure crosses disciplinary boundaries, in part because it is described by

a nonlinear field theory and also because it is readily observed Mathematicians, physicists,biologists, geologists, oceanographers, atmospheric scientists, engineers of many types,and even artists have been drawn to study, harness, and exploit fluid mechanics to developand test formal and computational techniques, to better understand the natural world, and

to attempt to improve the human condition The importance of fluid mechanics cannot beoverstated for applications involving transportation, power generation and conversion, mate-rials processing and manufacturing, food production, and civil infrastructure For example, inthe twentieth century, life expectancy in the United States approximately doubled About half

of this increase can be traced to advances in medical practice, particularly antibiotic therapies.The other half largely resulted from a steep decline in childhood mortality from water-bornediseases, a decline that occurred because of widespread delivery of clean water to nearly theentire populationda fluids-engineering and public-works achievement Yet, the pursuits ofmathematicians, scientists, and engineers are interconnected: Engineers need to understandnatural phenomena to be successful, scientists strive to provide this understanding, and math-ematicians pursue the formal and computational tools that support these efforts

Advances in fluid mechanics, like any other branch of physical science, may arise frommathematical analyses, computer simulations, or experiments Analytical approaches areoften successful for finding solutions to idealized and simplified problems and such solu-tions can be of immense value for developing insight and understanding, and for compari-sons with numerical and experimental results Thus, some fluency in mathematics, especiallymultivariable calculus, is helpful in the study of fluid mechanics In practice, drastic simpli-fications are frequently necessary to find analytical solutions because of the complexity ofreal fluid flow phenomena Furthermore, it is probably fair to say that some of the greatesttheoretical contributions have come from people who depended rather strongly on theirphysical intuition Ludwig Prandtl, one of the founders of modern fluid mechanics, firstconceived the idea of a boundary layer based solely on physical intuition His knowledge

of mathematics was rather limited, as his famous student Theodore von Karman (1954,page 50) testifies Interestingly, the boundary layer concept has since been expanded into

a general method in applied mathematics

As in other scientific fields, mankind’s mathematical abilities are often too limited to tacklethe full complexity of real fluid flows Therefore, whether we are primarily interested inunderstanding flow physics or in developing fluid-flow applications, we often must depend

on observations, computer simulations, or experimental measurements to test hypothesesand analyses, and develop insights into the phenomena under study This book is an intro-duction to fluid mechanics that should appeal to anyone pursuing fluid mechanical inquiry.Its emphasis is on fully presenting fundamental concepts and illustrating them with exam-ples drawn from various scientific and engineering fields Given its finite size, this bookprovidesdat bestdan incomplete description of the subject However, the purpose of this

book will be fulfilled if the reader becomes more curious and interested in fluid mechanics as

a result of its perusal

1.2 UNITS OF MEASUREMENT

For mechanical systems, the units of all physical variables can be expressed in terms of theunits of four basic variables, namely, length, mass, time, and temperature In this book, the inter-national system of units (Syste`me international d’unite´s) commonly referred to as SI (or MKS)units, is preferred The basic units of this system are meter for length, kilogram for mass, secondfor time, and Kelvin for temperature The units for other variables can be derived from thesebasic units Some of the common variables used in fluid mechanics, and their SI units, are listed

in Table 1.1 Some useful conversion factors between different systems of units are listed inAppendix A To avoid very large or very small numerical values, prefixes are used to indicatemultiples of the units given in Table 1.1 Some of the common prefixes are listed in Table 1.2.Strict adherence to the SI system is sometimes cumbersome and will be abandoned occa-sionally for simplicity For example, temperatures will be frequently quoted in degreesCelsius (
C), which is related to Kelvin (K) by the relation
C ¼ K  273.15 However, theEnglish system of units (foot, pound,
F) will not be used, even though this unit systemremains in use in some places in the world

1.3 SOLIDS, LIQUIDS, AND GASES

The various forms of matter may be broadly categorized as being fluid or solid A fluid is

a substance that deforms continuously under an applied shear stress or, equivalently, onethat does not have a preferred shape A solid is one that does not deform continuously under

an applied shear stress, and does have a preferred shape to which it relaxes when externalforces on it are withdrawn Consider a rectangular element of a solid ABCD (Figure 1.1a)

