Fluid Dynamics: An Introduction

517 Pages·2015·8.295 MB·English

by  Michel Rieutord (auth.)

Graduate Texts in Physics
Michel Rieutord
Fluid
Dynamics
An Introduction
Graduate Texts in Physics
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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
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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