The Oxford Handbook of Philosophy of Physics

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The Oxford Handbook of Philosophy of PhysicsEditedbyRobertBatterman

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CONTENTS

Contributors Introduction

Robert Batterman

1. For a Philosophy of Hydrodynamics Olivier Darrigol

2. What Is "Classical Mechanics" Anyway? Mark Wilson

3. Causation in Classical Mechanics

Sheldon R. Smith

4. Theories of Matter: Infinities and Renormalization Leo P. Kaodanoff

5. Turn and Face the Strange Ch-ch-changes: Philosophical

Questions Raised by Phase Transitions Tarun Menon and Craig Callender

6. Effective Field Theories

]onathan Bain

7. The Tyranny of Scales Robert Batterman

8. Symmetry

Sorin Bangu

9. Symmetry and Equivalence Gordon Belot

10. lndistinguishability

Simon Saunders

11. Unification in Physics

Margaret Morrison 12. Measurement and Classical Regime in Quantum Mechanics

Guido Bacciagaluppi 13. The Everett Interpretation

David Wallace 14. Unitary Equivalence and Physical Equivalence

Laura Ruetsche 15. Substantivalist and Relationalist Approaches to Spacetime

Oliver Pooley 16. Global Spacetime Structure

John Byron Manchak 17. Philosophy of Cosmology

Chris Smeenk Index

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ContributorsTheOxfordHandbookofPhilosophyofPhysicsEditedbyRobertBatterman

Contributors

GuidoBacciagaluppi

isReaderinPhilosophyattheUniversityofAberdeen.Hisfieldofresearchisthephilosophyofphysics,inparticularthephilosophyofquantumtheory.Healsoworksonthehistoryofquantumtheoryandhaspublishedabookonthe1927Solvayconference(togetherwithA.Valentini).Healsohasinterestsinthefoundationsofprobabilityandinissuesoftimesymmetryandasymmetry.

JonathanBain

isAssociateProfessorofPhilosophyofScienceatthePolytechnicInstituteofNewYorkUniversity.Hisresearchinterestsincludephilosophyofspace-time,scientificrealism,andphilosophyofquantumfieldtheory.

Contributors

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SorinBangu

isAssociateProfessorofPhilosophyattheUniversityofBergen,Norway.HereceivedhisPh.D.fromtheUniversityofTorontoandhaspreviouslybeenapostdoctoralfellowattheUniversityofWesternOntarioandafixed-termlecturerattheUniversityofCambridge,DepartmentofHistoryandPhilosophyofScience.Hismaininterestsareinphilosophyofscience(especiallyphilosophyofphysics,mathematics,andprobability)andlaterWittgenstein.Hehaspublishedextensivelyintheseareasandhasrecentlycompletedabookmanuscriptonthemetaphysicalandepistemologicalissuesarisingfromtheapplicabilityofmathematicstoscience.

RobertBatterman

isProfessorofPhilosophyattheUniversityofPittsburgh.HeisaFellowoftheRoyalSocietyofCanada.HeistheauthorofThedevilinthedetails:Asymptoticreasoninginexplanation,reduction,andemergence(Oxford,2002).Hisworkinphilosophyofphysicsfocusesprimarilyupontheareaofcondensedmatterbroadlyconstrued.Hisresearchinterestsincludethefoundationsofstatisticalphysics,dynamicalsystemsandchaos,asymptoticreasoning,mathematicalidealizations,thephilosophyofappliedmathematics,explanation,reduction,andemergence.

GordonBelot

isProfessorofPhilosophyattheUniversityofMichigan.Hehaspublishedanumberofarticlesonphilosophyofphysicsandrelatedareasandonesmallbook,Geometricpossibility(Oxford,2011).

CraigCallender

isProfessorofPhilosophyattheUniversityofCalifornia,SanDiego.Hehaswrittenwidelyinphilosophyofscience,metaphysics,andphilosophyofphysics.HeistheeditorofPhysicsmeetsphilosophyatthePlancklength(withHuggett)andtheOxfordhandbookofthephilosophyoftime.Heiscurrentlyworkingonabookmonographontherelationshipbetweenphysicaltimeandtimeasweexperienceit.

Contributors

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OlivierDarrigol

isaCNRSresearchdirectorintheSPHERE/RehseisresearchteaminParis.Heinvestigatesthehistoryofphysics,mostlynineteenthandtwentiethcentury,withastronginterestinrelatedphilosophicalquestions.HeistheauthorofseveralbooksincludingFromc-numberstoq-numbers:Theclassicalanalogyinthehistoryofquantumtheory(Berkeley:UniversityofCaliforniaPress,1992),ElectrodynamicsfromAmpretoEinstein(Oxford:OxfordUniversityPress,2000),Worldsofflow:AhistoryofhydrodynamicsfromtheBernoullistoPrandtl(Oxford:OxfordUniversityPress,2005),andAhistoryofopticsfromGreekantiquitytothenineteenthcentury(Oxford:OxfordUniversityPress,2012).

LeoP.Kadanoff

isatheoreticalphysicistandappliedmathematicianwhohascontributedwidelytoresearchinthepropertiesofmatter,thedevelopmentofurbanareas,statisticalmodelsofphysicalsystems,andthedevelopmentofchaosinsimplemechanicalandfluidsystems.Hisbest-knowncontributionwasinthedevelopmentoftheconceptsofscaleinvarianceanduniversalityastheyareappliedtophasetransitions.Morerecently,hehasbeeninvolvedintheunderstandingofsingularitiesinfluidflow.

JohnByronManchak

isanAssistantProfessorofPhilosophyattheUniversityofWashington.Hisprimaryresearchinterestsareinphilosophyofphysicsandphilosophyofscience.Hisresearchhasfocusedonfoundationalissuesingeneralrelativity.

TarunMenon

isagraduatestudentinPhilosophyattheUniversityofCalifornia,SanDiego.Hisresearchinterestsareinthephilosophyofphysicsandmetaphysics,particularlytime,probability,andthefoundationsofstatisticalmechanics.Heisalsointerestedinformalepistemologyandthecognitivestructureofscience.

Contributors

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MargaretMorrison

isProfessorofPhilosophyattheUniversityofToronto.Sheistheauthorofseveralarticlesonvariousaspectsofphilosophyofscienceincludingphysicsandbiology.SheisalsotheauthorofUnifyingscientifictheories:Physicalconceptsandmathematicalstructures(Cambridge,2000)andtheeditor(withMaryMorgan)ofModelsasmediators:Essaysonthephilosophyofnaturalandsocialscience(Cambridge,1999).

OliverPooley

isUniversityLecturerintheFacultyofPhilosophyattheUniversityofOxfordandaFellowandTutoratOrielCollege,Oxford.Heworksinthephilosophyofphysicsandinmetaphysics.Muchofhisresearchfocusesonthenatureofspace,time,andspacetime.

LauraRuetsche

isProfessorofPhilosophyattheUniversityofMichigan.HerInterpretingquantumtheories:Theartofthepossible(Oxford,2011)aimstoarticulatequestionsaboutthefoundationsofquantumfieldtheorieswhoseanswersmightholdinterestforphilosophymorebroadlyconstrued.

SimonSaunders

isProfessorinthePhilosophyofPhysicsandFellowofLinacreCollegeattheUniversityofOxford.Hehasworkedinthefoundationsofquantumfieldtheory,quantummechanics,symmetries,thermodynamics,andstatisticalmechanicsandinthephilosophyoftimeandspacetime.HewasanearlyproponentoftheviewofbranchingintheEverettinterpretationasaneffectiveprocessbasedondecoherence.Heisco-editor(withJonathanBarrett,AdrianKent,andDavidWallace)ofManyworlds?Everett,quantumtheory,andreality(OUP2010).

ChrisSmeenk

isAssociateProfessorofPhilosophyattheUniversityofWesternOntario.Hisresearchinterestsarehistoryandphilosophyofphysics,andseventeenth-centurynaturalphilosophy.

Contributors

SheldonR.Smith

isProfessorofPhilosophyatUCLA.Hehaswrittenarticlesonthephilosophyofclassicalmechanics,therelationshipbetweencausationandlaws,thephilosophyofappliedmathematics,andKant'sphilosophyofscience.

DavidWallace

studiedphysicsatOxfordUniversitybeforemovingintophilosophyofphysics.HeisnowTutorialFellowinPhilosophyofScienceatBalliolCollege,Oxford,anduniversitylecturerinPhilosophyatOxfordUniversity.Hisresearchinterestsincludetheinterpretationofquantummechanicsandthephilosophicalandconceptualproblemsofquantumfieldtheory,symmetry,andstatisticalphysics.

MarkWilson

isProfessorofPhilosophyattheUniversityofPittsburgh,aFellowoftheCenterforPhilosophyofScience,andaFellowattheAmericanAcademyofArtsandSciences.Hismainresearchinvestigatesthemannerinwhichphysicalandmathematicalconcernsbecomeentangledwithissuescharacteristicofmetaphysicsandphilosophyoflanguage.HeistheauthorofWanderingsignificance:Anessayonconceptualbehavior(Oxford,2006).Heiscurrentlywritingabookonexplanatorystructure.Heisalsointerestedinthehistoricaldimensionsofthisinterchange;inthisvein,hehaswrittenonDescartes,Frege,Duhem,andWittgenstein.HealsosupervisestheNorthAmericanTraditionsSeriesforRounderRecords.

Introduction

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IntroductionRobertBattermanTheOxfordHandbookofPhilosophyofPhysicsEditedbyRobertBatterman

AbstractandKeywords

Thischapterdiscussesthethemeofthisbook,whichisaboutthephilosophyofphysics.Thebookprovidesanoverviewofthetopicsbeingstudiedbyphilosophersofphysicsandidentifiestheoriesthatwouldnothavebeenconsideredfundamentalduringthe1980s.Itdescribesnewproblemsandissuesthatbecamethefocusofthephilosophyofphysicsinrecentyears,whichincludethephilosophyofhydrodynamics,classicalmechanics,effectivefieldtheories,andmeasurementinquantummechanics.Keywords:philosophyofphysics,hydrodynamics,classicalmechanics,effectivefieldtheories,quantummechanics

WhenIwasingraduateschoolinthe1980s,philosophyofphysicswasfocusedprimarilyontwodominantreasonablyself-containedtheories:Orthodoxnonrel-ativisiticquantummechanicsandrelativisticspacetimetheories.Ofcourse,therewereafewpaperspublishedoncertainquestionsinotherfieldsofphysicssuchasstatisticalmechanicsanditsrelationtothermodynamics.Theselatter,however,primarilytargetedtheextenttowhichthereductiverelationsbetweenthetwotheoriescouldbeconsideredastraightforwardimplementationoftheorthodoxstrategyoutlinedbyErnestNagel.

Philosophicalquestionsaboutthemeasurementproblem,thequestionofthepossibilityofhiddenvariables,andthenatureofquantumlocalitydominatedthephilosophyofphysicsliteratureonthequantumside.Questionsaboutrelationalismvs.substantivalism,thecausalandtemporalstructureoftheworld,aswellasissuesaboutunderdeterminationoftheoriesdominatedtheliteratureonthespacetimeside.Someworriesaboutdeterminismvs.indeterminismcrossedthedividebetweenthesetheoriesandplayedasignificantroleinshapingthedevelopmentofthefield.(HereIamthinkingofEarman'sAPrimeronDeterminism(1986)asaparticulardrivingforce.)

Theseissuesstillreceiveconsiderableattentionfromphilosophersofphysics.Butmanyphilosophershaveshiftedtheirattentiontootherquestionsrelatedtoquantummechanicsandtospacetimetheories.Inparticular,therehasbeenconsiderableworkonunderstandingquantumfieldtheory,particularlyfromthepointofviewofalgebraicoraxiomaticformulations.Newattentionhasalsobeengiventophilosophicalissuessurroundingquantuminformationtheoryandquantumcomputing.Andtherehas,naturally,beenconsiderableinterestinunderstandingtherelationsbetweenquantumtheoryandrelativitytheory.Questionsaboutthepossibilityofunifyingthesetwofundamentaltheoriesarise.Relatedly,therehasbeenafocusonunderstandinggaugeinvarianceandsymmetries.

However,Ibelievephilosophyofphysicshasevolvedevenfurther,andthisbeliefpromptsthepublicationofthisvolume.Recently,manyphilosophershavefocusedtheirattentionsontheoriesthat,forthemostpart,werelargelyignoredinthepast.Asnotedabove,therelationshipbetweenthermodynamicsandstatisticalmechanicsoncethoughttobeaparadigminstanceofunproblematictheoryreductionisnowahotlydebatedtopic.Philosophersandphysicistshavelongimplicitlyorexplicitlyadoptedareductionistmethodologicalbent.Yet,overtheyearsthismethodologicalslanthasbeenquestioneddramatically.Attentionhasbeenfocusedontheexplanatoryanddescriptiverolesofnon-fundamental,phenomenologicaltheories.Inlargepartbecauseofthisshiftoffocus,

Introduction

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oldtheoriessuchasclassicalmechanics,oncedeemedtobeoflittlephilosophicalinterest,haveincreasinglybecomethefocusofdeepmethodologicalinvestigations.

