Graduate Texts in Physics Michel Rieutord Fluid Dynamics An Introduction Graduate Texts in Physics Moreinformationaboutthisseriesat www.springer.com/series/8431 Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics,bothpureandapplied.ThesetextbooksservestudentsattheMS-orPhD-leveland theirinstructorsascomprehensivesourcesofprinciples,definitions,derivations,experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiringtimelyknowledgeof,aresearchfield. SeriesEditors ProfessorRichardNeeds CavendishLaboratory JJThomsonAvenue CambridgeCB30HE,UK [email protected] ProfessorWilliamT.Rhodes DepartmentofComputerandElectricalEngineeringandComputerScience ImagingScienceandTechnologyCenter FloridaAtlanticUniversity 777GladesRoadSE,Room456 BocaRaton,FL33431,USA [email protected] ProfessorSusanScott DepartmentofQuantumScience AustralianNationalUniversity ScienceRoad Acton0200,Australia [email protected] ProfessorH.EugeneStanley CenterforPolymerStudiesDepartmentofPhysics BostonUniversity 590CommonwealthAvenue,Room204B Boston,MA02215,USA [email protected] ProfessorMartinStutzmann WalterSchottkyInstitut TUMünchen 85748Garching,Germany [email protected] Michel Rieutord Fluid Dynamics An Introduction 123 MichelRieutord InstitutdeRechercheenAstrophysiqueetPlanétologie UniversitéPaulSabatier Toulouse France RevisedandexpandedtranslationfromtheFrenchlanguageeditionof:Uneintroductionàla DynamiquedesFluides,(cid:2)c 1997Masson,France. ISSN1868-4513 ISSN1868-4521(electronic) GraduateTextsinPhysics ISBN978-3-319-09350-5 ISBN978-3-319-09351-2(eBook) DOI10.1007/978-3-319-09351-2 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014958751 (cid:2)c SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The idea that guided the first French edition of the present book was to give to newcomers in Fluid Dynamics a presentation of the field that was anchored in Physics rather than in Applied Mathematics as it had been the case so often in thepast.Presently,however,connectionswithPhysicsaregettingstrongerandthis is fortunate. Indeed, Physics is, etymologically, the science of Nature and fluids occupya major place in Nature. They are everywherearoundus and their motion (their mechanics) influences our everyday life, at least through the weather. Any physicistcanhardlyescapebeingfascinatedbythesightofsomeremarkablefluid flowslikebreakingwavesorthegentlytravellingsmokering. TheconnectionbetweenFluidMechanicsandAppliedMathematicsiscertainly understandable by the very small number of equations that control a fluid flow. Thisisfascinatingforanappliedmathematician,especiallyifkeenonthetheoryof partialdifferentialequations.Actually,a few decadesago,expertisein asymptotic expansions,singularperturbations,andothermathematicaltechnicswasanecessary conditionto make progressin the theoryof fluid flows. But the pressureof maths hascertainlylessenedintherecenttimesbecauseofthestrong(exponential)growth ofnumericalsimulations.Itisnoweasiertoexperimentnumericallyafluidflowand getadetaileddescriptionofthesolutionsofNavier–Stokesequation.Interpretation oftheresultsmaychallengetheintuitionofthephysicistratherthantheskillofthe mathematician. But even in the pioneering times, when theoretical investigations of fluid flows were at the strength of the pencil, famous physicists like Newton, Maxwell,Kelvin,Rayleigh,Heisenberg,Landau,Chandrasekhar,andothersmade essential contributions to the field of Fluid Dynamics. As noted by Heisenberg himself,thetheoryofturbulenceawaitstobewritten,andthisisstillthecase. ThepresentbookisbasedonthelecturesIdeliveredatPaulSabatierUniversity in Toulouse during the last two decades. It is intended to beginners in the field andaimsatprovidingthemwiththenecessarybasisthatwillallowthemtoattack most of Fluid Dynamics questions. I have tried, as much as possible, to illustrate theconceptswithexamplestakeninnaturalsciences,ofteninAstrophysics,which is my playground. Some exercises are offered at the end of each chapter. The v vi Preface reader may thus check his/her understanding of the text. Some of the exercises are also meant to extend the subject in a different way. In that respect, I also give some references for further reading. As far as maths are concerned, the last chapter proposes some brief reminders or introduction to the mathematical tools that are used in the text. With the solutions of the exercises, the book should be self-contained. As far as teaching is concerned, the first four chapters constitute the bulk of aFluidMechanicsintroductiontothirdyearstudents.Thefourfollowingchapters