exact solution describing a shallow water flow in an extending stripe

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1、EXACTSOLUTIONDESCRIBINGASHALLOWWATERFLOWINANEXTENDINGSTRIPESergeyV.GolovinLavrentyevInstituteofHydrodynamicsofSBRAS,NovosibirskStateUniversityNovosibirsk,630090,Russiae-mail:sergey@hydro.nsc.ruPartiallyinvariantsolutionto(2+1)Dshallowwaterequationisconstructedandinv

2、estigated.Thesolutiondescribesanextensionofastripe,boundedbylinearsourceanddrainoffluid.Realizationsofsmoothflowandofhydraulicjumparepossible.Particletrajectoriesandsoniccharacteristicsontheobtainedsolutionarecalculated.1.Constructionofthesolution.Equationsdescribinga

3、thinlayerofwaterflowsoveraflatbottomareobserved.ut+uux+vuy+hx=0,vt+uvx+vvy+hy=0,(1.1)ht+(uh)x+(vh)y=0.Here(u,v)isavelocityvectorofaparticle,hisadepthoffluid.Thegravityaccelerationisscaledtog=1.Equations(1.1)admit9-dimensionalLiealgebraofinfinitesimaltransformationsL9[2]

4、,generatedbyoperators(notationsof[3]areadopted)X1=∂x,X2=∂y,X4=t∂x+∂u,X5=t∂y+∂v,X9=x∂y−y∂x+u∂v−v∂u,X10=∂t,X11=x∂x+y∂y+u∂u+v∂v+2h∂h,X=t2∂+tx∂+ty∂+(x−tu)∂+(y−tv)∂−2th∂.12txyuvhX13=2t∂t+x∂x+y∂y−u∂u−v∂v−2h∂h.WeobserveapartiallyinvariantsolutiongeneratedbyLiesubalgebraN={

5、X1,X4,X10+X12}⊂L9.InvariantsofNarethefollowingfunctionsy√λ=√,V=vt2+1−tλ,H=h(t2+1).t2+1arXiv:0802.4134v1[physics.flu-dyn]28Feb2008Accordingtothegeneralalgorithm[1]therepresentationofN-partiallyinvariantsolutioncanbewrittenasV=V(λ),H=H(λ),u=u(t,x,y).Therearetwoinvaria

6、ntfunctionsVandH,whicharesettodependontheonlyinvariantvariableλ,andonenon-invariantfunctionu,whichdependonallindependentvariables.WesubstituteanexpressionsforinvariantsVandHinthisrepresentation,andthensolveitwithrespecttounknownfunctionsvandh.Inwhatfollowsitisconven

7、ientto1representfunctionuasafunctionof(t,x,λ)insteadof(t,x,y).Thesegivethefollowingformulae:V(λ)+tλH(λ)yu=u(t,x,λ),v=√,h=,λ=√.(1.2)t2+1t2+1t2+1Substitutionoftherepresentation(1.2)into(1.1)givesu+uu+V(t2+1)−1u=0,txλVV′+H′=−λ,(1.3)
VH′+HV′+(t2+1)u−t=0.xHerethelowerin

8、dexesdenotepartialderivativeswithrespecttothecorrespondingvari-able;primeisthederivativewithrespecttoλ.Thefirstandthethirdequationsofsystem