Furthermore,somephilosophershavebecomemoreinterestedinlessfundamentalcontemporaryphysics.Forinstance,therearedeepquestionsthatariseincondensedmattertheory.Thesequestionshaveinterestingandimportantimplicationsforthenatureofmodels,idealizations,andexplanationinphysics.Forexample,modelsystems,suchastheIsingmodel,playimportantcomputationalandconceptualrolesinunderstandinghowtherecanbephasetransitionswithspecificcharacteristics.And,theuseofthethermodynamiclimitisanidealizationthat(somehaveargued)playsanessential,ineliminableroleinunderstandingandexplainingtheobserveduniversalityofcriticalphenomena.Thesespecificissuesarediscussedinseveralofthechaptersinthisvolume.

IntheUnitedStatesduringthe1970sand1980s,therewasagreatdebatebetweenparticlephysicistswhopushedforfundingofhigh-energyparticleacceleratorsandsolid-stateorcondensed-mattertheoristsforwhomthesiphoningoffofsomuchgovernmentfundingtofundamentalphysicswasunacceptable.AfamouspaperchampioningthelatterpositionisPhilipAnderson'sMoreIsDifferent(1972).Notonlywasthisadebateoverfunding,butitraisedissuesaboutexactlywhatshouldcountasfundamentalphysics.Whilehistoriansofphysicshavefocusedconsiderableattentiononthispublicdebate,philosophersofphysicshavereallyonlyrecentlybeguntoengagewiththeconceptualimplicationsofthepossibilitythatcondensedmattertheoryisinsomesensejustasfundamentalashigh-energyparticlephysics.

Thiscollectionaimstodotwothings.First,ittriestoprovideanoverviewofmanyofthetopicsthatcurrentlyengagephilosophersofphysics.Andsecond,itfocusesattentiononsometheoriesthatbyorthodox1980sstandardswouldnothavebeenconsideredfundamental.Itstrivestosurveysomeofthesenewissuesandtheproblemsthathavebecomeafocusofattentioninrecentyears.Additionally,itaimstoprovideup-to-datediscussionsofthedeepproblemsthatdominatedthefieldinthepast.

Inthefirstchapter,ForaPhilosophyofHydrodynamics,OlivierDarrigolfocusesattentiononlessonsthatcanbelearnedfromthehistoricaldevelopmentoffluidmechanics.Henotesthathydrodynamicshasprobablyreceivedtheleastattentionofanyphysicaltheoryfromphilosophersofphysics.Hydrodynamicsisnotafundamentaltheoryalongthelinesofquantummechanicsandrelativitytheory,anditsbasicformulationhasnotevolvedmuchfortwocenturies.Thesefacts,togetherwithalackofdetailedhistoricalstudiesofhydrodynamics,havekeptthetheoryofftheradar. DarrigolprovidesanaccountofthedevelopmentofhydrodynamicsasacomplextheoryonethatisnotfullycapturedbythebasicNavier-Stokesequations.Forthetheorytobeapplicable,particularlyforittoplayanexplanatoryrole,ahostoftechniquesidealizations,modelingstrategies,andempiricallydetermineddatamustcomeintoplay.Thisdiscussionshowsclearlyhowintricate,sophisticated,andmodernthetheoryofhydrodynamicsactuallyis.Darrigoldrawsanumberoflessonsaboutthestructuresofphenomenologicaltheoriesfromhisdetaileddiscussion,focusingparticularlyonwhathecallsthemodularstructureofhydrodynamics.

Continuingthediscussionofoldbutbynomeansdeadoreliminatedtheories,MarkWilsontakesontheformidabletaskoftryingtosayexactlywhatisthenatureofclassicalmechanics.Acommoninitialreactiontothistopicistodismissit:Surelyweallknowwhatclassicalmechanicsis!Justlookatanytextbook.ButasWilsonshowsinWhatIsClassicalMechanicsAnyway?,thisdismissiveattitudeismisleadingonanumberofimportantlevels.Classicalmechanicsislikeafive-leggedstoolonaveryunevenfloor.Itshiftsdramaticallyfromonefounda-tionalperspectivetoanotherdependingupontheproblemathand,whichinturnisoftenafunctionofthescalelengthatwhichthephenomenonisinvestigated.Inthecontextofplanetarymotions,billiards,andsimplifiedidealgasesinboxes,thepoint-particleinterpretationofclassicalmechanicswillmostlikelyprovideanappropriatetheoreticalsetting.However,assoonasonetriestoprovideamorerealisticdescriptionofwhatgoesoninsideactualbilliardballcollisions,onemustconsiderthefactthattheballswilldeformandbuildupinternalstressesuponcollision.Insuchsituations,thepoint-particlefoundationwillfailandonewillneedtoshifttoanalternativefoundation,providedbyclassicalcontinuummechanics.Yetathirdpotentialfoundationforclassicalmechanicscanbefoundwithinso-calledanalyticmechanics,inwhichthenotionofarigidbodybecomescentral.Hereconstraintforces(suchastheconnectionsthatallowaballtoroll,ratherthanskid,downaninclinedplane)playacrucialrole.Forcesofthistypearenotwhollyconsistentwiththesuppositionscentraltoeitherthepoint-particleorcontinuumpointsofview.AmajorlessonfromWilson'sdiscussionisthatclassicalmechanicsshouldbestbethoughtofasconstitutedbyvariousfoundationalmethodologiesthatdonotfitparticularlywellwithoneanother.Thisgoesagainstcurrentorthodoxythatatheorymustbeseenasaformallyaxiomatizableconsistentstructure.

1

Introduction

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Onthecontrary,toproperlyemployclassicalmechanicsfordescriptiveandexplanatorypurposes,onepushesafoundationalmethodologyappropriateatonescaleofinvestigationtoitslimitingutility,afterwhichoneshiftstoadifferentsetofclassicalmodelingtoolsinordertocapturethephysicsactiveatalowersizescale.Wilsonarguesthatagooddealofphilosophicalconfusionhasarisenfromfailingtorecognizethecomplicatedscale-dependentstructuresofclassicalphysics.

SheldonSmith'scontributionaddstoourunderstandingofaparticularaspectofclassicalphysics.InCausationinClassicalMechanics,headdressesskepticalargumentsinitiatedbyBertrandRusselltotheeffectthatcausationisnotafundamentalfeatureoftheworld.Inthecontextofclassicalphysics,onewayofmakingthisclaimmorepreciseistoarguethatthereisnoreasontoprivilegeretardedoveradvancedGreen'sfunctionsforasystem.Green'sfunctions,crudely,describetheeffectofaninstantaneous,localizeddisturbancethatactsuponthesystem.Itseemsthatthelawsofmotionforelectromagnetismorforthebehaviorofaharmonicoscillatordonotdistinguishbetweenretarded(presumablycausal)andadvanced(presumablyacausal)solutions.Ifthereistoberoomforaprincipleofcausalityinclassicalphysics,thenitlookslikeweneedtofindextra-nomologicalreasonstoprivilegetheretardedsolutions.SmithsurveysawiderangeofattemptstoanswerthecausalskepticinthecontextsoftheuseofGreen'sfunctionsandtheimpositionof(Sommerfeld)radiationconditions,amongotherattempts.Theupshotisthatitisremarkablydifficulttofindjustificationwithinphysicaltheoryforthemaximthatcausesprecedetheireffects.

Thenextchapter,byLeoKadanoff,focusesoncondensedmatterphysics.Inparticular,Kadanoffdiscussesprogressinphysicallyunderstandingthefactthatmattercanabruptlychangeitsqualitativestateasitundergoesaphasetransition.Aneverydayexampleoccurswiththeboilingwaterinateakettle.Asthetemperatureincreases,thewaterchangesfromitsliquidphasetoitsvaporphaseintheformofsteam.Mathematically,suchtransitionsaredescribedbyanimportantconceptcalledanorderparameter.Inafirst-orderphasetransition,suchastheliquidvaportransition,theorderparameterchangesdiscontinuously.Certainphasetransitions,however,arecontinuousinthesensethatthediscontinuityinthebehavioroftheorderparameterapproacheszeroatsomespecificcriticalvalueoftherelevantparameterssuchastemperatureandpressure.Foralongtimethereweretheoreticalattemptstounderstandthephysicsinvolvedinsuchcontinuoustransitionsthatfailedtoadequatelyrepresenttheactualbehavioroftheorderparameterasitapproacheditscriticalvalue.Thedevelopmentoftherenormalizationgroupinthe1970sremediedthissituation.Kadanoffplayedapivotalroleintheconceptualdevelopmentofrenormalizationgrouptheory.Inthischapter,hefocusesonthesedevelopments(particularly,theimprovementuponearlymeanfieldtheories)andonadeeplyinterestingfeaturehecallstheextendedsingularitytheorem.Thisistheideathatsharp,qualitativelydistinct,changesinphaseinvolvethepresenceofamathematicalsingularity.Thissingularitytypicallyemergesinthelimitinwhichthesystemsizebecomesinfinite.Theunderstandingofthebehaviorofsystemsatandnearphasetransitionsrequiresradicallydifferentconceptualapparatuses.Itinvolvesasynthesisbetweenstandardstatisticalmechanicalusesofprobabilitiesandconceptsfromdynamicalsystemstheoryparticularly,thetopologicalconceptionsofbasinsofattractionandfixedpointsofadynamicaltransformation.

ThediscussionoftherenormalizationgroupandphasetransitionscontinuesasTarunMenonandCraigCallenderexamineseveralphilosophicalquestionsraisedbyphasetransitions.Theirchapter,TurnandFacetheStrangeCh-ch-changes,focusesonthequestionofwhetherphasetransitionsaretobeunderstoodasgenuinelyemergentphenomena.ThetermemergentismuchabusedandconfusedinboththephilosophicalandphysicsliteraturesandsoMenonandCallenderprovideakindofroadmaptoseveralconceptsthathavebeeninvokedintheincreasingnumberofpapersonemergenceandphasetransitions.Inparticular,theydiscussconceptionsofreductionandcorrespondingnotionsofemergence:conceptualnovelty,explanatoryirreducibility,andontologicalirreducibility.Theirgoalistoestablishthatforanyreasonablesensesofreducibilityandemergence,phasetransitionsarenotemergentphenomena,andtheydonotpresentproblemsforthoseofareductionistexplanatorybent.Inasense,theirdiscussioncanbeseenaschallengingtheimportanceoftheextendedsingularitytheoremmentionedabove.MenonandCallenderalsoconsidersomerecentworkinphysicsthatattemptstoprovidewell-definednotionsofphasetransitionforfinitesystems.Theircontributionservestohighlightthecontroversialandevolvingnatureofourphilosophicalunderstandingofphasetransitions,emergence,andreductionism.

JonathanBain'scontributiononEffectiveFieldTheorieslooksatseveralphysicalandmethodologicalconsequencesofthefactthatsometheoriesatlow-energyscalesareeffectivelyindependentof,ordecoupledfrom,theoriesdescribingsystemsathigherenergies.Sometimesweknowwhatthehigh-energytheorylookslike

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andcanfollowarecipeforconstructinglow-energyeffectivetheoriesbysystematicallyeliminatinghigh-energyinteractionsthatareessentiallyunobservableatthelowerenergies.But,atothertimes,wesimplydonotknowthecorrecthigh-energytheory,yetnonetheless,westillcanhaveeffectivelow-energytheories.Broadlyconstrued,hydrodynamicsisanexampleofthelattertypeofeffectivetheory,ifweconsideritasanineteenthcenturytheoryconstructedbeforeweknewabouttheatomicconstitutionofmatter.Bain'sfocusisoneffectivetheoriesinquantumfieldtheoryandcondensedmatterphysics.Hisdiscussionconcentratesontheintertheoreticrelationsbetweenlow-energyeffectivetheoriesandtheirhigh-energycounterparts.Giventheeffectiveindependenceoftheformerfromthelatter,shouldonethinkofthisrelationasautonomousoremergent?Baincontendsthatananswertothisquestionisquitesubtleanddependsuponthetypeofrenormalizationschemeemployedinconstructingtheeffectivetheory.

