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FLUIDMECHANICS FluidMechanics AGeometricalPointofView S.G.Rajeev
DepartmentofPhysicsandAstronomy
DepartmentofMathematics
UniversityofRochester
Rochester,NY
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries
©S.G.Rajeev2018 Themoralrightsoftheauthorhavebeenasserted
FirstEditionpublishedin2018
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Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove
Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
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ISBN978-0-19-880502-1(hbk.)
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DOI:10.1093/oso/9780198805021.001.0001
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LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork.
Thisbookisdedicatedtomyteacher,A.P.Balachandran.
Preface Nexttocelestialmechanics,fluidmechanicsistheoldestpartoftheoreticalphysics. Eulerderivedthefundamentalequationsmorethan200yearsago.Yetitisfarfrom complete.WedonotyetknowwithmathematicalcertaintyifEuler’sequationshave regularsolutionsgivensmoothinitialdata.Moreimportanttophysics,thephenomenon ofturbulenceisstillmysterious.
Numericalmethodshavemademuchprogress(especiallyinapplicationstoengineering)inrecentyears.Theexponentialgrowthofcomputingpowerhasmadeitpossible todesignairplanes,submarinesandhouseholdapplianceswithoutcumbersometesting ofprototypes.Localweathercanbepredictedforaboutafortnight,afterwhicheventhe bestcomputersfail.Toolsfromstatisticalmechanicsandquantumfieldtheory(theareas thatroutinelydealwithaninfinitenumberofrandomvariables)oughttobeuseful.
Manyideasoftheoreticalphysics(e.g.,conformalinvariance)originatedinthestudy offluids.Abstractideas(suchasLiealgebras)appearhereinaconcreteandeasily visualizedsetting.InthisbookIwanttopresentfluidmechanicsasanonlinearclassical fieldtheory,anessentialpartoftheeducationofaphysicist.
Thecentralobjectoffluidmechanicsisavectorfield,thefluidvelocity.So,ageometric pointofviewisquitenatural.Atadeeperlevel,onecanunderstandEuler’sequationas ahamiltoniansystem,whosePoissonbracketsaredualtothecommutationrelationsof vectorfieldsandthehamiltonianisthekineticenergy(L 2 -norm).
ThisallowsonetothinkofEuler’sequationsforfluidsandhisequationforarigid bodyonthesamefooting:theybothdescribegeodesicsonaLiegroup.Ofcourse,the fluidismuchmorecomplicated:thegroupisinfinitedimensionalandthecurvature isnegative.Arnoldshowedthatthenotoriousinstabilitiesoffluidmechanicscanbe understoodintermsofthenegativecurvatureofitsgeometry.Althoughverynatural, thisneedssomemathematicalmachinery.Ihavetriedtopresentitinawayaccessibleto theoreticalphysicists.Clebsch’soldideaofusingcanonicalvariablesinfluidmechanics becomesusefulhere.
Moretraditionalsubjectslikeboundarylayertheory,vortexdynamics,andsurface wavesarealsoilluminatedbygeometry.ExpressingEulerandNavier–Stokesequations oncurvilinearcoordinatesisessential:mostboundariesofinterestarenotplanes.A littleRiemanniangeometrygoesalongwayhere.Thelineartheoryofinstabilities isunderstoodintermsofthespectrumofnon-normaloperators.HermannWeyl(of allpeople)foundanexcellentanalyticalapproximationfortheboundarycondition–Blasiustheoryofboundarylayers.Thevortexfilamentequationscanberelatedtothe Heisenbergmodelofmagnets;avortexinfluidmechanicsismappedtoasolitonofspin waves.
Throughout,Iwillpointtonumericalandanalyticalcalculations(mostlyusing Mathematica,butyoucanuseyourfavoritelanguage)togainunderstanding.Thisis notabookoncomputationalmethods.Youdon’tneedasupercomputeranymoretodo interestingwork,soeventhemosttheoreticalofuscanbenefitfromtheiruse.Switching fromananalytical/geometricalpointofviewtothemorepracticaldiscrete/numericalone andback,weunderstandbothbetter.
