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Dynamics of Suspensions of Rodlike Polymers with Hydrodynamic Interactions

Permanent Link: http://ufdc.ufl.edu/UFE0024219/00001

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Title: Dynamics of Suspensions of Rodlike Polymers with Hydrodynamic Interactions
Physical Description: 1 online resource (134 p.)
Language: english
Creator: Park, Joontaek
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: brownian, fiber, fluid, hydrodynamics, microchannel, polymer, rod, shear, slender
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Dynamics of suspensions of rigid rod-like polymers are studied by theoretical analysis and simulation considering hydrodynamic interactions. The motions of rigid rods in Newtonian suspending fluids are modeled using slender-body theory and and assumption of zero Reynolds number. Hydrodynamic interactions are included using methods by Butler & Shaqfeh. A kinetic theory is developed to investigate cross-stream migration of a rigid polymer undergoing rectilinear flow in the vicinity of a wall. Hydrodynamic interactions between the polymers and the boundary result in a cross-stream migration. A rigid polymer within shear flows at high Peclet numbers (weak Brownian motion) is simulated to enable comparisons with the steady and transient distributions predicted by the theory and simulation. The simulation and theoretical results are in good agreement at sufficiently high shear rates, validating approximations made in the theory. This study is directly applied to development of microfluidic devices or measurement of rheological properties.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joontaek Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Butler, Jason E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024219:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024219/00001

Material Information

Title: Dynamics of Suspensions of Rodlike Polymers with Hydrodynamic Interactions
Physical Description: 1 online resource (134 p.)
Language: english
Creator: Park, Joontaek
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: brownian, fiber, fluid, hydrodynamics, microchannel, polymer, rod, shear, slender
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Dynamics of suspensions of rigid rod-like polymers are studied by theoretical analysis and simulation considering hydrodynamic interactions. The motions of rigid rods in Newtonian suspending fluids are modeled using slender-body theory and and assumption of zero Reynolds number. Hydrodynamic interactions are included using methods by Butler & Shaqfeh. A kinetic theory is developed to investigate cross-stream migration of a rigid polymer undergoing rectilinear flow in the vicinity of a wall. Hydrodynamic interactions between the polymers and the boundary result in a cross-stream migration. A rigid polymer within shear flows at high Peclet numbers (weak Brownian motion) is simulated to enable comparisons with the steady and transient distributions predicted by the theory and simulation. The simulation and theoretical results are in good agreement at sufficiently high shear rates, validating approximations made in the theory. This study is directly applied to development of microfluidic devices or measurement of rheological properties.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joontaek Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Butler, Jason E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024219:00001


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andmy`great'brother,Jung-TaekPark. Ithankthemfortheirloveandsupport. 3

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Iwouldliketoacknowledgeallthepeoplewhohavebeeninstrumentalinhelpingmetocompletemydoctoraldegree.Firstofall,Idevotea`BIGTHANKS'tomyadvisor,Dr.JasonE.Butler,thegreatestteacher,boss,andmentorIhaveeverhad.Hispatientguidance,generousspirit,enthusiasm,and,aboveall,faithinmeenabledmetothriveasbothadoctoralstudentandaresearcher.Inaddition,hiswellspringofknowledgeandideas,engagingteachingstrategiesandresearchprojects,andhisintelligenceandwisdomhavebeenconstantsourcesofinspiration.Iwillalwayskeepinmindhisfavoriteproverb:\Bepositive."Iamalsoindebtedtomycommitteemembers,Dr.AnthonyJ.C.Ladd,Dr.DmitryI.Kopelevich,andDr.RenweiMei,fornotonlytheiradvice,feedback,andendeavorstoimprovethisdissertationbutalsoforservingasrolemodelsforthetypeofindependentresearcherandprofessorIwishtobecome.TheirgraduatecoursesareamongthebestcoursesIhaveevertakenthroughoutmycollegeyears.Iwishalsotogivethankstomycolleagues,Dr.JonathanM.Bricker,Dr.OsmanBerkUsta,Hyun-OkPark,MikeRoy,Xuan-ThinhPhan,ToddMock,Dr.JonghoonLee,Dr.Byung-JinChun,GauravMisra,RahulKekre,CarlosSilvera-Batista,Dr.PhilipD.CobbandDr.GunjanMohan,foralltheirhelpandpreciousandproductivediscussions.IamfortunatetohavehadtheopportunitytoworkwithandtoreceiveencouragementandadvicefromanumberofprofessorsinKorea:Dr.Ho-NamChang,Dr.Sang-YupLee,Dr.Hyun-GyuPark,Dr.Seung-ManYangattheKoreaAdvancedInstituteofScience&Technology;Dr.Cha-YongChoi,Dr.Byung-GeeKimatSeoulNationalUniversity;Dr.Sung-OkBaekatYeungnamUniversityandDr.Won-JiChungatChangwonUniversity.ThisworkwassupportedbytheNationalScienceFoundationthroughaCAREERAward(CTS-0348205),graduateandresearchassistantshipsfromtheUniversityofFlorida,theDepartmentofChemicalEngineering.IalsoacknowledgetheusageofcomputationaltimeintheChemicalEngineeringComputerSystematUF,RayW.Fahien 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 2INCLUSIONOFHYDRODYNAMICINTERACTIONSFORRIGIDRODSUSPENSIONS ......................... 17 2.1Introduction ................................... 17 2.2HydrodynamicInteractionsbetweenRigidRods ............... 18 2.3SedimentingOrbits ............................... 25 2.4Short-timeDiusivitiesofRigidRodSuspensions .............. 31 2.5Conclusion .................................... 36 3KINETICTHEORYFORCROSS-STREAMMIGRATIONOFARIGIDRODUNDERRECTILINEARFLOWNEARAWALL ................. 38 3.1Introduction ................................... 38 3.2Model ...................................... 40 3.2.1EquationofMotionnearaWall .................... 41 3.2.2DistributionFunction .......................... 43 3.3SteadyStateDistribution ........................... 48 3.3.1SimpleShearFlow ........................... 48 3.3.2Pressure-drivenFlow .......................... 49 3.3.3DiscussiononMigrationMechanisms ................. 52 3.4TransientDistribution ............................. 54 3.5Conclusion .................................... 56 4DYNAMICSIMULATIONOFARIGIDRODINSHEARINGFLOWSBETWEENTWOPARALLELWALLS ...................... 58 4.1Introduction ................................... 58 4.2Simulation .................................... 59 4.2.1GoverningEquation ........................... 59 4.2.2SimulationDetails ............................ 62 4.3Results ...................................... 65 4.3.1TransverseVelocityDuetoShearFlow ................ 65 4.3.2DistributionsinSimpleShearFlow .................. 68 4.3.3SteadyStateDistributioninParabolicFlow ............. 75 4.3.4SteadyStateDistributioninOscillatoryShearFlow ......... 78 6

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.................................... 81 5STRESSOFRIGIDRODSUSPENSIONSINSHEARINGFLOWSBETWEENTWOPARALLELWALLS ...................... 83 5.1Introduction ................................... 83 5.2CalculationofExtraParticleStressfromSimulations ............ 84 5.3TheoreticalPredictionofExtraParticleStress ................ 86 5.4Discussion:ComparisonofStressCalculations ................ 88 5.5Conclusion .................................... 91 6INHOMOGENEOUSDISTRIBUTIONSOFARIGIDRODINROTATIONALFLOWS ............................. 92 6.1Introduction ................................... 92 6.2MigrationintheRadialDirection ....................... 92 6.3SteadyStateDistributions ........................... 95 6.4StressCalculation ................................ 100 6.5Conclusion .................................... 101 7CONCLUSIONS ................................... 103 APPENDIX ACOMPONENTSOFGRANDMOBILITYTENSOR ............... 106 BDETAILSOFCALCULATIONSFORCHAPTER3 ............... 108 B.1LinearizationoftheGreen'sFunctionforaWall ............... 108 B.2AverageofBrownianContributions ...................... 109 B.3NumericalCalculationofOrientationMoments ............... 110 B.4AsymptoticSolutionsatLowPecletNumberLimit ............. 111 B.5Crank-NicolsonMethod ............................ 112 CADDITIONALTESTSFORCHAPTER4 ..................... 115 C.1TheEectofTorqueonMigration ...................... 115 C.2ExcludedVolumeEect ............................ 117 DASEDIMENTINGCLOUDOFRIGIDRODSINVISCOUSFLUID ...... 120 D.1Introduction ................................... 120 D.2SimulationMethods .............................. 121 D.3Results ...................................... 121 D.4OngoingWork .................................. 123 REFERENCES ....................................... 124 BIOGRAPHICALSKETCH ................................ 134 7

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Figure page 1-1Concentrationregimesforsuspensionsofrigidrods ................ 14 1-2Physicalmodelsforrigidrod-likeparticle ...................... 15 2-1AschematicdiagramforaStokeslet,f,actingattheorigin,O,inaninniteuid,whichinducesavelocityeld,u. ....................... 18 2-2AschematicdiagramforalinedistributionofStokesletsoveraslender-bodyinaninniteuid .................................... 19 2-3Rigidrodsandinaninniteuidwithimposedvelocityeld,u1. ..... 20 2-4RelationbetweenTand~F 22 2-5Sedimentationoftworigidrodswhicharealignedparalleltoappliedforceinaboxsize15L. ..................................... 28 2-6Demonstrationoftworodssedimentingparalleltogravity ............ 29 2-7Sedimentationoftworigidrodswhicharealignedperpendiculartogravity ... 30 2-8Motionsof2rodsonxy-plane.Gravityactsinthenegativez-direction. ..... 30 2-9Demonstrationoftworodssedimentingperpendiculartogravity ......... 31 2-10SimulationresultsofDST=DOTforsuspensionsofrigidrodswithA=50asafunctionofperiodicboxsize. ............................. 33 2-11SimulationresultsofDSR=DOTforsuspensionsofrigidrodswithA=50asafunctionofperiodicboxsize. ............................. 34 2-12PeriodicboxsizecorrectedDS=DOasafunctionofndL3. ............. 37 3-1Arigidpolymersuspendedinashearownearasolidboundary. ........ 40 3-2Transversevelocity,_ry,asafunctionofpolymerorientation. ........... 43 3-3Ensembleaveragesoforientationmomentsofaslender-bodyasfunctionsofPer. 46 3-4ContributionoftheshearowandBrowniantorquetothemigrationofarigidpolymerinsimpleshearowasafunctionofPer. ................. 47 3-5Ensembleaveragesoforientationmomentsofaslender-bodyasfunctionsofPer. 49 3-6Thesteadystatecenter-of-massdistributionsinsimpleshearow. ........ 50 3-7ThesteadystatedepletionlayerthicknessLd=Lofarigidber .......... 50 8

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........................ 52 3-9Comparisonofmigrationmechanismsinshearow ................ 53 3-10Thetransientcenter-of-massdistributionsofarigidberinsimpleshearows. 55 4-1Schematicdiagramsforsimulation:Arraysofcloselypackedbersformthewallsinaperiodicboundarysystem ............................ 59 4-2Velocityprolesofsimulatedshearows ...................... 65 4-3Transversevelocity_ryduetoshearowasafunctionofanglewithpz=0,ry=1:0L,andH=6L 66 4-4Transversevelocityduetoshearcontributionasafunctionofdistancefromawallforaxedangleunderaconstantshearrate ................. 67 4-5Thesimulationresultsandtheoreticalsolutionsforthetransientcenter-of-massdistributionsofarigidinsimpleshearowsofPe=1:2103 70 4-6Thesimulationresultsandtheoreticalsolutionsforthetransientcenter-of-massdistributionsofarigidberinsimpleshearowsofPe=1:2104 71 4-7Thesimulationresultsandtheoreticalsolutionsforthetransientcenter-of-massdistributionsofarigidberinsimpleshearowsofPe=4:8104 73 4-8Thesimulationresultswithandwithouthydrodynamicinteractionsforp2yofarigidberinsimpleshearows .......................... 74 4-9Rotationsofrigidrodsinshearow:Jeeryorbitmotionandpole-vaultmotion. 75 4-10Thesimulationresultsandtheoreticalsolutionsforthecenter-of-massdistributionsofarigidberinparabolicows .......................... 76 4-11Thesimulationresultswithhydrodynamicinteractionsforp2yofarigidberinparabolicows ................................... 77 4-12Thesimulationresultsforthecenter-of-massdistributionsofarigidberasinoscillatoryshearows ................................ 80 4-13Thesimulationresultsforp2yofarigidberinoscillatoryshearows ..... 81 5-1Theextraparticleshearstressesforrigidbersundersimpleshearowbetweentwowalls ....................................... 86 5-2TheparticleshearstressesPxy(ry)forarigidberwith=3,18,pz=0andA=10inagapofH=6L 90 6-1Schematicdiagramsforatorsionalow ....................... 93 9

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.............. 95 6-3Averageorientationmomentsofarigidrodintorsionalows ........... 98 6-4Center-of-massdistributionsofarigidrodintorsionalow ............ 99 6-5 ......................... 102 B-1SchematicdiagramofnitemeshesforsolvingEq. 3{27 usingCrank-Nicolsonscheme. ........................................ 113 C-1Transversevelocityduetotorquecontribution ................... 116 C-2Trajectoryofanon-Brownianrodnearawall ................... 118 D-1Sequentialimagesofasedimentingcloudofrigidrods ............... 122 10

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Dynamicsofsuspensionsofrigidrod-likepolymersarestudiedbytheoreticalanalysisandsimulationconsideringhydrodynamicinteractions.ThemotionsofrigidrodsinNewtoniansuspendinguidsaremodeledusingslender-bodytheoryandandassumptionofzeroReynoldsnumber.Hydrodynamicinteractionsareincludedusingthelinearizedforcedistributionalongarodaxis.Theorbitsintrajectoriesofapairofsedimentingrodsandtheconcentrationdependenttrendofshort-timediusivitiesaresimulatedtovalidatethesimulationmethodusedinthisstudy. Akinetictheoryisdevelopedtoinvestigatecross-streammigrationofarigidpolymerundergoingrectilinearowinthevicinityofawall.Hydrodynamicinteractionsbetweenthepolymersandtheboundaryresultinacross-streammigration.Insimpleshearow,polymersmigrateawayfromthewall,creatingadepletionlayerinthevicinityofthewallwhichthickensastheowstrengthincreasesrelativetotheBrownianforce.Inpressure-drivenow,ano-centermaximuminthecenter-of-massdistributionoccursduetoacompetitionbetweenhydrodynamicinteractionswiththewallandtheanisotropicdiusivityinducedbytheinhomogeneousoweld. Arigidpolymerwithinsimpleshearow,parabolicow,andoscillatoryowathighPecletnumbers(weakBrownianmotion)issimulatedtoenablecomparisonswiththesteadyandtransientdistributionspredictedbythetheoryandsimulation.Thesimulationandtheoreticalresultsareingoodagreementatsucientlyhighshearrates,validating 11

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Atheoreticalanalysisoncross-streammigrationofarigidpolymerintorsionalowisalsoinvestigated.Duetothecurvatureofradialshearow,arigidrodmigratestowardsthecenteraxis.Ananalogyismadebetweenexiblepolymersmigratingintorsionalowsandtheparticlestressisalsocalculatedforthepurposeofdeterminingthepracticalimpactofthemigrationuponrheologicalmeasurementinparallel-plategeometries. 12

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Rigid,rod-likepolymers,macromoleculesandparticles(simplyrigidrodsorrigidbers)canbefoundinmanyareasofcommonexistingandnewemergingtechnologies.Manymacromoleculesofbiologicaloriginhaverod-likestructure:ashortDNAfragmentlessthan100basepairs('50nm)( Newmanetal. 1974 ; Tiradoetal. 1984 ; Wangetal. 1991 ; Tinlandetal. 1997 ),xanthangumwhichisahelicalpolysaccharidechainofmolecularweight1:8106( Davidson 1980 ),actin( Schmidtetal. 1996 ),andcollagen( OhandPark 1992 ; ClaireandPecora 1997 ).Evenmicro-organisms,suchasfdbacteriophagesandtobaccomosaicviruses,haverod-likestructure( Caspar 1963 ; Chenetal. 1980 ; TracyandPecora 1992 ; CushandRusso 2002 ). Syntheticrigidpolymersornanotubesareusedforimprovementofmaterialproperties( Adamsetal. 1989 ).CompositesofKevlarandcarbonbergreatlyincreasetensilestrengthandstiness( Gustinetal. 2005 ).SemiconductorCdSenanorodsisusedinsolarcells( Huynhetal. 2002 ).Carbonnanotubeshaveenormousapplicationsduetothoseuniqueproperties( Salvetatetal. 1999 ; ColbertandSmalley 1999 ; Vigoloetal. 2000 ). Comparedtosuspensionsofsphericalparticles,suspensionsofrigidrodsexhibitamuchlargerrangeofbehaviorssincetheorientationofarigidrodmakesitscongurationmorelikelytobeaectedbyoweldsandinteractionswithotherparticlesatlowervolumefractionthanthatofasphericalparticle.Therefore,suspensionsofrigidrodsproducemuchstrongernon-Newtonianeects,suchasnormalstressdierences,shearthinningandthickening,thanasuspensionofsphericalparticlesatasimilarvolumefraction( Larson 1999 ).Comparedtoexiblepolymers( Larson 2005 ),rigidpolymershavedierentphysicalbehavior,suchasformationofliquidcrystalstructureathighconcentration.Thesecharacteristicphysicalpropertiesdependentonconcentrationmake 13

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Concentrationregimesforrod-likepolymer:A)dilute(ndL3<1),B)semi-dilute(1
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Physicalmodelsforarigidrod-likeparticle:A)rigid-dumbbell,B)prolatespheroid,C)slender-body,andD)shish-kebabmodel. nanotubesusingmicrochannelsusingdielectrophoresis( Krupkeetal. 2003 )andcapillaryelectrophoresis( Suarezetal. 2006 )areotherexamples. Inthiswork,rigidrodsaremodeledasslender-bodyassumingthataspectratio,A,aratiobetweenarodlengthandadiameter,ishigh.OthermodelsforarigidrodareillustratedinFigure 1-2 .Therigiddumbbellmodelissimple( Fixman 1985b a ; Bitsanisetal. 1988 1990 ),butimproperdistributionofresistanceresultsindierentdiusivitiesthanothermodels( CobbandButler 2005 ).Aprolatespheroidisthemostaccuratebutcomputationalloadpreventsdynamicsimulationofmultiparticlesystems( KimandKarrila 1991 ; ClaeysandBrady 1993a ).Theslender-bodymodelispopularbecausedistributedresistancealongthelengthandanisotropicmobilitycoecientsarerepresentedsimply.Thismodelisderivedbyintegratingresistancealongthelengthandonlytakingtheleadingorderterm( Batchelor 1970 ; Cox 1970 ).Therehavebeenmanysimilarapproachessuchasworksby Broersma ( 1960a b ); Moran ( 1963 ); HappelandBrenner ( 1965 )and`shish-kebab'modelby DoiandEdwards ( 1986 )inFigure 1-2 D. InChapter 2 ,dynamicsofhydrodynamicallyinteractingrigidrodsisdescribedbyderivingtheequationofmotionusedinsimulation.Thelengthandtheaspectratioofthe 15