Myowncontributiontothevolumeconcernsageneralprobleminphysicaltheorizing.Thisistheproblemofrelatingtheoriesormodelsofsystemsthatappearatwidelyseparatedscales.Ofcourse,therenormalizationgrouptheory(discussedbyKadanoff,MenonandCallender,andBaininthisvolume)isoneinstanceofbridgingacrossscales.Butmoregenerally,wemaytrytoaddresstherelationsbetweenfinitestatisticaltheoriesatatomicandnanoscalesandcontinuumtheoriesthatapplyatscales10+ordersofmagnitudehigher.Onecanask,forexample,whytheNavier-Cauchyequationsforisotropicelasticsolidsworksowelltodescribethebendingbehaviorofsteelbeamsatthemacroscale.Atthemicroscalethelatticestructureofironandcarbonatomslooksnothinglikethehomogeneousmacroscaletheory.Nevertheless,thelattertheoryisremarkablyrobustandsafe.Thechapterdiscussesstrategiesforupscalingfromtheoriesormodelsatsmallscalestothoseathigherscales.Itexaminesthephilosophicalconsequencesofhavingtoconsider,inone'smodelingpractice,structuresthatappearatscalesintermediatebetweenthemicroandthemacro.

Therehasbeenconsiderabledebateaboutthenatureofsymmetriesinphysicaltheories.Recentfocusongaugesymmetrieshasledphilosopherstoadeeperunderstandingoftheroleoflocalinvariancesinelectromagnetism,particlephysics,andthehuntfortheHiggsparticle.SorinBanguprovidesabroadandcomprehensivesurveyofconceptsofsymmetryandinvarianceinhiscontributiontothisvolume.OneofthemostseductivefeaturesofsymmetryconsiderationscomesoutofWigner'ssuggestionthatonemightbeabletounderstand,explain,orgroundlawsofnaturebyappealtoakindofsuperprincipleexpressingsymmetriesandinvariancesthatconstrainlawstohavetheformsthattheydo.Onthisconceptionsymmetriesare,perhaps,ontologicallyandepistemicallypriortolawsofnature.Thisraisesdeepquestionsforfurtherresearchontherelationshipbetweenformalmathematicalstructuresandourphysicalunderstandingoftheworld.

GordonBelotalsoconsidersissuesofsymmetryandinvariance.Hiscontributionexplorestheconnectionsbetweenbeingasymmetryofatheoryamapthatleavesinvariantcertainstructuresthatencodethelawsofthetheoryandwhatitisforsolutionstoatheorytobephysicallyequivalent.Itisfairlycommonplaceforphilosopherstoadopttheideathat,ineffect,thesetwonotionscoincide.Andiftheydo,thenwehavetightconnectionbetweenapurelyformalconceptionofthesymmetriesofatheoryandamethodological/interpretiveconceptionofwhatitisfortwosolutionstorepresentthesamephysicalstateofaffairs.Belotnotesthatinthecontextofspacetimetheoriesthereseemtobewell-establishedargumentssupportingthistightconnectionbetweensymmetriesandphysicalequivalence.However,heexploresthedifficultiesinattemptingtogeneralizethisconnectionincontextsthatincludeclassicaldynamicaltheories.Belotexaminesdifferentwaysonemightmakeprecisethenotionofthesymmetriesofaclassicaltheoryandshowsthattheydonotcomportwellwithreasonableconceptionsofphysicalequivalence.Thechallengetothereaderisthentofindappropriate,nontrivialnotionsofsymmetriesforclassicaltheoriesthatwillrespectreasonablenotionsofphysicalequivalence.

Yetanothertypeofsymmetry,permutationsymmetry,isthesubjectofthechapterbySimonSaunders,entitledIndistinguishability.Hefocusesontheproperunderstandingofparticleindistinguishabilityinclassicalstatisticalmechanicsandinquantumtheory.Intheclassicalcase,Gibbshadalready(priortoquantummechanics)recognizedaneedtotreatparticles,atleastsometimes,asindistinguishable.ThisisrelatedtotheinfamousGibbsparadoxthatSaundersdiscussesindetail.Theconceptofindistinguishabilityhadmeanwhileenteredphysicsinacompletelynewway,involvinganewkindofstatistics.ThiscamewiththederivationofPlanck'sspectraldistribution,inwhichPlanck'squantumofactionhfirstenteredphysics.Commonwisdomhaslongheldthatparticleindistinguisha-bilityisstrictlyaquantumconcept,inapplicabletotheclassicalrealm;andthatclassicalstatisticalmechanicsisanywayonlytheclassicallimitofaquantumtheory.Thisfitswiththestandardviewoftheexplanationofquantumstatistics(Bose-EinsteinorFermi-Diracstatistics):departuresfromclassical(Maxwell-

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Boltzmann)statisticsareexplainedbyparticleindistinguishability.WiththisSaunderstakesissue.Heshowshowitispossibletotreatthestatisticalmechanicalstatisticsforclassicalparticlesasinvariantunderpermutationsymmetryinexactlythesamewaythatitistreatedinthequantumcase.Hearguesthattheconceptionofpermutationsymmetrydeservesaplacealongsidealltheothersymmetriesandinvariancesofphysicaltheories.Specifically,hearguesthattheconceptofindistinguishable,permutationinvariant,classicalparticlesiscoherentandreasonablecontrarytomanyclaimsfoundintheliterature.

MargaretMorrison'stopicisUnificationinPhysics.Shearguesthatthereareanumberofdistinctsensesofunificationinphysics,eachofwhichhasdifferentimplicationsforhowweviewunifiedtheoriesandphenomena.Ontheonehand,thereisatypeofunificationthatisachievedviareductionistprograms.HereaparadigmexampleistheunificationprovidedbyMaxwellianelectrodynamics.Maxwell'semphasisonmechanicalmodelsinhisearlyworkinvolvedtheintroductionofthedisplacementcurrent,whichwasnecessaryforafieldtheoreticrepresentationofthephenomena.Thesemodelsalsoenabledhimtoidentifytheluminiferousaetherwiththemediumoftransmissionofelectromagneticphenomena.Twoaetherswereessentiallyreducedtoone.Whenthesemodelswereabandonedinhislaterderivationofthefieldequations,thedisplacementcurrentprovidedtheunifyingparameterortheoreticalquantitythatallowedfortheidentificationofelectromagneticandopticalphenomenawithintheframeworkofasinglefieldtheoreticaccount.ThistypeofunificationwasanalogoustoNewton'sunificationofthemotionsoftheplanetsandterrestrialtrajectoriesunderthesame(gravitational)theoreticalframework.However,notallcasesofunificationareofthistype.Morrisondiscussestheexampleoftheelectroweaktheoryinsomedetail,arguingthatthisunificatorysuccessrepresentsakindofsynthetic,ratherthanreductive,unity.Theelectroweaktheoryalsoinvolvesaunifyingparameter,namely,theWeinbergangle.However,theunityachievedthroughgaugesymmetryisasynthesisofstructure,ratherthanofsubstance,asexemplifiedbythereductivecases.Finally,incallingattentiontothedifficultieswiththeStandardModelmoregenerally,Morrisonnotesthatyetadifferentkindofunificationisachievedintheframeworkofeffectivefieldtheory.Thisprovidesanothervantagepointfromwhichtounderstandtheimportanceoftherenormalizationgroup.Morrisonarguesforathirdtypeofunificationintermsoftheuniversalityclasses,onethatfocusesonunificationofphenomenabutshouldbeunderstoodindependentlyofthetypeofmicro-reductioncharacteristicofunifiedfieldtheoryapproaches.

Asnotedearlier,therecontinuestobesignificantresearchonfoundationalproblemsinquantummechanics.GuidoBacciagaluppi'schapterprovidesanup-to-datediscussionofworkontwodistinctproblemsinthefoundationsofquantummechanicsthataretypicallyconflatedintheliterature.Thesearetheproblemoftheclassicalregimeandthemeasurementproblem.Bothproblemsarisefromdeepissuesinvolvingentanglementandthefailureofanignoranceinterpretationofreducedquantumstates.Bacciagaluppiprovidesacontemporaryandthoroughintroductiontotheseissues.Theproblemoftheclassicalregimeisthatofprovidingaquantummechanicalexplanationoraccountofthesuccessofclassicalphysicsatthemacroscale.Itis,inessence,aproblemofintertheoreticrelations.Contemporaryworkhasconcentratedontheroleofenvironmentaldecoherenceintheemergenceofclassicalkineticsanddynamics.Bacciagaluppiarguesthatthesuccessofappealstodecoherencetosolvethisproblemwilldependuponone'sinterpretationofquantummechanics.Hesurveysanontologicallyminimalistinstrumentalinterpretationandastandard,ontologicallymorerobustorrealisticinterpretation.

ThemeasurementproblemisthedistinctproblemofderivingthecollapsepostulateandtheBornrulefromthefirstprinciples(Schrdingerevolution)ofthequantumtheory.Inexaminingthemeasurementproblem,Bacciagaluppiprovidesadetailedpresentationofamodern,realistictheoryofmeasurementthatgoesbeyondtheusualidealizeddiscussionsofspinmeasurementsusingStern-Gerlachmagnets.ThisdiscussiongeneralizestheusualcollapsepostulateandtheBornruletotakeintoaccountthefactthatrealmeasurementsareunsharp.Itdoessobyemployingtheapparatusofpositiveoperatorvalue(POV)measuresandobservables.Theupshotisthatthemeasurementproblemremainsarealworryforsomeonewhowantstomaintainastandard,reasonablyorthodoxinterpretationofquantumtheory.PerhapsEveretttheories,GRW-likespontaneouscollapsetheories,andsoonarerequiredforasolution.

TheEverett,orManyWorlds,interpretationofquantummechanicsisthesubjectofDavidWallace'schapter.Itiswellknownthatthelinearityofquantummechanicsleads,viatheprincipleofsuperposition,tothepossibilitythatmacroscopicobjectssuchascatscanbefoundinbizarrestatessuperpositionsofbeingaliveandbeingdead.Wallacearguesthataproperunderstandingofwhatquantummechanicsactuallysayswillenableustounderstandsuchbizarresituationsinawaythatdoesnotinvolvechangingthephysics(e.g.,asinBohmianhidden

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variablemechanicsorGRWspontaneouscollapsetheories).Neither,heclaims,doesitinvolvechangingone'sphilosophyby,forexample,providinganoperationalistinterpretationthatimposessomespecialstatustotheobserverortowhatcountsasmeasurement,alongthelinesofBohr.Suchinterpretationsareatoddswithourunderstandingof,say,theroleoftheobserverintherestofscience.Wallacearguesforastraightforward,fullyrealistinterpretationofthebaremathematicalformalismofquantummechanicsandclaimsthatthisinterpretationwillmakesenseofsuperposedcats,andsoon,withoutchangingthetheoryandwithoutchangingouroverallviewofscience.ThestraightforwardrealistinterpretationthatistodoallofthisworkistheEverettinterpretation.Primafacie,thisclaimisitselfbizarre:afterall,theEverettinterpretationhasusmultiplyingworldsoruniversesuponmeasurements.Nevertheless,Wallacemakesastrongcasethatanunderstandingofsuperpositionasadescriptionofmultiplicity,ratherthanoftheindefinitenessofstates,isexactlywhatisneeded.Furthermore,thatisexactlywhattheEverettinterpretation(andnoother)provides.ThebulkofWallace'scontributionexaminesvariousproblemsthathavebeenraisedfortheEverettinterpretation.Inparticular,hefocuseson(1)theproblemofprovidingapreferredbasiswhatactuallyjustifiesourunderstandingofsuperpositionintermsofmultiplicityofworlds,and(2)theprobabilityproblemhowtounderstandtheprobabilisticnatureofquantummechanicsifonehasonlythefullydeterministicdynamicsprovidedbytheSchrdingerequation.HearguesthatthecontemporaryunderstandingoftheEverettinterpretationhastheresourcestoaddresstheseissues.

LauraRuetsche'schapterUnitaryEquivalenceandPhysicalEquivalenceinves-tigatesaquestionofdeepphysicalandphilosophicalimportance:Thedemandforcriteriaestablishingthephysicalequivalenceoftwoformulationsofaphysicaltheory.Inordinaryquantummechanicsthereceivedviewisthattwoquantumtheoriesarephysicallyequivalentjustincasetheyareunitarilyequivalent.Anypairoftheoriespurporting,say,todescribetwoentangledspin1/2systemsarereallyjustoneandthesamebecauseoftheJordanandWignertheoremshowingthatatheorythatrepresentsthecanonicalanticommutationrelationsforasystemofnspinsisuniqueuptounitaryequivalence.AsimilartheoremduetoStoneandvonNeumannguaranteesananalogousresultforanyHilbertspacerepresentationofthecanonicalcommutationrelationsforaHamiltoniansystem.Whataretheconsequencesofthebreakdownofunitaryequivalenceforthosequantumsystemsforwhichthesetheoremsfailtohold?Suchsystemsincludetheinfinitesystemsstudiedinquantumfieldtheory,quantumstatisticalmechanics,andevensimplerinfinitesystemslikeaninfiniteone-dimensionalchainofquantumspins.ShecallsthesetheoriescollectivelyQM .Theplethoraofunitarilyinequivalentrepresentationsintheseinfinitecasesdemandsthatwerevisitourassumptionsaboutphysicalequivalenceandthenatureofquantumtheories.Ruetscheexaminesvariouscompetingsuggestions,orcompetingprinciplesthatmayguidetheinvestigationintothisproblem.