Studyingchaoticadvectionisagoodwayofdevelopingintuitionforfluidflowsand fordynamicalsystems.Aref(1984)givesasimpleexample,whichhasturnedouttobe usefulinengineeringdevicesthatmixfluidsefficiently.TheSmalehorseshoe,although notusuallythoughtofaspartoffluidmechanics,givesasolidlyestablishedmathematical modelofchaos.Manychaoticsystems(includingAref’s)haveaSmalehorseshoe embeddedinsidethem.SoIhaveincludedadiscussionofthesetopicsinanappendix. Therearemanyreasonstobelievethatrenormalizationisusefultounderstand turbulence.Althoughadiscussionofthosetheoriesisbeyondthescopeofthisbook, Ihaveincludedanappendixonmoreelementaryapplicationsofrenormalization:the IsingmodelandFeigenbaum’sapproachtodynamicalsystemsinonedimension.
Ihavenottriedtosurveyeverysub-fieldindetail:suchacomprehensivesurveywould beaboutasusefulasamapofacountryonascaleof1:1.Afewcasesareexaminedin detail.Theoreticalphysicsisbasedongeneralprinciples(conservationlaws,variational principles)andspecialcasesthatcanbeunderstoodanalyticallyorbysimplenumerical computations.Thetechniquesdevelopedthiswaycanbeadaptedtothosethatarisein applications.Thatsaid,mychoiceoftopicsisnecessarilysubjective.Theemphasison Lietheoryanddynamicalsystemsisunusualforabookatthislevel.
Muchremainsuncovered.Itwouldhavebeennicetotalkofquantumfluids.The greatmysteryofturbulenceandattemptstomodelitcouldhavebeenreviewed.Models ofweatherprediction,oceanography,andastrophysicaljetsareallthriving.Thewhole fieldofmagnetohydrodynamicsisgivenshortshrift.Eachofthesetopicswouldrequire abookofitsown.
Thisisasortofsequelto AdvancedMechanics Rajeev(2013).Aknowledgeof mechanics,linearalgebra(eigenvalueproblems)andpartialdifferentialcalculusisthe mainprerequisite.Thesectionswitha ∗ intheirnamecanbeskippedonafirstreading; theycontainmoreadvancedmaterial.
Thebookisaimedatphysics/mathematicsgraduatestudents;someengineering studentswillalsofindituseful.Thereissomeoverlapandsomerepetition,becausesome ideas(e.g.,dynamicalsystems,Liealgebras)aresorecurrentinphysics.Othersareonly outlined,andworkingthemoutyourselfisanessentialpartoflearningthesubject.
Acknowledgments Icannotthankmywifeandchildrenenoughfortheirsupportovertheyears.A.P. Balachandrantaughtmuchmorethanphysics:theaudacitytogointonewareas.Thanks totheindulgenceofmycolleaguesinPhysicsandMathematicsDepartmentsforletting meexploredirectionsthatarenotprofitableintheshortterm.
IamgratefultoA.Kar,G.KrishnaswamiandM.Bhattacharyaforcommentingon partsofthemanuscript.ThanksfordiscussionswithA.Iosevichonfractals;D.Geba onregularityofNavier–Stokes;V.V.Sreedharonconformalinvariance;andV.P.Nair andT.Padmanabhanoncountlesstopicsofphysicsinourearlyyears.ThankstoSonke AdlungforencouragingmetowritethisbookandtoHarrietKonishiformanyhelpful suggestions.ThanksalsotoAlanSkullandLydiaShinojfortheexcellentediting.