Under the action of a shear force F the element assumes the shape ABC0D0 If the solid isperfectly elastic, it returns to its preferred shape ABCD when F is withdrawn In contrast,

a fluid deforms continuously under the action of a shear force, however small Thus, the element

of the fluid ABCD confined between parallel plates (Figure 1.1b) successively deforms toshapes such as ABC0D0 and ABC00D00, and keeps deforming, as long as the force F is main-tained on the upper plate When F is withdrawn, the fluid element’s final shape is retained;

it does not return to a prior shape Therefore, we say that a fluid flows

The qualification “however small” in the description of a fluid is significant This is becausesome solids also deform continuously if the shear stress exceeds a certain limiting value, cor-responding to the yield point of the solid A solid in such a state is known as plastic, and plasticdeformation changes the solid object’s unloaded shape Interestingly, the distinction betweensolids and fluids may not be well defined Substances like paints, jelly, pitch, putty, polymersolutions, and biological substances (for example, egg whites) may simultaneously displayboth solid and fluid properties If we say that an elastic solid has a perfect memory of itspreferred shape (because it always springs back to its preferred shape when unloaded) andthat an ordinary viscous fluid has zero memory (because it never springs back whenunloaded), then substances like egg whites can be called viscoelastic because they partiallyrebound when unloaded

FIGURE 1.1 Deformation of solid and fluid elements under a constant externally applied shear force (a) Solid; here the element deflects until its internal stress balances the externally applied force (b) Fluid; here the element deforms continuously as long as the shear force is applied.

TABLE 1.2 Common Prefixes

Although solids and fluids behave very differently when subjected to shear stresses, theybehave similarly under the action of compressive normal stresses However, tensile normalstresses again lead to differences in fluid and solid behavior Solids can support both tensileand compressive normal stresses, while fluids typically expand or change phase (i.e., boil)when subjected to tensile stresses Some liquids can support a small amount of tensile stress,the amount depending on the degree of molecular cohesion and the duration of the tensilestress.

Fluids generally fall into two classes, liquids and gases A gas always expands to fill theentire volume of its container In contrast, the volume of a liquid changes little, so that itcannot completely fill a large container; in a gravitational field, a free surface forms that sepa-rates a liquid from its vapor

1.4 CONTINUUM HYPOTHESIS

A fluid is composed of a large number of molecules in constant motion undergoing sions with each other, and is therefore discontinuous or discrete at the most microscopicscales In principle, it is possible to study the mechanics of a fluid by studying the motion

colli-of the molecules themselves, as is done in kinetic theory or statistical mechanics However,

we are generally interested in the average manifestation of the molecular motion For example,forces are exerted on the boundaries of a fluid’s container due to the constant bombardment

of the fluid molecules; the statistical average of these collision forces per unit area is calledpressure, a macroscopic property So long as we are not interested in the molecular mechanics

of the origin of pressure, we can ignore the molecular motion and think of pressure as simplythe average force per unit area exerted by the fluid

When the molecular density of the fluid and the size of the region of interest are largeenough, such average properties are sufficient for the explanation of macroscopicphenomena and the discrete molecular structure of matter may be ignored and replacedwith a continuous distribution, called a continuum In a continuum, fluid properties liketemperature, density, or velocity are defined at every point in space, and these propertiesare known to be appropriate averages of molecular characteristics in a small regionsurrounding the point of interest The continuum approximation is valid when the Knudsennumber, Kn ¼ l/L where l is the mean free path of the molecules and L is the length scale ofinterest (a body length, a pore diameter, a turning radius, etc.), is much less than unity Formost terrestrial situations, this is not a great restriction since l z 50 nm for air at room temper-ature and pressure, and l is more than two orders of magnitude smaller for water under thesame conditions However, a molecular-kinetic-theory approach may be necessary foranalyzing flows over very small objects or in very narrow flow paths, or in the tenuous gases

at the upper reaches of the atmosphere

1.5 MOLECULAR TRANSPORT PHENOMENA

Although the details of molecular motions may be locally averaged to compute ature, density, or velocity, random molecular motions still lead to diffusive transport of

temper-molecular species, temperature, or momentum that impact fluid properties at macroscopicscales.

Consider a surface area AB within a mixture of two gases, say, nitrogen and oxygen(Figure 1.2), and assume that the nitrogen mass fraction Y varies across AB Here the mass

of nitrogen per unit volume is rY (sometimes known as the nitrogen concentration or density),where r is the overall density of the gas mixture Random migration of molecules across AB

in both directions will result in a net flux of nitrogen across AB, from the region of higher Ytoward the region of lower Y To a good approximation, the flux of one constituent in

a mixture is proportional to its gradient:

FIGURE 1.2 Mass flux J m due to variation in the mass fraction Y(y) Here the mass fraction profile increases with increasing Y, so Fick’s law of diffusion states that the diffusive mass flux that acts to smooth out mass-fraction differences is downward across AB.

are called phenomenological laws Statistical mechanics can sometimes be used to derive suchlaws, but only for simple situations.