Thenextchapter,byOliverPooley,providesanup-to-date,comprehensivediscussionofsubstantivalistandrelationalistapproachestospacetime.Crudely,thisisadebateabouttheontologyofourtheoriesofspaceandspacetime.Thesubstantivalistsholdthatamongthefundamentalobjectsoftheworldisspace-timeitself.Relationists,tothecontrary,denythatpropositionsaboutspacetimeareultimatelytobeunderstoodintermsofclaimsaboutmaterialobjectsandpossiblespatiotemporalrelationsthatmayobtainbetweenthem.Pooleypresentsahistoricalintroduction,aswellasadetaileddiscussionofthecurrentlandscapeintheliterature.Specifically,heconsidersrecentrelationist,neo-MachianproposalsbyBarbour,aswellasdynamicalapproachesfavoredbyBrown,andPooleyandBrown,thataimtoprovideareductiveaccountofthespacetimesymmetriesintermsofthedynamicalsymmetriesoflawsgoverningthebehaviorofmatter.Inaddition,Pooleyprovidesacurrentassessmentoftheimpactoftheso-calledHoleArgumentagainstsubstantivalism.

InGlobalSpacetimeStructureJohnManchakexaminesthequalitative,primarilytopologicalandcausal,aspectsofgeneralrelativity.Heprovidesanabstractclassificationofvariouslocalandglobalspacetimeproperties.Intheglobalcausalcontextheexplicitlydefinesasetofcausalconditionsthatformastricthierarchyofpossiblecasualpropertiesofspacetime.Thestrongestistheconditionofglobalhyperbolicity,whichimpliesothersincludingcausalityandchronology.Anothersetofglobalpropertiesofspacetimeconcernsinwhatsenseaspacetimecanbesaidtopossesssingularities.Herehefocusesonthenotionofgeodesicincompleteness.Manchakthentakesupphilosophicalquestionsconcerningthephysicalreasonablenessofthesevariousspacetimeproperties.Inalocalcontext,beingasolutiontoEinstein'sFieldEquationistypicallytakentobephysicallyreasonable.But,globalpropertiesconcerningtheexistenceandnatureofsingularitiesandthepossibilityoftimetravelleadtoopenquestionsofphilosophicalinterestthatarecurrentlybeinginvestigated.

Last,butnotleast,ChrisSmeenk'scontributionconcernsphilosophicalissuesraisedincontemporaryworkoncosmology.Acommonviewisthatcosmologyrequiresadistinctivemethodologybecausetheuniverse-as-a-

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wholeisauniqueobject.Restrictionsonobservationalaccesstotheuniverseduetothefinitespeedoflightposeseverechallengestoestablishingglobalpropertiesoftheuniverse.Howcanweknowthatthelocalgeneralizationswetaketobelawfulinourlimitedregioncanbeextendedinaglobalfashion?Here,ofcourse,thereisoverlapwiththediscussionsofthepreviouschapter.Successesoftheso-calledStandardModelforcosmologyincludebig-bangnucleosynthesisandtheunderstandingofthecosmicbackgroundradiation,amongothers.ChallengestotheStandardModelresultfromgrowingevidencethatifitiscorrect,thenmostofthematterandenergypresentintheuniverseisnotwhatwewouldconsiderordinary.Instead,thereapparentlyneedstobedarkmatteranddarkenergy.Smeenkprovidesanoverviewofrecenthypothesesaboutdarkmatterandenergy,andrelatesthesediscussionstophilosophicaldebatesaboutunderdetermination.Adifferentkindofproblemarisesinassessingtheoriesregardingtheveryearlyuniverse.ThesetheoriesareoftenmotivatedbytheideathattheinitialstaterequiredbytheStandardModelishighlyimprobable.Thisdeficiencycanbeaddressedbyintroducingadynamicalphaseofevolution,suchasinflationarycosmology,thatalleviatesthisneedforaspecialinitialstate.Smeenknotesthatassessingthisresponsetofine-tuningisconnectedwithdebatesaboutexplanationandfoundationaldiscussionsregardingtime'sarrow.Oneveryimportantaspectofrecentworkincosmologyistheappealtoanthropicreasoningtohelpexplainfeaturesoftheearlyuniverse.Asecondrecentdevelopment,oftenrelatedtoanthropicconsiderations,isthemultiversehypothesistheexistenceofcausallyisolatedpocketuniverses.Thischapterbringsthesefascinatingissuestotheforeandraisesanumberofphilosophicalquestionsaboutthenatureofexplanationandconfirmationappropriateforcosmology.

Itismyhopethatreadersofthisvolumewillgainasenseofthewidevarietyofissuesthatconstitutethegeneralfieldofphilosophyofphysics.Thefocusofthefieldhasexpandedtremendouslyoverthelastthirtyyears.Newproblemshavecomeup,andoldproblemshavebeenrefocusedandrefined.Itisindeedmypleasuretothankalloftheauthorsfortheircontributions.Inaddition,IwouldliketothankPeterOhlinfromOxfordUniversityPress.Anumberofotherscontributedtothisprojectinvariousways.IamparticularlyindebtedtoGordonBelot,JuliaBursten,NicolasFillion,LauraRuetsche,ChrisSmeenk,andMarkWilsonforinvaluableadviceandsupport.

ReferencesP.W.Anderson.Moreisdifferent.Science,177(4047):393396,1972.

OlivierDarrigol.WorldsofFlow:AHistoryofHydrodynamicsfromtheBernoullistoPrandtl.OxfordUniversityPress,Oxford,2005.

JohnEarman.APrimeronDeterminism.Reidel,Dordrecht,1986.

Notes:(1)Darrigol'srecentWorldsofFlowfillsthislacunaprovidinganexceptionaldiscussionofthehistory(Darrigol2005).

RobertBattermanRobertBattermanisProfessorofPhilosophyatTheUniversityofPittsburgh.HeisaFellowoftheRoyalSocietyofCanada.Heistheauthorof_TheDevilintheDetails:AsymptoticReasoninginExplanation,Reduction,andEmergence_(Oxford,2002).Hisworkinphilosophyofphysicsfocusesprimarilyupontheareaofcondensedmatterbroadlyconstrued.Hisresearchinterestsincludethefoundationsofstatisticalphysics,dynamicalsystemsandchaos,asymptoticreasoning,mathematicalidealizations,thephilosophyofappliedmathematics,explanation,reduction,andemergence.

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ForaPhilosophyofHydrodynamicsOlivierDarrigolTheOxfordHandbookofPhilosophyofPhysicsEditedbyRobertBatterman

AbstractandKeywords

Thischapterdiscussestheneedforaphilosophyofhydrodynamicsandthelessonsthatcanbelearnedfromthehistoricaldevelopmentoffluidmechanics.Itexplainsthathydrodynamicshasbeennotgivenattentionbyphilosophersofphysicsbecauseofalackofdetailedhistoricalstudiesofhydrodynamics,andhighlightstheneedforidealizationsandmodelingstrategiesforthistheorytobeapplicable.Thechapteralsoconsidersthestructuresofphenomenologicaltheoriesandtheso-calledmodularstructureofhydrodynamics.Keywords:hydrodynamics,fluidmechanics,philosophersofphysics,historicalstudies,idealizations,modelingstrategies,phenomenologicaltheories,modularstructure

Amongthemajortheoriesofphysics,hydrodynamicsisprobablytheonethathasreceivedtheleastattentionfromphilosophersofscience.Untilrecently,threecircumstanceseasilyexplainedthisneglect.First,therewasverylittlehistoricalliteratureonwhichphilosopherscouldrely.Second,philosopherstendedtofocusonfundamentaltheoriessuchasrelativitytheoryandquantumtheoryandtoneglectmorephenomenologicaltheories.Third,theyharboredaneo-Hempelianconceptofexplanationfollowingwhichthefoundationsofatheoryimplicitlycontainallitsexplanatoryapparatus. EvenThomasKuhn,whobroughtthenormalphasesofsciencetothefore,restrictedconceptualinnovationtotherevolutionaryphases. Sincethefundamentalequationsofhydrodynamicshaveremainedessentiallythesameforabouttwocenturies,thisviewreducesthedevelopmentofthistheorytoamatteroftechnicalprowessinsolvingtheequations.

Inrecentyearsthesethreecircumstanceshavelostmuchoftheirweight.Wenowhavefairlydetailedhistoriesofhydrodynamics. Thesuperiorityoffundamentaltheoriesoverlowerscaleorphenomenologicaltheorieshasbeenmultiplychallenged,bothwithinscienceandinthephilosophyofscience. Andtherehasbeenagrowingawarenessthatexplanationmostlyresidesindevicesthatarenotcontainedinthebarefoundationsofatheory.Forexample,MaryMorganandMargaretMorrisonhaveemphasizedtheroleofmodelsasmediatorsbetweentheoryandexperiment;JeffryRamseyhasarguedtheconceptualsignificanceofapproximationsandtransformationreductions;RobertBattermanhasmadeexplanationdependonstrategiesfortheeliminationofirrelevantdetailsinthefoundations;PaulHumphreyshasplacedcomputabilityatthecenterofhisassessmentofthenatureandvalueofscientificknowledge.EricWinsberghasshowntheimportanceofextratheoreticalconsiderationsinjudgingthevalidityofnumericalsimulationsbasedonthefundamentalequations.Alreadyin1983,IanHackingandC.W.F.Everitt,whoweremoreintouchwiththeactualpracticeofphysiciststhanaveragephilosophers,introducedtheoryarticulationorcalculationasanessentialsemanticbridgebetweentheoryandobservation.

Grantingthattheoryarticulationisasphilosophicallyimportantasthebuildingoffoundations,hydrodynamicsbecomesatopicofexceptionalphilosophicalinterestlargelybecauseofthehugetimespanbetweentheestablishmentofitsfoundationsanditssuccessfulapplicationtosomeofthemostpressingengineeringproblems.Thisdelayisanindirectproofofthecreativityneededtoexpandtheexplanatorypoweroftheories.Itenablesus

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toobservearichsampleofthedevicesthroughwhichexplanatoryexpansionmayoccur.MargaretMorrison,MichaelHeidelberg,andMoritzEpplehaverecentlygivenphilosophicalstudiesoftwoofthesedevices:LudwigPrandtl'sboundary-layertheoryandhiswingtheory.Thepresentessayisconductedinthesamespirit.

Thefirstsectiongivesafewhistoricalexamplesofthemeansbywhichhydrodynamicsbecameapplicabletoagrowingnumberofconcretesituations.Thesecondprovidesatentativeclassificationofthesemeans.Thethirdcontainsadefinitionofphysicaltheoriesthatincludestheirevolvingexplanatoryapparatus.Specialemphasisisgiventoamodularstructureoftheoriesthatmakesthemmoreamenabletotests,comparisons,communication,andconstruction.

1.SomeHistoryInthemid-eighteenthcentury,JeanleRondd'AlembertandLeonhardEulerformulatedthegenerallawsofmotionofanonviscousfluid.InEuler'sform,callingvthevelocityofthefluid,Pitspressure,itsdensity,andfanimpressedforcedensity,theselawsaregivenbytheequationofmotion

theequationofcontinuity,

andtheboundaryconditionthatthefluidvelocitynexttothewallsofarigidcontainershouldbeparalleltothesewalls.Ifthefluidhasafreesurfaceatwhichittouchesanotherfluid,theboundaryconditions(laterprovidedbyLagrange)aretheequalityofthepressuresofthetwofluids,andtheconditionthataparticleofthesurfaceofonefluidshouldremainonitssurface.

Euler'sderivationoftheequationoffluidmotionassumesthepressurebetweentwocontiguousfluidpartstobeperpendiculartotheseparatingsurface,asisthecaseinhydrostatics.In1822ClaudeLouisNavierimplicitlydroppedthisassumptionbycomparingtheinternalfluidforceswiththemolecularforcesofhisgeneraltheoryofelasticity.TheresultingequationofmotionistheNavier-Stokesequation

whichinvolvestheviscosity.Thisequationwasreinventedseveraltimes.Therewasmuchhesitationontheproperboundaryconditions,althoughin1845GeorgeGabrielStokescorrectlyarguedforavanishingrelativevelocityofthefluidnexttorigidbodies.

