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2.TheNavier–StokesandEulerEquations–FluidandGasDynamics

Fluidandgasdynamicshaveadecisiveimpactonourdailylives.Therearethe nedropletsofwaterwhichsprinkledowninourmorningshower,thewaveswhichwefaceswimmingorsur ngintheocean,theriverwhichadaptstothetopographybyformingawaterfall,theturbulentaircurrentswhichoftendisturbourtransatlantic ightinajetplane,thetsunami1whichcanwreckanentireregionofourworld,theathmospheric owscreatingtornados2andhurricanes3,thelive-giving owofbloodinourarteriesandveins4…Allthese owshaveagreatcomplexityfromthegeometrical,(bio)physicaland(bio)mechanicalviewpointsandtheirmathematicalmodelingisahighlychallengingtask.

Clearly,thedynamicsof uidsandgasesisgovernedbytheinteractionoftheiratoms/molecules,whichtheoreticallycanbemodeledmicroscopically,i.e.byindividualparticledynamics,relyingonagrandHamiltonianfunctiondependingon3Nspacecoordinatesand3Nmomentumcoordinates,whereNisthenumberofparticlesinthe uid/gas.NotethattheNewtonianensembletrajectorieslivein6Ndimensionalphasespace!FormostpracticalpurposesthisisprohibitiveanditisessentialtocarryoutthethermodynamicBoltzmann–Gradlimit,which–undercertainhypothesisontheparticleinteractions–givestheBoltzmannequationofgasdynamics(seeChapter1onkineticequations)fortheevolutionoftheeffectivemassdensityfunctionin6-dimensionalphasespace.

Undertheassumptionofasmallparticlemeanfreepath(i.e.inthecolli-siondominatedregime)afurtherapproximationispossible,leadingtotime-dependentmacroscopicequationsinpositionspaceR3,referredtoasNavier–StokesandEulersystems.Thesesystemsofnonlinearpartialdifferentialequa-tionsareabsolutelycentralinthemodelingof uidandgas ows.

Formore(precise)informationonthismodelinghierarchywereferto[3].TheNavier–Stokessystem5waswrittendowninthe19thcentury.ItisnamedaftertheFrenchengineerandphysicistClaude–LuisNavier6andtheIrishmath-ematicianandphysicistGeorgeGabrielStokes7.

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Fig.2.1.IguassuFalls,BorderofBrazil-Argentina

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Fig.2.2.IguassuFalls,BorderofBrazil-Argentina

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Undertheassumptionofincompressibilityofthe uidtheNavier–Stokesequations,determiningthe uidvelocityuandthe uidpressurep,read:

u+(u·grad)u+gradp=νΔu+fdivu=0

HerexdenotesthespacevariableinR2orR3dependingonwhether2or3dimensional owsaretobemodeledandt>0isthetimevariable.Thevelocity eldu=u(x,t)(vector eldonR2or,resp.,R3)isinR2orR3,resp.,andthepressurep=p(x,t)isascalarfunction.f=f(x,t)isthe(given)externalforce eld(againtwoand,resp.three-dimensional)actingonthe uidandν>0thekinematicviscosityparameter.ThefunctionsuandparethesolutionsofthePDEsystem,the uiddensityisassumedtobeconstant(say,1)hereasconsistentwiththeincompressibilityassumption.ThenonlinearNavier–Stokessystemhastobesupplementedbyaninitialconditionforthevelocity eldandbyboundaryconditionsifspatiallycon ned uid owsareconsidered(orbydecayconditionsonwholespace).Atypicalboundaryconditionistheso-calledno-slipconditionwhichreads

u=0

ontheboundaryofthe uiddomain.

Theconstraintdivu=0enforcestheincompressibilityofthe uidandservestodeterminethepressurepfromtheevolutionequationforthe uidvelocityu.

Ifν=0thenthesocalledincompressibleEuler8equations,validforverysmallviscosity ows(ideal uids),areobtained.NotethattheviscousNavier–StokesequationsformaparabolicsystemwhiletheEulerequations(inviscidcase)arehyperbolic.TheNavier–StokesandEulerequationsarebasedonNew-ton’scelebratedsecondlaw:forceequalsmasstimesacceleration.Theyareconsistentwiththebasicphysicalrequirementsofmass,momentumandenergyconservation.

TheincompressibleNavier–StokesandEulerequationsallowaninterestingsimpleinterpretation,whentheyarewrittenintermsofthe uidvorticity,de nedby

ω:=curlu.

Clearly,theadvantageofapplyingthecurloperatortothevelocityequationistheeliminationofthepressure.Inthetwo-dimensionalcase(whenvorticitycanberegardedasascalarsinceitpointsintothex3directionwhenu3iszero)weobtain

Dω=νΔω+curlf,Dt

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whereDg

denotesthematerialderivativeofthescalarfunctiong:

Dg=gt+u.gradg.Dt

Thus,fortwo-dimensional ows,thevorticitygetsconvectedbythevelocity eld,isdiffusedwithdiffusioncoef cientνandexternallyproduced/destroyedbythecurloftheexternalforce.Forthreedimensional owsanadditionaltermappearsinthevorticityformulationoftheNavier–Stokesequations,whichcorrespondstovorticitydistortion.

