Citation
Dynamics and Rheology of Concentrated Suspensions of Rigid Fibers

Material Information

Title:
Dynamics and Rheology of Concentrated Suspensions of Rigid Fibers
Creator:
Shaikh, Saif Asif
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (95 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Chemical Engineering
Committee Chair:
BUTLER,JASON E
Committee Co-Chair:
NARAYANAN,RANGANATHA
Committee Members:
CHAUHAN,ANUJ
MEI,RENWEI

Subjects

Subjects / Keywords:
rheology -- suspensions
Chemical Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Chemical Engineering thesis, Ph.D.

Notes

Abstract:
A combined work of experiments and numerical simulations is proposed to investigate the dynamics and rheology of highly concentrated suspensions of non-colloidal rigid rods in a Newtonian fluid. Detailed measurements of the rheology and the microstructure are made using a variety of experimental devices with different geometries and imposed flows. Standard rheology experiments (volume-controlled rheology), as well as a novel method of rheometry, are carried out to measure torques, particle pressures, and volume fractions at high concentration (pressure-controlled rheology). Another experiment has been designed and constructed to study the microstructure (spatial and orientation distribution) of a suspension of rigid rods in an oscillatory parabolic flow. Though the flow is reversible in these systems, the changes in the microstructure are irreversibile in the case of concentrated suspensions due to particle interactions. The microstructure is affected by, and has an effect, on the imposed flows; this non-linear dependency includes hydrodynamic interactions. The purpose of these experiments is to gain insight into phenomena such as apparent shear-thinning at high shear rates and demixing due to shear-induced migration. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
Local:
Adviser: BUTLER,JASON E.
Local:
Co-adviser: NARAYANAN,RANGANATHA.
Statement of Responsibility:
by Saif Asif Shaikh.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
LD1780 2017 ( lcc )

Downloads

This item has the following downloads:


Full Text

PAGE 1

DYNAMICSANDRHEOLOGYOFCONCENTRATEDSUSPENSIONSOFRIGIDFIBERSBySAIFSHAIKHADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2017

PAGE 2

c2017SaifShaikh

PAGE 3

Tomyfamily,youarethepillarsuponwhichIstandtall

PAGE 4

ACKNOWLEDGMENTSTheworkpresentedinthisdissertationisinpartialfulllmentofadual-degreeprogrambetweentheCollegeofEngineeringattheUniversityofFloridaandAix-MarseilleUniversite,Marseille,France.Iamgratefultotheinstitutionsandfundingsourcesthathavefacilitatedthisprojectandcollaboration.Tobegin,Iamindebtedtomydoctoraladvisors,Dr.JasonE.ButlerandDr.ElisabethGuazzelliwhohavegreatlysupportedmethroughthisprogram.Dr.Butlerhasingrainedthecriticalthinkingandproblemsolvingskillsrequiredtoexcelasaresearcherandathinker.ThroughDr.Guazzelli'svastexperience,Ihavedevelopedasteadyhandandakeenmind;skillsthathavemademeacompetentexperimentalscientistalongwithgrantingmethefreedomtounderstandthatfailureisanintegralpartofthescienticprocess.Together,theyhavetaughtmetoneversettleforanythinglessthanthebestversionofmywork.Mypresentandfuturesuccessesarebecauseofthetimeandefforttheyhaveputintomyprojectandme.IwouldalsoliketothankmyFrenchandAmericancommittees.Dr.RangaNarayananattheUniversityofFloridacontinuallyprovidedinsightintomyproject,andalwaysremindedmetotakeastepbackfromthedetailstoaskbiggerquestions.Hisadvicehasbeeninstrumentalinhelpingmeadheretothelargescalecontributionsofthisproject.Dr.RenweiMeiandDr.AnujChauhanmustalsobethankedforkeepinganeyeonthesmallerdetailsandforhelpingmeapproachissuesfromadifferentper-spective.SpecialmentionmustbemadeofDr.ElisabethLemaireandDr.AnkeLindnerforagreeingtobereportersformydissertation.MyscienticjourneywouldnotbecompletewithoutthemanyhoursspentlearningaboutrelevantconceptsfromDr.OlivierPouliquen,Dr.BloenMetzger,andDr.HenriLhussieratAix-MarseilleUniversite.Theirsystematicinstructionhasdeepenedmyunderstandingofmyprojectinparticularandmyeldasawhole. 4