The analogous relation for heat transport via a temperature gradient VT is Fourier’s law,

where q is the heat flux (J m2s1), and k is the material’s thermal conductivity

The analogous relationship for momentum transport via a velocity gradient is tively similar to (1.1) and (1.2) but is more complicated because momentum and velocityare vectors So as a first step, consider the effect of a vertical gradient, du/dy, in the horizontalvelocity u (Figure 1.3) Molecular motion and collisions cause the faster fluid above AB to pullthe fluid underneath AB forward, thereby speeding it up Molecular motion and collisionsalso cause the slower fluid below AB to pull the upper faster fluid backward, thereby slowing

qualita-it down Thus, wqualita-ithout an external influence to maintain du/dy, the flow profile shown by thesolid curve will evolve toward a profile shown by the dashed curve This is analogous tosaying that u, the horizontal momentum per unit mass (a momentum concentration), diffusesdownward Here, the resulting momentum flux, from high to low u, is equivalent to a shearstress, s, existing in the fluid Experiments show that the magnitude of s along a surface such

as AB is, to a good approximation, proportional to the local velocity gradient,

FIGURE 1.3 Shear stress s on surface AB The diffusive action of fluid viscosity tends to decrease velocity gradients, so that the continuous line tends toward the dashed line.

where the constant of proportionality m (with units of kg m1s1) is known as the dynamicviscosity This is Newton’s law of friction It is analogous to (1.1) and (1.2) for the simple unidi-rectional shear flow depicted in Figure 1.3 However, it is an incomplete scalar statement ofmolecular momentum transport when compared to the more complete vector relationships(1.1) and (1.2) for species and thermal molecular transport A more general tensor form of(1.3) that accounts for three velocity components and three possible orientations of the surface

AB is presented in Chapter 4 after the mathematical and kinematical developments in ters 2 and 3 For gases and liquids, m depends on the local temperature T In ideal gases, therandom thermal speed is roughly proportional to T1/2, so molecular momentum transport,and consequently m, also vary approximately as T1/2 For liquids, shear stress is causedmore by the intermolecular cohesive forces than by the thermal motion of the molecules.These cohesive forces decrease with increasing T so m for a liquid decreases with increasing T.Although the shear stress is proportional to m, we will see in Chapter 4 that the tendency of

Chap-a fluid to trChap-ansport velocity grChap-adients is determined by the quChap-antity

where r is the density (kg m3) of the fluid The units of n (m2s1) do not involve the mass, so

n is frequently called the kinematic viscosity

Two points should be noticed about the transport laws (1.1), (1.2), and (1.3) First, only firstderivatives appear on the right side in each case This is because molecular transport iscarried out by a nearly uncountable number of molecular interactions at length scales thatare too small to be influenced by higher derivatives of the species mass fractions, tempera-ture, or velocity profiles Second, nonlinear terms involving higher powers of the firstderivatives, for example jVuj2, do not appear Although this is only expected for smallfirst-derivative magnitudes, experiments show that the linear relations are accurate enoughfor most practical situations involving mass fraction, temperature, or velocity gradients

1.6 SURFACE TENSION

A density discontinuity may exist whenever two immiscible fluids are in contact, forexample at the interface between water and air Here unbalanced attractive intermolecularforces cause the interface to behave as if it were a stretched membrane under tension, likethe surface of a balloon or soap bubble This is why small drops of liquid in air or smallgas bubbles in water tend to be spherical in shape Imagine a liquid drop surrounded by

an insoluble gas Near the interface, all the liquid molecules are trying to pull the molecules

on the interface inward toward the center of the drop The net effect of these attractive forces

is for the interface area to contract until equilibrium is reached with other surface forces Themagnitude of the tensile force that acts per unit length to open a line segment lying inthe surface like a seam is called surface tension s; its units are N m1 Alternatively, s can

be thought of as the energy needed to create a unit of interfacial area In general, s depends

on the pair of fluids in contact, the temperature, and the presence of surface-active chemicals(surfactants) or impurities, even at very low concentrations

An important consequence of surface tension is that it causes a pressure difference acrosscurved interfaces Consider a spherical interface having a radius of curvature R (Figure 1.4a)

If piand poare the pressures on the inner and outer sides of the interface, respectively, then

a static force balance gives

sð2pRÞ ¼ ðpi poÞpR2,from which the pressure jump is found to be

pi po¼ 2s=R;

showing that the pressure on the concave side (the inside) is higher

The curvature of a general surface can be specified by the radii of curvature along twoorthogonal directions, say, R1and R2(Figure 1.4b) A similar analysis shows that the pressuredifference across the interface is given by

pi po¼ s

1

1.7 FLUID STATICS

The magnitude of the force per unit area in a static fluid is called the pressure; pressure in

a moving medium will be defined in Chapter 4 Sometimes the ordinary pressure is called the

FIGURE 1.4 (a) Section of a spherical droplet, showing surface tension forces (b) An interface with radii of curvatures R 1 and R 2 along two orthogonal directions.