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InChapter 3 ,akinetictheoryforcross-streammigrationindilutesolutionsofrigidpolymersundergoingrectilinearownearasolidboundaryisdeveloped.Whilesphericalparticlesmigrateonlyinconcentratedsuspensions( LeightonandAcrivos 1987 ),asinglerigidrodcanmigrateinshearows( Ganatosetal. 1980 ; YangandLeal 1984 ; Saintillanetal. 2006a ).Acomparisonofmigrationmechanismofexiblepolymersandarigidrodisalsomade.Arigidrodnearaboundaryalsomigratesinaforceeld( Russeletal. 1977 ; DavisonandSharp 2006 ; SendnerandNetz 2007 ),howeverweonlyconsidercouplingofshearowandBrownianforcehere. InChapter 4 ,Browniandynamicssimulationofarigidrodinmicrochannelowisperformedtoobtaindistributionsinthechannel.TheresultsarecomparewiththeoreticalpredictionsinChapter 3 andthequantitativediscrepancybetweenprevioussimulationsby Saintillanetal. ( 2006a )areresolved.InChapter 5 ,thetheoryandsimulationareextendedforthecalculationofparticlestress.Theeectofinhomogeneousdistributiononthestressisdiscussed.InChapter 6 ,atheoreticalanlysisonradialmigrationofarigidrodintorsionalowismadetopredicttheinhomogeneousdistributionintheradialdirectionandtheresultedenhancedshearthinning.Finally,conclusionsaredrawninChapter 7 16

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2 .Previousstudiesonnon-Brownianbersuspensionshaveincludedhydrodynamicinteractions. Mackaplow ( 1995 )includedlong-rangehydrodynamicinteractionswhileperformingaMonteCarlosimulationofsedimentingrigidbers.However,theevaluationofhydrodynamicinteractionsinthismethodiscomputationallyexpensive,hencedynamicsimulationwaspossibleonlyusinganapproximatedpoint-particlealgorithm( MackaplowandShaqfeh 1998 ). Alternatively, Harlenetal. ( 1999 )usedalinearizedforcedistributiononabertosimulatenon-Brownianbersinow.Theforcedistributionontheberisrepresentedbyatotalforce,torque,andstressletbyformallylinearizingthevelocitydisturbance. ButlerandShaqfeh ( 2002 )appliedthemethodtothedynamicsimulationofthesedimentationofbersandincludedshort-rangehydrodynamicinteractions(lubricationforces)andalsoextendedthemethodtoexplicitlyconstructthemobilitymatrixwithhydrodynamicinteractionsbetweenBrownianbers( ButlerandShaqfeh 2005 ).Later,thesimulationmethodwasextensivelyusedinotherstudiestoexamineuctuatingpatternsofsedimentingrods( Saintillanetal. 2006b ),hydrodynamicallyinteractingchargedrods( Saintillanetal. 2006c ),andBrownianrodsinachannelow( Saintillanetal. 2006a ).Othermethodsofcalculatingdynamicsofrodswithhydrodynamicinteractionsexist,suchasthelattice-Boltzmanmethod( QiandLuo 2002 ; Qietal. 2002 )andthedissipativeparticledynamics(DPD)simulation( PryamitsynandGanesan 2008 ).However,thesemethodstreatsuspendingliquidsasadiscretzedphaseandinertiaisusallyincluded.Equationsderivedfordynamicsofslender-bodymodelwillbeusedinthisstudy. 17

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AschematicdiagramforaStokeslet,f,actingattheorigin,O,inaninniteuid,whichinducesavelocityeld,u. InSection 2.2 ,theequationofmotionofhydrodynamicallyinteractingrodsisderived.Asexamples,thedynamicsimulationofsedimentingrodsisdemonstratedinSection 2.3 andtheeectofconcentrationontheshort-timediusivitiesofrigidrodsarecalculatedinSection 2.4 usingtheequationofmotionderivedinSection 2.2 Moran 1963 ; Batchelor 1970 ; Cox 1970 ).Thebasicideainslender-bodytheoryforStokesowisthatthedisturbancevelocityduetothepresenceoftheparticleisapproximatelythesameasthatduetoasuitablychosenlinedistributionofpointforces,orStokeslets. AsinFigure 2-1 ,intheabsenceofinertia,apointforceappliedattheorigininaNewtoniansuspendinguidbalancesthehydrodynamicforcefH, Thevelocityelduatpositionx,whichisinducedbythepointforce,isgivenby whereG(x;O)istheGreen'sfunctionwiththeevaluationpointatxandthepointsourceattheorigin.Inaninnitebodyofuidfreefromboundaries,theGreen'sfunctionin 18

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AschematicdiagramforalinedistributionofStokesletsoveraslender-bodyinaninniteuid Eq. 2{2 issatisedbytheOseen-Burgertensor( Oseen 1927 ), 8I wherejxjisthedistancebetweenanevaluationpointandapointsource,istheuidviscosity,andIistheidentitytensor. AsinFigure 2-2 ,theStokesletsaredistributedoverthelengthofaslender-bodywheretheorientationisaunitvectorp,aunitvectorperpendiculartotheorientationisp?,lengthisL,diameterisd,andthecoordinatealongthelongaxisiss(s=0attheorigin).Ifthelinedensityoftheappliedforceisf(s),thentheresultinguidvelocityatpointxisexpressedasasuperpositionofStokesletsalongtheline, 8ZL=2L=2264f(s) Theapproximatesolutionofthevelocityintegralisoftheorderof2ln(2A).Theerrorinthisapproximationisoftheorder1,whichissmallincomparisonto2ln(2A)whentheaspectratioishigh.Theleadingorderapproximationinln(2A)resultsinthefollowing: 4(I+pp)f(s):(2{5) 19

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Rigidrodsandinaninniteuidwithimposedvelocityeld,u1. Theresultingsolutioniswrittentorelatethemotionofapointsonarigidrodtotheforcedistributionactingontherod, _r+s_p=u(s)+ln(2A) 4(I+pp)f(s);(2{6) wheresubscriptdenotestherod,_rand_parethevectorsspecifyingthecenterofmassandrotationalvelocitiesoftherod,andu(s)isthesumofuidvelocitiesonapointontherod,whichconsistoftheimposedvelocityeld,u1,andtheowdisturbances,u,causedbytheotherStokesletsources,suchastheotherparticlesorsolidboundariesasdemonstratedinFigure 2-3 .Thetensor(I+pp)inthisrelationshowsthatrigidbershaveananisotropicmobility. IntegratingEq. 2{6 overalengthofarodgives_rforarigidrod, _r=1 20

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4L,andFisthetotalforceactingontherodasgivenbytheintegratedlineforcedensity, Theexplicitexpressionforrotationalvelocity_pofarigidrodissolvedbytakingcrossproductofEq. 2{6 byspandintegrating, _p=12 wherethetotaltorqueactingontherodis, MultiplyingEq. 2{6 bys,integratingandprojectingusingtensor(Ipp)givestheexpression: _p=12 whichisequivalenttoEq. 2{9 .InsteadofTinEq. 2{9 ,Eq. 2{11 isexpressedintermsoftherstmomentoftheforcedistributionactingonarod,whichisdenedas TheformulasinthisworkarewrittenintermsofT,howeverthesimulationcodesaredevelopedbasedontheexpressionsusingeF.TherelationbetweenTandeFisdemonstratedinFigure 2-4 andalsogivenbythefollowingvectoridentity( Aris 1962 ), 21

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RelationbetweenTandeFactingonarigidrodwithorientationp,whereTp=(Ipp)eF. AsinEq. 2{1 ,thetotalsumofexternalforces,F,andtorques,T,onaberalsobalancesthehydrodynamicforces,FH,andtorques,TH, Inasuspensionofrigidrods,thevelocitydisturbance,u(s),onaberisthesumofdisturbancevelocitiesinducedbytheforcedistributionactingonallotherbers, whereNisthetotalnumberofbersinthesystem.Thecoordinatesisthepointofevaluationonberofthevelocitydisturbanceinducedbythelineforcedensityatpointsonber.Forisolatedrodsinaninniteuid,theGreen'sfunctionistheOseen-BurgertensorinEq. 2{3 .Fordynamicsimulationsofsuspensionsofhydrodynamicallyinteractingrigidbers,thesolutionof Hasimoto ( 1959 )fortheperiodicOseen-Burgertensorhasbeenwidelyused( Harlenetal. 1999 ; ButlerandShaqfeh 2002 ; Saintillanetal. 2006b ).Likewise,appropriateGreen'sfunctionswillbechosendependingontheconditionsofthesystems. ToincludehydrodynamicinteractionsbetweenrodsusingEq. 2{15 ,theforcedistributionf(s)shouldbedetermined.UsingEq. 2{6 ,Eq. 2{7 ,andEq. 2{9 ,theforcedistributioncanbesolvedintermsofthevelocitydisturbance,totalforce,andtotaltorque.ThevelocitydisturbancecanbeexpandedusingLegendrepolynomialsinthe 22

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Harlenetal. 1999 ; ButlerandShaqfeh 2002 ), wherethestressletcoecientisgivenby AStressletarisesfromtheinabilityofanindividualrodtostretchorcompressalongitsmajoraxis.Sincetherodspossessonlyonenitedimension,thestressletcoecientisascalarquantityrelatingtherateofstrainandtheparticlestress.ThemoredetailedrelationbetweenthestressletcoecientandtheparticlestressisdiscussedinChapter 5 .ThelinearizedforcedensitygivesthesimilarlevelofapproximationasusedinStokesiandynamics( BradyandBossis 1988 ). SubsequentsubstitutionsofEq. 2{15 withEq. 2{16 intoEq. 2{7 andEq. 2{9 givesalinearsetofequationsrelatingforcesandmotionsofthebers: _r=1 and _p=12 23

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Ineachmotionofrod,multibodyinteractionsareincludedthroughthestressletcoecientofrod.Therefore,theequationofmotionforrigidrodsinasuspensioniscompletedbysolvingthelinearsetofequationsforthestressletcoecients: whichisobtainedbysubstitutionofEq. 2{15 withEq. 2{16 intoEq. 2{17 .Equation 2{18 ,Eq. 2{19 ,andEq. 2{21 canbegeneralizedasamatrixequation: _eX=fMe;(2{22) where_eXisagrandvectorcontainingallsetsofvectorsforthemotionsofrigidrodsinthesystem, _eX=2666666666666664_r1 24

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andgrandmobilitymatrix, wherethecomponents,MinEq. 2{25 ,aretensorswhichrelatethemotionsandforcesofrods.Thesuperscriptsrepresentthetypeofmotionandforcewhicharerelatedbythematrixandthesubscriptsdesignatetheberswhichcorrespondtothemotionandforce.Forexample,MTTrepresentsthetensorwhichrelatesthetorqueonthebertothetranslationalmotionsoftheber.EachcomponentofamatrixMisadoubleintegraloftheGreen'sfunctioninEq. 2{20 overtwobers( ButlerandShaqfeh 2002 ).ThedetailedexpressionsarelistedinAppendix A 2{22 ClaeysandBrady ( 1993a )performedsimulationsshowingthatahydrodynamicallyinteractingpair 25

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Tosimulateaninnitesuspension,thesolutionof Hasimoto ( 1959 )fortheperiodicOseen-Burgertensor,HP,isusedfortheGreen'sfunctionin( 2{15 .Toconsidertheinteractionsofperiodicselfimagesproperly,subtractingtheOseen-Burgertensorfromtheperiodicsolutionby Hasimoto ( 1959 )shouldbeperformedinEq. 2{15 for=.Ifs=s,thelimitingform,H0P,derivedby Hasimoto ( 1959 )mustbeused: NumericalintegrationofEq. 2{15 isperformedbyGauss-Legendrequadrature.Thenumberofintegrationpointsonarodwaschosenasthree,whichgiveconvergentresultsforthissimulation. Animposedoweldu1isnotgiveninthissimulation,sothedisturbancevelocityoneachrodisinducedonlybythemotionoftheotherrods.Toapplygravitationalforceoneachrod,allthetorquesinthegrandforcevectorweresetto0,whileforcesaregivenby, 2gVpp^z;(2{27) wheregisthegravitationalaccelerationconstant,Vpisaparticlevolume,pisthereduceddensityofaparticleinthesuspendinguid,and^zisaunitvectorinthez-direction.TointegrateEq. 2{22 foru1=0,stressletcoecientsineshouldbesolvedintermsofthegivenforce,torque,andmobilitycomponentstogivethereducedformoftheequationofmotion, _X=M;(2{28) 26

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_X=266666664_r_r_p_p377777775;=266666664FFTpTp377777775;(2{29) and wheretheformertensorcontainsthedirectandtwobodycontributionsofforcesandtorquesandthelatterrelatesthemulti-bodycontributionsthroughthestressletcoecients.Theequationofmotion,Eq. 2{28 ,isthenintegratedusingthemid-pointmethodtogivethecongurationsoftherodsateachtime.ThemobilitytensorMisevaluatedateachcongurationexplicitly.Thetimestepistc=0:1,wherecharacteristictimeistc=L 2-5 areforrodsinitiallyalignedparalleltogravity,separatedbyadistanceof5Landinaboxofsize15L.Initially,theoweldscreatedbythesedimentingrodsrotatetherodsoutofalignmentandtherodsbegintoseparateduetothecouplingbetweenthegravitationalforce,orientation,andsedimentationvelocity.Atlongertimes,therodsbegintostronglyfeeltheinteractionwiththeneighboringimages,causingarotationoftherodstowardsparallelandeventuallytowardsthecenterofthecell.Consequently, 27

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Sedimentationoftworigidrodswhicharealignedparalleltoappliedforceinaboxsize15L.Initialcongurationsarep=(0;0;1)forbothbers,andcongurationsateachtime(tc=L theseparationdistancebetweentheoriginalpairbeginstodecreaseuntiltheorbitiscompletedandbeginstorepeat(Figure 2-6 ). Otherinterestingclosedorbitsfortwosedimentingberscanbefound,suchastheexampleshowninFigures 2-7 and 2-8 .Tworodsarealignedperpendiculartogravity,separatedbyadistanceof5L,withtheinitialcongurationofp=(1 Thesedimentingrodscreateaoweldthatmakesrodsrotateinamannerwhichcausesthedistancebetweentherodstodecrease,yettherodssedimentatthesamevelocitysothattheyremainonthesameplane.Atthestartofsedimentation,therodsareseparatedinthex-direction,whilehavingthesamey-coordinate.Thusastheysediment,theseparationdistancedecreasesinthex-directionduringtc=066.However,separatingmotionsinthey-directioninducethedisturbancethatinvertstherotation.Consequently,whenthedistanceinx-directionbecomes0andthedistanceiny-directionreachesthemaximumattc=66,therodsarealignedperpendiculartothe 28

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DemonstrationofFigure 2-5 forcedirectionagain.Thiscongurationisidenticalwiththeinitialcongurationexceptthatthedirectionshavebeenexchanged.Therefore,therodsedimentwiththeoppositedriftsuntiltimetc=132,whentheclosedorbitcompletesandbeginstorepeat(Figure 2-9 ). Inthissedimentationtest,themotionofoneberinuencesthedynamicsoftheothersbyattractingandrepellingeachotheraccordingtotheirrelativeorientations.Theirinteractionisgreaterasthedistancebecomessmaller,thereforethehydrodynamicinteractionsareexpectedtoinuencetheBrownianmotionofbersinnon-dilutesolution 29

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Sedimentationoftworigidrodswhicharealignedperpendiculartogravitywithinitialcongurationsofp=(1 Motionsof2rodsonxy-plane.Gravityactsinthenegativez-direction. 30

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DemonstrationofFigure 2-7 andresultindierentdynamicbehavior.Anothersedimentationtest,simulationofafallingdropofrigidrods,wasperformedanditsresultsaresummarizedinAppendix D 2{28 ,theshort-timeself-diusivitiesofasuspensionofrigidrodshavebeencalculatedconsideringhydrodynamicinteractions.Withouthydrodynamicinteractions,theshort-timediusivity,DS,isindependentofconcentrationandassumedequivalenttotheinnitedilutionvalue,DO.Theshort-timetranslationaldiusivity,DST,andtheshort-timerotationaldiusivity,DSR,ofaspheroidofA=57( ClaeysandBrady 1993b )hasbeencalculatednumerically.However,thecomputationalburdenpreventedenoughaveragingsothatonlyqualitativedecreasingtrendatincreasingconcentrationisshown.InSection 2.4 ,theshorttimediusivitiesofrigidrodswerecalculatedfornumberdensitiesbetween5and150byincludinghydrodynamicinteractions. 31

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wherefMisamobilitytensor(33)ofanindividualrod,oracomponentofagrandmobilityMwhichincorporatesmulti-bodyinteractionsthroughEq. 2{30 .Theanglebracketsdeneanaverageoverallcongurationsandtristhetraceoperator.Here,kBisBoltzmann'sconstantandTistheabsolutetemperature.Theinnitedilutionvaluesofarigidrodare 3L;DOR=3kBTln(2A) asshownin DoiandEdwards ( 1986 ). TocalculateDSusingEq. 2{31 ,randomcongurationsofsuspensionsofrodsweregeneratedandtracesofmobilitieswereaveragedoverallcongurations.Thesimulationprocedureissimilartotheworkof Phillipsetal. ( 1988 ).Anumberdensity,ndL3,waschosenandthenumberofrods,N,andsizeoftheperiodicbox,b,weredetermined.Positionsandorientationsofrodswererandomized.Theclosestdistancebetweenrodswerecalculatedusingtheformulasof FrenkelandMaguire ( 1983 )and Yamaneetal. ( 1994 )toavoidoverlaps.Themobilitytensoratthegeneratedcongurationswasconstructedconsideringhydrodynamicinteractions.HydrodynamicinteractionbetweenrodsareincludedatthestressletlevelbythemethodshowninSection 2.2 .Thediagonalcomponentswereaveragedtodeterminetheshort-timediusivities.Thisprocedurewasrepeateduntiltheaveragesofshort-timediusivitiesconverge.Averagesover20congurationsconvergedtowithin0.01%forsimulationswithN>400.Fiftycongurationswererequiredforconvergencetowithin0.01%forN<400.Theaspectratioofrodsinthesesimulationswasxedto50.ThenumberofGauss-Legendrequadraturepointsontherodsforintegratinghydrodynamicinteractionswassetto14toproduceaconvergencewithin0.02%atndL3=150. 32