TheNavier–StokesandEulerequationshadtremendousimpactonappliedmathematicsinthe20thcentury,e.g.theyhavegivenrisetoPrandtl’s9boundarylayertheorywhichisattheoriginofmodernsingularperturbationtheory.NeverthelesstheanalyticalunderstandingoftheNavier–Stokesequationsisstillsomewhatlimited:Inthreespacedimensions,withsmooth,decaying(inthefar eld)initialdatumandforce eld,aglobal-in-timeweaksolutionisknowntoexist(Leraysolution10),howeveritisnotknownwhetherthisweaksolutionisuniqueandtheexistence/uniquenessofglobal-in-timesmoothsolutionsisalsounknownforthree-dimensional owswitharbitrarilylargesmoothinitialdataandforcing elds,decayinginthefar eld.Infact,thisispreciselythecontentofaClayInstituteMillenniumProblem11withanawardofUSD1000000!!Averydeeptheorem(see[2])provesthatpossiblesingularitysetsofweaksolutionsofthethree-dimensionalNavier–Stokesequationsare‘small’(e.g.theycannotcontainaspace-timecurve)butithasnotbeenshownthattheyareempty…

Weremarkthatthetheoryoftwodimensionalincompressible owsismuchsimpler,infactsmoothglobal2 dsolutionsexistforarbitrarilylargesmoothdataintheviscidandinviscidcase(see[6]).

Whyisitsoimportanttoknowwhethertime-globalsmoothsolutionsoftheincompressibleNavier–Stokessystemexistforallsmoothdata?Ifsmoothnessbreaksdownin nitetimethen–closetobreak-downtime–thevelocity elduofthe uidbecomesunbounded.Obviously,weconceive owsofviscousreal uidsassmoothwithalocally nitevelocity eld,sobreakdownofsmoothnessin nitetimewouldbehighlycounterintuitive.Hereournaturalconceptionoftheworldsurroundingusisatstake!

ThetheoryofmathematicalhydrologyisadirectimportantconsequenceoftheNavier–Stokesor,resp.,Eulerequations.The owofriversingeneral–andinparticularinwaterfallslikethefamousonesoftheRioIguassuontheArgentinian-Brazilianborder,oftheOranjeriverintheSouthAfricanAugra-biesNationalParkandothersshownintheFigs.2.1–2.6,areoftenmodeledbythesocalledSaint–Venantsystem,namedaftertheFrenchcivilengineer9

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Fig.2.3.IguassuFalls,BorderofBrazil-Argentina

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Fig.2.4.AugrabiesFalls,SouthAfrica

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Adh´emarJeanClaudeBarr´edeSaint–Venant12.Themainissueistoincor-poratethefreeboundaryrepresentingtheheight-over-bottomh=h(x,t)ofthewater(measuredverticallyfromthebottomoftheriver).LetZ=Z(x)betheheightofthebottomoftherivermeasuredverticallyfromacon-stant0-levelbelowthebottom(thusdescribingtheriverbottomtopogra-phy),whichinthemostsimplesettingisassumedtohaveasmallvariation.NotethatherethespacevariablexinR1orR2denotesthehorizontaldi-rection(s)undu=u(x,t)thehorizontalvelocitycomponent(s),theverticalvelocitycomponentisassumedtovanish.Thedependenceontheverticalco-ordinateentersonlythroughthefreeboundaryh.Then,undercertainassump-tions,mostnotablyincompressibility,vanishingviscosity,smallvariationoftheriverbottomtopographyandsmallwaterheighth,theSaint–Venantsystemreads:

h+div(hu)=0 g (hu)+div(hu u)+gradh2+ghgradZ=02

Heregdenotesthegravityconstant.Notethath+Zisthelocallevelofthewatersurface,measuredverticallyagainfromtheconstant0-levelbelowthebottomoftheriver.Foranalyticalandnumericalworkon(evenmoregeneral)Saint–Venantsystemswerefertothepaper[4].Spectacularsimulationsofthebreakingofadamandofriver oodingusingSaint–VenantsystemscanbefoundinBenoitPerthame’swebpage13.

Manygas owscannotgenericallybeconsideredtobeincompressible,par-ticularlyatsuf cientlylargevelocities.Thentheincompressibilityconstraintdivu=0onthevelocity eldhastobedroppedandthecompressibleEulerorNavier–Stokessystems,dependingonwhethertheviscosityissmallornot,havetobeusedtomodelthe ow.

Herewestatethesesystemsunderthesimplifyingassumptionofanisentropic ow,i.e.thepressurepisagivenfunctionofthe(nonconstant!)gasdensity:p=p(ρ),wherepis,say,anincreasingdifferentiablefunctionofρ.UnderthisconstitutiveassumptionthecompressibleNavier–Stokesequationsread:

ForacomprehensivereviewofmodernresultsonthecompressibleNavier–Stokesequationswerefertothetext[5].

