Uniform shear flow far from equilibrium

140 Pages·1996·2.7 MB·English

by  Lee, Mirim

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