PAGE 5

ImmensethanksmustbegiveninparticulartoScottStrednakfromUFandFrancoTapiafromAMU.Francowasinstrumentalinassistingwiththeexperimentsusingpressure-imposedrheometerandwithhelpingmewithdataanalyses.Inasimilarvein,Scotthasbeenindispensablewithhisassistancewithrunningtheshear-inducedmigrationexperimentsandhislion'sshareinthesubsequentanalyses.Ihavelearnedalotbyworkingalongsidethemandhopetorepaytheminkindinthenearfuture.MyexperienceinFrancewouldbeanentirelylonesomeanddrearyexperiencewithoutthevariouspeoplewhomadeitalife-changingone.Firstly,ImusteffusivelythankDr.LaurenceBergougnouxwhotirelesslyheldmyhandwhilenavigatingtheFrenchlanguage,bureaucracy,andcultureasawhole.MathieuSouzyandhiswifeSandrahavebeenextremelypatientwithmynegligiblefrenchandhaveprovidedmewithmanyfrenchlessonsovertheyears.AnysuccessIhadingraspingtheirlanguageisattributedinlargeparttothem.Myfrenchcolleagues,bothpastandpresent,JorisChateau,CecileClavaud,AntoineBerut,Jean-FrancoisLouf,andothershavebecomelife-longfriendseventhoughtheyarecontinentsaway.BackinGainesville,mylabmatesRyanMontes,MertArca,PhongPham,andBradenSnookhaveservedasconstantinspirationforme.Ourtalks,bothscienticandcasualhavemademeabetterscientistandamorewholesomeperson.ThisspaceisnotenoughtomentioneveryonewhohashelpedthroughtheyearsofthisprogrambutIhopetheyareawareofmygratitude.ToDeborahSpiess,thankyouforbeingbymysidethroughthisjourneyandforbeingaconstantbastiononwhomIcouldalwaysdepend.Youhavemademygraduateprogramatrulyhappyexperience.Tomyfamily,yourconstantsupportandguidancehasnotbeenoverlooked.Iwouldliketothankmyfather,AsifShaikh,whotaughtmetoalwaysstriveforexcellenceinwhateverIdo.Thankyoutomymother,Zarinwhohasforeverinstilledinmeasenseofdeterminationtemperedwithempathy,andmybrother,MikailShaikh,whowasalwaysthereforme.Ihopetomakeyouproud. 5

PAGE 6

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 13 1.1OneFiberinFlow ................................ 17 1.2SuspensionswithInteractions ......................... 20 1.3MovingTowardConcentratedSuspensions ................. 23 1.3.1RheologyofConcentratedSuspensions ............... 24 1.3.2Shear-InducedMigrationofParticles ................. 26 2RHEOLOGYOFCONCENTRATEDSUSPENSIONSOFRIGIDFIBERS .... 28 2.1Experiments ................................... 29 2.1.1ParticlesandFluid ........................... 29 2.1.2PressureImposedRheometerSetup ................. 32 2.2ResultsandDiscussion ............................ 34 2.2.1ShearandNormalViscosity ...................... 34 2.2.2AnalysisofNear-JammingLimitRheology .............. 36 2.2.3ObservationofYieldStresses ..................... 41 2.3Conclusions ................................... 41 3DYNAMICSOFCONCENTRATEDSUSPENSIONSOFRIGIDFIBERS .... 43 3.1Experiments ................................... 47 3.1.1ParticlesandFluid ........................... 48 3.1.2ExperimentalApparatus ........................ 49 3.1.3ExperimentalProcedure ........................ 51 3.2ImageAnalyses ................................. 52 3.3ResultsandDiscussion ............................ 54 3.3.1ArealFractionDistribution ....................... 55 3.3.2Dynamicsofshearinducedmigration ................. 62 3.3.3Orientationdistribution ......................... 64 3.4Conclusions ................................... 66 4CONCLUSIONS ................................... 70 APPENDIX 6

PAGE 7

ARIGIDITYTESTSFORNON-SPHERICALPARTICLES ............. 74 BCALCULATIONSANDCALIBRATIONSOFPRESSURE-IMPOSEDRHEOL-OGYDATA ...................................... 77 B.1Height,VolumeFraction,andNumberDensity ................ 78 B.2TorqueandStressCalculations ........................ 78 B.2.1GeneralizedNewtonianuidmodel .................. 81 B.2.2GeneralStressModel .......................... 82 B.3NormalForceMeasurements ......................... 83 CUNIFYINGRHEOLOGYOFDENSESUSPENSIONSANDGRANULARME-DIA .......................................... 86 REFERENCES ....................................... 88 BIOGRAPHICALSKETCH ................................ 97 7

PAGE 8

LISTOFTABLES Table page 2-1Propertiesofeachbatchofbers.DatashownincludesthemeanvalueandstandarddeviationoftheaspectratioA,berlengthL,andberdiameterd.ValuesofthedimensionlessnumberSp,characterisingtherelativestrengthsoftheviscousandelasticforces,arealsoreported. ................ 30 3-1Concentration,nbulk,expressedvolumefraction,,foraspectratio,A=11.3and22.6. ....................................... 51 8