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SimulationresultsofDST=DOTforsuspensionsofrigidrodswithA=50dependentinverselywithperiodicboxsize,b=L.Valuesextrapolatedatinniteboxsizelimit(orboxsizecorrected)foreachnumberdensityandcorrespondingregressionstandarderrorsareshowninthelegend.A)ndL3=550andB)ndL3=70150. 33

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SimulationresultsofDSR=DORforsuspensionsofrigidrodswithA=50showadependenceontheinversecubeofperiodicboxsize,(b=L)3.Valuesextrapolatedatinniteboxsizelimit(orboxsizecorrected)foreachnumberdensityandcorrespondingregressionstandarderrorsareshowninthelegend.A)ndL3=550andB)ndL3=70150. 34

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2-10 forDSTandFigure 2-11 forDSR.Thoughdiusivitieswerecalculatedatthesamenumberdensity,theresultsvariedwithb,thesizeofthecubicperiodicbox.Thisboxsizeeectmustbecorrectedtoattainthevaluesforanunboundedsolution.Thetranslationaldiusivitiesshowlineardependenceontheinverseofboxsize,whileDSRdependsinverslyonthecubeoftheboxsize,approximately.Thesetrendswereusedtoapproximatelycorrecttheboxsizeeectbyextrapolationtoaninniteboxsize.Thelineartrendsofcurrentsimulationdatahavecorrelationcoecientsofmostlyover99:9%. Phillipsetal. ( 1988 )extrapolatedtoaninnitenumberofsphericalparticlesusingthescalingN1=3fortranslationaldiusivityandN1forrotationaldiusivity.SincethenumberdensityisndL3=N(L=b)3,scalingsusedinextrapolationsattheinnitebandtheinniteNunderthesamenumberdensityareidentical. Mackaplow ( 1995 )alsoreportedsimilartrendsforthemobilityofprolatespheroidsandexplainedthatthecontributiontotranslationalmobilityismainlythroughtheStokesletwhichisafunctionof1=r,whereristhedistancebetweencentersofmassofparticles.Therotationalmobilitydependsondipolemomentswhichareafunctionof1=r3.Otherformulaforcorrectingtheboxsizeeectofsuspensionsofsphericalparticlesusedratiosbetweensolventviscosityandeectiveviscosity( Ladd 1990 ). Mackaplow ( 1995 )triedtondamethodforsuspensionsofrigidrods,howeveraformulaonlyfortranslationalmobilityisderivedandtheresultscorrectedbythisformulalackconsistency.Therefore,themethodtocorrecttheboxsizeeectforsuspensionsofrigidrodsisnotdevelopedyet. Theshort-timediusivitiesconsideringhydrodynamicinteractionswithcorrectionoftheboxeectareplottedinFigure 2-12 .Theratio,L2DSR=DST,whichisexpectedtobeanimportantparameterforthelong-timediusivityscaling( CobbandButler 2005 )isalsoplotted.Asthenumberdensityincreases,bothdiusivitiesdecreasedbecausehydrodynamicinteractionshinderthemotionofrods.ThetranslationaldiusivitydecaysfasterwithincreaseofconcentrationthanDSR.Thiscanalsobeexplainedbythemobilitydependenceon1=randtherandomdistributionofrods.ThevalueofDSTwasderived 35

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CobbandButler ( 2005 )thatthescalingexponentofthelong-timediusivitydecreasedathigherratios,inclusionofhydrodynamicinteractionsisexpectedtodecreasetheexponent. ConvertingthenumberdensitydependentsimulationdataforDTSintovolumefraction,v,thevolumefractiondependentdatagivesascalingof12:66vatthedilutelimit.Comparedtothetheoreticalscaling11:83vforsphericalparticlesby Batchelor ( 1972 ),theDSTofrigidrodswithA=50hasastrongerdependenceonconcentration,asotherphysicalpropertiesofrigidrodsareallmoreconcentrationsensitivethanthoseofsphericalparticles.OthersimulationorexperimentaldataforDSforrod-likeparticlesarenotavailableyet.Asconcentrationincreasestoisotropicconcentrateregime,sensitivitiestoconcentrationarereduced.Thistrendsuggeststhathydrodynamicscreeningarisesathigherconcentrations. 2{22 areshownfordynamicsofapairofsedimenitngrodsandcalcualtionofshort-timediusivitiesindilutetoconcentratesuspensions.TheresultsconrmthevalidationofthemodelequationderivedinChapter 2 andthecodewhichisdevelopedusingtheequation. 36

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PeriodicboxsizecorrectedDS=DOconsideringhydrodynamicinteractionsforsuspensionsofrigidrodswithA=50asafunctionofndL3. 37

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3 ,akinetictheoryisdevelopedforcross-streammigrationofrigidpolymersindilutesolutionsundergoingrectilinearownearasolidboundary.Flexiblepolymersindilutesolutionmigrateacrossstreamlinesinsimpleshearandpressure-drivenows.Thoughtheoriginanddirectionofthemigrationwerecontroversial( Agarwaletal. 1994 ),recentwork( MaandGraham 2005 ; Khareetal. 2006 ; Ustaetal. 2006 2007 )hasclariedthatexiblepolymersprimarilymigrateawayfromboundingwallsduetoahydrodynamicliftforce.Thelocalshearowextendsthepolymer,generatingtensioninthechainandanadditionaloweldaroundthepolymer.Theoweldbecomesasymmetricnearano-slipboundaryandresultsinanetdriftawayfromthewallforbothsimpleshearandpressure-drivenow.Ininhomogeneousows,thevariationinthelocalshearratealtersthepositiondependentconformationandconsequentdiusivitytransversetotheow.Thisadditionalmechanismresultsinaweakdisplacementofthepolymersawayfromthecenterlineinpressure-drivenow,thoughthenetmigrationstilloccursawayfromthewallunlesshydrodynamicinteractionswiththewallarescreened,asoccursforhighlyconnedpolymers.Thisnetmigrationtowardsthecenterline,whichhasbeenobservedqualitativelyinexperimentsby Fangetal. ( 2005 ),hasimplicationsforthedesignandoperationofmicrouidicdevices( WhitesidesandStroock 2001 ; Stoneetal. 2004 ).Asoneeect,theliftforcehindersadsorptionofpolyelectrolytes( HodaandKumar 2007b a 2008 ). Themechanismanddirectionofmigrationremainsunclearforrigidpolymers,orBrownianrigidrods,indilutesolution.Measurementsonsemi-rigidxanthanmoleculesinpressure-drivenowindicatemigrationawayfromthewall,resultinginadepletionlayer( Ausserreetal. 1991 ).Mechanismsnotassociatedwithhydrodynamicinteractionswiththeboundingwallalsoalterthedepletionlayer.Simulationsandmodelswhichconsider 38

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dePabloetal. 1992 ; SchiekandShaqfeh 1995 ).Also,gradientsinthediusivity,causedbythecouplingoftheanisotropicmobilityofthecenterofmassofarodanddierencesintheshearrateacrossachannelwithpressure-drivenow,shiftthedistributionclosertothewalls( NitscheandHinch 1997 ; SchiekandShaqfeh 1997a ),similartothemechanismforexiblepolymers.Mostrecently,simulationsofrigidpolymersinpressure-drivenowpredictedmigrationawayfromthewall( Saintillanetal. 2006a ),thoughthedepletionlayerislargerthanpredictedbystericinteractionsalone.Theauthorsproposedthatasubtlecombinationoforientationeectsandhydrodynamicinteractionswiththewallsproducestheoverallmigration. Toconrmtheresultsof Saintillanetal. ( 2006a )andclarifytheoriginsoftheobservedmigration,atheoreticalmodelequationpredictingthedistributionofrigidpolymersnearsolidboundariesisdevelopedinSection 3.2 .Thetheorycontainsapproximationssimilartothosemadeforexiblepolymers( MaandGraham 2005 ):thepolymerdistributionfunctionisfactorizedintoaproductofacenter-of-massandorientationdistributionandafar-eldapproximationforthehydrodynamicinteractionwiththeboundingwallismade.Usingthemodelequation,distributionsofrigidpolymersinsimpleshearowandpressure-drivenowbetweenparallelwallsarepresentedinSection 3.3 .Resultsshowthatrigidpolymersmigrateawayfromthewallduetohydrodynamicinteractionswiththewallinthebothshearows.Forpressure-drivenow,theo-centermaximuminthecenter-of-massdistributionduetoacompetitionbetweenhydrodynamicinteractionswiththewallandtheanisotropicdiusivityinducedbytheinhomogeneousoweld.ThetheoryisalsodevelopedtogivethetimeevolutionofthedistributioninsimpleshearowinSection 3.4 39

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Arigidpolymersuspendedinashearowofstrength_.NotationsdescribinggeometryofarodaresameasdenedinChapter2exceptthatsandsindicateanevaluationpointandasourcepointonarod,respectively.Ano-slipboundaryislocatedaty=0. 3-1 )isgovernedbyacontinuityequationforthedistributionfunction,(r;p;t): @r(_r)(Ipp):@ @p(_p):(3{1) Fortherigidber,isfactorizedintoaproductofacenter-of-mass,n,andorientation,,distribution, (r;p;t)=n(r;t)(r;p;t);(3{2) wheren=Rdp.IntegratingEq. 3{1 overtheorientationdistributionandsolvingforthesteadystategives 0=@ @r(nh_ri);(3{3) 40

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3{3 necessitatesdevelopmentofanexplicitexpressionforthetransversevelocity,_ry. 2{7 givesanexpressionforthecenter-of-massvelocity,_r,approximatedbyslender-bodytheory.InEq. 2{7 ,alocaluidvelocityonarodu(s)undershearownearasolidboundaryincludescontributionsfromtheimposedshearow,_,andthedisturbancevelocitygeneratedbytheforceonthepolymerwhichisreectedbythewall, where^xisaunitvectorinthex-direction.FortheGreen'sfunctioninEq. 3{4 ,thesolutionof Blake ( 1971 )foraplanarwallwiththeOseen-Burgertensorremovedisused.Thecoordinatesindicatesthelocationofapointforceonthepolymer,whereassisthepointofevaluationofthereecteddisturbanceinvelocity. CombiningEq. 3{4 andthelinearizedforcedensity,Eq. 2{16 ,followedbyasubstitutionintoEq. 2{7 givesacompleteexpressionfor_r,approximatedatthestressletlevel: _r="1 csL5G(1b)ppZL=2L=2u1(s)sds#+1(I+pp)+1 csL5G(1b)ppG(1a)F+12 csL8G(1b)ppG(2)(Tp); where L4pG(2)p:(3{6) RecallthatdoubleintegralofGreen'sfunctionsareindicatedasG,asdenedinEq. 2{20 Undertheassumptionthatadistanceofapolymerfromawalliscomparabletoitsownlength(ry&L),theGreen'sfunctionislinearizedaboutthecenterofthepolymer. 41

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3{5 withthelinearizedexpressions,showninAppendix B ,retainingleadingordersineachcontributionsandextractingthetransversecomponentofthevelocitygive, _ry=_(ry)pxpy13p2y+1(^y+pyp)F+24 whereryandpyindicatey-componentsofvectorsrandp,respectively,^yisaunitvectoriny-directionand (ryH)2#;(3{8) whichlinearlysuperposestheeectoftheotherwallthatisatadistanceofH.ThereexistsaGreen'sfunctionfortwowalls( LironandMochon 1976 ),howeveritsanalyticalmanipulationistoocomplicatedandsuperpositionapproximationqualitativelycorrectforH>5L( Ustaetal. 2007 ). ThersttermontherighthandsideofEq. 3{7 correspondstotheshearowcontributiontothetransversemotion.TheBrownianforceonthecenter-of-massresultsinauctuationinthey-position,butlessobviousisthepredictionofatransversevelocityduetotheBrowniantorquewhichappearsasthethirdcontributioninEq. 3{5 ThestressletcoecientwithinEq. 2{16 couplestheshearowtoatransversemotion.Inthissense,thestressletfortherigidpolymerisanalogoustothespringforceinthetheoryforaexiblepolymer;theinabilityoftherigidpolymertostretchorcompressalongitsmajoraxisgeneratesanadditionaloweldwithintheuid.Thisoweldisreectedbythewallandcreatesatransversemotionofthepolymer.ThiscontributiontothepolymermotionappearsinthersttermofEq. 3{7 : _ry=_(ry)pxpy13p2y;(3{9) whichmatchespreviousresultsgivenwithoutderivation Saintillanetal. ( 2006a ).Figure 3-2 shows_ry=_asafunctionoforientationwithrespecttothewallforapolymerinthe 42

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Transversevelocity,_ry,asafunctionofpolymerorientation,=cos1px=(p planeofshear(pz=0).Foraforceandtorque-freepolymerrotatinginshearow,Eq. 3{9 suggeststhecenter-of-masswilloscillateperpendiculartothewall( YangandLeal 1984 ; HsuandGanatos 1994 ; Olla 1999 ).Symmetryoftheorientationdistributionmustbebrokentoproduceanetvelocityeitherawayortowardsthewall( Olla 1999 ).Inthecaseofarigidpolymer,evenveryweakBrownianuctuationsbreakthesymmetry. 3{7 areexpressedintermsofthedistributionfunction, Afterperforminganaverageof_ryovertheorientationdistributionfunction,theaveragetransversevelocityofarigidber,h_ryi,ismultipliedbyntogivetheparticleuxasa 43

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"1+p2y@n @ry+n@p2y L2(ry)3p2y1; ofwhichdetailedderivationsareinAppendix B IntegratingEq. 3{3 ,usingEq. 3{11 andtheconditionofnoparticleuxthroughtheboundingwalls(nh_ryi=0),givesanequationforthesteadystatedistribution, @ry=(ry)KS+KB+hi hDi:(3{12) Here,hKicontainsthecontributionstomigrationduetohydrodynamicinteractionwiththeboundingwalls, wherethePecletnumber,Pe,isdenedas kBT(3{14) andDOTisthetranslationaldiusivityofaslender-bodyinaninniteuid( DoiandEdwards 1986 ).ThequantityKSisthecontributiontotransversemotionfromtheimposedshearowduetoincludingthestressletcoecientandKBisthecontributionarisingfromtheBrowniantorque.AtPe=0orPe=1,wheretheorientationdistributionissymmetric,hKibecomes0.However,wewillshowthattheasymmetricorientationdistributionathighvaluesofPeresultsinapositivevalueforhKiandamigrationawayfromtheboundingwalls.ThetermhDirepresentsthediusioncomponentactingperpendiculartothewall, 44

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@yhpy2i thatwaspreviouslyidentiedby NitscheandHinch ( 1997 )and SchiekandShaqfeh ( 1997a ). TheensembleaveragesoforientationmomentswithinEq. 3{12 canbecalculatedbynumericallysolvingthegoverningequationfortheorientationdistribution.TheequationisderivedbyapplyingEq. 3{2 toEq. 3{1 atsteadystate, @p(_p)=@ @r(n_r):(3{17) Approximating_rbyh_riinEq. 3{17 andapplyingEq. 3{3 gives (Ipp):@ @p(_p)=0:(3{18) Equation 3{18 assumesthattheorientationdistributionequilibratesmuchfasterthanparticlesmigrateordiuseacrossstreamlines. Furtherassumptionsthatrotationisnotinuencedbyhydrodynamicorstericinteractionswiththeboundingwallsgivesapositionindependentexpressionfortherotationalvelocity, _p=_py(^xpxp)12 @p:(3{19) Replacing_pinEq. 3{18 withEq. 3{19 andsimplifyinggives, (Ipp):@2 @p@p=Per(Ipp):@ @p[py(^xpxp)];(3{20) wheretherotaryPecletnumber,Per,isdenedas DOR=_L2 andDORistherotationaldiusivityofaslender-bodyinanunboundeduid( DoiandEdwards 1986 ). 45

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Ensembleaveragesoforientationmomentsofaslender-bodyasfunctionsofPer.SymbolsdenotethenumericalsolutionsandlinesaretheasymptoticsolutionsatlowPerlimit. NumericalsolutionofEq. 3{20 asafunctionofPergivestheorientationmomentsshowninFigure 3-3 .OurnumericalsolutionwasobtainedfromaseriesofBrowniandynamicssimulationsforasingleberinaninniteuidusingthealgorithmof CobbandButler ( 2005 ),whichisdescribedinAppendix B .TheasymptoticsolutionsatlowPerarealsocalculatedandplottedonFigure 3-3 toconrmthenumericalresults.UsingthenumericalresultsinFigure 3-3 ,thecontributionsofhKSiandhKBitothecenter-of-massdistributionfunctionareplottedinFigure 3-4 .ThevalueofhKBi=hDiapproachesalimitingvalueof72,whereashKSi=hDiincreasesindenitely.Sincetheorientationmomentsareindependentofpositionforsimpleshearow,hKiandhDiareconstantsdependingonlyonPeand@ @yhpypyiiszero. ThescalingswithPeroftheorientationmomentsathighPershowninFigure 3-5 agreecloselywiththosederivedforslenderbodiesinshearathighPer( LealandHinch 46