Fromamathematicalpointofview,themostevidentgoalofthetheoryistointegratetheequationsofmotionforanygiveninitialconditionsandboundaryconditions.Thereareatleastthreereasonsnottoconfinefluidmechanicstothisgoal:

1.Inthecaseofacompressiblefluid,thesystemofequationsisnotcompletebecauseoneneedstherelationbetweenpressureanddensity.Thisrelationimpliesthermodynamicconsiderations,andthereforeforcesustoleavethenarrowcontextoffluidmechanics.2.Itisgenerallyimpossibletosolvetheequationsbyanalyticalmeansbecauseoftheirnonlinearcharacter.Moreover,thefewrestrictedcasesinwhichthisispossiblemayhavelittleornoresemblancewithactualflowbecauseofinstabilities.Nowadays,numericalintegrationisoftenpossibleandis,indeed,sufficientforsomeengineeringproblems.Thisleadsustothethirdcaveat.3.Theanswertomostphysicalquestionsregardingfluidbehaviorisnottobefoundinthesolutionofspecificboundary-valueproblems.Rather,thephysicistisofteninterestedingenericpropertiesofclassesofsolutions.Inmathematicalterms,weneedtohaveahandleonthestructureofthespaceofsolutions.

Whatdophysicistsdowhenthesolutionofboundaryproblemsnolongerservestheirinterests?Inordertoanswerthisquestion,wewillconsultsomeofthehistoricalevolutionofhydrodynamics.

1.1Bernoulli'sLaw

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Fromapracticalpointofview,themainresultthatEulercouldderivefromhisnewhydrodynamicswasthelaw

relatingthepressureP,thepositionr,andthevelocityvforthesteadymotionsofanincompressiblefluidthatadmitavelocitypotential(gistheaccelerationofgravity).Thisachievementmayseemmeagerforthefollowingreasons: thelawhadalreadybeenderivedbyDanielBernoulliinthe1730sasanapplicationoftheconservationofliveforce(energy)tosteady,parallel-slice,incompressiblefluidmotion;thelawrequiresanarrowspecializationofthetheory;oneaspectofthisspecialization,theexistenceofavelocitypotential,is(orwas)physicallyobscure(itsoriginalpurposewastosimplifytheequationsofmotionandtopermittheirintegration);underthisspecialization,thelawisastraightforwardmathematicalconsequenceofEuler'sequations.

Fromtheseremarks,onemightbetemptedtojudgethatBernoulli'slawaddsnothingsignificanttothefundamentalequationsofhydrodynamics.Yetthepracticeofphysicistsandengineerssuggeststhecontrary:Thislawisusedinmanycircumstances,surelymoreoftenthanEuler'sequationsthemselves.Thereareseveralgoodreasonsforthis:

(1)Bernoulli'slawrelateseasilyaccessibleparametersoffluidmotioninasimplemanner,withoutanyreferencetothesubtletiesoftheunderlyingdynamics;(2)itisrelatedtothegeneralprincipleofenergyconservation,whichbridgeshydrodynamicswithmechanics;(3)itprovidesthebasisforthehydraulicianslanguageofpressurehead,velocityhead,andgravityhead;and(4)thislanguageisstillusedwhenthelawisviolated.

Althoughthislastpointmayseemparadoxical,itillustratesahighlyimportantmodeofconceptformationinthepost-foundationallifeofatheory:thesolutionsofthegeneraltheoryarecharacterizedwithreferencetothesolutionsofamoreworkablespecializationofthistheory.Theconceptsengenderedbythespecializationthusenrichthelanguageofthegeneraltheory.Theyareusefulaslongasthelawisvalidinpartsoftheinvestigatedsystemandaslongasthelociofitsviolationsaresufficientlyunderstood.Intypicalhydraulicsystems,thereareregularpipesandreservoirsinwhichthelawapplieswithaknowncorrection(viscousorboundary-layerretardationinpipes)andtherearephenomenologicallyortheoreticallyknownlossesofheadwhensomeaccidents,suchaspipe-to-pipeconnectionsorsuddenenlargementsofthesectionofapipe,occur.

1.2SurfaceWavesHistorically,thesecondsuccessfulapplicationofEuler'sequationswastotheproblemofwaterwaves.Inthiscase,specializationisalsonecessary:thefluidistakentobeincompressibleandavelocitypotentialisassumed.Moreover,someapproximationsmustbeintroducedtocircumventthenonlinearityoftheequations.Inamemoirof1781,JosephLouisLagrangeoriginallyassumedwavesofsmallamplitudeandoflengthmuchlargerthanthedepthofthewater.Inthemid-1810s,SimonDenisPoissonandAugustinCauchydidwithoutthelatterapproximation.Theresultingdifferentialequationforthedeformationofthewatersurfaceislinear,anditadmitssine-wavesolutionswhosepropagationvelocitydependsonthewavelength.Atthis(first-order)approximation,onemayuseanautonomouslanguageofsinewavesthatisnolongerreminiscentoftheunderlyingfluiddynamicsandthatisequallyapplicabletootherkindsoflinearwaves.Alloneneedstoknowishowtocombine(superpose)varioussinewavesinordertoaccommodategiveninitialshapesorperturbationsofthewatersurface.Wehereencounterasecondcaseofbridgingofhydrodynamicswithothertheories:theintroductionofconceptsthatapplytosimilarmodesofmotionindifferenttheories(optics,hydrodynamics,acoustics).

Thisisnottosaythatalllinearwaveproblemsareunderstoodonceweknowthedispersionlaw(howthevelocityofasinewavedependsonitswavelength).Historically,mucheffortwasneededtounderstandthestructureofasuperpositionofsinewaves.Employingstrictlymathematicalmethods,PoissonandCauchyonlysucceededindescribingthewavecreatedbyastonethrownintoapond.JohnScottRussell(in1844)andWilliamFroude(in1873)laterobservedthatthefrontofagroupofwavestraveledatasmallervelocitythanindividualwavesinthegroup.In1876,Stokesgavethemoderntheoreticalexplanationintermsofphaseandgroupvelocity.Tenyearslater,WilliamThomson(LordKelvin)determinedtheformofshipwavesbyacleverapplicationoftheseconcepts.Onthephysicalsideofhisdeduction,hereliedontheopticalprincipleofinterference.Onthemathematicalside,

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heinventedthemethodofthestationaryphase,whichisnowcommonlyusedinvariousdomainsofphysics.Again,wehaveacaseofconceptsandtoolsgeneratedinaregionofagiventheorybutultimatelyappliedtoregionsofmanyothertheories(byregion,Imeanarestrictionofthetheorytoalimitedclassofsystemsandboundaryconditions).Theseconceptswerepartlyderivedbyamathematicalprocessofspecializationandapproximation,partlybyobservation,partlybyanalogywithotherdomainsofphysics.

Similarremarksapplytothecaseofnonlinearwaves.GeorgeBiddellAiryandStokestamednonlinearperiodicwavesbysuccessiveapproximationstothefundamentalequations,withapplicationstooceanwavesandrivertides.Thiswasamostlymathematicalprocessofacumbersomebutfairlyautomaticnature.Incontrast,ScottRussellobservedsolitarywaves(isolatedswells)ofinvariableshapelongbeforetheoristsadmittedtheirpossibility.WhenJosephBoussinesqandLordRayleighatlastdeducedsuchwavesfromtheory,itbecameclearthatqualitativeresults(suchasthedeformationoftravelingwaves)derivedbyconsideringseparatelyasmall-depth(nondispersive)approximationandasmall-amplitude(linear)approximation,nolongerobtainedwhenthedepthandamplitudewerebothlarge.Thecompensationofthedispersiveandnonlinearcausesofdeformationforwavesofaproperlyselectedshapeisamechanismwhich,again,appliestomanyotherdomainsofphysics.

1.3VortexMotionEarlyfluidmechanicsusuallyassumedtheexistenceofavelocitypotentialbecauseitgreatlysimplifiedthefundamentalequationsandalsobecauseLagrangehadshownthatitresultedfromtheequationsofmotionforalargeclassofboundaryconditions(motionstartedfromrestandcausedbymovingsolids).Anotherreason,emphasizedbyBritishfluidtheorists,wasthefactthatthevelocitypotentialofanincompressiblefluidobeysthesamedifferentialequation(Laplace'sequation)asthegravitational,electric,andmagneticpotentials.ThisformalanalogywasaconstantsourceofinspirationforStokes,Thomson,andJamesClerkMaxwell.Itpermittedanintuitivedemonstrationofsomebasictheoremsoftheabstractpotentialtheory,anditprovidedfluid-mechanicalanalogsofelectrostatic,electrokinetic,andmagnetostaticphenomena.

Figure1.1 Aportionofavortexfilament.Theproductofthevorticity(indicatedbythearrows)bythenormalsectionofthefilamentisaconstantalongthefilament.Itisalsoinvariableduringthemotionofthefluid.

Thegeneralcaseinwhichnovelocitypotentialexistswasjudgedintractableuntil1858whenHermannHelmholtzdiscoveredafewremarkabletheoremsthatpushedthiscasetotheforefrontofthetheory.AsCauchyandStokeshadearlierproved,theinfinitesimalevolutionofafluidelementcanberegardedasthesuperpositionofthreekindsofmotion:atranslationofthecenterofgravityoftheelement,adilationoftheelementalongthreemutuallyorthogonalaxes,andarotation.Formally,therotationperunittimeishalfthevector=vwhichhasthecomponents /y /zetc.Thisvector,nowcalledvorticity,vanishesifandonlyifthereexistsavelocitypotential(inaconnecteddomain).Thiskinematicanalysisofinfinitesimalfluidmotionispartoftheconceptualfurnitureofmodernfluidmechanics.Maxwellusedittodevelopthephysico-mathematicalconceptsofcurlanddivergencethatapplytoanyfieldtheory.Helmholtzreinventedittointerpretthenon-existenceofthevelocitypotentialandthevector=vgeometrically.

Helmholtzextendedthegeometricalinterpretationtothevorticityequation,

whichderivesfromEuler'sequationswhenthefluidisincompressible.Forthispurpose,hedefinedvortexfilamentsasthinbundlesoflineseverywheretangenttothevorticity,andtheintensityofafilamentastheproductofanormalsectionofthisfilamentbythevalueofthevorticityinthesection(seefigure1.1).Hethenshowedthattheintensityofafilamentwasaconstantalongafilamentandthatthevorticityequationwasequivalenttothestatementthatthevortexfilamentsmovedtogetherwiththefluidwithoutalteringtheirintensity.Thistheoremimpliesthatthedistributionofvorticityinaperfectliquidisinasenseinvariant:ittravelstogetherwiththefluidwithoutanyalteration.

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Inthislight,Helmholtzarguedthatthevorticityfield(astoday'sphysicistssay)betterrepresentedarbitraryflowsthanthevelocityfield:itsinvariantpropertiescompletelydeterminetherotationalcomponentoftheflow,whiletheirrotationalcomponentisruledbythetheoremsofpotentialtheory.Withthehelpofanelectromagneticanalogy,Helmholtzthendeterminedthevelocityfieldsassociatedwithsimpledistributionsofvorticity:straightvortexlines,vortexsheets,andvortexrings.Healsocalculatedtheinteractionsofvorticesandverifiedhispredictionsexperimentally.

ThevortexsheetsplayedanimportantroleinHelmholtz'slaterwritings.Theyaremathematicallyequivalenttoafiniteslideoffluidoverfluid,andtheyshouldoccur,accordingtoHelmholtz,wheneverafluidisforcedtopasstheedgeofanimmersedbody.Asanillustrationofthisprocess,Helmholtzgavetheformationofsmokejetswhenheblewthesmokeofacigarthroughhislips.Throughingeniousreasoning,heprovedtheinstabilityofthediscontinuitysurfacesorvortexsheets:anysmallbumponthemmustrollupspirally.Thismechanism,nowcalledHelmholtz-Kelvininstability,playsanimportantroleinmanyhydraulicandmeteorologicalphenomena,asHelmholtzhimselfforesaw.

Helmholtznotonlymeanttoimprovetheapplicabilityofhydrodynamicsbutalsotoequipthistheorywithanewmodeofdescriptionforfluidmotioninwhichvorticesanddiscontinuityweretheleadingstructuralfeatures.Theenormoussuccessofthisprojectinthelaterhistoryofhydrodynamicsissomewhatparadoxical,becauseHelmholtz'stheoremsonlyholdintheunrealisticcaseofaperfectliquid.ThephysicistsuseofthevorticityconceptinmuchmoregeneralsituationsiscomparabletothehydrauliciansuseoftheconceptofhydraulicheadinsituationsinwhichBernoulli'stheoremdoesnotapply.Insomecasesofvortexmotion,theeffectsofcompressibilityandviscositycanbeshowntobenegligible.Inallcases,onecantakeHelmholtz'stheoremsasareferenceandcorrectthemthroughtermsderivedfromtheNavier-Stokesequation,asVilhelmBjerknesdidinthelatenineteenthcentury.Asforthevortexsheets,wewillseeinamomentthatintheearlytwentiethcenturyLudwigPrandtlusedthemtoapproximatelydescribeimportantaspectsoffluidresistanceathighReynoldsnumber(lowviscosity).