1.1Thevelocityfield1
1.2Space-timeapproach2
1.3EulerianvsLagrangianpicture3
1.4Integralcurves3
1.5Themethodofcharacteristics4
1.6Conservationlaw5
1.7Densitiesvsscalars6
1.8Steadyflows8
1.9Incompressibleflow8
1.10Irrotationalflow8
1.11Irrotationalandincompressibleflow9
1.12Jacobimatrix10
1.13Flownearafixedpoint11 2Euler’sEquations 14
2.1Conservationofmomentum14
2.2Thestresstensor15
2.3Incompressibleflow16
2.4Conservationofenergy17
2.5Helmholtzequation18
2.6Steadyflows18
2.7Equationsofstate19
2.8TransportformofEuler’sequation21
2.9LinearizationofEuler’sequations:sound24
2.10InviscidBurgersequation25
2.11Scaleinvariance26
2.12d’Alembert’sparadox:limitationstheidealfluidmodel28
3TheNavier–StokesEquations 30
3.1Viscosity30
3.2Viscousincompressibleflow33
3.3Dissipationofenergyatconstantdensity33
3.4Dissipationofenergyforcompressibleflows34
xii Contents
3.5Scaleinvariance:Reynoldsnumber35
3.6Navier–Stokesincurvilinearcoordinates36
3.7Diffusionandadvection37
3.8Thediffusionkernel38
3.9Growthofentropyindiffusion41
3.10Theadvection–diffusionkernel42
4IdealFluidFlows 45
4.1Statics45
4.2SolutionsofLaplace’sequation49
4.3Complexanalyticmethods51
4.4Fluidwithastirrer52
4.5Flowpastacylinder56
4.6Thed’Alembertparadox58
4.7Joukowskiairfoil59 4.8Surfacewaves60
5ViscousFlows 64
5.1PipePoiseuilleflow64
5.2CircularCouetteflow65
5.3Stokesflow66
5.4Stokesflowpastasphere66
5.5Vortexwithdissipatingcore71
6Shocks 72
6.1TheBurgersequations72
6.2TheCole–Hopftransformation74
6.3Thelimitofsmallviscosity78
6.4Maxwell–Lax–Oleneikminimumprinciple78
6.5TheRiemannproblem80
7BoundaryLayers 83
7.1Prandtl’stheory83
7.2TheBlasiusreduction84
7.3Weyl’smethod88
7.4Dragonaflatplate89
7.5Limitationsofboundarylayertheory90
8Instabilities 91
8.1TheRayleigh–Taylorinstability91
8.2LinearizationofNavier–Stokesequations94
8.3Orr–Sommerfeldequation95
8.4Transientsolutionsoflinearequations99
8.5Normaloperators100
8.6Anon-normaloperator101
8.7Anonlinearmodelwithtransients102
8.8Stabilityregained104
8.9Rapidlychangingexternalforce105
8.10TheKapitzapendulum107
9IntegrableModels 109
9.1KdV109
9.2Thesolitonsolution110
9.3Multi-solitonsolutions111
9.4Laxpair111
9.5HamiltonianformalismofFadeevandZakharov113
9.6ThehamiltonianformalismofMagri114
9.7Thevortexfilament114
9.8Geometryofcurves115
9.9Velocityfromvorticity:theBiot–Savartlaw116
9.10Thevelocityfieldofavortexfilament117
9.11Regularizationandrenormalization118
9.12RelationtotheHeisenbergmodel121
10HamiltonianSystemsBasedonaLieAlgebra 123
10.1Rigidbodymechanics123
10.2Vectorspaces124
10.3Liealgebra126
10.4TheVirasoroalgebra133
10.5Hamiltonianforthetwo-dimensionalEulerequations135
10.6Spectraldiscretizationoftwo-dimensionalEuler138
10.7Clebschvariables139
10.8Hamiltonianformof3DEulerequations140
10.9Poissonbracketsofvelocity142
10.10Analogywithangularmomentum143
11CurvatureandInstability 144
11.1Riemanniangeometry144
11.2Covariantderivative145
11.3Geodesicdeviationandcurvature146
11.4Curvatureasabi-quadratic148
11.5Addingdissipation149
11.6Liegroups151
11.7InfinitedimensionalLiegroups152
11.8Geometryofleft-invariantmetrics152
11.9Geodesicsonagroupmanifold153
xiv Contents
11.10Covariantderivativeonagroup155
11.11Curvatureofaleft-invariantmetric156
11.12Geodesicson SO(3)157
11.13Thediffeomorphismgroup158 11.14Liealgebraofvectorfields160 11.15Diffeomorphismsofthecircle160
11.16The L2 -metricforvectorfieldson R3 161
11.17Incompressiblediffeomorphisms162 11.18Curvatureofthediffeomorphismgroup164
12Singularities 166
12.1Norms: L 2 , L p ,Sobolev167 12.2Thedissipationofenergy169 12.3SolutionofNavier–Stokesbyperturbationtheory170 12.4Leray:finitetimeregularity171 12.5Scaleinvariantsolutions173