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ContributionofthetotalhKi,theshearowKS,andtheBrowniantorqueKBtothemigrationofarigidpolymerinsimpleshearowplottedasafunctionofPer. 1971 ; HinchandLeal 1972 ; Brenner 1974 ).Followingtheapproachof ChenandKoch ( 1996 ),thecoecientsofthescalingsaredeterminedfromabestttothenumericaldataovertherangeof102Per4104.AlthoughwetawiderrangethanofPerthanthatttedby ChenandKoch ( 1996 )(102Per103),thereportedcoecientsfortheorientationmomentshpxpyiandp2xp2ymatch.AgreementwasalsofoundbetweenthenumericaldataproducedbytheBrowniandynamicsalgorithmandcalculationsforPer103by ChenandJiang ( 1999 )andfor0
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hDi;(3{22) wherethenormalizationconstantCisdeterminedsothatRndry=1(0:1Lry(H0:1)Landn(0:1)=n(H0:1)=0).ThevaluesofhDiandhKiaregivenby and wherethecorrelationsfortheorientationmomentshavebeenused.NotethatallPerhavebeenconvertedtoPeusingEq. 3{21 .Equation 3{22 isafullyanalyticexpressionforsimpleshearbetweentwoboundingwallswhichcorrespondstoanumericalresultforasingleboundingwallwithsimpleshear. 3.3.1SimpleShearFlow 3{22 ;theresultsaredisplayedinFigure 3-6 forH=6Land10L.TheresultshowsthatanetmigrationawayfromthewallexistsatlargePe.ThelargedepletionlayercanexceedthatpredictedforrigidpolymersathighPewhichinteractwiththeboundingwallonlythroughexcludedvolume( dePabloetal. 1992 ).AsmentionedafterEq. 3{13 ,thesymmetryoftheorientationdistributionisbrokenandhKiispositive.AsPeincreases,theratiobetweenthemigrationandthediusion,hKi=hDi,increases,thereforetherigidbersdistributemorestronglytowardsthecenterofthechannel.Comparingthedistributionsinthewiderandnarrowerchanneldemonstratesthatreducingthelevelofconnementincreasesthedepletionlayer.ThedepletionlayerLd,denedasthepointryatwhichnreturnstothebulkvalue,iscalculatedfromEq. 3{22 andplottedinFigure 3-7 48

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Ensembleaveragesoforientationmomentsofaslender-bodyasfunctionsofPer:(2)p2y0:41Pe1=3r,()hpxpyi0:36Pe1=3rand()hpxpy3i0:83Pe1r.Symbolsdenotethenumericalsolutionsandlinesarethescalingswithcoecientsdeterminedbythebestttothenumericaldata. forA=10anddierentvaluesofPeandH.Theincreasedhydrodynamicliftassociatedwithincreasingtheshearrate,togiveahighervalueofPe,generatesalargerdepletionlayer.Furthermore,increasingthechannelwidthwhileholdingPeconstantextendsthedepletionlayerfurtherintothechannel.Thoughnotplotted,thedepletionlayeralsodependsupontheaspectratiosincethehydrodynamicliftvariesas1=ln(2A)asseeninEq. 3{7 .Consequently,increasingtheaspectratioreducesthedepletionsincethehydrodynamicinteractionbetweenthewallandberbecomescreened. 3{12 isintegratednumerically,withtheorientationmomentscalculatedfromEq. 3{20 ateachpositionaccordingtothelocalshearrate, _(ry)=kBT 49

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Thesteadystatecenter-of-massdistributionsn(ry)ofarigidberpredictedbyEq. 3{22 asafunctionofdistancefromawall,ry=L,withA=10andA)H=6LandB)H=10Linsimpleshearow. ThesteadystatedepletionlayerthicknessLd=LofarigidberofaspectratioA=10predictedbyEq. 3{22 asafunctionofPeandHinsimpleshearow. 50

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3-8 showstheresultsofthecalculationsforA=10.Nearthecenterofthechannel,wheretheshearrateissmall,thecontributionduetotheanisotropicdiusivityresultsinaweakmigrationofpolymerstowardthewall.Specically,hpypyidecreasesfromavalueof1=3atthecenterofthechannelandapproacheszeronearthewall.Asaresult,themigrationisbalancedbythehydrodynamicinteractionsofpolymerswiththeboundariestogiveano-centermaximumforn(ry).Theeectoftheanisotropicdiusivitybecomesincreasinglyimportantrelativetothewallinteractionsforpolymerswithlargeraspectratios.Consequently,themaximumvalueofn(ry)willmovetowardsthewallasAincreases. Whenneglectingbothstericandhydrodynamicinteractionswiththewallhowever,amigrationofpolymerstowardthewallispredictedforarbitraryvaluesofPe( NitscheandHinch 1997 ).Similarly,underconditionsofhighlyconnedpolymerscoupledwithweakpressure-drivenow,modelsthatincludestericwalleects( SchiekandShaqfeh 1997a )predictmigrationtowardsthewall.Inthesecases,themigrationoccurssolelybecauseoftheanisotropicdiusivityofthepolymerasaresultoftheinhomogeneousoweld.Suchamechanismpredictsnomigrationinsimpleshearowwheretheorientationdistributionisspatiallyuniform,contrarytoourresults.Furthermore,thequalitativedierenceinpredictedmigrationinpressure-drivenowhighlightstheimportanceofincludinglong-rangehydrodynamicinteractions. Thetheoreticalresultsforpressure-drivenowareonlyqualitativelysimilartothosefromsimulationswhichincludehydrodynamicinteractionswiththewall( Saintillanetal. 2006a ).While Saintillanetal. ( 2006a )obtainedLd>Lat 51

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Center-of-massdistributionforarigidpolymerwithA=10inpressure-drivenow.Theheightofthechannelis8L. NitscheandRoy 1996 )foundthathydrodynamicinteractionsactinguponaBrownianbershearednearawallhavelittleeectonthedepletionlayer.Thattheoryconsideredthelimitofweakshearowsonly,however.Athighratesofshear,thekinetictheoryinChapter 3 predictsmigrationarisingfromthreemechanismsasshowninEq. 3{12 ,twoofthemrelatedtohydrodynamicinteractions.Thethirdcorrespondstothegradientmechanismidentiedby NitscheandHinch ( 1997 )and SchiekandShaqfeh ( 1997a )thatdoesnotdependuponfar-eldhydrodynamicinteractionsandmovestherigidbersclosertotheboundingwallsonaverage.Mechanismswhichscreenhydrodynamicinteractionsbetweentheberandwallwouldrenderthistermdominant.Screeningcanoccurforbersofveryhighaspectratioandforaseparationbetweenboundingwallswhichisofthelengthoftheberorsmaller. 52

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Comparisonofmigrationmechanismsinshearow:A)Stressletcontribution,B)BrowniantorquecontributionofrigidrodandC)restoringspringforcecontributionofexibledumbbell.Thethinarrowsindicateactionsonpolymersbytheshearowandthethickarrowsrepresentcorrespondingresistancesofpolymers. AscomparedinFigure 3-9 ,thestronger(Figure 3-9 A)ofthetwohydrodynamicmechanismsidentiedinthischapterissimilartothemechanismdescribedforexiblepolymers(Figure 3-9 C).Theinabilityoftherigidbertodeformwiththeuidcreatesanadditionaloweldwhichisreectedbyaboundingwalltocreateatransversemotion.Thistransversemotioncanoccurtowardsorawayfromtheboundingwalldependingupontheinstantaneousorientationoftherod.Thecenter-of-massofaforceandtorquefreerodtumblinginaJeeryorbit( Jeery 1922 )duetoanimposedshearowwilloscillatetransversetothewallsincethesymmetryoftheorientationdistributionispreserved.Thiscouplingofashearingowandatransversemotionhasbeennotedandinvestigatedinpasttheoriesandsimulationsforrod-likeparticles( YangandLeal 1984 ),prolatespheroids( HsuandGanatos 1976 ; Olla 1999 ),andevenforoblatespheroids( ModyandKing 2005 ).Producinganetmigrationrequiresbreakingthesymmetryoftheorientationdistributionfortheparticletumblingintheshearow.KinetictheorypredictsthatweakBrownianmotionalterstherotationaldynamicsofarigidberinshearow( Leal 53

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, 1971 ; Stoveretal. 1992 ; HijaziandZoaeter 2002 ),changingtheorientationdistributioninamannerthatresultsinanetmigrationoftheberawayfromthewall. Theweaker(Figure 3-9 B)ofthetwohydrodynamicmechanismsofmigrationnewlypredictedbythekinetictheorydoesnothaveananalogwithinthetheoryfortheexibledumbbell.ThissecondarymigrationarisesfromahydrodynamiccouplingoftheBrownianrotationoftherigidberandboundingwall.Aparticlewithanisotropicorientationdistributionuctuatestransverselytothewallinresponsetothisinteraction,buttheshearowcreatesapreferentialorientationandaconsequentliftawayfromthewall. 3{3 @t=@ @y(nh_ryi):(3{26) UsingtheformulaforaverageberuxderivedinEq. 3{11 withtheorientationmomentsobtainedinFigure 3-3 underthesameassumptionsmadeinthepreviousSection,thegoverningequationbecomes @t=hDi@2n @y2hKi@ @y[n(ry)]:(3{27) Thisequationissimilartotheevolutionequationforthecenter-of-massdistributionofaexiblepolymerasderivedby MaandGraham ( 2005 ).Fortheexiblepolymers,hKiisrelatedtothecouplingoftherestoringspringforceandshearowwhichinducesaowdisturbanceandmigrationawayfromthewallwhilehKiinEq. 3{27 relatesthecouplingofthestressletcoecient,orinabilityofthebertostretch,andBrowniantorquewiththeshearow. Equation 3{27 issolvednumericallyusingtheCrank-Nicolsonschemewithaninitiallyuniformdistributionandtheboundaryconditionofnouxofthecenter-of-massthroughawall.ThedetailednumericalformulasaredescribedinAppendix B .Thetime 54

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Thetransientcenter-of-massdistributionsofarigidberpredictedbyEq. 3{27 asafunctionofdistancefromawall,ry=L,withA=10andH=6LinsimpleshearowsofA)Pe=1:2103andB)Pe=4:8104.Theunitsoftimetare_1. 55

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3-10 .Atshorttimes,apeakappearsneareachwall.Astimeproceeds,thepeakgraduallymovestowardsthecenterandbecomesmoreblunt.AtatimecomparabletoPe_1,thepeaksfromthetwowallsmergeandthedistributionisfullydevelopedatatimeofabout2Pe_1.Sincethemigrationvelocityduetohydrodynamicinteractionsismuchlargernearawall,therigidbersaccumulatenearthewallfasterthandiusioncansmooththeprole.AthigherPe,migrationismuchstrongerthandiusion;therefore,thepeakismuchsharpernearthewallforPe=4:8104thanPe=1:2103.Astimepasses,themigrationanddiusionofbersbalancesandthedistributionreachessteadystate. AnupperboundonthetimetStoachievesteadystatecanbeestimatedasthetimeforarigidbertodiusefromthewalltothecenterofthechannel, 2hDiOPeH2 wherehDiisapproximatedusingtheleadingorderofPe1inexpression,Eq. 3{23 .ThisestimatefortSatH=6LisabouttwiceaslargeasindicatedbytheresultsinFigure 3-10 MaandGraham ( 2005 )arguedthataexiblepolymerneedstodiuseonlyadistanceequivalenttothedepletionwidth,whereaswendthatthedistributionsatt=2Pe_1arevirtuallyidenticaltothesteadydistributions. 56

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SchiekandShaqfeh 1997a )aswellasimprovementstothewallinteractions( LironandMochon 1976 )canbeincluded.ValidationofmultipleapproximationsmadeinthetheorywillbeconrmedbycomparisonwithsimulationinChapter 4 57

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3 qualitativelypredictsthemigrationobservedinthesimulationsof Saintillanetal. ( 2006a ),therearequantitativedisagreements.Thereforemultipleapproximationsmadeinthetheoryshouldbeinvestigatedtovalidatetheiruse.Theseincludesolvingfortheorientationdistributionseparatelyfromthecenter-of-massandanevaluationofthehydrodynamicinteractionwiththewallthatlacksrigorinfavorofsimplicity.Excludedvolumeinteractionsoftherigidberwiththewallwerealsoneglectedinthetheory. JustasBrowniandynamicssimulationshavebeenperformedtotestkinetictheoriesforthemigrationofexiblepolymers( Hernandez-Ortizetal. 2006 ; HodaandKumar 2007a ),wetestthetheorybyperformingsimulationswithandwithouthydrodynamicinteractionsofarigidberwithweakBrownianmotionsuspendedinrectilinearowsbetweentwowalls.Comparisonsbetweensimulationsdemonstratethatthedepletionlayerisaectedbyhydrodynamicsonlyforverylargeratesofshear;atmoderateratesofshearwherethehydrodynamicmechanismformigrationisrelativelyweak,thethicknessofthedepletionlayeriscontrolledbyexcludedvolumeinteractionsoftherodwiththeboundingwalls.Thedistributionsobtainedbythesimulationwithhydrodynamicinteractionsareingoodagreementwiththosepredictedbythetheoryforsucientlyhighratesofshearratherthanbythesimulationof Saintillanetal. ( 2006a ). ArigidberconnedbetweentwoboundingwallsissimulatedasshowninFigure 4-1 .Toincludehydrodynamicinteractionsofthebulkparticlewithboundingwalls,werepresenteachwallasaperiodicarrayofslenderbodiesandcalculateinteractionsbetweenallparticlesusingmethodsdescribedinChapter 2 .ThisapproachdiersfromcalculatingtheGreen'sfunctionfortwoplanarwallsofinniteextent( LironandMochon 1976 ; Stabenetal. 2003 )asdoneby Saintillanetal. ( 2006a ).However,representing 58

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Arraysofcloselypackedbersformthewallsinaperiodicboundarysystemofdimensionbx,byandbz.A)Aviewfromthevorticity-gradient(zy)-plane,B)viewofsimpleshearow,andC)parabolicowfromtheow-gradient(xy)-plane. thewallsbycloselypackedarraysofparticleshasbeenusedwithsuccessinStokesiandynamics( NottandBrady 1994 ; SinghandNott 2000 ; BrickerandButler 2007 )andthemethodcanbeextendedtoperformsimulationsofnon-dilutesystemswhichisafocusofongoingwork.InSection 4.2 ,theequationsgoverningthemotionoftherigidbersaremodiedfromEq. 2{22 andadditionaldetailsofthesimulationmethodaregiven.InSection 4.3 ,simulateddistributionsinvariousshearingowsarepresented. 4.2.1GoverningEquation 4-1 ,particlesinthebulk,aswellaswallsaresimulatedasrigidrodsusinggoverningequation,Eq. 2{22 ,whichisforrigidrodsinperiodicsuspensionsisalsousedforthissimulation.Inthissimulation,theGreen'sfunctiongivenby Beenakker ( 1986 ),BP,whichistheEwald-summedexpressionofGreen'sfunctionby RotneandPrager ( 1969 )and Yamakawa ( 1970 )inaperiodicsystem,isused.Therefore, 59

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2{26 shouldbechangedaccordingly, whereB0isthelimitingformgivenby Beenakker ( 1986 )andBis 6ach19jxj whereacisaneectiveradius( RotneandPrager 1969 ; Yamakawa 1970 ). Fordynamicsimulationsofsuspensionsofhydrodynamicallyinteractingrigidbers,thesolutionof Hasimoto ( 1959 )fortheperiodicOseen-Burgertensorhasbeenwidelyused( Harlenetal. 1999 ; ButlerandShaqfeh 2002 ; Saintillanetal. 2006b ).However,thisresultsinamobilitytensorwhichsometimeslacksthepropertyofbeingpositivedenitewhenapairofbersisincloseproximity.Thesolutionof Beenakker ( 1986 )ensuresapositivedenitematrixforanycongurationoftheinteractingparticlesregardlessofseparationdistance,soithasbeenappliedtothesesimulationsduetothepresenceofcloselypackedarraysofberswhichformthewalls. TornbergandGustavsson ( 2006 )alsousedBeenakker'ssolutioninthesimulationofsedimentingrigidbers.ComparingthevelocitydisturbancegeneratedbyaforceonaberbyintegratingOseen-BurgertensorandBeenakker'ssolutionalongthelengthoftheroddemonstratesthattheerrorisnegligibleatadistanceofthreetimestheberdiameter.Forthespeciccaseofaforceactingperpendicularlytoarodcenteredattheorigin,thevelocitydisturbancesintheperpendiculardirectionagreewithin5%atadistanceoftwodiametersforapointlocatedontheplanelyingperpendiculartotherodandcenteredontherod. Weidentifytwotypesofparticlesinthissimulation:wallbers(W)andabulkber(B).CloselypackedarraysofthewallbersformthetwoplanarboundariesasshowninFigure 4-1 .Thebulkberiscontainedinthespacebetweentheupperwallandthe 60

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2{22 arereplacedwiththetypesofparticles. Theshearowisnotgivenastheimposedoweldu1,butisgeneratedbytheowdisturbanceu,inducedbymovingwallbers.Hence,solvingEq. 2{22 requiresthefollowingmanipulationofaresistanceproblem: wherestressletcoecientsonwallbersaredesignatedbythevectorSW.Sincetheuidmotionisgeneratedbymovingthewallparticles,theratesofstraincorrespondingtoanimposedowhavebeensettozero,0,inEq. 4{3 ;thisdoesnotmeanthatthebersexperiencenostrainingow,butthatthestrainingowisgeneratedbytherelativemotionoftheparticlesonly.UsingtheresultsfromEq. 4{3 ,themotionofthebulkberconnedbetweenthetwoparallelwallsisgivenby 61

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andQisthemobilityofthebulkrod, TheBrownianforcesandtorquesactingonthebulkbercanbeobtainedfromtheinverseofthemobilitymatrixQbysatisfyingtheuctuation-dissipationtheorem, where thethermalenergyiskBT,andthevectorWcontainsrandomnumbersgeneratedfromauniformdistributionofzeromeanandavarianceofone.CholeskydecompositionisusedtoevaluateBasexplainedby ButlerandShaqfeh ( 2005 )and Saintillanetal. ( 2006c ).