PAGE 9

LISTOFFIGURES Figure page 1-1MeasurementsandpredictionsofthenormalstressdifferencesN1andN2insuspensionsofrigidbersoverarangeofaspectratiosAandconnements,H. ........................................... 16 1-2Arigidberinashearowshowingthedentionofitsorientation,includingJefferyorbitsforasingleberinow. ........................ 19 1-3RegimesofconcentrationsofsuspensionsofrigidrodsassuggestedbyDoiandEdwards. ..................................... 20 1-4Schematicofthevelocitydisturbancecausedbyasinglerodonapointx,intheuidandtheeffectofthedisturbancevelocityonthemotionofaneigh-boringrod. ..................................... 21 1-5Illustrativeexampleoftheeffectofhydrodynamicinteractionsontheorienta-tiondistributionofaconcentratedsuspensionofrigidrods. ........... 25 2-1Experimentalsetupformeasuringrheologicalparametersnearthejamminglimit,microscopeimageofthebers,andzoomed-inimageoftheporoustop-plate. ......................................... 30 2-2Examplerawdataobtainedfrompressureimposedrheometerforaspectra-tio,A=11.3 ...................................... 34 2-3Shear(=f_)andnormal(P=f_)viscosities,aswellasthecorrectedvaluesofshearandnormalviscositiesaftertheremovaloftheyieldstressvalues,asafunctionofvolumefraction,. ......................... 35 2-4Shearstress()andparticlepressure(P)versusshearrate,_,alongwiththecorrespondingyieldvalues,0andP0,forthebersuspensionofbatch(II)atdifferent. ....................................... 36 2-5Comparisonsofshearviscosity(s)vs.volumefraction()andnormalvis-cosity(n)todimensionlessshearrateJ. ..................... 37 2-6Criticalvaluesofmaximumpackingfraction,m()andfrictioncoefcient,s()atthejammingpointversusberaspectratio,A,togetherwiththedata( F )obtainedbyBoyeretal.forsuspensionsofspheres(A=1). ........ 38 2-7Divergenceofshear(s)andnormal(n)viscosityasafunctionofthevolumefractionscaledwiththemaximumpackingfraction,m. .............. 39 3-1SamplemicroscopephotographforPMMAcoreberopticbers. ........ 48 9

PAGE 10

3-2Experimentalsetuptostudymigrationofsuspendedparticlesinpipe-ows,similartothesetupusedtostudythemigrationofsphericalparticlesperformedbySnooketal.. .................................... 50 3-3Exampleschematicshowingthevariousimageprocessingtechniquesforasampleimageofbulkconcentration,nbulk=0.84. ................. 53 3-4Setofprocessedimagesthatqualitativelyillustrateshearinducedmigrationinconcentratedbersuspensions.Theberconcentrationisnbulk=0.84,thestrainamplitude0=15,bershaveanaspectratioA11. ........... 56 3-5Localnumberdensity,4A(r)=,plottedasafunctionoftheradialpositionr=R,forinitialbulkconcentrationnbulk=0.84,0=15,andA=11.3. ...... 57 3-6Thedependenceofthestrainamplitudeonmigrationisshownbylookingatthelocalnumberdensity,4A(r)=,forexperimentswiththesameberas-pectratioA11,nbulk=0.84,butforadifferentstrainamplitude,0=6ascomparedtoFigure3-5where0=15. ....................... 58 3-7Forabulkinitialconcentrationnbulk=0.84,0=15,andaspectratioA=22.6,thedependenceoftheberaspectratiobycomparinglocalnumberdensityascomparedtoFigure3-5foridenticalconditionsbutforbershavinganas-pectratioA=11.3. .................................. 59 3-8ForA11and0=15,nosignicantmigrationwasseenfornbulk=0.5andnbulk=3. ....................................... 60 3-9Extentofmigration(nr=0)]TJ /F3 11.9552 Tf 12.484 0 Td[(nr=1)=nbulk,asafunctionofthebulkparticlecon-centrationnbulk. .................................... 61 3-10Dynamicsofshearinducedmigrationofbersofaspectratio,A=11.3andA=22.6. .......................................... 63 3-11Denitionoftheorientationofasingleberinshearowdescribedbythean-glesandz. ..................................... 64 3-12Sampleorientationdistribution,forA11,atconcentrationnbulk=0.84,and0=15showingthepreferentialalignmentofthebersinthevorticitydirec-tion. .......................................... 65 3-13ForA11atbulkinitialconcentrationnbulk=0.84,and0=15,theprobabil-ityP()ofndingaparticleatanangleshowsapreferentialalignmentfortheowdirection,where=90. .......................... 66 3-14Sampleorientationdistribution,forA11,atconcentrationnbulk=0.84,and0=15showingthepreferentialalignmentofthebersintheowdirection,andthedistributionoftheowalignmentacrossthetube. ............ 67 10