weretypicallytaughttofourthyearstudents,whilepartofthelastonesarecurrently taughttostudentsabouttostarta Ph.D.Asthereaderwillnote,somesectionsare taggedwith(cid:3). Theycan beskippedatfirst readingandpresentotherillustrations ofthesubjectofthechapter. Endingthisshortpreface,Iwouldliketothankthemanycolleagueswhohave, byvariousmeans,contributedtotheachievementthatabookwritingrepresents.I wouldliketospeciallythankAlainVincentandHervéWillaimewhoprovidedme with original data of turbulent flows. I have much benefitted from the remarks of ArnaudAntkowiak,Pierre-LouisBlelly,BorisDintrans,KatiaFerrièreandThierry Roudier. They helped me very much at improving various parts of the work. I cannotforgetthatthis adventureof writingstarted,thanksto the supportand help of José-Philippe Pérez. I know that my wife Geneviève and my children Clément and Sylvain will forgiveme for the many hoursspent outside the real world. The realizationofthepresentbookowesmuchtothekindsupportofDr.RamonKhanna ofSpringer;Ithankhimverymuchforhisfaithintheproject.Finally,Ishouldthank themanystudentswhoattendedtheperformancewrittenbelow,theirquestionswere alwaysbeneficial,theirenthusiasmalwaysstimulatingandtheirfearchallengingfor theteacher. Toulouse,France MichelRieutord May2014 Contents 1 TheFoundationsofFluidMechanics..................................... 1 1.1 AShortHistoricalPerspective....................................... 1 1.2 TheConceptofaFluid............................................... 2 1.2.1 Introduction................................................. 2 1.2.2 ContinuousMedia.......................................... 2 1.3 FluidKinematics..................................................... 3 1.3.1 TheConceptofFluidParticle.............................. 3 1.3.2 TheLagrangianView....................................... 3 1.3.3 TheEulerianView.......................................... 4 1.3.4 MaterialDerivatives........................................ 4 1.3.5 DistortionofaFluidElement.............................. 5 1.3.6 IncompressibleFluids...................................... 8 1.3.7 TheStreamFunction ....................................... 9 1.3.8 EvolutionofanIntegralQuantityCarried bytheFluid ................................................. 10 1.4 TheLawsofFluidMotion........................................... 11 1.4.1 MassConservation ......................................... 11 1.4.2 MomentumConservation .................................. 14 1.4.3 EnergyConservation ....................................... 17 1.4.4 TheConstitutiveRelations................................. 19 1.5 TheRheologicalLaws ............................................... 19 1.5.1 ThePressureStress......................................... 19 1.5.2 ThePerfectFluid ........................................... 21 1.5.3 NewtonianFluids........................................... 22 1.6 TheThermalBehaviour.............................................. 26 1.6.1 TheHeatFluxSurfaceDensity ............................ 26 1.6.2 TheEquationsofInternalEnergyandEntropy ........... 27 vii viii Contents 1.7 Thermodynamics..................................................... 29 1.7.1 TheIdealGas............................................... 30 1.7.2 Liquids...................................................... 31 1.7.3 BarotropicFluids........................................... 31 1.8 BoundaryConditions................................................. 32 1.8.1 BoundaryConditionsontheVelocityField............... 32 1.8.2 BoundaryConditionsonTemperature..................... 35 1.8.3 SurfaceTension............................................. 35 1.8.4 InitialConditions........................................... 37 1.9 MoreAboutRheologicalLaws:Non-NewtonianFluids(cid:3)......... 37 1.9.1 TheLimitsofNewtonianRheology....................... 37 1.9.2 TheNon-NewtonianRheologicalLaws................... 38 1.9.3 LinearViscoelasticity ...................................... 39 1.9.4 TheNonlinearEffects...................................... 40 1.9.5 ExtensionalViscosities..................................... 41 1.9.6 TheSolid–FluidTransition................................. 45 1.10 AnIntroductiontotheLagrangianFormalism(cid:3) ................... 45 1.10.1 TheEquationsofMotion................................... 46 1.10.2 AnExampleoftheUseoftheLagrangian Formulation................................................. 47 1.11 Exercises.............................................................. 48 FurtherReading.............................................................. 