Inthehistoricalexamplesdiscussedsofar,itbecameincreasinglydifficulttoproducetheneedednewconceptualapparatus.ThedegreeofdifficultycanbetakentobeproportionaltothetimeelapsedbetweentheinventionofEuler'sequationsandtheintroductionofthisapparatus.Forexample,Bernoulli'slawwaseasiesttoderive,asitonlyrequiresasimpleintegration.Butpuremathematicsdidnotsufficetodiscoverthelawsofwavepropagationonawatersurface.Someintuitionofinterferenceprocesses(borrowedfromoptics),andalsoafewexperimentalobservations(groupsofwaves,solitarywaves),wereinstrumental.Thediscoveryofthelawsofvortexmotionwasevenmoredifficult.Acenturyelapsedfromthetimewhend'AlembertandEulergavethevorticityequationtothetimewhenHelmholtzinterpreteditthroughhistheorem.Experimentsorobservationsdidnotbythemselvessuggestthisinterpretation,thoughHelmholtz'seffortswere,infact,partofaprojectforimprovingthetheoreticalunderstandingoforganpipes.Helmholtz'ssuccessprimarilydependedonhisabilitytocombinevariousheuristicdevicesincludingalgebraicmanipulationinthestyleofLagrange,geometricvisualizationinthestyleofThomsonandMaxwell,andafocusoninvariantquantitiesasexemplifiedinhisownworkonenergyconservation.

1.4InstabilitiesExactsolutionsofEuler'sorNavier'sequationsundergivenboundaryconditionsmaydifferwidelyfromtheflowobservedinaconcreterealizationoftheseconditions.Forinstance,theflowofwaterinapipeofrapidlyincreasingdiameterneverhasthesmooth,laminarcharacterofexactsteadysolutionsoftheNavier-Stokesequationinthiscase.AsStokesalreadysuspectedinthe1840s,thisdiscrepancyhastodowiththeinstabilityoftheexactsteadysolutions:anysmallperturbationofthesesolutionswillinducewidedeparturesfromtheoriginalmotion.Consequently,theknowledgeofexactsolutionsofthefundamentalequationsor(morerealistically)theknowledgeofsomefeaturesofthesesolutionsundergivenboundaryconditionsisnotsufficientforthepredictionofobservedflows.Onemustalsodeterminewhetherthesesolutionsorfeaturesarestable.

Inprinciplethisquestioncanbemathematicallydecided,byexamininghowaslightlyperturbedsolutionoftheequationsevolvesintime.Aswesaw,afirstsuccessinthisdirectionwasHelmholtz'spredictionofthespiralrollingupofabumponadiscontinuitysurface.Laterinthecentury,LordRayleighandLordKelvintreatedthemoredifficultproblemofthestabilityofplaneparallelflow.Theirresultswereonlypartial(Rayleigh'sinflectiontheoreminthenonviscouscase),orwrong(Kelvin'spredictionofstabilityfortheplanePoiseuilleflow).Mostofthesequestions

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exceededthemathematicalcapacityofnineteenth-centurytheorists,andsomeofthemhaveremainedunresolvedtothisday.TheeffortsofRayleighandothersnonethelessyieldedageneralmethodandlanguageofperturbativestabilityanalysis.Rayleighlinearizedtheequationofevolutionoftheperturbation,andsoughtplane-wavesolutions.Thesesolutionsarepropermodeswhoseoscillatoryorgrowingcharacterdependsontherealorimaginarycharacterofthefrequency.Thisproper-modeanalysisofstabilitygoesbeyondhydrodynamics:itoriginatedinLagrange'scelestialmechanicsanditcanbefoundinmanyotherpartsofphysics.

Asthemathematicaldiscussionofstabilitywasnearlyasdifficultasthefindingofexactsolutionsofthefundamentalequations,themostimportantresultsinthisdomainwerereachedbyempiricalmeans.Plausibly,theobservedinstabilityofjetsmotivatedHelmholtz'sderivationoftheinstabilityofdiscontinuitysurfaces.Certainly,Tyndall'sobservationsofthiskindmotivatedRayleigh'scalculationsforparallelflow.Mostimportant,GotthilfHagen(1839)andOsborneReynolds(1883)discoveredthatpipeflow,foragivendiameterandagivenviscosity,suddenlychangeditscharacterfromlaminartoturbulentwhenthevelocitypassedacertaincriticalvalue.Thesharpnessofthistransitionwasasurprisetoalltheorists.FromReynoldstothepresent,attemptstomathematicallydeterminethecriticalvelocity(orReynoldsnumber)incylindricalpipeshavefailed.Thisisaquestionofacademicinterestonly,becauseunpredictableentranceeffects(thewaythefluidisintroducedintothepipe),nottheinherentinstabilityinapipeofinfinitelength,usuallydeterminethetransition.

Inthetwentiethcentury,significantprogresshasbeenmadeinunderstandingthetransitionfromlaminartoturbulentflow.Inthefirsthalfofthecentury,LudwigPrandtl,WalterTollmien,WernerHeisenberg,andChiaChiaoLinprovedtheinstabilityoftheplanePoiseuilleflowandunveiledthespatialperiodicityofthemechanismofthisinstability. Inthesecondhalfofthecentury,developmentsinthetheoryofdynamicalsystemsattheintersectionbetweenpuremathematics,meteorology,andhydrodynamicspermittedadetailedqualitativeunderstandingofthetransitiontoturbulence,withintermediateoscillatoryregimes,bifurcations,andstrangeattractors. Itremainstruethatmostofthepracticalapplicationsofhydrodynamicsonlyrequirearoughknowledgeoftheconditionsunderwhichturbulenceoccurs.Thesourceofthisknowledgeispartlytheoreticalandpartlyempirical.Thereisnoeasywaytogatheritfromthefundamentalequations.Inmostcases,thebestthatcanbedoneistorepeatReynolds'sroughargumentthatthefullvorticityequationhastwoterms,aviscoustermthattendstodampanyeddyingmotion,andaninertialtermwhichpreservestheglobalamountofvorticity.Thelaminarorturbulentcharacterofthemotiondependsontheratioofthesetwoterms,whoseorderofmagnitudeisgivenbytheReynoldsnumber.

1.5TurbulenceThestateofmotionthatfollowstheturbulenttransitionisevenmoredifficulttoanalyzethanthetransitionitself.Casualobservationofturbulentflowrevealsitschaoticandmulti-scalecharacter.Thedetaileddescriptionofanymotionofthiskindseemstorequireahugeamountofinformation,muchmorethanishumanlyaccessible(withoutcomputersatleast).AsReynoldspondered,weareherefacingasituationsimilartothatofthekinetictheoryofgases:theeffectivedegreesoffreedomaretoonumeroustobehandledbyahumancalculator.Unfortunately,turbulentmotionismoreoftenencounteredinnatureandinmanmadehydraulicdevicesthanlaminarmotion.Engineersandphysicistshavehadtoinventwaysofcopingwiththisdifficulty.

Onestrategyistodesignthehydraulicoraeronauticartifactssothatturbulencedoesnotoccur.Whenturbulencecannotbeavoided,onemayadoptapurelyempiricalapproachandseekrelationsbetweenmeasuredquantitiesofinterest.Forinstance,nineteenth-centuryengineersgaveempiricallawsfortheretardation(lossofhead)inhydraulicpipes.Asecondapproachistofindrulesallowingthetransferoftheresultsofmeasurementsdoneatonescaletoanotherscale.Stokes,Helmholtz,andFroudepioneeredthisapproachinthecontextsofpendulumdamping,balloonsteering,andshipresistance,respectively.TheyderivedtheneededscalingrulesfromthescalingsymmetriesoftheNavier-Stokesequationoroftheunderlyingdynamicalprinciples.Thisisanexampleofahybridapproach,foundedpartlyonthefundamentalequations,andpartlyonmeasurementsoftheoreticallyunpredictableproperties.

Inathirdapproach,onemaycompletelyignorethefoundationsoffluidmechanicsandcookupamodelbasedonagrosslysimplifiedpictureoftheflow.Animportantexampleisthelawsforopenchannelflowdiscoveredinthe1830sand1840sbyafewFrenchPolytechnique-trainedengineers:JeanBaptisteBlanger,JeanVictorPoncelet,andGaspardCoriolis.Theyassumedtheflowtooccurthroughparallelslicesthatrubbedagainstthebottomofthechannelaccordingtoaphenomenologicalfrictionlaw,andtheyappliedmomentumorenergybalancetoeach

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slice.

Inthe1840sAdhmarBarrdeSaint-Venantemphasizedthetumultuouscharacterofthefluidmotioninopenchannelflowandsuggestedadistinctionbetweenthelarge-scaleaveragemotionofthefluidandthesmaller-scaletumultuousmotion.Themaineffectofthelattermotionontheformer,Saint-Venantargued,wastoenhancemomentumexchangebetweensuccessive(large-scale)fluidlayers.Basedonthisintuition,hereplacedtheviscosityintheNavier-Stokesequationwithaneffectiveviscositythatdependedonvariousmacroscopiccircumstancessuchasthedistancefromawall.Inthe1870s,BoussinesqsolvedtheresultingequationforopenchannelsofsimplesectionandthusobtainedlawsthatresembledBlanger'sandCoriolis'slaws,withdifferentinterpretationsoftherelevantparameters.

In1895,Reynoldsreliedonanalogywiththekinetictheoryofgasestodevelopanexplicitlystatisticalapproachtoturbulentflow.InthespiritofMaxwell'skinetic-molecularderivationoftheNavier-Stokesequation,hederivedalarge-scaleequationoffluidmotionbyaveragingoverthesmall-scalemotionsgovernedbytheNavier-Stokesequation.Reynolds'sequationdependsontheReynoldsstress,whichdescribestheturbulentexchangebetweensuccessivemacrolayersofthefluid.LikeSaint-Venant'seffectiveviscosity,theReynoldsstresscannotbedeterminedwithoutfurtherassumptionsconcerningtheturbulentfluctuationaroundthelarge-scalemotion.Therehavebeenmanyattemptstofillthisgapinthetwentiethcentury.ThemostusefuloneswereKrmn'sandPrandtl'sderivationsofthelogarithmicvelocityprofileofaturbulentboundarylayer.Theassumptionsmadein(improved)versionsofthesederivationsaresimpleandnatural(uniformstress,matchingbetweentheturbulentlayerandalaminarsublayernexttothewall),andtheresultingprofilefitsexperimentsextremelywell(muchbetterthanearlierphenomenologicallaws).Thelogarithmicprofileisthebasisofeverymodernengineeringcalculationofretardationinpipesoropenchannels.

DespitepowerfulstudiesbyGeoffreyTaylor,AndreyNikolaevichKolmogorov,andmanyothers,theprecisemannerinwhichturbulencedistributesenergybetweendifferentscalesoffluidmotionremainsamystery. Thereisnodoubt,however,thatthegeneralideaofdescribingturbulentflowstatisticallyhasbeenfruitfulsinceitsfirstintimationsbySaint-Venant,Boussinesq,andReynolds.Inthecaseofturbulentfluidmechanics,asinstatisticalmechanics,anewconceptualstructureemergesatthemacroscaleofdescription.Similarquestionscanberaisedinbothcasesconcerningthenatureofthereductionoremergence.Doesthemicroscaletheorytrulyimplythemacroscalestructure?Isthisstructureuniquelydefined?Canthisstructurebeusedwithoutfurtherreferencetothemicroscale?Aretheresingularsituationsinwhichthereductionfails?Theanswerstothesequestionstendtobemorepositiveinthecaseofstatisticalmechanicsthaninthecaseofthestatisticaltheoryofturbulence,becausetherelevantstatisticsarebetterknownintheformerthaninthelattercase.

1.6BoundaryLayersFromapracticalpointofview,twooutstandingproblemsoffluidmechanicsarefluidresistanceandfluidretardation.Fluidresistanceisthedeceleratingforceexperiencedbyarigidbodymovingthroughafluid.Fluidretardationisthefallofpressureorlossofheadexperiencedbyafluidduringitstravelalongpipesorchannels.Thetwoproblemsarerelated,sincetheybothinvolvethemutualactionofafluidandanimmersedsolid.In1768,d'Alembertchallengedthesagaciousgeometerswiththeparadoxthatresistancevanishedforaperfectliquidinhisnewhydrodynamics.Therewerevariousstrategiestocircumventthistheoreticalfailure.Someengineersdeterminedbypurelyempiricalmeanshowtheresistancedependedonthevelocityandshapeoftheimmersedbody.OthersretreatedtoIsaacNewton'snavetheorybytheimpactoffluidparticlesonthefrontofthebody,althoughsomeconsequencesofthistheory(suchastheirrelevanceoftheshapeoftheendofthebody)hadalreadybeenrefuted.Inthemid-nineteenthcentury,Saint-Venant,Poncelet,andStokestracedresistancetoviscosityandtheproductionofeddies.Withthedampingofpendulumsinmind,StokessuccessfullydeterminedtheresistanceofsmallspheresandcylindersbyfindingsolutionstothelinearizedNavier-Stokesequation.Formostpracticalproblems,thelargersizeoftheimmersedbodyandthesmallnessoftheviscositiesofairandwaterimplythatthenonlineartermofthisequationcannotbeneglected(theReynoldsnumberistoohigh).Stokeshadnothingtosayinsuchcasesbeyondthequalitativeideaofdissipationbytheproductionofeddies.