PAGE 11

B-1Schematicoftherheometercellshowingtheporoustopplateandthepressure-imposedfeedbackloop. ............................... 79 B-2Schematicshowingforcesexertedbytherodsonthetopplate. ......... 84 11

PAGE 12

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDYNAMICSANDRHEOLOGYOFCONCENTRATEDSUSPENSIONSOFRIGIDFIBERSBySaifShaikhDecember2017Chair:JasonE.ButlerMajor:ChemicalEngineeringAcombinedworkofexperimentsanddataanalysesisproposedtoinvestigatethedynamicsandrheologyofhighlyconcentratedsuspensionsofnon-colloidalrigidrodsinaNewtonianuid.Detailedmeasurementsoftherheologyandthemicrostructurearemadeusingavarietyofexperimentaldeviceswithdifferentgeometriesandimposedows.Standardrheologyexperiments(volume-controlledrheology),aswellasanovelmethodofrheometry,arecarriedouttomeasuretorques,particlepressures,andvolumefractionsathighconcentration(pressure-controlledrheology).Anotherexperimenthasbeendesignedandconstructedtostudythemicrostructure(spatialandorientationdistribution)ofasuspensionofrigidrodsinanoscillatoryparabolicow.Thoughtheowisreversibleinthesesystems,thechangesinthemicrostructureareirreversibileinthecaseofconcentratedsuspensionsduetoparticleinteractions.Themicrostructureisaffectedby,andhasaneffect,ontheimposedows;thisnon-lineardependencyincludeshydrodynamicinteractions.Thepurposeoftheseexperimentsistogaininsightintophenomenasuchasapparentshear-thinningathighshearratesanddemixingduetoshear-inducedmigration. 12

PAGE 13

CHAPTER1INTRODUCTIONSuspensionsareaclassofcomplexuidinwhichinsolubleparticles,eithersolidorliquid,aredispersedinaliquidphase[ 1 ].Highlyconcentratedparticle-liquidsuspen-sionsareubiquitousinnatureandinindustry.Innature,lavaandsilt-ladenriversaremacroscaleexamplesofsuspensions.Atsmallerscales,bloodisanexampleofasus-pensionwheredisc-likeplateletsaresuspendedinplasma.Elongatedparticlescanbefoundinwidevarietyofapplications.Forexample,theyareaddedtoconcreteslurriestoreinforceitsstrengthandenhanceitsperformance[ 2 ],rheologicalpropertiesofdrillinguidsarealteredbytheadditionofrod-likeparticles[ 3 4 ],andtheproductionofpaperfromwoodbersisanimportantapplicationofnon-sphericalparticlesuspensions.Frompharmaceuticalstooilreneries,paper-mills,manufacturing,andwastedisposal,theindustrialmanifestationsandapplicationsofsuspensionsarecommonplace.Fromanengineeringstandpoint,betterpredictionsofthedynamicsofparticlesuspensionshelpindesigningpumps,piping,mixers,andotherowandprocessequipment,withthegoalofoptimizingperformanceandenergyconsumptionwhilealsoreducingthecapitalandoperatingcosts.Asastartingpoint,rheologicalpropertiesofsuspensionsareneededasaninputforeventhemostbasicmodelsoftheirmacro-scopicow,andmodernrheometershavemadeitpossibletoquicklygeneratealargeamountofdataoncesampleshavebeenprepared.Yet,accuratelyinterpretingandsuccessfullyutilizingthedatafromrheometersremainsasignicantproblemforawideclassofsuspensions.Asanexampleofthedifculties,considertheevaluationoftheshearrheologyofnon-colloidalspheressuspendedinaNewtonianuid,wherethedensityoftheuid Author'snote:Thisdissertationisinpartialfulllmentofadual-degreedoctoralprogrambetweentheCollegeofEngineeringattheUniversityofFlorida,Gainesville,FL.USAandAix-MarseilleUniversite,Marseille,France. 13

