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UNIFORMSHEARFLOWFARFROMEQUILIBRIUM By MIRIMLEE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 1996 ACKNOWLEDGMENTS IwouldliketothankProfessorJamesW.Duftyforhisguidance,encour- agementandpatienceforlastfiveyears.Ifeelmyselfveryfortunatenotonlyto havebenefitedfromhisacademicguidancebutalsotohaveenjoyedhisinvaluable friendship. IalsothankProfessorsKlauder,Muttalib,Obukhov,Reitze,Simmons,Brey, SantosandLutskofortheiracademicadvice. Inaddition,mygratitudegoestomyfellowgraduatestudentswhohavebeen goodfriendsthroughouttheseyears,inrandomordersDave,Beth,Thor,Chema, Steve,Mike,Namkyoung,Matthias,Gwyneth,WeonwooKim,andDon. AsformyfamilymembersinKorea,Icouldnotthankthemenough.Their loveandsupportmadeitpossibleformetoaccomplishwhatIhave. Special thanksgotomyparentsforalwaysbelievinginme. Finally,Iextendmydeepgratitudetomybestfriendandmyhusband,Jaewan Kim,forhisencouragementandsupport. TABLEOFCONTENTS ACKNOWLEDGMENTS ii LISTOFFIGURES v LISTOFSYMBOLS vi ABSTRACT xi CHAPTERS 1 1 INTRODUCTION 1 2 NONEQUILIBRIUMSTATISTICALMECHANICS 10 2.1 FormalSolutiontotheLiouvilleEquation 10 2.2 DensityFluctuations 15 3 KINETICMODELSFORCORRELATIONS 21 3.1 KineticEquationsforCorrelations 22 3.2 KineticModelsforCorrelations 28 3.3 SteadyStateCorrelations 33 3.4 Examples 36 3.4.1 EquilibriumStates 37 3.4.2 StationaryStatesNearEquilibrium 40 4 TRANSPORTEQUATIONSFORUNIFORMSHEARFLOW 45 4.1 StationaryStateSolution 45 4.2 NormalSolution 53 4.3 HydrodynamicEquations 58 5 TWOTIMECORRELATIONEQUATION 62 5.1 LinearizedHydrodynamicEquations 63 5.2 HydrodynamicModes 67 6 EQUALTIMECORRELATIONFUNCTION 83 7 CONCLUSIONS 87 APPENDICES 91 A DENSITYFLUCTUATIONS 91 B DERIVATIONOFKINETICEQUATIONS 96 C DERIVATIONOFKINETICMODEL 101 D EQUILIBRIUMEXCITATION 103 E FIRSTORDERAPPROXIMATION 106 F TRANSPORTCOEFFICIENTS 109 G THEHYDRODYNAMICMODES 118 H CORRELATIONFUNCTIONS 121 BIBLIOGRAPHY 122 BIOGRAPHICALSKETCH 125 IV LISTOFFIGURES 3.1 Realandimaginarypartsoftheextendedhydrodynamicmodes fromtheBGKmodel(inunitsofv).±indicatessoundmodes,H theheatmodeandStheshearmodes 39 4.1 Shearviscosityrj(a)(solidline)andviscometricfunction*(a)(dashed curve)areshownasfunctionofa.r\isinunitofpjv,*isinunit ofPs/v2andaisinunitofv 52 4.2 Thediagonalpartsofthegeneralizedthermalconductivity,\jj(a) alirneesihsofwornjas=axf,unacntdiotnhoefdaa.sTh-hdeotsolliindeliisnefoisrfjor=jz=.y\,jjthiesdinasuhneidt ofksPa/Tni/andaisinunitofv 60 5.1 Criticallinesforstabilityareshown.TheBGK-kineticmodelgives tfisohrefodsrioflnfi*edrel=innte0d.ae2nnsdainttdiheesd.aotsDhhaeersdhtechdur-redevoeltiinselsifnoaerriens*ffro=ormn0*.N4a.=viAe0l,rl-dSaotrtoetkeeidsncouurnrdivetesr ofthemeanfreepathandmeanfreetime 71 5.2 Tk*im=e0e.v1olutionof(a.)Sux(t')and(b)Suy(t')fora*=0.5and 77 v LISTOFSYMBOLS ashearrate A(t) physicalobservableattimet Aag matrix bij scatteringoperator Bag matrix B(t) physicalobservableattimet csoundvelocity Ccorrelationfunctionattwophasepointsandtimes C FourierrepresentationofC Cv specificheat Cag hydrodynamiccorrelationfunction Cag matrix Dap matrix Dt thermaldiffusivity /'s' s-particlereduceddistributionfunction /'s' s-particlemicroscopicphasedensity vi /solutiontotheBoltzmannequation / Fourierrepresentationof/ f,stationarysolutiontotheBoltzmannequation fi localequilibriumdistributionfunction /e Maxwell-Boltzmanndistributionfunction Fag matrix ga eigenfunctionsofinhomogeneouslinearizedBoltzmannoperator 9; velocityfunctionsinAppendixE Gsingletimepaircorrelationfunction Gap hydrodynamicpaircorrelationfunction Gn densityfluctuation H%] matrix ha complexconjugatesofga ha integralfunctioninAppendixE / linearizedBoltzmanncollisionoperator Ia matrix JBoltzmanncollisionoperator k wavevector KGreen'sfunction Kap matrix fcg Boltzmannconstant Imfp meanfreepath LLiouvilleoperator Lvelocityoperatorfromchapter4 Cinhomogeneouslinearoperator C FourierrepresentationofC m massoftheparticle ndensity Nap matrix ppressure p equilibriumpressure P projectionoperator qheatflux Q constant Rresolventfunction Si eigenvaluesofthematrixF Sdynamicstructurefactor ttime tij monentumflux,pressuretensor Tij volumeintegralofiy Ttemperature T(l,2) binaryscatteringoperator uflowvelocity vvelocityvariable v' velocityvariablerelativetotheuniformshearflowvelocity v thermalvelocity Vpeculiarvelocity V(r) potentialenergy w inhomogeneousterminthetemperatureequation xdimensionlessvariable Xaicoefficientsofthehydrodynamicgradients Yaj coefficientsofthehydrodynamicgradients 2/o hydrodynamicvariables z rootsofadeterminantequation Zcomplexprobabilityintegral adimensionlessparameter inversetemperaturemultipliedtokg1 Tsounddampingconstant 7sourcetothepaircorrelationfunctionG fH transportcoefficients eenergydensityinChapter2 £ universalparameter Q'' eigenvectorsofthematrixF rf^ biorthogonalsetofC*'' r\shearviscosity t\bshearviscosityfromtheBoltzmannequationforhardspheres 8dimensionlessparameter k, thermalconductivity kb thermalconductivityfromtheBoltzmannequationforhardspheres A linearoperator Aproportionalconstantintheexternalforce \a eigenvaluesof£ Xjjdiagonalpartsofgeneralizedthermalconductivity p.bulkviscosity vcollisionfrequency t?ajtransportcoefficients pmassdensity p(t) N—bodydistributionfunction Pi localequilibriumdistributionfunction adiameterofthehardsphere <j>a linearcombinationofthesummationalinvariants Xequilibriumpaircorrelationfunction ipa summationalinvariants *i viscometricfunctions