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Figure1.2 Discontinuitysurface(ee)formedwhenadownwardflowencountersthediskA.FromThomson(1894,220).

Intheidealcaseofvanishingviscosity,theproofofd'Alembert'sparadoximplicitlyassumesthecontinuityofthefluidmotion.However,Helmholtz'sstudyofvortexmotionimpliesthatfiniteslipoffluidoverfluidisperfectlycompatiblewithEuler'sequations.Around1870,KirchhoffandRayleighrealizedthatHelmholtz'sdiscontinuitysurfacesyieldedafiniteresistanceforanimmersedplate.AccordingtoHelmholtz,atubulardiscontinuitysurfaceisindeedproducedatthesharpedgesoftheplate.Thewaterbehindtheplateandwithinthissurfaceisstagnant,sothatitspressurevanishes(whenmeasuredinreferencetoitsuniformvalueatinfinitedistancesfromtheplate)(seefigure1.2).Sincethepressureatthefrontoftheplateispositive,thereisafiniteresistance,whichKirchhoffandRayleighdeterminedbyanalyticalmeans.Theresultroughlyagreedwiththemeasuredresistance.

Inthecaseofships,theresistanceproblemiscomplicatedbythefactthatshipsarenotsupposedtobecompletelyimmersed.Consequently,waveformationatthewatersurfaceisasignificantcontributiontotheresistance.Theleadingnineteenthcenturyexpertsonthisquestion,WilliamJohnMacquornRankineandWilliamFroude,distinguishedthreecausesofresistance:waveresistance,skinresistance,andeddyresistance.Skinresistancecorrespondstosomesortoffrictionofthewaterwhenittravelsalongthehull.Eddyresistancecorrespondstotheformationofeddiesatthesternoftheship;itisusuallyavoidedbyproperprofilingofthehull.RankineandFroudetracedskinresistancetotheformationofaneddyingfluidlayernexttothehull.Theyderivedthisnotionfromtheobservationthattheflowofwateraroundtheship,whenseenfromthedeck,appearstobesmootheverywhereexpectforanarrowtumultuouslayernexttothehullandforthewake.RankineassumedthevalidityofEuler'sequationsinthesmoothpartoftheflowandsolvedittodeterminethehullshapesthatminimizedwaveformation.Froudegaveafairlydetaileddescriptionofthemechanismofretardationintheeddyinglayer,althoughhewasnotabletodrawquantitativeconclusions.Intheend,Froudemeasuredskinfrictiononplates,totalresistanceonsmall-scaleshipmodels,andthenusedseparatescalinglawsforskinandwaveresistanceinordertodeterminetheresistanceofaprospectiveshiphull.

Figure1.3 Formationofadiscontinuitysurfacebehindacylinder.FromPrandtl(1905,57980).

Insum,RankineandFroudedistinguishedtwodifferentregionsofflowamenabletodifferenttheoreticalorsemi-empiricaltreatmentsandcombinedtheresultinginsightstodeterminethetotalresistance.FroudethusobtainedthefirstquantitativesuccessesintheproblemoffluidresistanceatahighReynoldsnumber.Althoughhisand Rank-ine'sconsiderationsappealedtohighertheoryinseveralmanners,theyalsorequiredconsiderableempirical input.

ThenextandmostfamousprogressinthehighReynolds-numberresistanceproblemoccurredinGttingen,undertheleadershipofLudwigPrandtl.ImpressedbythequalitativesuccessofHelmholtz'ssurfacesofdiscontinuity,PrandtlassumedthatthesolutionoftheNavier-Stokesequationforhigh-ReynoldsflowaroundabodysomewhatresembledasolutionofEuler'sequation(withstrictlyvanishingviscosity).Inthelattersolution,thefluidslidesalongthesurfaceofthebody,whereasforaviscousfluidtherelativevelocityofthefluidmustvanishatthesurfaceofthebody.Consequently,fortherealflowPrandtlassumedathin(invisible)layerofintenseshearthat

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imitatedthefiniteslideoftheEuleriansolution.HealsoassumedthatinsomecasesthislayercouldshootoffthesurfaceofthebodytomimicaHelmholtziansurfaceofdiscontinuity(withitscharacteristicinstabilityresultinginaneddyingtrail).Thisistheso-calledseparationprocess.Outsidetheboundarylayer,PrandtlnaturallyappliedEuler'sequations.Withintheboundarylayer,theintenseshearallowedhimtouseanapproximationoftheNavier-Stokesequationthatcouldbeintegratedtodeterminetheevolutionofthevelocityprofilealongthebody.Forsufficientlycurvedbodies,Prandtlfoundthatatsomepointtheflowwasinvertedinthepartoftheboundarylayerclosesttothebody.Heinterpretedthispointastheseparationpointfromwhicha(quasi)discontinuitysurfacewasformed.Inthecaseofaflatorlittlecurvedsurface(forwhichseparationdoesnotoccur),hedeterminedtheresistancebyintegrationofthesheerstressalongthesurfaceofthebody.Heillustratedtheseparationprocessthroughexperimentsdonewithatankandapaddle-wheelmachine(figure1.3).

ComparisonwithFroude'searlierconceptofeddyinglayerleadstothefollowingremarks.UnlikeFroude,Prandtlwasabletodeterminetheoreticallyandpreciselytheflowwithintheboundarylayer.ThisdeterminationrequiresaprevioussolutionoftheEulerianflowproblemaroundthebody,becausetheevolutionoftheboundarylayerdependsonthepressureatitsconfines.Conversely,thisevolutionmayinduceseparation,whichnecessarilyaffectstheEulerianpartoftheflow.PrandtlhimselfemphasizedthisinteractionbetweentheEulerianflowandtheboundarylayer.WhereasFroudehadnointerestinseparation(whichshipbuilderssystematicallyavoided),Prandtlhadaprecisecriterionforitsoccurrence.WhereasFroudecouldonlymeasurethesheerstressoftheboundarylayer,Prandtlcoulddetermineittheoretically.

SofarthecomparisonseemstofavorPrandtl.Inreality,inmanycasesincludingshipresistance,theboundarylayerhasaninternalturbulencethatisnottakenintoaccountinPrandtl'soriginaltheory.In1913,Prandtl'sformerstudentHeinrichBlasiusfoundthatbeyondacertaincriticalReynoldsnumber,theedgewiseresistanceofaplateobeyedFroude'sempiricallawandnotPrandtl'stheoreticallaw.PrandtlexplainedthattheprofileofalaminarboundarylayercouldbecomeunstableandthusleadtoaturbulentboundarylayerlaFroude.HeusedthisnotiontoexplainthebizarrereductionoftheresistanceofspheresthatGustaveEiffelhadobservedatacertaincriticalvelocity:turbulenceinaboundarylayer,Prandtlexplained,delaystheseparationprocessandthussharplydecreasestheresistance.Paradoxically,itiswhentheboundarylayeristurbulentthattheglobalflowmostlyresemblesthesmoothEulerianflow.

Astheboundarylayersaroundairplanewingsareturbulent,Prandtlneededtoknowthesheerstressalongsuchlayersinordertodeterminethedragofthewings.Heoriginallyreliedonplateresistancemeasurements,asFroudehaddoneinthepast.Aswasalreadymentioned,itbecamepossibletocalculatethisstressinthe1830swhenKrmnandPrandtldiscoveredthelogarithmicvelocityprofileofturbulentlayers.

Itisnowtimetoreflectontherelationthatboundary-layertheoryhastothefoundationaltheoryofNavier-Stokes.Prandtl'sidea(ifwebelievehisownplausibleaccount)hasitstheoreticaloriginintheideaofusingsolutionstoEuler'sequationsasaguideforsolvingtheNavier-StokesequationatahighReynoldsnumber.Thisisonlyaheuristic,becausePrandtlhadnomathematicalproofthatthelow-viscositylimitofasolutionoftheNavier-StokesequationisasolutionofEuler'sequation.YetthemotionimaginedbyPrandtl,withitsEulerian,high-sheer,andstagnantregions,clearlyisanapproximatesolutionoftheNavier-Stokesequation.Whatismissingisaproofoftheuniquenessofthissolution(undergivenboundaryconditions),aswellasageneralproofofitsexistenceforanyshapeoftheimmersedbody.Withthisconcession,theboundary-layertheorycanlegitimatelyberegardedasanapproximationoftheNavier-Stokestheory.

Aninterestingfeatureoftheboundary-layertheoryisitsuseofdifferentapproximateequationsindifferentregionsoftheflow.OurdiscussionofBernoulli'slawshowedthatthislawisoftenusedregionally(i.e.,inlaminarregionsoftheflow)withheadlosseslocalizedinturbulentregions.Boundary-layertheorysimilarlyintroducesdifferentregionsofflow,althoughitdoessoinamoreinteractivemanner.Eachregionisdescribedthroughcomputablesolutionsofappropriateequationsofmotion,andthepreciseconditionsforthematchingoftheregionalsolutionsareknown(continuityofpressure,stress,andvelocity).Thesematchingconditionsimplycausalrelationsbetweenfeaturesofthetworegions:forinstance,thepressuredistributionintheEulerianregiondeterminestheevolutionofthevelocityprofileintheboundarylayer,andinthecaseofseparatedflow,thepositionoftheseparatingsurfaceaffectstheEulerianregion.

Inqualitativeapplications,Prandtl'stheorymayberestrictedtothegeneralideasofaboundarylayer,afreefluid,

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andtheirinteractionsometimesleadingtoseparation.Inquantitativeengineeringapplications,thispicturemustbesupplementedwithalawfortheevolutionofthesheerstressalongaboundarylayer(laminarorturbulent),andwithquantitativecriteriaforseparationandforthetransitionbetweenlaminarandturbulentlayer.Grantedthatthissupplementaryinformationisavailable,thetheorycanbeusedwithoutreferencetotheNavier-Stokestheory.Thegaininpredictiveefficiencyisenormous,asverifiedbytheimmensesuccessofPrandtl'stheoryinengineeringapplications.Yetoneshouldnotforgetthatmuchofthesupplementaryinformationcomesfromtheintimateconnectionbetweentheboundary-layertheoryandtheNavier-Stokestheory.Infactthelegitimacyofthewholepicturedependsuponthisintimateconnection.Theboundary-layertheory,unliketheearlyFrenchmodelsofopenchannelflow,isnotanadhocmodelthatowesitssimplicitytocounterfactualassumptions.ItisalegitimatearticulationoftheNavier-Stokestheory.

2.ExplanatoryProgressTheaboveexamplesmakeclearthatinthecourseofitshistory,hydrodynamicshasacquiredasophisticatedexplanatoryapparatuswithoutwhichitwouldremainmerelyapapertheory.Theexplanatoryapparatusispresentedinvariouschaptersinmoderntextbooks.Wewillnowreflectonthewaysthisapparatuswasobtained,onitscomponents,andonitsfunctions.

2.1TheSourcesofExplanatoryProgressInsomecases,explanationwasimprovedthroughblindmathematicalmethods.Forinstance,asimpleintegrationyieldedBernoulli'slaw(afterproperspecialization),thesymmetriesoftheNavier-Stokesequationyieldedscalinglaws,andstandardapproximationproceduresyieldedthetheoryofwavesofsmallamplitude.Despitetherelativelyeasyandautomaticwayinwhichtheseresultswereobtained,theyconsiderablyimprovedtheexplanatorypowerofthetheorybydirectlyrelatingquantitiesofphysicalinterest.

Inothercases,moreintra-orintertheoreticalheuristicswasneeded.KinematicanalysisofthevorticityequationledtoHelmholtz'svortextheorems;asymptoticreasoningledtoPrandtl'snotionsoflaminarboundarylayerandseparation;scalingandmatchingargumentsledtothelogarithmicvelocityprofileofturbulentboundarylayers.Theseheuristicsrequiredanunusualamountofcreativity;theyinvolvedintuitionsboundtopersonalstylesofthinking.Suchintuitionsaretentativeandmayleadtoerroneousguesses.Forinstance,thegreatKelvinerredinhisstabilityanalysisofparallelflow.Arigorouscheckofthecompatibilityoftheconclusionswiththefundamentalequationsisalwaysneeded.