PAGE 14

matchesthedensityoftheparticles.AssummarizedinhisBinghamawardlecture,Acrivos[ 5 ]foundlargediscrepanciesreportedintheliteraturefortheviscosityathighvaluesoftheparticleconcentration.Atleastoneclearissuewasidentiedasasourceforthediscrepencies:forasuspensionundershearinaCouettegeometry,theparticleswerefoundtomigrateoverlongtimesfromthegapbetweentheshearingsurfacestotheregionbelowthebobwheretheshearratevanishes[ 6 ].Asaresultofthemigration,theconcentrationinthegapislowerthanthebulk,andthetorquerequiredtorotatetheshearingsurfacesataxedratedrops.Theviscosityconsequentlyappearstobelowerthanitshould,andthevaluethatismeasuredwilldependonboththetimeofthemeasurementandthespecicdimensionsofthegeometryused.Evenatshorttimes,theviscosityappearstoincreaseduetomigrationacrossthegapoftheCouettecell[ 6 ].Themigrationisdrivenbythenon-uniformityofthesheargradientanditscouplingwiththeshear-induceddiffusionoftheparticles[ 7 8 ],accordingtoonetheory,andnormalstresses[ 9 ]inthesuspensions,accordingtoanother.Consequently,rheologicalmeasurementsofthisclassofsuspensionmustbeinterpretedwithcare,asunavoidableshear-gradientsmaycauseaninhomogeneousdistributionofparticlesanderroneousvaluesoftheviscosityasafunctionofconcentration.Similarly,characterizingtherheologyofelongatedparticles,suchasrigidberswhicharenon-colloidalandneutrallybuoyant,presentsarangeofchallenges.Littleworkhasbeendoneexaminingtheshear-inducedmigrationofrigidbersinows,thoughexperimentsbyMondyetal.[ 10 ]haveveriedthatmigrationoccursinCouettecellsatsufcientlyhighconcentrations.Forsuspensionsofrods,additionalcomplexitiesariseduetotheinteractionoftherodswiththeboundaries.Forexample,Figure 1-1 showsmeasurementsofthenormalstressdifferencesforrodsuspensionsatconcentrationsofn>1=L2d,wherenisthenumberdensityandthelengthanddiameteroftherodareLanddrespectively.ThenormalstressdifferencesmeasuredfromfreesurfaceowsbySnooketal.[ 11 ]wereperformedinshearingowswherethebounding 14

PAGE 15

wallseparation,H,wasmuchgreaterthantheparticlelength.Measurementsinmoreconnedgeometriesusingtraditionalrheologicalequipment,alsoshowninFigure 1-1 ,giveresultsthatcanbedifferentbylargefactors.Resultsfromsimulations[ 11 ],whichagreeatleastqualitativelywiththemeasurements,revealthatthedifferencesareduedirectlytothemicrostructure:theboundingwallspreventparticlesfromrotatingandmovingfreely,alteringthestructureand,consequently,therheology.Thedisturbanceinthestructureimposedbythewallpropagatesintotheuidforadistancemuchlargerthanexpected,andthemeasuredrheologywillbethatofthebulkmaterialonlyforverylargevaluesofH=L.Thisdissertationpresentsresultsfrominvestigationsofthedynamicsandrheologyofnon-colloidalsuspensionsofrigidrods.Multipleconstraintswereplacedonboththeuidandparticlepropertiesintheworkdiscussedhere.InadditiontohavinganaspectratioA=L=dmuchlargerthanone,thebers(orrods)arestraightandrigid.Theparticlesofinterestarelarge,sohydrodynamicforcesaredominantandBrownianuctuationssafelycanbeassumedtohavenoimpactonthedynamicsandrheologyofthesuspensions.Allofthebersinasuspensionareidentical(monodisperse)withregardtotheirsizeandshape,andtheeffectsofgravityareremovedbymatchingtheuiddensitytothedensityoftheparticles.ThesuspendinguiditselfisNewtonianandsufcientlyviscoussothatinertiaisnegligible.Theinvestigationswereaimedataddressingquestionsintroducedabove,regardingthepropercharacterizationandinterpretationoftherheologywhentheconcentrationishighandphenomenasuchasboundaryeffectsandshear-inducedmigrationbecomerelevant.Beforeintroducingthespecicinvestigationsthatwereperformedinsubsec-tions 1.3.1 and 1.3.2 ,thefollowingsectionsbrieyreviewthemotionofarodindilutesuspensionsandthenthephysicalmechanismsthatinuencethemotionofrodsinsuspensionathigherconcentrations. 15

PAGE 16

Figure1-1.MeasurementsandpredictionsofthenormalstressdifferencesN1andN2insuspensionsofrigidbersoverarangeofaspectratiosAandconnements,H.A)ThevalueofN1)]TJ /F3 11.9552 Tf 11.955 0 Td[(N2,normalizedbythestresso_ofthesuspendinguidofviscosity0shearedatarateof_,isshownfromsimulations[ 11 ](opensymbols)andexperiments(solidsymbols)ofKeshtkaretal.[ 12 ](green),Bounouaetal.[ 13 ](red),andSnooketal.[ 11 ](black)fordifferentlevelsofconnementH.B)Experimentalmeasurementsoftherstnormalstresscoefcient1=N1=(_)forbersoflowaspectratiooverarangeofconcentrationsandconnements. 16

