2The Navier-Stokes and Euler Equations(6)

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2TheNavier–StokesandEulerEquations–FluidandGasDynamics

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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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