49 References.................................................................... 49 2 TheStaticofFluids......................................................... 51 2.1 TheEquationsofStatic .............................................. 51 2.2 EquilibriuminaGravitationalField................................. 52 2.2.1 PascalTheorem............................................. 53 2.2.2 Atmospheres................................................ 54 2.2.3 AStratifiedLiquidBetweenTwoHorizontalPlates...... 56 2.2.4 RotatingSelf-gravitatingFluids(cid:3) ......................... 57 2.3 SomePropertiesoftheResultantPressureForce................... 60 2.3.1 ArchimedesTheorem....................................... 61 2.3.2 TheCentreofBuoyancy ................................... 62 2.3.3 TheTotalPressureonaWall............................... 63 2.4 EquilibriawithSurfaceTension..................................... 63 2.4.1 SomeSpecificFiguresofEquilibrium..................... 64 2.4.2 EquilibriumofLiquidWettingaSolid .................... 65 2.5 Exercises.............................................................. 66 FurtherReading.............................................................. 70 References.................................................................... 70 3 FlowsofPerfectFluids..................................................... 71 3.1 EquationsofMotions ................................................ 71 3.1.1 OtherFormsofEuler’sEquation .......................... 72 Contents ix 3.2 SomePropertiesofPerfectFluidMotions.......................... 72 3.2.1 Bernoulli’sTheorem........................................ 72 3.2.2 ThePressureField.......................................... 74 3.2.3 TwoExamplesUsingBernoulli’sTheorem............... 75 3.2.4 Kelvin’sTheorem........................................... 77 3.2.5 InfluenceofCompressibility............................... 79 3.3 IrrotationalFlows .................................................... 80 3.3.1 DefinitionandBasicProperties............................ 80 3.3.2 RoleofTopologyforanIrrotationalFlow ................ 81 3.3.3 Lagrange’sTheorem........................................ 82 3.3.4 TheoremofMinimumKineticEnergy .................... 83 3.3.5 ElectrostaticAnalogy....................................... 84 3.3.6 PlaneIrrotationalFlowofanIncompressibleFluid....... 85 3.3.7 ForcesExertedbyaPerfectFluid.......................... 88 3.4 FlowswithVorticity.................................................. 95 3.4.1 TheDynamicsofVorticity................................. 95 3.4.2 FlowGeneratedbyaDistributionofVorticity: AnalogywithMagnetism .................................. 97 3.4.3 ExamplesofVortexFlows ................................. 99 3.5 Problems.............................................................. 105 FurtherReading.............................................................. 109 References.................................................................... 109 4 FlowsofIncompressibleViscousFluids.................................. 111 4.1 SomeGeneralProperties............................................. 111 4.1.1 TheEquationsofMotion................................... 111 4.1.2 LawofSimilarity........................................... 112 4.1.3 Discussion .................................................. 114 4.2 CreepingFlows....................................................... 114 4.2.1 Stokes’Equation............................................ 114 4.2.2 VariationalPrinciple(cid:3)...................................... 115 4.2.3 FlowAroundaSphere...................................... 117 4.2.4 Oseen’sEquation........................................... 121 4.2.5 TheLubricationLayer...................................... 121 4.3 BoundaryLayerTheory.............................................. 125 4.3.1 PerfectFluidsandViscousFluids ......................... 125 4.3.2 MethodofResolution ...................................... 127 4.3.3 FlowOutsidetheBoundaryLayer......................... 127 4.3.4 FlowInsidetheBoundaryLayer........................... 128 4.3.5 SeparationoftheBoundaryLayer......................... 130 4.3.6 ExampleoftheLaminarBoundaryLayer: Blasius’Equation........................................... 131 4.4 SomeClassicExamples.............................................. 134 4.4.1 Poiseuille’sFlow............................................ 134 4.4.2 HeadLossinaPipe......................................... 137 4.4.3 FlowsAroundSolids....................................... 139