PAGE 17

1.1OneFiberinFlowThecenterofmassxofarigidberthatisforce-freemoveswiththeoweldasevaluatedatitscenteras _x=u(x);(1)where_xisthecenterofmassmotionforimposedoweldsu(x)thatarelinearandwheretheparticleisfarfromboundaries.Additionalcorrectionstotheoweldmustbeincludedfornon-linearowsandtoaccountforboundingwalls, _x=1 LZL=2)]TJ /F4 7.9701 Tf 6.587 0 Td[(L=2u(x+sp)ds;(1)whereuisthesumoftheimposedowandanyadditionalvelocitydisturbancesintheuid,pistheunitvectoralongthebermajoraxis,andsisthepositionalongthataxis.Forinstance,theoweldforaparabolicowbetweentwoparallelwallsseparatedbydistance2H,withmaximumvelocityUmatthecenterbetweenthewallsis, u=Um H2(H2)]TJ /F3 11.9552 Tf 11.956 0 Td[(y2)x;(1)whereyistheco-ordinateinthegradientdirectionmeasuredfromthemid-planebetweenthewalls,andxisthedirectionoftheow.TheresultantmotionoftherodcanbecalculatedfromEquation 1 _x=Um H2(H2)]TJ /F7 11.9552 Tf 11.955 0 Td[((xy)2)x)]TJ /F3 11.9552 Tf 13.15 8.088 Td[(UmL2 12H2(py)x;(1)where(xy)isthecenter-of-masscoordinateoftherodinthegradientdirectiony.Thersttermisthemotionoftherodcorrespondingtothevelocityoftheuidevaluatedatthecenteroftherod,andthesecondtermcorrespondstothecorrectiontotherodvelocityduetothecurvatureoftheoweld.Inequation 1 ,theeffectsoftheboundingwallonthecenter-of-massmotionhavebeenignored,thoughmethodsexistforincorporatingthoseeffectshavebeendeveloped[ 14 15 ].AsoriginallyshownbyGanatosetal.[ 16 ],asinglerodcanmigrateacrossstreamlines,withtherateand 17

PAGE 18

directionofthemigrationdependingontheorientation.Thisisverydifferentfromasinglesphereinasimpleshearownearaboundingwall,whichsimplytranslatesparalleltothewallwithoutcrossingstreamlines.Tocalculatethevelocitiesofaparticle,whetherinthecaseofEquation 1 oranyothermotion,theorientationisgenerallyneededandcanbecalculatedfromaninitialconditionandvelocity.Foranisolatedellipsoidinashearingow,theellipsoidrotateswithanorientationvelocityrstdescribedbyJeffery[ 17 ]as _p=p+A2)]TJ /F7 11.9552 Tf 11.955 0 Td[(1 A2+1(I)]TJ /F6 11.9552 Tf 11.955 0 Td[(pp)Ep:(1)TheJefferyorbit,asderivedfromthisequation,isafamilyofclosedrotationsthatde-pendontheinitialorientationoftherod.Theorbitcanbedenedbyanorbitconstant, C=1 Atan(z))]TJ /F3 11.9552 Tf 5.48 -9.684 Td[(A2cos2(y)+sin2(y)1=2;(1)wheretheangleszandyarespeciedinFigure 1-2 .AccordingtoEquation 1 ,therodsimplyrotateswiththerateofrotation,=1 2[(ru)-222(ru)T],andafractionoftherateofextension,E=1 2[(ru+ru)T].Figure 1-2 showstherotationalmotionofanellipsoidinasimpleshearow,u=_yx,wheretheaspectratioAoftheellipsoidwassetto10.Equation 1 isnotlimitedtojustellipsoids,asoriginallyderivedbyJeffery[ 17 ].ByreplacingAwithaneffectiveaspectratio(Ae),elongatedparticleswithalargerangeofspecicshapescanbecalculated.Forexample,Bretherton[ 18 ]showedthatsettingAe=0:8Agivesanaccuratecalculationfortherotationaldynamicsacylindricalrod,andwaslaterconrmedmyMasonetal.[ 19 20 ].Thoughalloftheworkreportedinthisdissertationregardsbersthatareneutrallybuoyant,insuspensionsathighconcentrationinter-particlecontactforcescanbepresent.Hence,itisinstructivetoconsiderthemotionofanisolatedrodinaquiescentuid,whichdiffersqualitativelyfromthatofasphere.Here,thecenterofmassmotion 18

PAGE 19

Figure1-2.Arigidberinashearow.A)Theshearow_x=_(xy)x,isshown,wherezistheangleoftheberwiththevorticityaxisandyistheanglebetweenthegradientaxisyandtheprojection(grayoutline)oftheberontotheow-gradient(x)]TJ /F3 11.9552 Tf 11.955 0 Td[(y)plane.B)Jefferyorbitsforthreeinitialconditions(C=0:1,0.8,and3.0)areshownforanellipsoidofaspectratioA=10.Theellipsoidrotatesfastestwhenalignedinthegradient-vorticityplane(x==2)androtationslowswhentheorientationisneartheow-vorticityplane;therelativerateofrotationisindicatedbythespacingofthepointswhichareplottedatequivalentintervalsintime. _xisduetoaforce,F,andisgivenby _x=ln(2A) 4fL(I+pp)F;(1)wheretheuidviscosityisf.Thisistheleading-orderresultpredictedbyslenderbodytheory[ 21 22 ].Thedependenceofthemotionoftherodontheorientationisclearlyseenbylookingatthevelocityoftherodwhenitisparalleltotheforce,Uk=2Fln(2A)=4fL,versusthemotionwhentherodisorientedperpendiculartotheforce,U?=Uk=2.Alsoforrodsorientedatatanglewiththedirectionoftheappliedforce,componentsofmotionexistinthedirectionperpendiculartotheforce.Forspheres,themotionisalwaysinthesamedirectionoftheforceforzeroReynoldsnumberconditions. 19