absolute pressure, in order to distinguish it from the gauge pressure, which is defined as theabsolute pressure minus the atmospheric pressure:

pgauge¼ p  patm:The standard value for atmospheric pressure patmis 101.3 kPa ¼ 1.013 bar where 1 bar ¼ 105

Pa An absolute pressure of zero implies vacuum while a gauge pressure of zero impliesatmospheric pressure

In a fluid at rest, tangential viscous stresses are absent and the only force between adjacentsurfaces is normal to the surface We shall now demonstrate that in such a case the surfaceforce per unit area (or pressure) is equal in all directions Consider a small volume of fluidwith a triangular cross section (Figure 1.5) of unit thickness normal to the paper, and let

p1, p2, and p3be the pressures on the three faces The z-axis is taken vertically upward Theonly forces acting on the element are the pressure forces normal to the faces and the weight

of the element Because there is no acceleration of the element in the x direction, a balance offorces in that direction gives

p1¼ p2¼ p3, (1.6)

so that the force per unit area is independent of the angular orientation of the surface Thepressure is therefore a scalar quantity

We now proceed to determine the spatial distribution of pressure in a static fluid Consider

an infinitesimal cube of sides dx, dy, and dz, with the z-axis vertically upward (Figure 1.6)

A balance of forces in the x direction shows that the pressures on the two sides perpendicular

to the x-axis are equal A similar result holds in the y direction, so that

This fact is expressed by Pascal’s law, which states that all points in a resting fluid medium(and connected by the same fluid) are at the same pressure if they are at the same depth.For example, the pressure at points F and G in Figure 1.7 are the same

Vertical equilibrium of the element in Figure 1.6 requires that

p dx dy  ðp þ dpÞ dx dy  rg dx dy dz ¼ 0,which simplifies to

the pressure rise at a depth h below the free surface of a liquid is equal to rgh, which is theweight of a column of liquid of height h and unit cross section.

EXAMPLE 1.1

Using Figure 1.7, show that the rise of a liquid in a narrow tube of radius R is given by

h ¼2s sinargR ,where s is the surface tension and a is the contact angle between the fluid and the tube’s innersurface

Solution

Since the free surface is concave upward and exposed to the atmosphere, the pressure just belowthe interface at point E is below atmospheric The pressure then increases linearly along EF At F thepressure again equals the atmospheric pressure, since F is at the same level as G where the pressure

is atmospheric The pressure forces on faces AB and CD therefore balance each other Verticalequilibrium of the element ABCD then requires that the weight of the element balances the verticalcomponent of the surface tension force, so that

s2pRsina ¼ rghpR2

,which gives the required result

the basic thermodynamic concepts, so this section merely reviews the main ideas and themost commonly used relations in this book.

A thermodynamic system is a quantity of matter that exchanges heat and work, but nomass, with its surroundings A system in equilibrium is free of fluctuations, such as thosegenerated during heat or work input from, or output to, its surroundings After any suchthermodynamic change, fluctuations die out or relax, a new equilibrium is reached, andonce again the system’s properties, such as pressure and temperature, are well defined.Here, the system’s relaxation time is defined as the time taken by the system to adjust to

a new thermodynamic state

This thermodynamic system concept is obviously not directly applicable to a scopic volume of a moving fluid in which pressure and temperature may vary consider-ably However, experiments show that classical thermodynamics does apply to small fluidvolumes commonly called fluid particles A fluid particle is a small deforming volumecarried by the flow that: 1) always contains the same fluid molecules, 2) is large enough

macro-so that its thermodynamic properties are well defined when it is at equilibrium, but 3) issmall enough so that its relaxation time is short compared to the time scales of fluid-motion-induced thermodynamic changes Under ordinary conditions (the emphasis inthis text), molecular densities, speeds, and collision rates are high enough so that theconditions for the existence of fluid particles are met, and classical thermodynamicscan be directly applied to flowing fluids However, there are circumstances involving rari-fied gases, shock waves, and high-frequency acoustic waves where one or more of thefluid particle requirements are not met and molecular-kinetic and quantum theories areneeded

The basic laws of classical thermodynamics are empirical, and cannot be derived fromanything more fundamental These laws essentially establish definitions, upon which thesubject is built The first law of thermodynamics can be regarded as a principle that definesthe internal energy of a system, and the second law can be regarded as the principle thatdefines the entropy of a system

First Law of Thermodynamics

The first law of thermodynamics states that the energy of a system is conserved;

is a manifestation of the random molecular motion of the system’s constituents In fluidflows, the kinetic energy of the fluid particles’ macroscopic motion has to be included inthe e-term in (1.10) in order that the principle of conservation of energy is satisfied For devel-oping the relations of classical thermodynamics, however, we shall only include the thermalenergy in the term e