13SpectralMethods 175
13.1TheChebychevbasis175 13.2Spectraldiscretization176 13.3Sampling177 13.4Interpolation178 13.5Differentiation178 13.6Integration179 13.7ThebasicODE180 13.8Downsampling181 13.9SpectralsolutionoftheOrr–Sommerfeldequation183 13.10Higherdimensions185
14FiniteDifferenceMethods 186
14.1Differentialanddifferenceoperatorsinonedimension186 14.2Padéapproximant188 14.3Boundaryvalueproblems189 14.4Explicitschemeforthediffusionequation190 14.5Numericalstability192 14.6Implicitschemes192 14.7Physicalexplanation194 14.8ThePoissonequation195 14.9DiscreteversionoftheClebschformulation197 14.10Radialbasisfunctions199
14.11ALagrangiandiscretization201
15GeometricIntegrators 203
15.1Liegroupmethods203 15.2Exponentialcoordinates204
AppendixADynamicalSystems 209
A.1Jacobimatrixatafixedpoint209
A.2Stableandunstablemanifolds210
A.3Arnold’scatmap211
A.4Thehomoclinictangle213
A.5TheSmalehorseshoe214
A.6Binarycode215
A.7Iteratedfunctionsystemsontheinterval217
A.8Normalformofthehorseshoe219
C.1TheIsingmodel228
C.2Transfermatrix230
C.3Renormalizationdynamics231
C.4Cayleytree234
C.5SpontaneousmagnetizationoftheIsingmodelontheCayleytree236
C.6TheIsingmodelonsquareandcubiclattices239
C.7Iterationsofafunction239
C.8Feigenbaum’srenormalizationdynamics241
C.9TheFeigenbaum–Cvitanonicequation241
Listoffigures 1.1Integralcurveofafoliation13
4.1Densityofaself-gravitatingfluid48
4.2Flowaroundastirrerinthehalf-plane53
4.3Flowaroundastirrerinsideacircularboundary55
4.4Flowaroundacylinder57
4.5FlowaroundaJoukowskiairfoil60
4.6Groupvelocityofsurfacewaves63
6.1Velocityatashock75
6.2Densityatthesameshock75
6.3SolutiontotheBurgersequation80
7.1TheBlasiussolution87
8.1Solutiontoanon-normaloperator102
8.2Transients103
9.1Vortexfilamentsoliton121
13.1Spectralmethodforanoscillator182
13.2Orr–Sommerfeldspectrum184
14.1TheAdams–Bashforthmethod189
14.2FDMforaboundaryvalueproblem191
14.3Thestableregioninimplicitschemes193
14.4Implicitvsexplicitschemes194
14.5ExactsolutionofaPoissonequationcomparedtothenumericalsolution196
14.6RBFinterpolation200
A.1ManifoldsofArnold’scatmap212
A.2Aglimpseofahomoclinictangle213
A.3Thehorseshoeofthecatmap214
A.4Thehorseshoemap216
A.5Periodicorbitofasequence223
B.1Chaoticadvection226
C.1RenormalizationoftheIsingmodel233
C.2Cayleytree235
C.3SpontaneousmagnetizationoftheCayley–Isingmodel237
VectorFields Afluidiscomposedofalargenumberofmolecules.Thesemoleculesareinrapidmotion, eachinadifferentdirection.Theycollidewitheachother,whichtendstorandomizethe molecularvelocities.Inthefluidapproximation,wethinkofthesystemascomposedof “fluidelements”whicharelargeenoughtocontainamultitudeofmoleculesbutstill smallcomparedtothesizeofthevesselcontainingthem.
Theaveragevelocityofafluidelementwillbemuchsmallerthantheindividual molecularvelocities.Thedistancebetweenmoleculesissosmallthatwecanregardthe densityandvelocityofafluidascontinuousfunctionsofspaceandtime.Afunction inspace-timeiscalledafield.Thus,thefluidpressureisascalarfieldwhilefluid velocityisavectorfield.Otherexamplesofscalarfieldsaretemperature,entropy,and theconcentrationofsomechemicalpollutantcarriedbythefluid.
1.1Thevelocityfield Let xi for i = 1,2,3bethecoordinatesofsomepointwithinthefluid.Tobeginwith, thinkofthemasCartesiancoordinates.Later,wewillseethatcurvilinearco-ordinates workjustaswell.Wewillfollowtheconventionofgeometryinwritingtheindexon coordinatesassuperscripts. vi (x, t ) arethecomponentsofthefluidvelocityattime t and position x (sometimeswewillomittheindexonthecoordinate).Thismeansthatafluid elementat xi attime t willmoveto xi + vi (x, t ) atthenextinstant t + ,where isan infinitesimallysmalltimeinterval.