PAGE 20

Figure1-3.RegimesofconcentrationsofsuspensionsofrigidrodsassuggestedbyDoiandEdwards.Intheconcentratedregime,nL2d1,contactsbetweenrodsbecomesignicant.AtvaluesofnL2dc>1,therodsmustaligninordertotwithinthespeciedvolume.Attheseconcentrations,therodscrystallizeintoahighlyorderedstate. 1.2SuspensionswithInteractionsAstheconcentrationinasuspensionincreasesbeyonddilute,interactionsbetweenparticlesbegintosignicantlyimpactthedynamics.Regimesofconcentrationsforsuspensionsofrods,asdenedbyDoiandEdwards[ 23 ],areshowninFigure 1-3 intermsofthenumberdensityofthesuspensionn=N=V,whereNisthenumberofbersandVisthetotalvolumeofthesuspension.ConcentratedsuspensionsarethoseforwhichnL2d1;here,Listhelengthoftherodanddisitsdiameter.AtnL2d1,thefreerotationofanyrodishinderedbysurroundingrods,givingrisetorod-rodcontactsthatsubstantiallyinuencethedynamicsandmicrostructureofthesuspension.Thesechangesinmicrostructureaffectthemacroscopicmeasurementsofthesuspension,suchasviscosity.Evenwidelyseparatedparticlescanbeaffectedbyeachother.Whenaforceisappliedtoaparticlewithinaviscousuid,thereisadisturbanceintheuidvelocityinresponsetotheforce.Thesedisturbancesaffectthemotionofotherparticlesintheuid,givingrisetonon-lineardependenciesthatareknownashydrodynamicinteractions.Tomodelmulti-bodyhydrodynamicinteractions,themoststraightforward 20

PAGE 21

Figure1-4.Schematicofthevelocitydisturbancecausedbyasinglerodonapointx.A)ThevelocitydisturbancecausedbyarodcanbecalculatedatanypointxbyintegratingtheGreensfunctionandlineforcedensityoverthelengthoftherodandindicatedinEquation 1 .B)Tocalculatethemotionofarodinthepresenceoftherod,thevelocitydisturbancecausedbyrodcanbeevaluatedateverypointalongthecenterlineofandthenintegratedtogivethemotion. caseistorstconsidertheinuenceoftheforcesofoneberonanotherasillustratedinFigure 1-4 (a).Usingtheslenderbodyhypothesis[ 21 ],i.e,foraveryhighaspectratio,therodcanbeapproximatedbyalineofpointforces,andthedisturbancevelocity,u0(x),duetoanyrod,calculatedatanypoint,xcanbecalculatedfrom u0(x)=ZL=2)]TJ /F4 7.9701 Tf 6.586 0 Td[(L=2G(x;x+sp)f(x+sp)ds;(1)whichisthesummation(integration)ofthedisturbancescreatedbyeachpointovertheentirelengthoftheber.Here,G(x;x+sp)istheGreensfunctionforStokesowevaluatedatanypointatadistancesalongtheaxisoftheber.Thelineforcedensityisgivenbyf(x+sp).Here,theGreensfunctionG(x;x+sp)isgivenby G(x;x+sp)=1 8)]TJ /F6 11.9552 Tf 6.675 -1.596 Td[(I r)]TJ /F6 11.9552 Tf 13.15 8.087 Td[(rr r3;(1)whereIistheidentitymatrix,r=(x+sp))]TJ /F6 11.9552 Tf 11.955 0 Td[(x,andr=jrj.FurthermoreasillustratedinFigure 1-4 (b),thedisturbancevelocitycanbecalcu-latedatanypointonanotherberfromEquation 1 andtheresultingmotionofber 21