Instillothercases,observationsorexperimentssuggestednewconceptssuchasgroupvelocity,solitarywaves,thestabilityorinstabilityoflaminarflow,andturbulentboundarylayers.Theveryfactthatpuretheorywashistoricallyunabletoleadtotheseconcepts(andsometimesevenresistedtheirintroduction)showsthevanityofregardingthemasimplicitconsequencesofthefundamentalequations.Theyneverthelessbelongtofundamentalhydrodynamicsinasmuchastheircompatibilitywiththefundamentalequationscanbeverifiedaposteriori.

Lastly,theimpossibilityofsolvingthefundamentalequationandtheevidentcomplexityofobservedflowssometimesforcedengineersandevenphysiciststoarbitrarilyanddrasticallysimplifyaspectsoftheflow.Thishappenedforinstanceinearlymodelsofopenchannelflow.Thesemodelscannotbestrictlyregardedaspartsoffundamentalhydrodynamics,sincesomeoftheirassumptionscontradictbothobservedandtheoreticalpropertiesoftheflow.YettheirsuccesssuggestsaloosersortofrelationwiththeNavier-Stokestheory.Inthecaseofopen-channelflow,themodelscanbereinterpretedasre-parametrizationsofthetrueequationsfortheapproximate,large-scalemotionderivedfromturbulentsolutionsoftheNavier-Stokesequations.

Ineverycase,thetheoreticaldevelopmentsoccurredwithspecificapplicationsinmind:somekindofflowfrequentlyobservedinnatureneededtobeexplainedorthefunctioningofsomeinstrumentsordevicesneededtobeunderstood.Purelymathematicalmethodsbroadlyappliedtogeneralflowwereoflittleavail.Insightwasgainedasaresultofinvestigationdirectedatconcreteandrestrictedgoals.Thisiswhytheheroesofnineteenth-centuryandearlytwentieth-centuryfluidmechanicswereeithermathematicallyfluentengineersorphysicistswhohadafootintheengineeringworld.

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2.2TheComponentsofExplanationAfirstalleytowardbetterexplanationinvolvestherestrictionofthescopeofatheory.TheNavier-Stokesequations,regardedasthegeneralfoundationofhydrodynamics,canbespecializedinvariousways.Therearehomogeneousspecializationsoridealizationsinwhichtherestrictedchoiceofparametersandkindsofsystems(boundaryconditions)leadstomoretractableintegrationproblemsorsuccessfulstatisticalapproaches.TypicalexamplesareirrotationalEulerianflow,lowReynolds-numberflow,andfullyturbulentflow.Therearealsoheterogeneousspecializationsinwhichtherestrictionsonparametersandsystemsleadtoflowsthathavedifferentregions,eachofwhichdependsuponaspecificsimplificationoftheNavier-Stokesequations.Thisisthecaseforthehigh-ReynoldsresistanceproblemandtheairplanewingproblemaccordingtoPrandtl.Aswasalreadymentioned,successhererequirespropermatchingbetweenthedifferentregions.

Anotherexplanatoryresourceistheidentificationofinvariantstructuresofaflowbelongingtoagivenclass.ThemostimpressiveexampleofthissortisHelmholtz'sdemonstrationoftheconservationofvortexfilaments.Asthemindtendstofocusoninvariantaspectsofourenvironment,theidentificationofnewinvariantsoftenshapeourdescriptivelanguage.AsHelmholtzpredicted,thishas,infact,happenedinfluidmechanics:thevorticityfieldisnowoftenpreferredtothevelocityfieldasadescriptionofflow.

Third,insteadofseekingstructureinagivensolution,wemayattendtothestructureofthespaceofsolutionsofthefundamentalequationwhentheboundaryconditionsvary.Forinstance,wemayaskwhetherlaminarsolutionsaretypical,whethersmallperturbationsleadtodifferentsortsofsolutions:thisistheissueofstability.Wemayalsoaskwhethersomeclassesofsolutionsharecommonlarge-scalefeatures,aswedointhestatisticaltheoriesofturbulence.And,wemayaskwhethersomepropertiesorlawsaregenericinsomeregimeofflow:thisistheissueofuniversality,whichwebrieflytouchedwiththelogarithmicprofileofturbulentboundarylayers.

Lastly,explanationandunderstandingmaycomefromlinkinghydrodynamicstoothertheories.Wehaveencounteredafewexamplesofthiskind:potentialtheory,waveinterference,groupvelocity,solitarywaves,fieldkinematics,andproper-modeanalysisofstability.Inhalfofthesecases,conceptsofhydrodynamicoriginwerebroughttobearonothertheoriesandnotviceversa.Thecross-theoreticalsharingofconceptsnonethelessremainsatokenoftheirexplanatoryvalue.

2.3APragmaticDefinitionofExplanationAswasstatedabove,thegoaloffluidmechanicscannotbereducedtofindingintegralsofthefundamentalequationsthatsatisfygivenboundaryconditions.Thisisusuallyimpossiblebyanalyticalmeans,andmodernnumericalmeansrequireadifferentsimulationforeachchoiceintheinfinitevarietyofboundaryconditions.AsBatterman,Ramsey,andHeidelbergerhaveargued,barefoundationsdonotanswerthequestionsthattrulyinterestphysicistsandengineers.Practitionerswanttobeabletocharacterizeaphysicalsituationbyahumanlyaccessiblenumberofphysicalparametersandtopossessapictureofthesituationthatenablesthemtoderiverelationsbetweentheseparametersinareasonableamountoftime.Inotherwords,theyneedaconceptofexplanationthatintegratesourhumancapacityatrepresentingandintervening.AsBattermanemphasizes,thisrequiresmeansforeliminatingirrelevantdetailsinourdescriptionofsystems.Thisalsoimpliestheelaborationofadescriptivelanguage,theconceptsofwhichdirectlyrefertocontrollablefeaturesofthesystem.

Withthispragmaticdefinitionofexplanation,itbecomesclearthattheearlierdescribeddevelopmentsofhydrodynamicsservedthepurposeofincreasingtheexplanatorypowerofthetheory.Homogeneousspecializationsdosobyofferingadequateconceptsandmethodsforcertainkindsofflow.Heterogeneousspecializationsdosobycombiningtheformerspecializationstodescribeflowsthatoccurinproblemsofgreatpracticalimport.Theidentificationofinvariantstructuresforcertainclassesofmotionimprovestheeconomyoftherepresentation.Attentiontostructureinthespaceofsolutionsenablesustodecidetowhatextentsmallerdetailsofthemotionaffectthefeaturesofpracticalinterest,andtowhatextenttheireffectcanbesmoothedoutbysomeaveragingprocess.Intertheoreticallinksproducefamiliarconceptsthatcanindifferentlybeusedinvariousdomainsofphysics.

Inthislight,thepracticeofphysicshasmoresimilaritywithengineeringthanisusuallyassumed.Theremarkisnotuncommoninrecentwritingsinthephilosophyofscience.Forinstance,RamseyrevivesJ.J.Thomson'sold

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characterizationoftheoriesastoolsforsolvingphysicsorengineeringproblems;EpplecomparestheformationofPrandtl'swingtheorytoanengineeringprocesscombiningmultipletheoreticalandexperimentalresources.Inthesescholarsview,theengineeronlydiffersfromthephysicistby(usually)notparticipatingintheinventionofthetheoriesandbyhismoresystematicappealtoextra-theoreticalcomponents.Physicistsandengineersnotonlysharethegoalofefficientintervention,theyalsosharesomeofthemeans.

RamseyandHeidelbergerinsistthatthearticulationoftheoriesimpliestheformationofnew,adequateconcepts.OnecouldevenarguethatthebareNavier-Stokestheoryhasnophysicalconcepts.Itharborsonlymathematicalconceptssuchasthevelocityfieldthatcorrespondtoanidealdescriptionoftheflow,ignoringmolecularstructureandpresumingindefiniteresolution.Aconcept,intheetymologicalsenseoftheword(concipioinLatin,orbegreifeninGerman),isamentalmeanstograspsomeconcreteobjectorsituation.Hydraulichead,vortices,wavegroups,solitarywaves,thelaminar-turbulenttransition,boundarylayers,separation,etc.areconceptsinthispracticalsense.ThedetailedvelocityfieldorthevarioustermsoftheNavier-Stokesequationarenot.WhatThomasKuhnoncebelittledasthemoppingupoftheoriesinthenormalphasesofsciencetrulyisconceptformation.

3.TheoriesAndModules

3.1DefiningPhysicalTheoriesOncewerecognizethecognitiveimpotenceofthebarefoundationsofatheory,weneedageneraldefinitionoftheorythatisnotlimitedtothefundamentalequationsandafewnaverulesofapplication.Thedefinitionmustallowforevolvingcomponents,sincethecognitiveefficiencyofanygoodtheoryalwaysincreasesintime.Itmustincludeexplanatorydevicesanditmustallowtheintertheoreticalconnectivityfoundinmaturetheories.Thefollowingisasketchofsuchanenricheddefinition.

Aphysicaltheoryisamathematicalconstructincluding:

(a)asymbolicuniverseinwhichsystems,states,transformations,andevolutionsaredefinedbymeansofvariousmagnitudesbasedonCartesianpowersofR(orC)andonderivedfunctionalspaces.(b)theoreticallawsthatrestrictthebehaviorofsystemsinthesymbolicuniverse.(c)interpretiveschemesthatrelatethesymbolicuniversetoidealizedexperiments.(d)methodsofapproximationandconsiderationsofstabilitythatenableustoderiveandjudgetheconsequencesthatthetheoreticallawshaveontheinterpretiveschemes.

Thesymbolicuniverseandthetheoreticallawsarepermanentlygiven.Theycorrespondtothefamilyofmodelsofthesemanticviewofphysicaltheories.Inthecaseofhydrodynamics,thesymbolicuniverseconsistsinthevelocity,pressure,anddensityfieldsforeachfluidofthesystem,intheboundariesofrigidbodiesthatmayormaynotmove,andinforcedensitiessuchasgravity.ThetheoreticallawsaretheNavier-Stokesequations,boundaryconditions,and(forcompressiblefluids)arelationbetweendensityandpressurethatmayinvolvemodularcouplingwiththermodynamics(wewillreturntothispoint).

Inthesemanticviewoftheories,theempiricalcontentofatheoryisdefinedbyanisomorphismbetweenpartsofthesymbolicuniverseandempiricaldata;althoughthemeansbywhichthisisomorphismisdeterminedareusuallyleftinthedark.Thenotionofaninterpretiveschemeisintendedtofillpartofthisgap.Bydefinitionaninterpretiveschemeconsistsinagivensystemofthesymbolicuniversetogetherwithalistofcharacteristicquantitiesthatsatisfythethreefollowingproperties.(1)Theyareselectedamongorderivedfromthe(symbolic)quantitiesthatdefinethestateofthissystem.(2)Atleastforsomeofthem,idealmeasuringproceduresareknown.(3)Thelawsofthesymbolicuniverseimplyrelationsofafunctionalorastatisticalnatureamongthem.Morespecifically,interpretiveschemesareblueprintsofconceivableexperimentswhoseoutcomesdependonlyonrelationsbetweenafinitesetofmutuallyrelatedquantities,asufficientnumberofwhicharemeasurable.Insomecases,theintendedexperimentsmaybedesignedtodeterminesometheoreticalparametersfromthemeasuredquantities.Inothercases,thetheoreticalparametersaregiven,andtheoreticalrelationsbetweenthemeasuredquantitiesareverified.Inallcases,theinterpretiveschemesdonotcontainrigidlinguisticconnectionsbetweentheoreticaltermsandphysicalquantities;theirconcreteimplementationisanalogical,historical,andsubjecttorevisions.

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Theintroductionofinterpretiveschemesimpliesaselectionofsystemsandquantitiesfromtheinfinitevarietyofelementsinthesymbolicuniverseofthetheory.Thisselectioncanevolvedramaticallywiththenumberandnatureoftheimaginedapplicationsofthetheory.Thetwomainclassesofinterpretiveschemesofearlyhydrodynamicswerethepiercedvessel,inwhichtheeffluxofwaterisrelatedtotheheightofthewatersurface;andtheresistanceschemeinwhichasolidbodyimmersedinastreamofwaterexperiencesaforcerelatedtothevelocityofthestream.Anotherinterestingscheme,Bernoulli'spipeofvariablesection,impliedpressuremeasurementthroughverticalcolumnsofwater.Asampleoflaterschemesincludesthedeterminationofthevelocityofsurfacewavesasafunctionofdepthandwavelength,thevisualizedmotionofvorticesasafunctionoftheirrelativeconfiguration,thevisualizedlinesofflowaroundanimmersedbodyasafunctionoftheasymptoticvelocity,thedragandliftofawingasafunctionofasymptoticvelocityandangleofattack.Someschemeswerereactionstowell-iden

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The Oxford Handbook of Philosophy of Physics