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4.3.2 .Theboundingwallsareconstructedof2bxbz=Ldbersofaspectratio10.Themobilitymatrices,consistingofdualintegralsovertheparticlelengths,aregeneratedusingavepointandsixpointGaussianquadratureforthecasesofsimpleshearandparabolicow,respectively.ThenumberofsummationforperiodicGreen'sfunctionissetto3withac=1:66dforsimpleshearowandac=2:05dforparabolicow. Togeneratesimpleshearow,thewallparticlesareassignedavelocityinthex-direction,_rx,butwithoppositesignsforparticlesinthe`top'and`bottom'wall;allothervelocitiesaresettozero.Forasimpleshearow,thevelocity,_rx=H_=2,generatesashearowof_.Bothwallstranslateinthepositivex-directionatthesamevelocityofH _(y)=2 _2y H1;(4{9) where _isthemeanshearrate.Inbothcasestheresultcorrespondscloselytothedesired,theoreticalprolesandthereisnonetowthroughtheyz-planesincetheperiodicsumswereconstructedtoensurezeronetowthroughanyplane.Theowproles,showninFigure 4-2 ,weregeneratedusingEq. 2{15 withthelinearizedforcedistributioninEq. 2{16 andintheabsenceofabulkparticle.Inaddition,weconrmedthatuiddoesnotowperpendiculartothewalls. Lubricationinteractionsdominatewhenthebulkparticleapproacheswithinoneberdiameterofthewall.Thisshortrangeinteractionisaddeddirectlytotheresistancematrixusingmethodsidenticaltothoseof ButlerandShaqfeh ( 2002 ).Theinteractionitselfdependsupontherelativeorientationoftherodwithrespecttothewallandisevaluatedusingthemethodof ClaeysandBrady ( 1989 ).Thelubricationinteractionis 63

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1exp(10h=L)^y;(4{10) wherehistheclosestdistancebetweentheberandtheboundingwall,theforceactstopushtheberawayfromthewall,and^yistheunitvectorinthey-direction.Whenevaluatingtheseparationbetweenthebulkberandthewall,theendtipoftheberisregardedasahardspherehavingthesamediameterastheber.Thisrepulsiveforceisincludedwiththeforcesandtorquesactingonthebulkberandthewallbers.Simulationresultsconrmthattheparticledoesnotcrossthewall.Wealsotestedotheralgorithmsformaintainingtheexcludedvolume( dePabloetal. 1992 ; HijaziandKhater 2001 )inAppendix C.2 ,butthedierencesinthedistributionsweresmall. Todistinguishtheeectsofthehydrodynamicandexcludedvolumeinteractionsonthedepletionlayer,simulationsarealsoperformedintheabsenceofhydrodynamicinteractionswiththewalls.ThesesimulationsareperformedusingEq. 2{7 andEq. 2{9 withthevelocityuB(sB)replacedbythetheoreticalexpressionforsimpleshearoworparabolicowasappropriate.TheBrownianforcesandtorquesarecalculatedusingtheuctuationdissipationtheory,butwiththemobilityofasinglerodsuspendedinaninniteuid.ThesamerepulsiveforcegiveninEq. 4{10 preventstheberfromcrossingthewall. Equation 4{4 isintegratedusingthemodiedmidpointmethod;thismethodintegratesthetrajectoriesofthestochasticmotionaccuratelyatrstorderwithoutrequiringanexplicitevaluationofthedivergenceofthemobility( Fixman 1978 ; Grassiaetal. 1995 ; Morse 2004 ).Thetimestepforintegratingpositionsissetto0:001 _1.NotethatmanyofthesubmatricesinEq. 4{5 needtobecalculatedonlyonce.Theself-mobilitiesandanylubricationinteractionsforthebulkberareupdatedateachtimestep,butthelong-rangehydrodynamicinteractionsbetweenthebulkparticleandwallparticlesareupdatedlessfrequentlyatintervalsof0:05 _1.Testsonthis 64

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Simulationresults(brokenlines)ofA)thevelocityproleinthex-direction(unitsofL _)oftheshearowgeneratedbymovingwallsasfunctionsofdistancefromthebottomwall,y=L.FibershaveanaspectratioofA=10andthewallsareseparatedbyagapofH=6Linaperiodicboxofdimensionsbx:by:bz=3L:8L:3Lforsimpleshearowandbx:by:bz=3L:12L:3Lforparabolicow.Theoreticalvaluesareplottedassolidlines. integrationschemedemonstratedconvergenceoftheresultswhilegreatlydecreasingthecomputationalexpenseofthecalculations. 4.3 .Wedemonstratethatthecalculationsareingoodagreementforthepredictionofthecongurationdependentliftvelocity(Section 4.3.1 ),thesteadyandunsteadydistributioninsimpleshearow(Section 4.3.2 ),andthesteadydistributioninparabolicow(Section 4.3.3 ). 65

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Transversevelocity_ryduetoshearowasafunctionofanglewithpz=0,ry=1:0L,andH=6L.SimulationconditionsarethesameasthoseinFigure 4-2 Bwithsimpleshearow.Theinsetillustratesarigidberwithanangleandpz=0. asafunctionofcongurationisembodiedinthersttermofEq. 3{7 _ry=_(ry)pxpy13p2y:(4{11) TheresponseofabertotheshearowcanbedeterminedfromthesimulationusingEq. 4{4 bysettingFBto0andextractingthey-componentofthevelocity, _ry=^yP0BBBBBBB@_rW0W0W0B1CCCCCCCA:(4{12)ComparisonsbetweenthetheoreticalanalysisEq. 4{11 andcomputationalresultEq. 4{12 appearinFigures 4-3 and 4-4 66

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Transversevelocity_ryduetoshearcontributionasafunctionofdistancefromawall,ry=L,foraxedangle(=3andpz=0)underaconstantshearrate_.Resultsareshownforthetheoryandthreesimulationswithdierentperiodicdimensions. Figure 4-3 comparesthedependenceofthetransversevelocityonangle(anglebetweenthex-axisandprojectionofpBontothexy-planeasillustrated)foraberxedatadistance1Lfromthewall.Theresultsmatchqualitatively,withthelargestquantitativediscrepanciesoccurringatorientationscorrespondingtothesecondarymaximumof_ry.ThequantitativedierencesbetweenthecalculationsoftransversevelocityalsodependuponthepositionoftheberandsizeoftheperiodicboxasshowninFigure 4-4 CausesofthequantitativediscrepanciesclearlyincludetheapproximationsmadefortheevaluationoftheGreen'sfunctionusedintheanalyticexpression.Thetheoryoverestimatesthetransversevelocitynearthecenterduetosuperimposingtheeectofthetwoboundingwalls.ThetheorywasderivedbylinearizingtheGreen'sfunction,assumingthattheberisfarfromthewallincomparisontoitslength;theincreasingdiscrepancy 67

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4-4 .Duetoperiodicity,thetransversevelocitypredictedbythesimulationislowerthanexpectedforasinglerodsuspendedbetweentwoplatesofinniteextent. Wealsoperformedtestsonthecomponentsof_ryarisingfromforcesandtorques(Appendix C.1 )giveninEq. 3{7 .Comparisonsofthesetermswiththeresponsesfromsimulationsalsodemonstrateaqualitativematchandquantitativedierencessimilartothoseobservedfortheshearcontribution. 3 Figure 4-5 showsthetimeevolutionofcenter-of-massdistributioninsimpleshearowforPe=1:2103.Thedistributionsexhibitamaximumnearthewallatshorttimes(Figure 4-5 A),withalargermaximumproducedbythesimulationthanthetheory.Whilethetheorypredictsattainmentofasteadystatedistributionaroundt=2Pe_1,thedistributionaveragedovertimesoft=25005000_1(Figure 4-5 B)stillshowssignicantvariationsaroundthepredicteddistributionwhichisatsteadystate.Thesimulateddistributionconformscloselytothetheoreticaldistributionatsteadystate(Figure 4-5 C)whenaveragedoverlongtimes. 68

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4-5 betweentheresultsofthetheoryandsimulationwithhydrodynamicinteractionisfortuitous.SimulationswithandwithouthydrodynamicinteractionswiththewallsatPe=1:2103produceanearlyidenticaldistributionasshowninFigure 4-5 C.Thesimilarityestablishesthatexcludedvolume,whichthetheorydoesnotconsider,iscontrollingthedistribution;theowstrengthatPe=1:2103istoolowtogenerateclearevidenceofthehydrodynamiclift.TheresultsinFigure 4-5 Carealsoconsistentwithpreviousinvestigations( dePabloetal. 1992 ; SchiekandShaqfeh 1995 )ofthechangesinthedepletionlayerduetoshearandexcludedvolume.Wenotethatthedistributionbecomesuniformatavalueofabout0:6L0:7L,ratherthan0:5L,sincetheendoftherodisahemispherethatextendsanadditionaldistanceof0:1Lbeyond0:5Landthesurfaceofthewalliscorrugated. SimulateddistributionswithhydrodynamicinteractionsatPe=1:2104areshowninFigure 4-6 .Thetransientmaximumnearthewallappearsearlyinthesimulationresults(Figure 4-6 A)andagreeswiththetheoreticalpredictionwithintheaveragingerrors.Theintermediatedistribution(Figure 4-6 B)uctuatesaroundthecorrespondingtheoreticaldistributionandthedistributionfort>17000_1(Figure 4-6 C)closelycorrespondstothetheoreticalsteadystate. ThesteadydistributionfromsimulationswithouthydrodynamicinteractionatPe=1:2104isplottedinFigure 4-6 C.AscomparedtothecorrespondingresultatPe=1:2103inFigure 4-5 C,thedepletionlayerincreasesbecauseofthehigherrateofshear,butstilldoesnotextendbeyondry0:65L,wherethedistributionbecomesuniform.ComparingthedistributionsfromsimulationsatPe=1:2104withandwithouthydrodynamicinteractiondemonstratesthatthehydrodynamicinteractionsstronglyinuencethedistributioninamannerconsistentwiththetheoreticalprediction. Figure 4-7 showsdistributionsfromsimulationwithhydrodynamicinteractionatPe=4:8104.Althoughtheaveragingerrorsarelargeatshorttimes,thetransient,o-centermaximumisdetectableinthemeanvalues.Thesimulationwasterminatedat 69

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Thesimulationresults(symbols)andtheoreticalsolutions(lines)forthetransientcenter-of-massdistributionsofarigidberasafunctionofdistancefromawall,ry=L,withA=10andH=6LinsimpleshearowsofPe=1:2103.Simulationresultsareaveragedoverthetimeranges(unitsof_1)asindicatedineachlegend. 70

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Thesimulationresults(symbols)andtheoreticalsolutions(lines)forthetransientcenter-of-massdistributionsofarigidberasafunctionofdistancefromawall,ry=L,withA=10andH=6LinsimpleshearowsofPe=1:2104.Simulationresultsareaveragedoverthetimeranges(unitsof_1)asindicatedineachlegend. 71

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4-7 Cdemonstratesacleardierenceascomparedtotheonewithhydrodynamicinteraction,indicatingagaintheroleplayedbythehydrodynamicliftforce. Figure 4-8 showstheorientationmomentp2ycalculatedfromthesimulationsasafunctionofpositionandPe;theorientationmomentsforunboundedow,whichareusedtomakethetheoreticalprediction,arealsoshown.Thevaluesofp2yfromsimulationswithandwithouthydrodynamicinteractionhavenodiscernibledierencesregardlessofdistancefromthewall,conrmingtheassumptionmadewithinthetheorythathydrodynamicinteractionwiththewallsdoesnotaecttheorientationdistribution.Thewallaectstheorientationthroughexcludedvolumeinteractionsforry<0:7L,resultinginsubstantialdierenceswiththeorientationdistributionassumedwithinthetheoreticalcalculations. Theovershootofp2yaround0:65LforniteshearratesisaninterestingfeatureobservableinFigure 4-8 .Theactionoftheshortrangerepulsiveforcewiththewallontheendofarodrotatingintheshearowcreatesa`pole-vault'typeofmotionthatdisplacestherods,orientednearlyperpendiculartothewall,outwardstogivetheenhancedprobabilityofp2yintherangeofry=0:6L0:7L.Thoughtheeectislikelycausedbytherepulsiveforce,attemptstoeliminatetheovershootbyalteringthestrengthandrangeoftherepulsiveforcewereunsuccessful.Wenotethat`pole-vault'motionshavebeenobservedbyothersinsimulationsintheabsenceofBrownianmotion( StoverandCohen 1990 ; ModyandKing 2005 )andeveninexperiments( HolmandSoderberg 2007 ). Inthevicinityofawall(ry.0:7L),thestericeectscontroltheorientationdistributionandignoringtheexcludedvolumewithinthetheoryisclearlyinerror.However,theneglectofexcludedvolumehasalimitedeectonthetheoreticalpredictionofthecenter-of-massdistributionforvaluesofPeabove1:2103.Theprimaryreasonis 72

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Thesimulationresults(symbols)andtheoreticalsolutions(lines)forthetransientcenter-of-massdistributionsofarigidberasafunctionofdistancefromawall,ry=L,withA=10andH=6LinsimpleshearowsofPe=4:8104.Simulationresultsareaveragedoverthetimeranges(unitsof_1)asindicatedineachlegend. 73

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Thesimulationresultswith(lledsymbols)andwithout(opensymbols)hydrodynamicinteractionsfortheorientationmomentp2yofarigidberasafunctionofdistancefromawall,ry=L,withA=10andH=6L.Theexpectedvaluesofp2yforunboundedowareplottedaslines. thatfewbersremainwithinadistanceof0:7Loftheboundingwallsforthehighershearratesduetothehydrodynamicmigrationawayfromthewall. Thequalitativefeaturesofthecenter-of-massdistributionspredictedbythetheoryappearinthesimulationsforPe=1:2104and4:8104:thetransientmaximumnearthewallsformsatshorttimes,thenmovestowardsthecenterwhilebroadeninganddisappearsattPe_1.Quantitativecomparisonsoftransientprolesarehinderedbythelargererrorsinsomeoftheresults,buttheoverallgoodagreementveriesthattheerrorsassociatedwithassumptionsmadeinthetheorydonothavesevere,adverseaectsonthepredictionsolongasthedepletionlayerextendsbeyondtherangeatwhichtherotationoftherodishinderedbythewall. ThismechanismwhichinducesthedepletionlayersolelybyexcludedvolumeeectofthewallisdemonstratedinFigure 4-9 .Arigidrodinshearowisfoundtoberotate 74

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Rotationsofrigidrodsinshearowofstrength_:A)JeeryorbitmotionfarfromawallandB)pole-vaultmotionignoringhydrodynamicinteractionswithawall.Thedottedarrowindicatesatrajectoryofcenter-of-massofarodundereachrotation. followingJeeryorbit(Figure 4-9 A).However,ifthedistanceofarodfromawallisclosesothatitsrotationisrestrictedbythewall,therotationpushestherodgivingaslightincreaseofcenter-of-massinadirectionnormaltothewall.Thistypeofmotioniscalled`pole-vault'motion(Figure 4-9 )( StoverandCohen 1990 ; ModyandKing 2005 ). _L2=DTasshowninFigure 4-10 .Thenumberofruns,initialconguration,andsamplingmethodmatchesthoseofsimulationsforsimpleshearowpresentedinSection 4.3.2 .ThedistributionsarecomparedwiththesteadydistributionsobtainedbynumericallyintegratingEq. 3{12 .TheorientationmomentsusedwithinEq. 3{12 dependuponpositionaccordingtothelocalshearrateasgivenbysolutiontoEq. 3{20 ;alterationsintheorientationduetohydrodynamicinteractionsorexcludedvolumeareignoredaswasdoneforthecalculationsinsimpleshear. At 75

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Thesimulationresults(symbols)andtheoreticalsolutions(lines)forthecenter-of-massdistributionsofarigidberasafunctionofdistancefromawall,ry=L,withA=10andH=6LinparabolicowsofA) _1)rangesatwhichthesimulationdataweretakenareindicatedineachlegend. 76

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Thesimulationresultswithhydrodynamicinteractions(symbols)forp2yofarigidberasafunctionofdistancefromawall,ry=L,withA=10andH=6Linparabolicowsof hydrodynamiclifthaslittleeectuponthenear-walldistributionforthisandlowervaluesofPe.Cleardierencesareobservedbetweensimulationswithandwithouthydrodynamicinteractionfor ThesimulationsofaBrownian,rigidberinparabolicowby Saintillanetal. ( 2006a )wereperformedat 3 aswellas Parketal. ( 2007 ).Thecurrentresultsimply 77

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Saintillanetal. ( 2006a )wasmostlyduetostericeects.However,thesimulationsof Saintillanetal. ( 2006a )werealsoperformedinawiderchannelofH=8Landwithoutperiodicboundaries,wherethehydrodynamicliftwouldbeexpectedtobestronger. Theresultsofthesimulationswithouthydrodynamicinteractionsandtheoryexhibitano-centermaximumduetothegradientindiusivityinducedbythevariationinshearrateacrossthechannel( NitscheandHinch 1997 ; SchiekandShaqfeh 1997a ).Evidenceoftheo-centermaximumisdiculttodetectinthesimulationswithinteractionsduetolargeaveragingerrors,thoughFigure 4-10 Bseemstodemonstrateareductioninthedistributionnearthecenterlineofthechannel.Thelackofaclear,o-centermaximummaybedueinparttotheorientationdistribution.Figure 4-11 showsthatthegradientinp2y,whichcausesthemigrationtowardsthewallthroughthetermhiinEq. 3{12 ,issmallerthanassumedinthetheoreticaldescription.NotethatFigure 4-11 alsoshowsanovershootinp2yatry=0:6L0:7L,similartotheresultsforthecaseofsimpleshearow. 4.3.2 and 4.3.3 providethatEq. 3{7 ,whichmodelsthemotionofarodnearawall,isvalidatthebulkregioninachannel(ry&0:5L)whileexcludedvolumeeectdominatesnearawall.ItsuggeststhatnumericallyintegratingEq. 3{7 consideringexcludedvolumeeectrepresentedbyEq. 4{10 enablessimulationofarigidrodnearawallatreducedcomputationalburden. Here,wethisconcepttostudyrodsinoscillatoryshear.Themotionofanupperwalloscillatingwithfrequency!inthex-directionisgivenby 78

