Fluid Mechanics : A Geometrical Point of View PDF
Preview Fluid Mechanics : A Geometrical Point of View
OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi FLUID MECHANICS OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi Fluid Mechanics A Geometrical Point of View S. G. Rajeev DepartmentofPhysicsandAstronomy DepartmentofMathematics UniversityofRochester Rochester,NY 1 OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©S.G.Rajeev2018 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2018 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2018932827 ISBN978-0-19-880502-1(hbk.) ISBN978-0-19-880503-8(pbk.) DOI:10.1093/oso/9780198805021.001.0001 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi Thisbookisdedicatedtomyteacher,A.P.Balachandran. OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi Preface Next to celestial mechanics, fluid mechanics is the oldest part of theoretical physics. Euler derived the fundamental equations more than 200 years ago. Yet it is far from complete. We do not yet know with mathematical certainty if Euler’s equations have regularsolutionsgivensmoothinitialdata.Moreimportanttophysics,thephenomenon ofturbulenceisstillmysterious. Numericalmethodshavemademuchprogress(especiallyinapplicationstoengineer- ing) in recent years.The exponential growth of computing power has made it possible todesignairplanes,submarinesandhouseholdapplianceswithoutcumbersometesting ofprototypes.Localweathercanbepredictedforaboutafortnight,afterwhicheventhe bestcomputersfail.Toolsfromstatisticalmechanicsandquantumfieldtheory(theareas thatroutinelydealwithaninfinitenumberofrandomvariables)oughttobeuseful. Manyideasoftheoreticalphysics(e.g.,conformalinvariance)originatedinthestudy of fluids. Abstract ideas (such as Lie algebras) appear here in a concrete and easily visualizedsetting.InthisbookIwanttopresentfluidmechanicsasanonlinearclassical fieldtheory,anessentialpartoftheeducationofaphysicist. Thecentralobjectoffluidmechanicsisavectorfield,thefluidvelocity.So,ageometric pointofviewisquitenatural.Atadeeperlevel,onecanunderstandEuler’sequationas ahamiltoniansystem,whosePoissonbracketsaredualtothecommutationrelationsof vectorfieldsandthehamiltonianisthekineticenergy(L2-norm). This allows one to think of Euler’s equations for fluids and his equation for a rigid body on the same footing:they both describe geodesics on a Lie group.Of course,the fluid is much more complicated: the group is infinite dimensional and the curvature is negative. Arnold showed that the notorious instabilities of fluid mechanics can be understood in terms of the negative curvature of its geometry. Although very natural, thisneedssomemathematicalmachinery.Ihavetriedtopresentitinawayaccessibleto theoreticalphysicists.Clebsch’soldideaofusingcanonicalvariablesinfluidmechanics becomesusefulhere. More traditional subjects like boundary layer theory, vortex dynamics, and surface wavesarealsoilluminatedbygeometry.ExpressingEulerandNavier–Stokesequations on curvilinear coordinates is essential: most boundaries of interest are not planes. A little Riemannian geometry goes a long way here. The linear theory of instabilities is understood in terms of the spectrum of non-normal operators. Hermann Weyl (of all people) found an excellent analytical approximation for the boundary condition– Blasius theory of boundary layers.The vortex filament equations can be related to the Heisenbergmodelofmagnets;avortexinfluidmechanicsismappedtoasolitonofspin waves. OUPCORRECTEDPROOF – FINAL,28/7/2018,SPi viii Preface Throughout, I will point to numerical and analytical calculations (mostly using Mathematica, but you can use your favorite language) to gain understanding. This is notabookoncomputationalmethods.Youdon’tneedasupercomputeranymoretodo interestingwork,soeventhemosttheoreticalofuscanbenefitfromtheiruse.Switching fromananalytical/geometricalpointofviewtothemorepracticaldiscrete/numericalone andback,weunderstandbothbetter. Studyingchaotic advectionis a good way of developingintuitionfor fluidflows and fordynamicalsystems.Aref(1984)givesasimpleexample,whichhasturnedouttobe usefulinengineeringdevicesthatmixfluidsefficiently.TheSmalehorseshoe,although notusuallythoughtofaspartoffluidmechanics,givesasolidlyestablishedmathematical model of chaos. Many chaotic systems (including Aref’s) have a Smale horse shoe embeddedinsidethem.SoIhaveincludedadiscussionofthesetopicsinanappendix. There are many reasons to believe that renormalization is useful to understand turbulence. Although a discussion of those theories is beyond the scope of this book, I have included an appendix on more elementary applications of renormalization: the IsingmodelandFeigenbaum’sapproachtodynamicalsystemsinonedimension. Ihavenottriedtosurveyeverysub-fieldindetail:suchacomprehensivesurveywould beaboutasusefulasamapofacountryonascaleof1:1.Afewcasesareexaminedin detail.Theoreticalphysicsisbasedongeneralprinciples(conservationlaws,variational principles)andspecialcasesthatcanbeunderstoodanalyticallyorbysimplenumerical computations.Thetechniquesdevelopedthiswaycanbeadaptedtothosethatarisein applications.That said,my choice of topics is necessarily subjective.The emphasis on Lietheoryanddynamicalsystemsisunusualforabookatthislevel. Much remains uncovered. It would have been nice to talk of quantum fluids. The greatmysteryofturbulenceandattemptstomodelitcouldhavebeenreviewed.Models of weather prediction,oceanography,and astrophysical jets are all thriving.The whole fieldofmagnetohydrodynamicsisgivenshortshrift.Eachofthesetopicswouldrequire abookofitsown. This is a sort of sequel to Advanced Mechanics Rajeev (2013). A knowledge of mechanics,linear algebra (eigenvalue problems) and partial differential calculus is the mainprerequisite.Thesectionswitha∗intheirnamecanbeskippedonafirstreading; theycontainmoreadvancedmaterial. The book is aimed at physics/mathematics graduate students; some engineering studentswillalsofindituseful.Thereissomeoverlapandsomerepetition,becausesome ideas(e.g.,dynamicalsystems,Liealgebras)aresorecurrentinphysics.Othersareonly outlined,andworkingthemoutyourselfisanessentialpartoflearningthesubject. Acknowledgments I cannot thank my wife and children enough for their support over the years. A. P. Balachandrantaughtmuchmorethanphysics:theaudacitytogointonewareas.Thanks totheindulgenceofmycolleaguesinPhysicsandMathematicsDepartmentsforletting meexploredirectionsthatarenotprofitableintheshortterm. I am grateful to A.Kar,G.Krishnaswami and M.Bhattacharya for commenting on parts of the manuscript. Thanks for discussions with A. Iosevich on fractals; D.Geba on regularity of Navier–Stokes;V.V.Sreedhar on conformal invariance;and V.P.Nair andT.Padmanabhanoncountlesstopicsofphysicsinourearlyyears.ThankstoSonke AdlungforencouragingmetowritethisbookandtoHarrietKonishiformanyhelpful suggestions.ThanksalsotoAlanSkullandLydiaShinojfortheexcellentediting.
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