Givenascalarfield f (x, t ),therearetwonotionsoftimederivativethatareimportant influidmechanics.Theobviousoneisthepartialderivative,
inwhichwelookatthechangeintimeatafixedlocationinthefluidelement.Theother isthetotalderivativeormaterialderivative
1.2Space-timeapproach Inrelativisticphysics,wemustthinkofphysicalquantitiesasfunctionsofspace-time. Thisisafour-dimensionalmanifoldwhosecoordinates xμ arethethreecoordinatesof space xi , i = 1,2,3plustime t ,whichisusuallythoughtofasthezerothcoordinate ofspace-time x0 = t .Thispointofviewisalsoconvenientinsomenon-relativistic situations,1 includingfluidmechanics.Ifweset v0 = 1:
Thusthevelocityfieldcanbethoughtasafirstorderdifferentialoperator(i.e.,the materialderivative)inspace-time:
Evenifthefluidelementsareatrest, vi = 0,theyaremovingforwardintime.Evenif thefluidismoving,therateatwhichitmovesforwardintimeisunaffected.(Thisisthe non-relativisticapproximation).Thecoordinates (t , x) donotneedtobeCartesian;and theydonotneedtobemeasuredinaninertialframe.Wecanmakeany(possiblytimedependent)transformationofthespacecoordinates.Butthetimecoordinateshouldnot change,becauseinthenon-relativisticlimitallobserversagreeonthedefinitionoftime. Ofcourse,thereisatheoryofrelativisticfluids(PoissonandWill,2014).Thisbook happensnottobeaboutthat.
Applyingthechainruleofdifferentiation
1 RecallthetitleofFeynman’sclassicpaper,
Integralcurves 3 and
Thustime-dependenttransformationsofthetypeweareconsideringpreservethe conditionthatthetimecomponentofvelocityisone.
1.3EulerianvsLagrangianpicture Innon-relativisticmechanicsthereissomethingspecialaboutaninertialreferenceframe: Newton’slawsasoriginallystatedholdinsuchframes.Influidmechanicsthisiscalled theEulerframeortheEulerianpicture.AtransformationfromoneEulerianframeto anotheristime-independent.Itcanstilltransformspacecoordinatesinany(possibly nonlinear)way.
Ifweallowtime-dependenttransformations,wecanevenchoose φ i suchthatthefluid velocityiszeroeverywhereinthenewsystem.Thenewsystemwouldbeco-movingwith thefluid:anon-inertialreferenceframe.ThisistheLagrangianpicture.TheEulerianand Lagrangianpicturesarerelatedbythetransformationsatisfyingthedifferentialequation obtainedbysetting ˜ vi = 0ineqn(1.1):
WewillmostlysticktotheEulerianpicture,asdynamicallawsoffluidmotion(suchas theEulerequations)areeasiesttounderstandfromthispointofview.Butsomepersistent structuresoffluids(vortices,jetstreams)areeasiertoseeintheLagrangianpicture. Sometimeswewillswitchtothispicture.
1.4Integralcurves Imagineaspeckofdustcarriedalong(oradvected)bythefluid.Itsposition ξ i (t ) will changewithtimeaccordingtothedifferentialequation
Givenitsinitialposition, ξ i (0),thisdifferentialequationcanbesolvedtodetermine thepathfollowedbythedustparticle.Geometrically,thisisacurvestartingat ξ i (0) whosetangentateachpointisthevelocityvectoratthatpoint(andinstantoftime). Theprocessofsolvingadifferentialequationisakindofintegration.So ξ i (t ) iscalled
theintegralcurveofthevectorfield vi (x, t ).Exactlyonesuchcurvepassesthrough everypointinspace.Piecingtogetheralltheintegralcurves,wehaveafunction i (x, t ) satisfying
andtheinitialcondition
1.5Themethodofcharacteristics Ascalarfieldissaidtobeadvectedbytheflow vi (x, t ) ifitsatisfies
Thatis,itisconstantalongthepathofaparticlecarriedalongbytheflow.Supposewe aregiventheinitialvalueofthisscalarfield f0 (x) = f (x,0) andwewanttopredictwhat willbeitsvalueatsomelatertime t .Wemustthenstartatthepoint x attime t andtrace backalongtheintegralcurveof v tofindthepoint η(x, t ) whereitwasattimezero.You canseethatthisistheinverseoftheproblemwediscussedinthesectiononintegral curves:
(η(x, t ), t ) = x
Soifweknowalltheintegralcurves (x, t ) andcanfindtheinversefunctionabove, wecansolvetheequationforadvectedscalarfields
f (x, t ) = f0 (η(x, t ))
Since (x, t ) isdeterminedby v(x, t ) ,soisitsinverse η(x, t ).Forexample,if v(x, t ) = v aconstant,thesolutionis f (x, t ) = f0 (x vt ) ascanbeeasilyverified.