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Distributionsinoscillatoryshearowsareobtainedbyamodiedsimulation.TheoreticalanalysesonrigidrodsuspensionsinoscillatoryshearhavebeenlimitedtolowPeOandsmallamplitudelimits( KirkwoodandAuer 1951 ; Ullman 1969 ). ParkandFuller ( 1985 )and SchiekandShaqfeh ( 1997b )addedtheeectofconnementbythewallbutnothydrodynamicinteractions.Here,simulationsareperformedathighPeOwithconsideringhydrodynamicinteractionswiththewall.Tocomparetheeectofoscillatoryshearondistributionswiththepreviousonesofothershearows,owsofPeO=1200andPeO=48000arechosen. ResultsatvariousmaxareshowninFigure 4-12 .AtPeO=1200,distributionsareverysimilarwithonesshowninFigure 4-5 Cforsimpleshearow.Thereforemigrationduetohydrodynamicinteractionwithawallisstilltooweaktogiveanyeects.Itisobservedthatincreaseofmaxslightlythickensdepletionlayer.EvenathigherPeOof48000,wheremigrationduetohydrodynamicinteractionisclear,distributionsareverysimilarwithonewithouthydrodynamicinteractionsshowninFigure 4-7 C.Thisresultdemonstratesthatoscillatoryowremoveshydrodynamiclifteect.AlimitedincreaseofdepletionlayerwithincreasingamplitudeisalsoobservedatPeO=48000. Thelackofmigrationinoscillatoryshearcanbeexplainedbychangesintheorientationdistribution.InChapter 3 ,itwasstatedthatbrokensymmetryoftheorientationdistributionduetoBrownianrotationprovidesthecongurationwhichfavorsmigrationawayfromawall.Theaverageorientationmoment,hpxpyi,whichcontrolsthemigrationinsteadyshear,isfoundtobeO(105)forPeO=1200andO(103)forPeO=48000,whicharemuchsmallerthanvaluesforsimpleshearows>102foundinFigure 3-5 .Oscillatoryshearseemstorecoverthesymmetryoftheorientationdistributionagainandnullifythemigrationduetohydrodynamicinteractions. 79

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Thesimulationresultsforthecenter-of-massdistributionsofarigidberasafunctionofdistancefromawall,ry=L,withA=10,H=6L,andvaryingmaxinoscillatoryshearowsofA)PeO=1:2103,andB)4:8104. 80

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Thesimulationresultsforp2yofarigidberasafunctionofdistancefromawall,ry=L,withA=10andH=6LinoscillatoryshearowsofPeO=1:2103(opensymbols)andPeO=4:8104(closedsymbol).Eachvalueofmaxisindicatedinlegend. Figure 4-13 showsthedistributionofp2ywithvaryingamplitude.Astheamplitudedecreases,p2yapproaches1=3,whichisfornoow.Smallamplitudeorsmallfrequencyimpliesthataroddoesnothaveenoughtimetobealignedbytheow.Thereforethedepletionlayerissimilartothatofsimpleshearowwithouthydrodynamicinteractions( dePabloetal. 1992 ). 3 canadequatelypredictthecenter-of-massdistributionforarigidbersuspendedbetweentwoplanewallsineithersimpleshearorparabolicowofsucientlyhighstrength,orPe.FortheselargePe,thegoodagreementvalidatesthatthemultipleapproximationsmadeinthetheorydonothaveasevere,adverseaectuponthepredictionsfortheconditionsstudied.Thesuccessofthetheoryathighratesofshearis 81

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4.3 )andtheparticleorientationdistributionconformstothatofanunboundedow(Figures 4-8 and 4-11 ). ThetheorycannotbeappliedwithcondenceifPe.1:2103forthecasestudiedhereofA=10andH=6L.Thestericinteractionsbetweentherodandboundingwalls,whichareignoredinthecurrentversionofthekinetictheory,controlthethicknessandshapeofthedepletionlayerattheselowerratesofshearwherethehydrodynamicliftforceistooweaktomovetheparticleoutofrangeofthewall.AtheorycapableofmakingaccuratepredictionsoverawiderrangePeshouldincludethestericinteractionsasdoneinpreviousinvestigations( dePabloetal. 1992 ; HijaziandKhater 2001 ). SimulationspresentedinthecurrentworkwerelimitedtotheconditionsofA=10andH=6L.Thetheory,whichreliesonasuperpositionoftheGreen'sfunctionforplanewalls,willbecomecontinuallylessaccurateasthegapwidthHbecomessmallerwithregardtotheparticlelengthL.Simulationsofthemigrationofexiblepolymers( Ustaetal. 2007 )indicatethattheoreticalpredictionsfortherigidroddistributionwouldbegreatlyinerrorforH.5L.Conversely,thetheoryisexpectedtobecomemoreaccurateastheratioH=Lincreasesandcomparisonswithsimulationsof Saintillanetal. ( 2006a )onparabolicowsuggestthatthehydrodynamicliftforceremainsofimportanceatlowerPewithinlargerchannels,againconformingtoexpectations. 82

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3 and 4 thatthedynamicbehaviorofrigidrodssuspendedinaconneddomainisaectedbythewallthroughhydrodynamicinteractionsandexcludedvolumetogiveaninhomogeneousdistributioninthechannel.Asaconsequence,therheologicalpropertiesoftheconnedsuspensioncandierfromthoseofunboundedsuspensionsduetotheinteractionsbetweenthesolidboundariesandthebulkparticles.Anon-localtheoryforcalculationofparticlestressofrigidrodsinaconnedsystemhasbeendevelopedby SchiekandShaqfeh ( 1995 ).HoweverhydrodynamicinteractionswiththeboundarieswerenotconsideredandthecalculationislimitedtolowPe,orweakow.Sincetheviscositychangeswithparticleconcentrationduetohydrodynamicinteraction( ShaqfehandFredrickson 1990 ),theeectofhydrodynamicinteractionwiththeboundariesonparticlestresswillbeinvestigatedinChapter 5 .AneectoftheinhomogeneousdistributionsobtainedathighPeonthestressisalsoexpected. TheeectivestressofaNewtoniansuspensionofviscosity,shearedatarate_,is wherePistheuidpressure,Eistherateofstraintensorwhichsatises_=1 2(E:E),and Batchelor 1971 ) 3IffB(sB)pBgsBdsB(5{2) 83

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2{16 andintegratingalongtheaxisgives 3I:(5{3) Notethatweignorethedirectcontributionofrepulsiveforcesbetweenthewallandbulkparticlewhencalculatingthestress.Consequently,thetorqueTBinEq. 5{3 containscontributionsonlyfromBrownianmotion.TherelationbetweenthestressletcoecientSBandtheparticlestressisalsomadeclearinEq. 5{3 CalculatingtheeectivestresscontributionoftherigidberrequiresaveragingEq. 5{3 overtheorientationandcenter-of-massdistributions.Thisaverageiscalculatedfromthesimulationsbysamplingthedistributionsatsteadystate(Section 5.2 )andthenfromtheoryatthesamelevelofapproximationasusedinChapter 3 forpredictingthecenter-of-massdistribution(Section 5.3 ).Resultsarepresentedcomparingthecalculatedvalueofthetotalcontribution,aswellasindividualcontributions(Section 5.4 ). 5{2 issampledNRtimesfromthesimulationresultsandaveraged, togivethemeanexpectedvaluefortheextraparticlestress.HerendisthenumberdensityN=V.ThenumberofrodsineachperiodiccellisN=1andthevolumeVrepresentsthespacebetweentheboundingwallswherethebulkparticleresides,ratherthantheentirevolumeoftheperiodiccell.Calculatingthisvolumewhileaccountingforthebumpywallsgives 84

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Notethattheshearstressfortheconnedowcanbecalculatedfromthesimulationresultsintwoways:asabovefromtheparticlestressletsorbynormalizingthetotalforceactingonthewallsinthex-directionbythearea.Theforcesonthewallduetotheuidintheabsenceofaparticlemustbesubtractedfromthetotalvaluesinordertoisolatethecontributionfromtheparticles.Theexpectedvalueoftheuidstressmatcheswithin1:38%oftheresultfromsimulationsintheabsenceofparticlesandthemeanvalueoftheparticlecontributiontotheshearstressascalculatedfromthestressletsandthewallforcesagreeswithinatleast0.04%forallcases. Figure 5-1 AshowsthetotalcontributionoftheparticlestressforvaluesofPebetween1:2103and1:2105;thevaluesof arisingfromtheBrowniantorque 5{3 and thenaveragedasdescribedinEq. 5{4 .TheresultforeachcontributionisshowninFigures 5-1 Band 5-1 C.NotsurprisinglysincePeislarge,thecontributionfromshearexceedsthatfromtheBrownianrotationbyordersofmagnitude. 85

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TheextraparticleshearstressesforrigidberswithA=10undersimpleshearowbetweentwowallswithagapofH=6Lcalculatedfromsimulation(),simulatedorientationmoments(H),andtheory().ResultsshownincludeA)thetotalparticlestress 5.2 withthetheoreticalcalculation,theapproachoftheprevioustheoryinChapter 3 whichusestheapproximate,linearizedGreen'sfunctionisapplieddirectlytocalculatethestressusingEq. 5{3 .FortheBrowniantorque,theexpressionasgiveninEq. 3{10 isused.Theexplicitresultforthestressletcoecientisgivenby 86

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3{10 fortheBrownianforce, itbecomesclearthatthesecondtermisrelatedtothegradientinthecenter-of-massdistributionwhichiscreatedbythehydrodynamicinteractionoftherodandboundingwalls. ReplacingTBandSBinEq. 5{3 withEq. 3{10 andEq. 5{10 givesanexpressionforthestresslet, 3IkBT(ry)pBpB1 3I2pypB+p2y1^y@ln whichmustbeaveragedtogivethemeanstress.TheaverageisperformedovertheexpectedcongurationinthesystemofvolumeV, Byfactorizingthedistributionfunction=nunderthesameassumptionthatorientationdistributionisnotinuencedbythewallandequilibratesfasterthann,thisequationiscompletedintermsoftheensembleaverageovertheorientationdistribution,, 3Ihpxpyi +4 @rydryhpBpBi+Ip2y1 33p2ypBpB: HinchandLeal ( 1976 ).Therst(detailsofitsderivationisinAppendix B.4 )representstheaectoftheBrowniantorqueontheeectivestress, 87

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5{13 isacontributiontotheeectivestressfromthenon-uniformdistributionofthecenter-of-massoftherodscausedbythemigration. Theshearstresscomponent(xy)of 5{13 usingtheanalyticalexpressionobtainedinChapter 3 forn(ry)andnumericallyintegrating(ry)@n=@ryoverthechannelheight.TheresultsareplottedinFigure 5-1 A.Theadditionalcontributionfromtheinhomogeneousdistributionislabeled 5{13 )andissmallasseeninFigure 5-1 D. 5-1 Ashowsthattheeectiveparticlestressascalculatedfromthesimulationresultsandthetheoreticaldescriptioncloselyagree.Thecloseagreement,giventhesmallcontributionfromtheinhomogeneousdistributioninthetheory,impliesthattheeectiveparticlestressisclosetothatofadilutesuspensionofBrownianbersintheabsenceofboundingwalls. Toinvestigatetheeectsoftheboundingwalls,aswellasperiodicimages,ontheresultscalculatedfromthesimulations,wecalculatethedilutecontributionevaluateddirectlyfromthesimulations.ThecontributionfromtheBrowniantorque,givenby iscalculatedusingthemomentofpxpyassampledfromthesimulationsandaveragedoverthechannelheight.Theresult,plottedinFigure 5-1 B,showsthatthedilutecontributionisslightlylowerthanthevaluecalculateddirectlyfromthesimulation.Likewise,Figure 5-1 Cshowsthecontributionfromtheshearwherethevalue 88

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5{8 ThedierenceinthetotalparticlestressfromthesimulationsasgivenbyEq. 5{6 andthedilutevalueascalculatedfromtheorientationmomentsproducedbythesimulationsbycombiningEq. 5{14 andEq. 5{15 isgiveninFigure 5-1 D.Thisquantityrepresentstheadditionalstressduetothewalls,inhomogeneousdistributionforthecenterofmass,andinteractionswiththeperiodicimages.Figure 5-1 Dshowsthatthevaluepredictedfromthetheoryforthecontributionbeyondthatofadilutesuspensionintheabsenceofwallsismuchsmallerthanthatpredictedbythesimulations. ThelackofagreementshowninFigure 5-1 DclearlyarisesfromtheapproximationsmadewithinthetheoryfortheGreen'sfunctions.Thetheorypredictsanadditionalcontributiontothestressfromthepresenceoftheinhomogeneousdistribution,butdoesnotfullyaccountforthereductioninthestressletcoecientduetodirectinteractionswiththeboundingwallsandothermechanisms.ImprovedapproximationsfortheevaluationoftheGreen'sfunctioncouldimprovetheagreementwiththesimulationresults. Thatthepresenceofwallscanaecttherheologyofsuspensionsystemsisawellknownresult( HappelandBrenner 1965 ).ForthespeciccaseofBrownianrods, SchiekandShaqfeh ( 1995 )calculatedthestressforaboundsuspensionatlowvaluesofPeusinganonlocaltheoryandaccountingfortheexcludedvolume,butnothydrodynamicinteractionswhichaltertheprobabilitydistribution.Theincreasingconnementoftherodsreducedtheeectiveviscosityfurtherascomparedtoanunboundedsuspension.TheanalysispresentedinEq. 5{1 5{3 isbaseduponalocalapproximation,butdoesaccountforthedependenceofthestressuponthepositionoftherod.Fortheconditionsstudiedhere,thedierencebetweentheboundedandunboundedresultsisabout2%atbest.Adirectinvestigationofthedependenceofthestressuponparticleposition 89

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TheparticleshearstressesPxy(ry)forarigidberwith=3,18,pz=0andA=10inagapofH=6L,normalizedbythetheoreticalvalueswiththesamecongurationsinunboundedow,asafunctionofadistancefromawall,ry=L.Calculationsarefromthesimulationwithsimpleshearowwithaforceandtorquefreeber. isgiveninFigure 5-2 ,whichshowstheinstantaneousvalueoftheeectiveparticlestressascalculatedfromthesimulationsnormalizedbythedilutevaluefortwodierentorientations.Theparticlestressisonlyabout1%higherthantheunboundedvalueatry=0:5Landvanishestolessthan0:3%ofthevalueatthecenter.Onlyasmallamountofadditionalstressisfoundtoexistevenforparticlesverynearawall,consequentlythefactthattheaverageparticlestresscloselymatchesthedilutevaluesisnotsurprisingathighPewheretheparticlemigratesawayfromtheboundingwalls.Thelimitedeectofthewallsontherheologyofdilutesuspensionsofrodshasbeennotedbyothers( Attansioetal. 1972 ; GananiandPowell 1985 ; Petrie 1999 ; Mosesetal. 2001 ; Zurita-Gotoretal. 2007 ). 90

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91

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AubertandTirrell 1980 ; Aubertetal. 1980 ; Brunn 1984 ; BrunnandChi 1984 ),haveshownthatexiblepolymersinrotationalowsmigratetowardtheaxisofrotation,resultingininhomogeneousdistributions.Forrigidpolymers, AubertandTirrell ( 1980 )and Aubertetal. ( 1980 )speculatedthatasimilarmigrationwouldhappenforellipsoidsininhomogeneousvelocityelds,citingaformulapredictingmotionsofrigidellipsoids( HappelandBrenner 1965 ; Brenner 1966 ).However,quantitativeandqualitativeanalysesofthedynamicsofmigration,center-of-massdistribution,andparticlestresscontributionofrigidpolymersininhomogeneousvelocityelds,suchasrotationalows,havenotbeenmade. Arigidpolymerinonetypeofrotationalow,atorsionalow,isstudiedinChapter 6 .TheequationofmotionwhichconsiderstheinhomogeneousoweldisderivedinSection 6.2 .Predictionofthecenter-of-massdistributionsintheradialdirectionaremadeinSection 6.3 andtheparticlestresscontributionisevaluatedfromtheresultingdistributioninSection 6.4 2xx:rru+:(6{1) ThevelocityofarigidrodisapproximatedbyapplyingEq. 6{1 atitscenter-of-mass.IntegratingtheuidvelocityalongtherodusingEq. 2{7 givesthecenter-of-massvelocity 92

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SchematicdiagramsforatorsionalowinA)Cartesiancoordinatesandcylindricalcoordinates,andinviewsfromB)thexy-plane,C)andthexz-plane.Arigidrodislocatedatr=(0;R;0:5H)sothat^x=^and^y=^R. intheinhomogeneousvelocityeld, _r=1(I+pp)F+1 whichwillbeusedtopredictthevelocityofarodinatorsionalow. Figure 6-1 illustratesatorsionalow,wherealiquidbetweentwoparalleldisksofequalradiiofRmaxandseparatedbyagapofHisshearedbyrotatingthetopdiskwithangularvelocity.Incylindricalcoordinates,theaxisofrotationisinthez-direction,theunitvectors^Rand^indicatetheradialandangulardirections,respectively.TheoriginOislocatedatthecenterofbottomdiskwhichisstationary.CartesiancoordinatesarealsoshowninFigure 6-1 .Forconvenience,therodispositionedsothatthey-directioncorrespondstotheradialdirectionandthex-directiontotheowdirection. Therearenoowsintheradialandz-directions.Theowatapositionxisonlyintheangulardirection, 93

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whichisthensubstitutedintoEq. 6{2 togetthequadratictermwhichcontributestomigration, Ifthecenter-of-massofarigidrodisxedonthey-axis,suchthatr=(0;R;0:5H)asinFigure 6-1 ,theradialvelocityatagivenH,Rmax,and_max=_(Rmax)=Rmax=Hbecomes _ry=L2_max AnidenticalresulttoEq. 6{6 canbederivedfromdirectintegrationoflocalvelocities,u(s),alongarod.Foraforceandtorquefreerodatr=(0;R;0:5H),thevelocityintheradialdirectionbecomes 1 whichshowsthatEq. 6{6 isvalidaslongastheslender-bodyapproximationholds.UsingEq. 6{7 ,thedependenceofmigrationvelocityonpolymerorientation,theanglebetweenthex-directionandprojectionofpontothexz-plane,isdemonstratedinFigure 6-2 .Arigidpolymermovestowardstheaxisortheedgedependingupontheinstantaneousorientation.Sincetheorientationdistributionissymmetric,anon-Brownianroddoesnotexperienceanynetmigration.However,Brownianrotationbreaksthesymmetryto 94

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Adriftvelocity_ry,unitsofRmax=_maxL2,ofaforce-freerigidrodinatorsionalowasafunctionof.Negativevelocityindicatesamigrationtowardstheaxis. produceanaverageangleofsmallpositivevalue,whichalsocorrespondstomigrationtowardtheaxis,asinrectilinearshearowsinChapter 3 Aslongasthepositionofthecenter-of-massisnottooclosetoeithertheaxisortheedge(0
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3{2 .Withnonetmigrationintheangularandz-direction,thecenter-of-massdistributionintheradialdirectionsatises 0=1 @RnRD_RE:(6{8) Applyingaboundaryconditionofnouxattheedgeoftheplatesintheradialdirectiongives 0=nD_RE;(6{9) whichcanbeobtainedfromEq. 6{6 .ReplacingFinEq. 6{6 withEq. 3{10 ,followedbyintegratingovertheorientationdistributiongives "1+p2y@n @R+n@p2y Furthermanipulationwithsubstitutionof givesadierentialequation, @R=KI+hi where whichrelatesthemigrationduetotheinhomogeneousvelocityofatorsionalow,and whichistheanisotropicdiusivitytermwhichcausesmigrationfromlowtohighshearregions( NitscheandHinch 1997 ; SchiekandShaqfeh 1997a ). IntegrationofEq. 6{12 overRgivesacenter-of-massdistributionintheradialdirection,requiringcalculationofaverageorientationmoments.Theorientation 96

