2The Navier-Stokes and Euler Equations(11)

math

2TheNavier–StokesandEulerEquations–FluidandGasDynamics

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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/