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1、ExactsolutionsoftheunsteadyNaYier-StokesequationsCYWangDepartmentofMathematics,MichiganStateUniversity,EastLansing,Michigan48824TheunsteadyNavier-Stokesequationsareasetofnonlinearpartialdifferentialequationswithveryfewexactsolutions.Thispaperattemptstoclassifyandre
2、viewtheexistingunsteadyexactsolutions.Therearethreemaincategories:parallel,concentricandrelatedsolutions,Beltramiandrelatedsolutions,andsimilaritysolutions.Physicallysignificantexamplesareemphasized.i.IntroductionAtthispointweshoulddefinewhatismeantbyan"exactsoluti
3、on".ExactsolutionisoneThebasicequationsoffluidmechanicsarethethatsatisfiesEqs(1,2)forallx,iandallcon•Navier-Stokesequations.Duetotheirinherentstantp,v.Althoughpcanbeeliminatedfromnonlinearity,exactsolutionsarerare.Exactso•Eq(1)bytakingthecurl,theparameterv(orlution
4、sareimportantnotonlyinitsownrightinnondimensionalform,theReynoldsnumber)assolutionsofparticularflows,butalsoserveisabasicgoverningparameter.Exactsolutionsasaccuracychecksfornumericalsolutions.shouldhavethepropertythatvisleftasanThelastreviewofexactsolutionswaspub•a
5、rbitraryconstant.Closedformsolutionswithlished26yearsagobyBerker(1963).Sincethenexplicitvareobviouslyexactsolutions.Sim•manynewsolutionshaveappearedindiverseilaritysolutionswithimplicituanduniversaljournalsandanewreviewisnecessary.Theprofilesarealsoexactsolutions.H
6、owever,di•presentreviewwillbealimitedattempttocom•rectnumericalsolutionsofEqs(1,2)arenotplementBerker'swork.Insteadofacompre•exactsolutionsbecauseforeachsolutionsomehensivereview,weshallconcentrateonlyonthespecificnumericalvalueofvmustbeassigned.exactsolutionsofthe
7、unsteadyNavier-OnedegeneratesolutionoftheNavier-Stokesequations.EvenwithsuchmodestaimsStokesequationsispotentialflowwhereq=ourreviewcouldnotbeall-inclusive.ThereV-ThentheNavier-Stokesequationsreducemaybemethodsorsolutionswhich,inspiteoftotheLaplaceequationoureff
8、orts,havenotbeendulyincluded.Tothoseauthorsweofferoursincereappologies.V2$=0(3)TheequationsinquestionaretheconstantThereareinfinitelymanysteadyan