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3{20 duetosimilarapproximationsmadeforfactorizationofdistributionfunctionsinEq. 3{2 TosolveEq. 3{20 ,theequationofmotionforrotationofarigidslender-bodyintorsionalowisrequired,whichcanbeobtainedbysubstitutionofEq. 2{9 withEq. 6{4 forarodwithpositionr=(0;R;0:5H).Theresultingrotationis _p=_max Thersttermontherighthandsiderepresentstranslationoforientation:Theorientationrelativetothecenter-of-mass,owingintheangulardirection,doesnotchange.Thesecondtermcorrespondstothechangeoforientationduetoshearowsofrate_(R)inthex-directionwiththegradientinthez-direction.ThesetwotermsimplythatorientationdistributiondependsonlydependentonRandissamewiththatofsimpleshearow. AsinChapters 3 and 4 ,averageorientationmomentsofarigidrodofr=(0;R;0:5H)intorsionalowsofvariousPemaxrarealsoobtainedbyperformingthesamenumericalmethod.ResultsareshownasafunctionofR=RmaxinFigure 6-3 .Threeorientationmomentsarechosen:p2yandhpxpziareusedforintegrationofEq. 6{12 andhp2xp2ziwillbeusedinthecalculationofstress.AsPemaxrchanges,theaverageorientationmomentsshiftaccordingtothelocalPecletnumber,Per(R)=RPemaxr=Rmax. Figure 6-4 showsthecenter-of-massdistributionsintheradialdirectionintorsionalowsofvariousPemaxr,whichisobtainedfromintegrationofEq. 6{12 usingorientationmomentsfromFigure 6-3 .ThedistributionfunctionnisnormalizedsuchthatRndR=1andscaledwithRmax.AsPemaxrincreases,n(r)distributestowardstheaxis.Asinsimpleshearow,Brownianrotationbreaksthesymmetryoforientationdistributiontogivepositivevaluesofhpxpzi,whichinducemigrationtowardstheaxisaspredictedinEq. 6{6 .Comparedtoexiblepolymers( Brunn 1984 ),rigidpolymersrequirelargerPemaxrforeectivemigration.Sincethecurvatureaectorientationmomentslittleasshownin 97

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AverageorientationmomentsofarigidrodintorsionalowsatthreevaluesofPemaxrasafunctionofR=Rmax. 98

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Center-of-massdistributionsofarigidrodintorsionalowofvariousPemaxrasafunctionofR=Rmax.Insetshowsthedistributionwithoutinhomogeneousvelocityeldmigration. Figure 6-3 andKIdoesnotdependonRexplicitly,allsteadystatedistributionsatvariousRmaxarematchedifscaledwithRmax. TheanisotropicdiusivitytermhiinEq. 6{12 ,whichinducesanopposingmigrationinpressure-drivenow( NitscheandHinch 1997 ; SchiekandShaqfeh 1997a ; Parketal. 2007 ),isexpectedtogeneratemigrationawayfromtheaxissincetheshearrateincreasesintheradialdirectionfortorsionalows.However,itseectistooweaktogiveanyapparenteect.Theinsetof 6-4 showsthedistributionobtainedbyintegratingEq. 6{12 withouttheinhomogeneousvelocityeldtermKI.Inpressure-drivenow,theo-centermaximumduetocompetitionbetweentwooppositemigrationsonlyappearsnearthecenterlinewherehydrodynamicliftdecayswithr2y( Saintillanetal. 2006a ; Parketal. 2007 ).However,asshowninEq. 6{7 themigrationeectduetotheinhomogeneousvelocityelddoesnotdecayandovercomesthecompetition. 99

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Pooleetal. 2007 ; Pipeetal. 2008 )andmayhaverelavancetostudiesondynamicsinmicrochannelswithcurvature,suchasmicrobends( DavisonandSharp 2008 ; Gulatietal. 2008 ). InChapter 6 ,hydrodynamicinteractionswiththewallarenotincluded.However,thebrokensymmetryoforientationduetoBrownianmotionshouldbecoupledtocausemigration.Therefore,hydrodynamicinteractionswillinduceonlymigrationinthez-directionandnotaectthemigrationinradialdirectionindiluteconcentration. 6.3 .Equation 5{3 ,whichrepresentsastressletSofarigidslender-body,isrewrittenusingEq. 3{10 forashearowinthex-directiongeneratingstressinthez-direction; 3I:(6{16) AveragingSovertheexpectedcongurationinthesystemofvolumeVgivesthemeanparticlestress, Factorizingthedistributionfunction=nandtakingtheaverageovertheorientationdistributiongives 100

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Rmaxhpxpzppi1 3Ihpxpzi: ExtractingtheshearstresscomponentxzfromEq. 6{18 givesparticlestresscontributions whichisfactorizedintoshearstresscontribution 6-3 andn(R)obtainedfromEq. 6{12 areusedfortheintegrationinEq. 6{20 .AsinFigure 6-4 ,usingtheentiretermsofEq. 6{12 givesaninhomogeneousdistributionduetotheinhomogeneousvelocityelds,whileneglectingKIgivesanalmostuniformdistribution.Thestresscontributionsarealsocalculatedusingthebothdistributionsforcomparison. Figure 6-5 showstheresultingparticlestresscontributions 6{20 .As Brickeretal. 2008 ). 6 .Therelationbetweenmigrationdirectionandpolymerorientationisclaried,showingthatnet 101

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Particlestresscontributions migrationrequiresbreakingofthesymmetryoftheorientationdistributioninthesheardirection.Thecenter-of-massdistributionintheradialdirectionisalsopredictedfromtheequationofmotion.RigidpolymersaredistributedtowardtheaxisandmorestronglyathigherPemaxr.Thisdistributionaectstheparticlestresscontributionandresultsinanenhancedshearthinning.Thistheoryisbeingimprovedforthetransientdistributionandstressnow. 102

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Rigidrod-likepolymerssuspendedinNewtonianuidswerestudiedtheoreticallyaswellasnumerically.Applyingamethodofsimulationadaptedfrompreviouswork( Harlenetal. 1999 ; ButlerandShaqfeh 2002 2005 ),itwasfoundthatapairofsedimentingrods,whichhaveacertaininitialconguration,haveorbitsintheirtrajectoriesduetohydrodynamicinteractions.Theshort-timediusivityofrigidrodsindiluteandconcentratedsuspensions(ndL3=0150)werealsocalculatedbythesimulationmethodtodemonstratetheconcentration-dependentreductionduetohydrodynamicinteractions. Akinetictheorywasdevelopedforthemigrationofarigidrodinrectilinearowsnearsolidboundaries.ThecouplingofabrokensymmetryoftheorientationdistributionduetoBrownianmotionandthetransversemotioninducedbyhydrodynamicinteractionswithawallcausesnetmigrationnormaltothewalltogiveaninhomogeneousdistribution. Threemechanismswereidentiedforthemigrationofarigidrod.Amechanismduetoshearowisamajorcontributiontomigrationawayfromawallandhasananalogywiththemigrationmechanismforexiblepolymers.ThesecondmechanismduetoBrowniantorqueisnewlyfound,isuniquetorigidrods,andisaweakercontribution.Theanisotropicdiusivitymechanism,whichhasbeenalsoidentiedby NitscheandHinch ( 1997 )and SchiekandShaqfeh ( 1997a ),givestheoppositemigration.Thesemechanismarecombinedtoaectthedistributionsinthechannel. Forsimpleshearowsbetweentwowalls,thekinetictheorypredictedthatthecenter-of-masswoulddistributetowardsthecenterlineofthechannelandthedepletionlayerisextendedintothebulkregionathighershearrates.Forpressure-drivenows,thepredictionshowedano-centermaximumforthecenter-of-massdistribution.Thetheoryalsopredictedthatfortransientbehavior,aninitiallyuniformdistributionwouldshowapeaknearawallatshorttimes.Astimeproceeds,thepeakwouldgraduallymovetowardsthecenterandnallyvanishbeforereachingthesteadystatedistribution. 103

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Theparticlestressofarigidrodinshearowsbetweentwoparallelwallswascalculatedfromthesimulationaswellasthetheoreticalpredictionatthesamelevelofapproximationusedinthekinetictheory.Theinhomogeneousdistributioninducedbythehydrodynamicinteractionswiththewallmadethevaluesoftheparticlestressveryclosetothoseofanunboundedsystemsincetheadditionalwallcontributiontotheparticlestressisnegligiblefarfromawall.ThevaluesoftheadditionalwallcontributionevaluatedfromthesimulationwerelargerthanthosefromthetheoreticalpredictionowingtotheapproximateevaluationoftheGreen'sfunctionandcanlikelybeimprovedupon. Atheoreticalanalysisfordynamicsofarigidrodinatorsionalowwasdeveloped.ThecombinedeectsoftheaverageorientationduetoBrownianuctuationandtheinhomogeneousvelocityeldgenerateanetmigrationtowardtheaxis.Thecenter-of-massdistributionintheradialdirectionwasalsopredictedtoshowthatrigidrodsaredistributedmorestronglytowardtheaxisathigherPemaxr.Theparticlestressevaluatedfromtheinhomogeneousdistributionresultedinanenhancedshearthinningascomparedtoauniformdistribution. 104

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SundararajakumarandKoch ( 1997 )and PryamitsynandGanesan ( 2008 ),examinedtherheologyofunboundedshearow.Thenetmigrationcausedbyhydrodynamicinteractionwiththewallsinanon-dilutesystemwouldbeexpectedtoalterthemicrostructure,whichisdirectlyrelatedtotherheologicalproperties( ShaqfehandFredrickson 1990 ; DhontandBriels 2003 ).Theresultsmaybeusefulinexplainingtheshearthinningbehaviorobservedforsemi-dilutesuspensionsofbersathighPe( GananiandPowell 1985 ; ChaoucheandKoch 2001 ; Brickeretal. 2008 ).ThesimulationmethodisalsobeingappliedtothestudiesonowsofDNAfragmentsinnanochannelsunderelectrophoresisandtheseparationofsinglewallcarbonnanotubesinamicrochannelusingdielectrophoresis. 105

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MobilitycomponentsofthegrandmobilitymatrixfMinEq. 2{25 areshowninAppendix A .TheyareevaluatedforinnitesuspensionsusingthesolutionsforperiodicStokeslets.ForisolatedparticlesusingaGreen'sfunctionforfreespace,theintegratedGreen'sfunctionsinself-terms(or),whichrelatesinteractionsbetweenselfimages,are0. Amobilitycomponent, relatestranslationalmotionofarigidrod,_r,tothetotalforceactingonitselfanditsself-periodicimages,F.Itisamatrixofdimension33. isa33matrixandrelates_rtoF. areboth33tensorsandrelate_rtoTandT,respectively. are31vectorswhichrelate_rtoS. Therotationofarigidrodisdeterminedusing33tensors, 106

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and31vectors, Theprojectiontensor(Ipp)isincorporatedafterdecompositionorinversionoffMtoavoiditssingularity( ButlerandShaqfeh 2005 ; Saintillanetal. 2006c ). Theimposedratesofstrainarerelatedtoforcesby13vectors, L4pG(2);andMET=6 L4pG(2);(A{10) andthescalarquantities, L4pG(2)p;andMES=6 L4pG(2)p:(A{11) 107

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Blake 1971 )is, whereZrelatesthepointforceandreecteddisturbance.Inmoredetail, wherethersttermcancelsnormalowsonthewall,andthesecondterm,PD(potentialdipole),andthethirdterm,Sd(Stokesdipole),canceltangentialowsonthewall.Linearizationaroundthecenter-of-massgives, @x[Z(x;r)]x=r+sp@ @x[Z(r;x)]x=r=3 32ry(I+^y^y)+3s IntegratingtheresultedlinearizedformtwiceoverthelengthofarodandincludingtheeectfromthetwoboundingwallswithagapofHusingsuperpositionunderan 108

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321 (ryH)(I+^y^y)G(1a)=L4 (ryH)2#[(I+^y^y)py+^ypp^y]G(1b)=L4 (ryH)2#[(I+^y^y)py^yp+p^y]G(2)=0: 3{7 overorientationis Z(I+pp)1 @ Z(I+pp)1 Z(I+pp)1 @r+@ @rndp=kBT @n @rZ(I+pp)dpnkBT Z(I+pp)@ @rdp=kBT @n @rZ(I+pp)dpnkBT @n @rZ(I+pp)dpZ@ @r(I+pp)dp=kBT (I+hppi)@n @rnkBT @hppi ofwhichthey-componentisusedinEq. 3{11 TheaverageoftheBrowniantorquecontributioninEq. 3{7 overorientationbecomes L4ZG(1b)rplndp=12nkBT L4ZG(1b)rpdp=12nkBT L4ZrpG(1b)dp+12nkBT L4ZrpG(1b)dp; where @p:(B{7) 109

PAGE 110

L4ZrpG(1b)dp=24nkBT L4ZpG(1b)dp=3nkBT (ryH)2#2hppyi+p2y1^y; whereStokes'sspecialtheoremonasphericalsurfaceisused.Thesecondtermbecomes 12nkBT L4ZrpG(1b)dp=3nkBT (ryH)2#2hppyip2y1^y:(B{9) AddingEq. B{8 andEq. B{9 completesEq. B{6 : (ryH)2#2hppyi+p2y1^y=72nkBT L2(ry)2hppyi+p2y1^y; ofwhichthey-componentisalsousedinEq. 3{11 Likewise,theBrowniantorquecontributiontotheparticlestressin( 5{13 )isderivedusingStokes'sspecialtheoremonasphericalsurface: VZZn[prp(ln)]dpdr=ndkBTZprpdp=ndkBTZrp(p)dpZrppdp=ndkBT2ZppdpZ(Ipp)=ndkBT(3hppiI): 3{20 throughBrowniandynamicssimulation.Sinceorientationofarodisassumedtobeindependentofposition,rotationofasingle 110

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_p=_py(^xpxp)+121 whichisderivedfromEq. 2{9 .BrowniantorqueisgeneratedbyarandomvectorofunitvarianceWwhichsatisestheuctuation-dissipationtheoreminthediscretetimestep, Equation( B{12 )isthenmadedimensionlessusingthecharacteristictimescale _p=py(^xpxp)+2 wherethethirdtermontherighthandsideisacorrectiontermfornumericalintegrationbyamodiedEulermethod.Theorientationsattime Specically,theaveragingisperformedover1000simulationtimeincrementsof 3{20 issolvedatthelowPerbyassuming, Rearrangementofthesubstitutedexpressiongivesequationsatthe0thorder, (Ipp):@20 111

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(Ipp):@21 @p[py(^xpxp)0];(B{18) thesecondorder, (Ipp):@22 @p[py(^xpxp)1];(B{19) ofPerequations,andsoon.SolutionsofeachequationdeterminethecoecientsoftheseriestogiveasymptoticexpressionfororientationdistributionfunctionatlowPerlimit, 4+Per Averagingtheorientationsoverusingintegrationoverasphericalsurface( Birdetal. 1987 )givestheensembleaverageorientationmomentsatthelimitoflowPer,suchas, 3Pe2r 3{27 issolvednumericallyusingtheCrank-Nicolsonscheme,anitedierencemethod.Figure B-1 illustratesmeshesusedinthisscheme.AchannelofheightHisdividedintoatotalnumberofY+2pointsandsizeofy.Asolutionforcenter-of-massatthem-thtimestepandthei-thpositionstepisdenotedasnmi,whichisequivalentton(iy)att=mt.Boundaryvaluesareindicatedasi=0andi=Y+1.TheCrank-Nicolsonschemeisanimplicitmethod,whichisstableatallratiosoft=y.Itisalsoamid-pointmethodintimestep,thereforeitserrorordersareO(t2)andO(y2).ManipulationofEq. 3{27 givesexpressionsrelatingnitemeshes.Mid-point,Rmi,markedasingure B-1 ,issolvedexplicitlyusingvaluesatm-thtimestep, tnmiui;3nmi+1;fori=1;;Y;(B{23) 112

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SchematicdiagramofnitemeshesforsolvingEq. 3{27 usingCrank-Nicolsonscheme. where tui;3=hDi where (ryH)3#:(B{25) Mid-pointsadjacenttoboundariesaresolvedusingno-uxthroughawallboundarycondition.Adiscretizedexpressionforaboundaryconditionaty=0becomes tnm1u1;3nm2;(B{26) where hDiy(ry)hDi t;andu1;3=hDi 113

PAGE 114

tnmY;(B{28) where hDiy(ry)hDi t: Finally,center-of-massdistribution,evolvedatnewtimestep,m+1,areobtainedimplicitlybysolvingthematrixequation, whichissolvedbyThomasalgorithm,simpliedformofGaussianeliminationforatridiagonalmatrix.ThesolutionsateachtimesteparealsonormalizedwithRndryasinsteadystatedistributions. 114

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4.3.1 .WhileonlytheorientationdependentmigrationduetoshearowwithoutforceandtorqueisdemonstratedinFigure 4-3 ,theeectoftorqueunderconditionsofnoowandforceisinvestigatedhereinSection C.1 AtheoreticalpredictionoftheBrowniantorquecontributiontothetransversemotioncanbemadeusinganexpressionextractedfromEq. 3{7 : _ry=24 Thecorrespondingmotionisextractedfromthesimulation(Eq. 4{4 )with_andFsettozero, Thetransversevelocities_ryarecalculatedboththeoreticallyusingEq. C{1 andnumericallyusingEq. C{2 asafunctionoforientationwithaconstanttorqueof whereatorqueofthepositivevaluerotatestheorientationoftherodtowardsparalleltothey-direction,whilethenegativevaluerotatestherodperpendicularlytothey-direction.ThedirectionsoftheappliedtorquesareillustratedinFigure C-1 C.TheresultsareplottedasfunctionsofaswellasryinFigure C-1 SimilarlytothecomparisonoftheoreticalandnumericalcalculationsofshearcontributionsinSection 4.3.1 ,simulationresultsandtheoreticalpredictionsarein 115