ForthecompressibleEulerequations,obtainedbysettingλ=0andν=0,globallysmoothsolutionsdonotexistingeneral.Considertheone-dimensional12

13ρt+div(ρu)=0(ρu)t+div(ρu u)+gradp(ρ)=νΔu+(λ+ν)grad(divu)+ρf.Hereλisthesocalledshearviscosityandν+λisnon-negative.http://www-groups.dcs.st-and.ac.uk/～history/Mathematicians/Saint–Venant.htmlhttp://www.dma.ens.fr/users/perthame/

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case,thesocalledp-system,withoutexternalforce:

ρt+(ρu)x=0 2 (ρu)t+ρu+p(ρ)x=0

Thisisanonlinearhyperbolicsystem,degenerateatthevacuumstateρ=0.ForanextensivestudyoftheRiemannproblemwereferto[7]andforthepioneeringproofofglobalweaksolutions,usingentropywavesandcompensatedcompactness,to[8].

Finally,weremarkthattheincompressibleinviscidSaint–VenantsystemofhydrologyhasthemathematicalstructureofanisotropiccompressibleEulersystemwithquadraticpressurelawin1or,resp.,2dimensions,wherethespatialground uctuationsplaytheroleofanexternalforce eld.

CommentsontheFigs.2.1–2.4AnimportantassumptioninthederivationoftheSaint–VenantsystemfromtheEuleror,resp.,Navier–Stokesequations–apartfromtheshallowwatercondition–isasmallnessassumptiononthevariationofthebottomtopography,i.e.gradZhastobesmall.Clearly,thisrestrictstheapplicabilityofthemodel,inparticularitsuseforwaterfallmodelling.Recently,anextensionoftheSaint–Venantsystemwaspresentedin[1],whicheliminatesallassumptionsonthebottomtopography.TherethecurvatureoftheriverbottomistakenintoaccountexplicitelyinthederivationfromthehydrostaticEulersystem(assumingasmall uidvelocityinorthogonaldirectiontothe uidbottom).WeremarkthatextensionsoftheSaint–Venantmodelstogranular ows(likedebrisavalanches)existintheliterature,seealso[1].

CommentsontheFigs.2.5–2.6Turbulent ows14arecharcacterizedbyseem-inglychaotic,randomchangesofvelocities,withvorticesappearingonavarietyofscales,occurringatsuf cientlylargeReynoldsnumber15.Non-turbulent owsarecalledlaminar,representedbystreamline ow,wheredifferentlayersofthe uidarenotdisturbedbyscaleinteraction.Simulationsofturbulent owsarehighlycomplicatedandexpensivesincesmallandlargescalesinthesolutionsoftheNavier–Stokesequationshavetoberesolvedcontemporarily.Varioussimpli-fyingattempts(‘turbulencemodeling’)exist,typicallybasedontime-averagingtheNavier–Stokesequationsandusing(moreorless)empiricalclosurecon-ditionsforthecorrelationsofvelocity uctuations.The owsdepictedintheFigs.2.5and2.6arehighlyturbulent,withapparentmicro-scales.

Weremarkthattheturbulentpartsofthe owsdepictedintheFigs.2.1–2.6aretwo-phase ows,duetotheairbubblesentrainedclosetothefreewater-airsurfaceinteractingwiththeturbulentwater ow.

Fig.2.5.TurbulentFlow,CascadadeAguaAzul,Chiapas,Mexico

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Fig.2.6.Turbulent

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References

[1]F.Bouchut,A.Mangeney-Castelnau,B.PerthameandJ.-P.Vilotte,Anew

modelofSaintVenantandSavage–Huttertypeforgravitydrivenshallowwater ows,C.R.Acad.Sci.Paris,Ser.I336,pp.531–536,2003

[2]L.Caffarelli,R.Kohn,andL.Nirenberg,Partialregularityofsuitableweak

solutionsoftheNavier–Stokesequations,Comm.Pure&Appl.Math.35,pp.771–831,1982

[3]C.Cercignani,TheBoltzmannequationanditsApplication,Springer-Verlag,

1988

[4]J.-F.GerbeauandB.Pertame,DerivationofviscousSaint–Venantsystemfor

laminarshallowwater;numericalvalidation.INRIARR-4084

[5]P-L.Lions,MathematicalTopicsinFluidDynamics,Vol.2,Compressible

Models,OxfordSciencePublication,1998

[6]dyzhenskaya,TheMathematicalTheoryofViscousIncompressibleFlows

(2ndedition),GordonandBreach,1969

[7]J.Smoller,ShockWavesandReaction-DiffusionEquations,(secondedition),

Springer-Verlag,Vol.258,GrundlehrenSeries,1994

[8]R.DiPerna,ConvergenceoftheViscosityMethodforIsentropicGasDynamics,

Comm.Math.Phys.,Vol.91,Nr.1,1983

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