Fluid Mechanics, Fourth Edition Founders of Modern Fluid Dynamics LudwigPrandtl G.I.Taylor (1875–1953) (1886–1975) (BiographicalsketchesofPrandtlandTayloraregiveninAppendixC.) PhotographofLudwigPrandtlisreprintedwithpermissionfromtheAnnualReviewofFluid Mechanics,Vol.19,Copyright1987byAnnualReviewswww.AnnualReviews.org. PhotographofGeoffreyIngramTayloratage69inhislaboratoryreprintedwithpermission fromtheAIPEmilioSegre`VisualArchieves.Copyright,AmericanInstituteofPhysics,2000. Fluid Mechanics Fourth Edition Pijush K. Kundu OceanographicCenter NovaSoutheasternUniversity Dania,Florida Ira M. Cohen DepartmentofMechanicalEngineeringand AppliedMechanics UniversityofPennsylvania Philadelphia,Pennsylvania withcontributionsbyP.S.AyyaswamyandH.H.Hu AMSTERDAM•BOSTON•HEIDELBERG•LONDON NEWYORK•OXFORD•PARIS•SANDIEGO SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO AcademicPressisanimprintofElsevier Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400 Burlington, MA 01803, USA Elsevier, The Boulevard, Langford Lane Kidlington, Oxford, OX5 1GB, UK © 2008 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. 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For information on all Academic Press publications visit our Web site at www.elsevierdirect.com Printed in the United States 10 11 12 13 14 10 9 8 7 6 5 4 3 2 1 The fourth edition is dedicated to the memory of Pijush K. Kundu and also to my wifeLindaanddaughtersSusanandNancywhohavegreatlyenrichedmylife. “Everythingshouldbemadeassimpleaspossible, butnotsimpler.” —AlbertEinstein “Ifnaturewerenotbeautiful,itwouldnotbeworthstudyingit. Andlifewouldnotbeworthliving.” —HenryPoincare´ In memory of Pijush Kundu Pijush Kanti Kundu was born in Calcutta, India, on October 31, 1941. He received a B.S. degree in Mechanical Engineering in 1963 from Shibpur Engineering College of Calcutta University, earned an M.S. degree in Engineering from Roorkee University in 1965,andwasalecturerinMechanicalEngi- neeringattheIndianInstituteofTechnology in Delhi from 1965 to 1968. Pijush came to the United States in 1968, as a doctoral stu- dentatPennStateUniversity.WithDr.John L.Lumleyashisadvisor,hestudiedinstabili- tiesofviscoelasticfluids,receivinghisdoctor- atein1972.Hebeganhislifelonginterestin oceanographysoonafterhisgraduation,workingasResearchAssociateinOceano- graphy at Oregon State University from 1968 until 1972. After spending a year at theUniversitydeOrienteinVenezuela,hejoinedthefacultyoftheOceanographic CenterofNovaSoutheasternUniversity,whereheremaineduntilhisdeathin1994. Duringhiscareer,Pijushcontributedtoanumberofsub-disciplinesinphysical oceanography,mostnotablyinthefieldsofcoastaldynamics,mixed-layerphysics, internal waves, and Indian-Ocean dynamics. He was a skilled data analyst, and, in thisregard,oneofhisaccomplishmentswastointroducethe“empiricalorthogonal eigenfunction”statisticaltechniquetotheoceanographiccommunity. IarrivedatNovaSoutheasternUniversityshortlyafterPijush,andheandIworked closelytogetherthereafter.Iwasimmediatelyimpressedwiththeclarityofhisscien- tificthinkingandhisthoroughness.Hismostimpressiveandobviousquality,though, was his love of science, which pervaded all his activities. Some time after we met, Pijush opened a drawer in a desk in his home office, showing me drafts of several chapters to a book he had always wanted to write. A decade later, this manuscript becamethefirsteditionof“FluidMechanics,”theculminationofhislifelongdream; which he dedicated to the memory of his mother, and to his wife Shikha, daughter Tonushree,andsonJoydip. JulianP.McCreary,Jr., UniversityofHawaii Contents New in This Book ......................................... xvi Preface................................................... xvii PrefacetoThirdEdition.................................... xix PrefacetoSecondEdition .................................. xxi PrefacetoFirstEdition .................................... xxiii Author’sNotes............................................ xxvii Chapter 1 Introduction 1. FluidMechanics ................................................. 1 2. UnitsofMeasurement ............................................ 2 3. Solids,Liquids,andGases ........................................ 3 4. ContinuumHypothesis ........................................... 4 5. TransportPhenomena ............................................ 5 6. SurfaceTension.................................................. 8 7. FluidStatics..................................................... 9 8. ClassicalThermodynamics........................................ 12 9. PerfectGas...................................................... 16 10. StaticEquilibriumofaCompressibleMedium ...................... 18 Exercises........................................................ 22 LiteratureCited.................................................. 24 SupplementalReading............................................ 24 Chapter 2 Cartesian Tensors 1. ScalarsandVectors............................................... 25 2. RotationofAxes:FormalDefinitionofaVector..................... 26 3. MultiplicationofMatrices ........................................ 29 4. Second-OrderTensor............................................. 