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Transversevelocities_rywithunitsofkBT Lofarigidrodwithpz=0andA=10inagapofH=6LduetothetorquecontributionA)asafunctionofanglewithry=1:0LandB)asafunctionofdistancefromawallry=Lwith=3.SimulationconditionsarethesameasthoseinFigure 4-2 except_=0andF=0.C)Directionsofappliedtorquesandresultingmigrationsareillustrated. qualitativeagreementforboththeangleandpositiondependenciesandhavequantitativedierencesinthatsimulationunderestimatesthetransversemotion(Figure C-1 ). Thetransversemotionsresultingfromthetorqueappliedwiththepositivevaluearealwaystowardthewall,whilethosefromthetorquewiththenegativevalueareawayfromthewall.TheangledependenceofmigrationinFigure C-1 alsoshowsthatthereisnonetmigrationforsymmetricorientationdistributions.AswiththeshearcontributioninSection 4.3.1 ,theBrownianmotionalsocontributestothenetmigrationifthedistributionisassymetric.TheBrowniantorqueisproportionaltothenegativegradientofthe 116

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3{10 .Sincetheorientationofrigidslender-bodytendstobealignedtotheshearowdirection,Browniantorqueundershearowhasanaveragenegativevaluesothatitsdirectionbecomesdominantinoppositiontotheowdirection. AsdiscussedinSection 3.3.3 ,duetotheresistanceofarigidrodtobeingstretchedbyshearow,shearcontributioninducesmigrationawayfromthewall.Likewise,theresistancetobeingalignedbyshearow,theBrowniancontributioninducesmigrationawayfromthewall. 4{10 Notethathydrodynamicinteractionswiththewallarenotincludedinthiscomparison.Startingfromthesameinitialcongurationofry=0:3L,=0:6,andpz=0,threetrajectoriesfromeachalgorithmarecomparedinFigure C-2 .Theresultshavenosignicantdierences:alltrajectoriesshowsuddenjumpsofryto0:6L(consideringbumpywallandsphericaltipofarodwithd=0:1L)aftercontactwithawall.ThesimilartrajectoriesshowthattherepulsiveforceusedinthesimulationsofChapter 4 ensuresthepole-vaultmotionofarod,nopenetrationthroughawall,andnounrealistically 117

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TimeevolutionsofA)center-of-massdistancefromawallry=LandB)orientationpyofanon-BrownianrigidrodofA=10undershearof_nearawall.Onsetillustratesthepole-vaultmotionoftherod. largedisplacementfromawall.Simulationswerealsoperformedatsmallertthan0:001_1toconrmtheconsistencyoftusedforthesimulationsinChapter 4 Theeectofhydrodynamicinteractionsonthepole-vaultmotionisalsoinvestigatedbyaddinghydrodynamicinteractionstothesimulationofanon-Brownianrodwithrepulsiveforces.Atthestartofrotation,ryisgraduallyreducedduetothecouplingofshearandhydrodynamicinteractionswithawall,whichcanbeexplainedbytherelationbetweentheangleandtransversemotiondemonstratedinEq. 3{9 andFigure 4-3 .However,thejumpofryduringthepole-vaultrotationisdelayedascomparedtocaseswithouthydrodynamicinteractions. ThetimeevolutionoforientationinFigure C-2 Bshowsthatrotationbeforethewallcontactisretardedbyhydrodynamicinteractions.Afterthejump,ryiselevatedjustalittlebitduetothecoupling.Thesesimulationresultsshowsthatrotationofanon-Brownianrigidrodundershearownearawallisinuencedbyhydrodynamicinteractions.However,theresultsfromsimulationsofaBrownianrodinChapter 4 donotshowanysignicantdierencefromthetheoreticalpredictionwhichignoredtheinuenceofhydrodynamicinteractionsonrotation.InChapter 3 andChapter 4 ,thetimescalefor 118

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C-2 .Therefore,approximationthatrotationisnotinuencedbyhydrodynamicinteractionswithawall,madeforthetheoryinChapter 3 doesnotseverelyimpactpredictionsofcenter-of-massdistributionathighPe. 119

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Thedispersionofswarmsofparticlesoccursinnaturalprocessesofsedimentationofsiltinriversandonthecontinentalshelf.Theprocesshasimportforthedeliveryofinjectedparticles,suchascatalysts,drugsandinoculatedcells,intoabulkmedium.Extensivereviewsofthestudiesregardingthedeformationofdropshavebeenconductedby Stone ( 1994 )and Machuetal. ( 2001 ).Thedeformationofswarmsofparticles,whichisanalogoustothatofliquiddrops,hasbeenfocusedonsince Adachietal. ( 1978 )reportedonthistopicforthersttime.Thestudiesonparticledrops,whichareassumedtobespherical,havesuggestedthatthebreakupisduetoinertiaoraninstabilityinducedbytheuctuationoftheparticlevelocitiesandpositions( NitscheandBatchelor 1997 ; SchaingerandMachu 1999 ; Machuetal. 2001 ; Bosseetal. 2005 ; Metzgeretal. 2007 ).AtlowReynoldsnumber,theuctuationalonecausesthebreakup. InAppendix D ,thedeformationofaswarmofnonsphericalparticles,suchasrodlikeparticles,isdemonstratedbysimulationforthersttime.ThesimulationmethodisdescribedinSection D.2 .ThetimeevolutionofthedeformationisillustratedinSection D.3 .OngoingworkissummarizedinSection D.4 120

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Harlenetal. ( 1999 )and ButlerandShaqfeh ( 2002 ).Forfar-eldinteractionsofrigidrodsinasingledrop,asolutionby RotneandPrager ( 1969 )and Yamakawa ( 1970 )isusedinsteadofoneforperiodicsystem( Hasimoto 1959 ; Beenakker 1986 ).ThenumberofGauss-Legendrequadratureforintegratingthesolutionalongtherodissetto5.Forshort-rangeinteractions,lubricationforceandrepulsiveforcehavebeenconsideredasdoneby ButlerandShaqfeh ( 2002 ),withtheadjustedparameters.Oneoftheparametersusedintheformulasforthelubricationforce,A,isreplacedwithA2=2toensurethepositivedenitemobilitymatrixforlocallydensecongurationsduringdeformation.Theparametersforrepulsiveforce,relatedtoitsmagnitudeandrange,aresettoln2A Aninitialcongurationismadebyplacingthecenter-of-massofrigidrodsinasphericalregionwitharadiusofRCbysettingthenumberoftherodsNCandA.Thecenter-of-massandorientationofrodsarerandomized.Theequationofmotionofsedimentingrodsisequivalenttothatof ButlerandShaqfeh ( 2002 ),ofwhichamoregeneralexpressionisEq. 2{28 .Itissolvedateachinstantintimeandintegratedovertimeinthesamewayas ButlerandShaqfeh ( 2002 ).Congurationsateachtimearesampledandsomeofthem,whichrepresentdistinguishingfeaturesofthedeformation,areillustratedinFigure D-1 D-1 withtimeincreasingfromAtoE.Thecharacteristictimeistc=RC=vC,wherevC=NCFg 121

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SequentialimagesofasedimentingcloudofrigidrodswithRC=5:007L,NC=750andA=10atA)tc=23:6,B)tc=119:5,C)tc=191:4,D)tc=263:3,andE)tc=335:2.Imagesinthetoprowsareviewsnormaltothedirectionofgravity;therowofimagesonthebottomareviewsfrominthedirectionofgravity. Thedrop,withaninitiallysphericalsymmetry,emanatesparticlesinaverticaltailduetothesurroundingows,whichalsogenerateatorsionalvortexinsideofthedrop.Thehydrodynamicstagnationowattheheadandhydrodynamicdispersionattenthedrop(Figure D-1 A).Theparticlelossfromthetailandthedeformationdepopulatethecenterofthedropsothatittransformsintoatoroidalshape(Figure D-1 B).Theparticleleakageandthedispersionmaketheinnerradiusofthetoruslargersothatthehydrodynamicstagnationowpenetratesthecenteroftorus.Thistorusisunstable(Figure D-1 C)sothatitshattersintosecondarydroplets(Figure D-1 D),whichthemselves 122

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D-1 E). NCLcontrolstb.Anotherinvestigationontheaverageradialvelocityofrodssuggeststhattheuctuationsinthevelocitiesandpositionsoftherods,intheabsenceofinterfacialtension,acceleratesthebreakupprocessascomparedtodropsofliquidsorsphericalparticles.Thisresultalsosuggeststhatinjectingrodlikeparticlesreducesdispersiontimeascomparedtosphericalparticles. Experimentsbeingperformedbycollaboratingresearchgrouphavequalitativelyobservedthedeformationprocedureasshownbythesimulation.Quantitativecomparisonofsimulationandtheexperimentalresultswillbeperformedsoon. 123

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Adachi,K.,Kiriyama,S.,andKoshioka,N.\Thebehaviourofaswamofparticlesmovinginaviscousuid."Chem.Engng.Sci.33(1978):115{121. Adams,W.W.,Eby,R.K.,andMcLemore,D.E.TheMaterialsScienceandEngineeringofRigid-RodPolymers.Pittsburgh,PA:MaterialsResearchSociety:Pittsburgh,1989. Agarwal,U.S.,Dutta,A.,andMashelkar,R.A.\Migrationofmacromoleculesunderow:Thephysicaloriginandengineeringimplications."Chem.Engng.Sci.49(1994):1693{1717. Aris,R.Vectors,Tensors,andtheBasicEquationofFluidMechanics.NewYork,NY:DoverPublicationsInc.,1962. Asokan,K.,Ramamohan,T.R.,andKumaran,V.\AnovelapproachtocomputingtheorientationmomentsofspheroidsinsimpleshearowatarbitraryPecletnumber."Phys.Fluid14(2002):75{84. Attansio,A.,Bernini,U.,Gallopo,P.,andSegre,G.\SignicanceofViscosityMeasurementsinMacroscopicSuspensionsofElongatedParticles."Trans.Soc.Rheol.16(1972):147{154. Aubert,J.H.,Prager,S.,andTirrell,M.\Macromoleculesinnonhomogeneousvelocitygradientelds.II."J.Chem.Phys.73(1980):4103{4112. Aubert,J.H.andTirrell,M.\Macromoleculesinnonhomogeneousvelocitygradientelds."J.Chem.Phys.72(1980):2694{2701. Ausserre,D.,Edwards,J.,Lecourtier,J.,Hervet,H.,andRondelex,F.\Hydro-dynamicthickeningofdepletionlayersincolloidalsolutions."Europhys.Lett.14(1991):33{38. Batchelor,G.K.\Slender-bodytheoryforparticlesofarbitrarycross-sectioninStokesow."J.FluidMech.44(1970):419{440. |||.\Thestressgeneratedinanon-dilutesuspensionofelongatedparticlesbypurestrainingmotion."J.FluidMech.46(1971):813{829. |||.\Browniandiusionofparticleswithhydrodynamicinteraction."J.FluidMech.74(1972):1{29. Beenakker,C.W.J.\EwaldsumoftheRotne-Pragertensor."J.Chem.Phys.85(1986):1581{1582. Bird,R.B.,Curtiss,C.F.,Armstrong,R.C.,andHassager,O.DynamicsofPolymericLiquidsVolume2.KineticTheory.NewYork,NY:JohnWiley&SonsInc.,1987. Bitsanis,I.,Davis,H.T.,andTirrell,M.\Browniandynamicsofnondilutesolutionsofrodlikepolymers.1.Lowconcentrations."Macromolecules21(1988):2824{2835. 124

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Chen,S.B.andJiang,L.\Orientationdistributioninadilutesuspensionofberssubjecttosimpleshearow."Phys.Fluids11(1999):2878{2890. Chen,S.B.andKoch,D.L.\Rheologyofdilutesuspensionsofchargedbers."Phys.Fluids8(1996):2792{2807. Claeys,I.L.andBrady,J.F.\Lubricationsingularitesofthegrandresistancetensorfortwoarbitraryparticles."Physico.Chem.Hydrodyn.11(1989):261{293. |||.\SuspensionsofprolatespheroidsinStokesFlow.Part1.Dynamicsofanitenumberofparticlesinanunboundeduid."J.FluidMech.251(1993a):411{442. |||.\SuspensionsofprolatespheroidsinStokesFlow.Part2.Statisticallyhomogeneousdispersions."J.FluidMech.251(1993b):443{477. Claire,K.andPecora,R.\Translationalandrotationaldynamicsofcollagenindilutesolution."J.Phys.Chem.B101(1997):746{753. Cobb,P.D.andButler,J.E.\SimulationsofconcentratedsuspensionsofrigidFibers:Relationshipbetweenshort-timediusivitiesandthelong-Timerotationaldiusion."J.Chem.Phys.123(2005):054908. Colbert,D.T.andSmalley,R.E.\Fullerenenanotubesformolecualrelectronics."TrendsBiotechnol.17(1999):46{50. Cox,R.G.\Themotionoflongslenderbodiesinaviscousuid.Part1.Generaltheory."J.FluidMech.44(1970):791{810. Cush,R.C.andRusso,P.S.\Self-Diusionofarodlikevirusintheisotropicphase."Macromolecules35(2002):8659{8662. Davidson,R.L.HandbookofWater-SolubleGumsandResins.McGraw-Hill,1980. Davison,S.M.andSharp,K.V.\Boundaryeectsontheelectrophoreticmotionofcylindricalparticles:Concentricallyandeccentrically-positionedparticlesinacapillary."J.ColloidInterfaceSci.303(2006):288{297. |||.\Transientsimulationsoftheelectrophoreticmotionofacylindricalparticlethrougha90corner."MicrouidNanouid4(2008):409{418. dePablo,J.J.,Ottinger,H.C.,andRabin,Y.\Hydrodynamicchangesofthedepletionlayerofdilutepolymersolutionsnearawall."AIChEJ.38(1992):273{283. Dhont,J.K.G.andBriels,W.J.\Inhomogeneoussuspensionsofrigidrodsinow."J.Chem.Phys.118(2003):1466. 126

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Fang,L.,Hu,H.,andLarson,R.\DNAcongurationandconcentrationinshearingownearaglasssurfaceinamicrochannel."J.Rheol.49(2005):127. Fixman,M.\Simulationofpolymerdynamics.I.Generaltheory."J.Chem.Phys.69(1978):1527{1537. |||.\Dynamicsofsemidilutepolymerrods-Analternativetocages."Phys.Rev.Lett.55(1985a):2429{2432. |||.\Entanglementsofsemidilutepolymerrods."Phys.Rev.Lett.54(1985b):337{339. Frenkel,D.andMaguire,J.F.\Molecular-dynamicsstudyofthedynamicalpropertiesofanassemblyofinnitelythinhard-rods."Mol.Phys.49(1983):503{541. Ganani,E.andPowell,R.L.\Suspensionsofrodlikeparticles:Literaturereviewanddatacorrelations."J.CompositeMater.19(1985):194{215. Ganatos,P.,Weinbaum,S.,andPfeer,R.\Astronginteractiontheoryforthecreepingmotionofaspherebetweenplaneparallelboundaries.Part1.Perpendicularmotion."J.FluidMech.99(1980):739{753. Grassia,P.S.,Hinch,E.J.,andNitsche,L.C.\ComputersimulationsofBrownianmotionofcomplexsystems."J.FluidMech.282(1995):373{403. Gulati,S.,Liepmann,D.,andMuller,S.J.\ElasticsecondaryowsofsemidiluteDNAsolutionsinabrupt90microbends."Phys.Rev.E78(2008):036314. Gustin,J.,Joneson,A.,Mahinfalah,M.,andStone,J.\LowvelocityimpactofcombinationKevlar/carbonbersandwichcomposites."Comp.Struct.69(2005):396{406. Happel,J.andBrenner,H.LowReynoldsNumberHydrodynamics.Prentice-Hall,1965. Harlen,O.G.,Sundararajakumar,R.R.,andKoch,D.L.\Numericalsimulationofaspheresettlingthroughasuspensionofneutrallybuoyantbers."J.FluidMech.388(1999):355{388. Hasimoto,H.\OntheperiodicfundamentalsolutionsoftheStokesequationsandtheirapplicationtoviscousowpastacubicarrayofspheres."J.FluidMech.5(1959):317{328. Hernandez-Ortiz,J.P.,dePablo,J.J.,andGraham,M.D.\Cross-stream-linemigrationinconnedowingpolymersolutions:Theoryandsimulation."Phys.Fluids18(2006):123101. 127

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JoontaekParkwasborninSeoul,RepublicofKorea,1973toJong-OhParkandWon-SunPaek.Withthehighestscoreontheentranceexam,hewasadmittedtoDaeilForeignLanguageHighSchoolfromwhichhegraduatedatthetopofhisclassin1992.JoontaekattendedSeoulNationalUniversityin1992.HereceivedaBachelorofScienceinchemicaltechnologyin1996.HecontinuedhiseducationintheKoreaAdvancedInstituteofScience&Technologyin1996.HeundertookresearchwithDr.Ho-NamChangintheareaofbiochemicalengineering.Heperformedresearchontheproductionandseparationofbacteriorhodopsin,amembraneproteinappliedtoalightsensitivebiochip,fromHalobacteria.HereceivedaMasterofScienceinchemicalengineeringin1998.JoontaekjoinedSKEngineering&ConstructionLtd.asaresearchengineertogainindustrialexperiencesaswellastofulllmilitaryduty.Heparticipatedinprocesssimulation,start-up,troubleshootinginethanoldistillationplantsandtheoverallplanningofreneries.In2000,SKECawardedhimthe\Super-Excellent"awardforoutstandingresearch.Duringthattime,healsoexperimentedwithelectricguitarplayingandwrotemusicreviewsforanon-lineCDstore.Oneofhiscompositionsistitled\PhaseTransition".Aftercompletinghismilitaryduty,JoontaekreturnedtoacademiatopursueaPhD.HeacceptedanoerfromUniversityofFloridaasaresearchassistantin2004.HejoinedDr.JasonE.Butler'sresearchgroup,whichfocusesoncomplexuids.Heperformedresearchondynamicsofrigidrodsuspensionswithhydrodynamicinteractionswhichcanbeappliedtomicrouidicdevices.HewonRayW.FahienGraduateTeachingAwardsforthebestteachingassistantin2008.HereceivedaPhDinchemicalengineeringin2009.Joontaekisreadytostepforwardintoanewworldwhereheutilizesallhislearningforhisownresearch. 134