Fluid Mechanics, Third 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 Third Edition Pijush K. Kundu OceanographicCenter NovaUniversity Dania,Florida Ira M. Cohen DepartmentofMechanicalEngineeringand AppliedMechanics UniversityofPennsylvania Philadelphia,Pennsylvania withachapteronComputationalFluidDynamicsbyHowardH.Hu AMSTERDAM (cid:1) BOSTON (cid:1) HEIDELBERG (cid:1) LONDON NEW YORK (cid:1) OXFORD (cid:1) PARIS (cid:1) SAN DIEGO SAN FRANCISCO (cid:1) SINGAPORE (cid:1) SYDNEY (cid:1) TOKYO ElsevierAcademicPress 525BStreet,Suite1900,SanDiego,California92101-4495,USA 84Theobald’sRoad,LondonWC1X8RR,UK Thisbookisprintedonacid-freepaper.(cid:1) ∞ Copyright©2004,ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyany means,electronicormechanical,includingphotocopy,recording,oranyinformation storageandretrievalsystem,withoutpermissioninwritingfromthepublisher. PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRights DepartmentinOxford,UK:phone:(+44)1865843830,fax:(+44)1865853333, e-mail:permissions@elsevier.com.uk.Youmayalsocompleteyourrequeston-linevia theElsevierhomepage(http://elsevier.com),byselecting“CustomerSupport” andthen“ObtainingPermissions.” LibraryofCongressCataloging-in-PublicationData AcataloguerecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN 0-12-178253-0 ForallinformationonallAcademicPresspublications visitourWebsiteatwww.academicpress.com PrintedintheUnitedStatesofAmerica 04 05 06 07 08 9 8 7 6 5 4 3 2 1 ThethirdeditionisdedicatedtothememoryofPijushK.Kunduandalsotomywife LindaanddaughtersSusanandNancywhohavegreatlyenrichedmylife. “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- dent at Penn State University.With Dr. John L.Lumleyashisadvisor,hestudiedinstabili- tiesofviscoelasticfluids,receivinghisdoctor- atein1972.Hebeganhislifelonginterestin oceanographysoonafterhisgraduation,workingasResearchAssociateinOceanog- raphyatOregonStateUniversityfrom1968until1972.Afterspendingayearatthe UniversitydeOrienteinVenezuela,hejoinedthefacultyoftheOceanographicCenter ofNovaSoutheasternUniversity,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 Preface ................................................... xvii PrefacetoSecondEdition ................................. xviii PrefacetoFirstEdition ..................................... xx Author’sNotes ........................................... xxiii 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................... 17 Exercises .................................................... 22 LiteratureCited .............................................. 23 SupplementalReading ........................................ 23 Chapter 2 Cartesian Tensors 1. ScalarsandVectors ........................................... 24 2. RotationofAxes:FormalDefinitionofaVector.................. 25 vii viii Contents 3. MultiplicationofMatrices..................................... 28 4. Second-OrderTensor ......................................... 29 5. ContractionandMultiplication................................. 31 6. ForceonaSurface............................................ 32 7. KroneckerDeltaandAlternatingTensor......................... 35 8. DotProduct.................................................. 36 9. CrossProduct ................................................ 36 10. Operator :Gradient,Divergence,andCurl..................... 37 ∇ 11. SymmetricandAntisymmetricTensors ......................... 38 12. EigenvaluesandEigenvectorsofaSymmetricTensor............. 40 13. Gauss’Theorem.............................................. 42 14. Stokes’Theorem.............................................. 45 15. CommaNotation ............................................. 46 16. BoldfacevsIndicialNotation .................................. 47 Exercises .................................................... 47 LiteratureCited .............................................. 49 SupplementalReading ........................................ 49 Chapter 3 Kinematics 1. Introduction.................................................. 50 2. LagrangianandEulerianSpecifications ......................... 51 3. EulerianandLagrangianDescriptions:TheParticleDerivative .... 53 4. Streamline,PathLine,andStreakLine.......................... 54 5. ReferenceFrameandStreamlinePattern ........................ 56 6. LinearStrainRate ............................................ 57 7. ShearStrainRate............................................. 58 8. VorticityandCirculation ...................................... 59 9. RelativeMotionnearaPoint:PrincipalAxes .................... 61 10. KinematicConsiderationsofParallelShearFlows................ 64 11. KinematicConsiderationsofVortexFlows ...................... 65 12. One-,Two-,andThree-DimensionalFlows...................... 68 13. TheStreamfunction........................................... 69 14. PolarCoordinates............................................. 72 Exercises .................................................... 73 SupplementalReading ........................................ 75 Contents ix Chapter 4 Conservation Laws 1. Introduction.................................................. 77 2. TimeDerivativesofVolumeIntegrals........................... 77 3. ConservationofMass......................................... 79 4. Streamfunctions:RevisitedandGeneralized..................... 81 5. OriginofForcesinFluid ...................................... 82 6. StressataPoint .............................................. 84 7. ConservationofMomentum ................................... 86 8. MomentumPrincipleforaFixedVolume........................ 88 9. AngularMomentumPrincipleforaFixedVolume................ 92 10. ConstitutiveEquationforNewtonianFluid ...................... 94 11. Navier–StokesEquation....................................... 97 12. RotatingFrame............................................... 99 13. MechanicalEnergyEquation .................................. 104 14. FirstLawofThermodynamics:ThermalEnergyEquation......... 108 15. SecondLawofThermodynamics:EntropyProduction............ 109 16. BernoulliEquation............................................ 110 17. ApplicationsofBernoulli’sEquation............................ 114 18. BoussinesqApproximation .................................... 117 19. BoundaryConditions ......................................... 121 Exercises .................................................... 126 LiteratureCited .............................................. 128 SupplementalReading ........................................ 128 Chapter 5 Vorticity Dynamics 1. Introduction.................................................. 129 2. VortexLinesandVortexTubes ................................. 130 3. RoleofViscosityinRotationalandIrrotationalVortices .......... 130 4. Kelvin’sCirculationTheorem.................................. 134 5. VorticityEquationinaNonrotatingFrame....................... 138 6. VelocityInducedbyaVortexFilament:LawofBiotandSavart. ... 140 7. VorticityEquationinaRotatingFrame.......................... 141 8. InteractionofVortices......................................... 146 9. VortexSheet ................................................. 149 Exercises .................................................... 150