Thismethodofusingasystemofordinarydifferentialequations(ODEs)tosolve apartialdifferentialequation(PDE)isaparticularcaseofthemethodofcharacteristics.See(CourantandHilbert,1962),Chapter1foranexpositionofthe generalcase.
Whenthevectorfieldistime-independent(steady)wecanexpectasolutiontothe advectionequationthatisalsosteady.Thatis,
Example1.1 If v(x) = x2 , x1 theintegralcurvesarecircles.Asteadysolutiontotheadvectionequation isanyfunctionthatisrotationinvariant.
Thepathsofadvectedparticlescanbequitecomplicated;ingeneraltheycanbe determinedonlybynumericalintegration.
Example1.2 Thedoublegyre Theparameter controlsthetimedependence;ifitiszerothevelocityisindependentoftime.
Exercise1.1 WriteaprograminMathematica(oryourownfavoritelanguage)which plotstheintegralcurvesofthedoublegyreforvariousvaluesof andtheinitial point.Asmallchangeintheinitialconditioncanleadtoabigchangeintheoutcome afterenoughtime.Ananimationwillbemoreinterestingthanastaticplot.The values α = 1.1, = 0.25aregoodchoices.
1.6Conservationlaw Thetotalmasscontainedwithinsomeregion V ofthefluidis V ρ(x, t )dx.Sincemass isconservedinnon-relativisticphysics,thechangeofthisquantitymustbeequaltothe inflowofmassintotheregionthroughitsboundary.
Here dSi istheareaofaninfinitesimalelementontheboundary,whichisthoughtof asavectorpointingalongtheoutwardnormal.AppealingtoGauss’theorem
Thustheconservationofmassbecomes(sometimeswewillsuppressthearguments offunctionsforsimplicityofnotation)
Sincethisholdsforanyregion V wemusthave
Thesameargumentholdsforanyscalarquantitythatisconserved.Forexample,supposeourfluidisasolutionoftwokindofmoleculesthatdonotinteractchemicallywith eachother.Thentheirnumberdensities ρ1 (x, t ), ρ2 (x, t ) willbeseparatelyconserved.
Suppose
istherelativeconcentrationofonetypeofmolecule comparedtotheother.Fromtheconservationlawswecandeducethat
Thatis,therelativeconcentrationisanadvectedscalarfield.Thisisoneofthereasons whyadvectedscalarsareinterestinginfluidmechanics.
1.7Densitiesvsscalars Notallphysicalquantitiesdescribedbyasinglerealnumberateachpointarescalars.For example,theycouldbedensities.Thedifferenceisinhowtheychangeunderacoordinate transformation.Ifwemakeachangeofcoordinates(assumedtobetime-independent forsimplicity)
ascalartransformsinasimpleway:
where φ 1 istheinversetransformation.
Ifthetransformationisinfinitesimal(i.e.,differsfromtheidentitybyaninfinitesimally smallquantitytimesavectorfield),
thechangeinthefunctionisalsosmall
Notethatthechangeis(uptoafactor ) thedirectionalderivativeofthefunction alongavectorfieldwhosecomponentsare vi .Vectorfieldsareinfinitesimaltransformationsofspace.Thechangeofanyquantityundertheactionofavectorfieldiscalledits Liederivative.Forascalaritisjustthedirectionalderivative:
Adensitydescribestheamountofsomephysicalquantity(mass,charge,etc.)in asmallvolume.Soitisthecombination ρ(x)d 3 x whichisinvariantundercoordinate transformations.
Recallthat
.Infinitesimally,
Thus,theinfinitesimalchangeofadensityunderavectorfieldisthecoefficientof in theabove