30 5. ContractionandMultiplication .................................... 32 vii viii Contents 6. ForceonaSurface ............................................... 33 7. KroneckerDeltaandAlternatingTensor............................ 36 8. DotProduct ..................................................... 37 9. CrossProduct.................................................... 38 10. Operator∇:Gradient,Divergence,andCurl ........................ 38 11. SymmetricandAntisymmetricTensors............................. 40 12. EigenvaluesandEigenvectorsofaSymmetricTensor ................ 41 13. Gauss’Theorem ................................................. 44 14. Stokes’Theorem................................................. 47 15. CommaNotation................................................. 49 16. BoldfacevsIndicialNotation...................................... 49 Exercises........................................................ 50 LiteratureCited.................................................. 51 SupplementalReading............................................ 51 Chapter 3 Kinematics 1. Introduction ..................................................... 53 2. LagrangianandEulerianSpecifications............................. 54 3. EulerianandLagrangianDescriptions:TheParticleDerivative........ 55 4. Streamline,PathLine,andStreakLine ............................. 57 5. ReferenceFrameandStreamlinePattern............................ 59 6. LinearStrainRate................................................ 60 7. ShearStrainRate ................................................ 61 8. VorticityandCirculation.......................................... 62 9. RelativeMotionnearaPoint:PrincipalAxes........................ 64 10. KinematicConsiderationsofParallelShearFlows ................... 67 11. KinematicConsiderationsofVortexFlows.......................... 68 12. One-,Two-,andThree-DimensionalFlows ......................... 71 13. TheStreamfunction .............................................. 73 14. PolarCoordinates ................................................ 75 Exercises........................................................ 77 SupplementalReading............................................ 79 Chapter 4 Conservation Laws 1. Introduction ..................................................... 82 2. TimeDerivativesofVolumeIntegrals .............................. 82 3. ConservationofMass ............................................ 84 4. Streamfunctions:RevisitedandGeneralized ........................ 87 5. OriginofForcesinFluid.......................................... 88 6. StressataPoint.................................................. 90 Contents ix 7. ConservationofMomentum....................................... 92 8. MomentumPrincipleforaFixedVolume........................... 93 9. AngularMomentumPrincipleforaFixedVolume................... 98 10. ConstitutiveEquationforNewtonianFluid.......................... 100 11. Navier–StokesEquation .......................................... 104 12. RotatingFrame .................................................. 105 13. MechanicalEnergyEquation...................................... 111 14. FirstLawofThermodynamics:ThermalEnergyEquation ............ 115 15. SecondLawofThermodynamics:EntropyProduction ............... 116 16. BernoulliEquation ............................................... 118 17. ApplicationsofBernoulli’sEquation............................... 122 18. BoussinesqApproximation........................................ 124 19. BoundaryConditions............................................. 129 Exercises........................................................ 134 LiteratureCited.................................................. 136 SupplementalReading............................................ 137 Chapter 5 Vorticity Dynamics 1. Introduction ..................................................... 139 2. VortexLinesandVortexTubes .................................... 140 3. RoleofViscosityinRotationalandIrrotationalVortices.............. 141 4. Kelvin’sCirculationTheorem ..................................... 144 5. VorticityEquationinaNonrotatingFrame.......................... 149 6. VelocityInducedbyaVortexFilament:LawofBiotandSavart ....... 151 7. VorticityEquationinaRotatingFrame ............................. 152 8. InteractionofVortices ............................................ 157 9. VortexSheet..................................................... 161 Exercises........................................................ 161 LiteratureCited.................................................. 163 SupplementalReading............................................ 163 Chapter 6 Irrotational Flow 1. RelevanceofIrrotationalFlowTheory.............................. 165 2. VelocityPotential:LaplaceEquation ............................... 167 3. ApplicationofComplexVariables ................................. 169 4. FlowataWallAngle ............................................. 171 5. SourcesandSinks................................................ 173 6. IrrotationalVortex................................................ 174 7. Doublet ......................................................... 174 8. FlowpastaHalf-Body............................................ 175