PAGE 22

calculatedfromEquation 1 _x=1 LZL=2)]TJ /F4 7.9701 Tf 6.587 0 Td[(L=2ZL=2)]TJ /F4 7.9701 Tf 6.586 0 Td[(L=2G(x+sp;x+sp)f(x+sp)dsds;(1)wheresisapositionalongtherod.Considerthemotionofonlythreerods,eachacteduponbyaforce:thedistur-bancegeneratedbytheforceoneachrodresultsinadisturbanceontheothertwo.Thepresenceofthedisturbanceonanyonerodduetotheothertwoalterstheforcedistri-bution,andthisalterationoftheforcedistributionmustbeconsideredwhencalculatingthedisturbancevelocity.Thiscycleofreectingthevelocitydisturbancesandforcedistributionscontinuesendlesslyandwouldappeartobeinsolvable.Consequently,simulatingthecollectivedynamicsoftherodsrequiresanapproximation,andaveryconvenientoneisavailable.Eachreectionofaninteractionissignicantlyweakerthantheprevious,byafactorof 1 ln(2A)L r;(1)whererisameasureoftheseparationdistancebetweentherods[ 24 ].FortheMthreection,thechangetothemotionofarodwouldbe 1 ln(2A)L rM:(1)Forwidelyspacedrodsofhighaspectratio,theerrorintruncatingtheinteractionsafterM=2or3issmall.Suchanapproachiscommonlyusedwhensimulatingthecollectivemotionofsphereswherethereectionsarestronger.Forspheresofradiusa,eachreectionMoftheinteractionscontributesavelocitythatscalesasafraction, a rM;(1)oftheleadingcontributiontothevelocity[ 25 ].NotethattherapiddecayoftheinteractionsgivenbyEquation 1 isthebasisofthefrequentclaimthathydrodynamicinteractionscanbeignoredinsuspensionsof 22

PAGE 23

slenderbodies.Whiletrueforrelativelydilutesuspensions(i.e.L=rsmall)forrodsastheaspectratiogoestoinnity,theapproximationmustbeusedwithcaution.Eveninthislimitofdiluteconcentrations,uctuationsintheconcentrationthatgeneratepairsofnearbyrodscanmaketheapproximationinvalid[ 26 27 ].Althoughhydrodynamicinteractionsarecrucialtostudiesinthesemi-diluteregime,studiesinthedenserregimeofsuspensionsclaimthatlong-rangehydrodynamics,andevenshortrangelubricationinteractions,arenotnecessarytocapturethedynamicsandrheologyofbersystems.Oneoftheprimaryaimsofthisworkistotestthisidea. 1.3MovingTowardConcentratedSuspensionsAthigherconcentrations,fornL2d>1,hydrodynamicinteractionsbecomeincreasinglyinsignicantandparticlecontactsbecomedominant.AnexampleofthisisshowninFigure 1-5 .Here,thesimulationoutcomesshownincludethoseofSalahuddinetal.[ 28 ]andFanetal.[ 29 ].Thesesimulationsincorporatelong-rangehydrodynamicandlubricationinteractions,andparticlecontacts.ResultsofsimulationsbySnooketal.[ 30 ],alsoshowninFigure 1-5 indicatethattheimportanceofincludinghydrodynamicinteractionsinthecalculationsofthemicrostructurelessensasthenumberdensityandaspectratioincrease.Collisionsbetweentherodsincreasesatthehigherconcentrationsanddominatethehydrodynamicforces,whichalsobecomeweakerathighvaluesofA.Forthesemi-diluteregime(nL2d1),hydrodynamicinteractionsmustbeincludedinthesimulations.Forexample,examiningFigure 1-5 atnL2d=1showsthatthesim-ulationpredictionswithouthydrodynamicinteractionsarewellbelowtheexperimentalmeasurementandtheothersimulations.Simulationswithshort-range(lubrication),butwithoutlong-range,hydrodynamicinteractions[ 31 ]arealsoshown.Interestingly,theseresultscloselyagreewiththoseofSalahuddinetal.[ 28 ],suggestingthatthelong-rangeinteractionshaveaweakeffectontheorientationdistributionseveninthesemi-diluteregime.Overall,moreworkisneededtodenitivelyestablishtherelativeeffectsof 23

PAGE 24

thevariousinteractions(hydrodynamic,shortandlong-range,andcontacts)onthepredictionsofthemicrostructureinshearows.Theworkdescribedinthisdissertationaimstoprovideinsightintothedynamicsandrheologyofconcentratedsuspensionsofrigidrodswiththeaimofprovidingdatatohelpresolvethisquestion.Thisdocumentisdividedintotwotopics:comprehensivemeasurementsofbulkrheologicalpropertiesofaconcentratedsuspensionofrigidrodsusingacustom-builtrheometer(seeChapter 2 ),introducedinSubsection 1.3.1 ,andtheinvestigationofcollectivemotionofrigidrodsduetoanimposedoscillatorypipeow(seeChapter 3 ),introducedinSubsection 1.3.2 ,showingtherstexperimentalevidenceofshear-inducedmigrationofrodsinpipeows. 1.3.1RheologyofConcentratedSuspensionsThoughtherehavebeennumericalstudiesperformedtounderstandthedynamicsofconcentratedsuspensions,theresultshavebeeninconsistent.Thereisalsoasmallamountofexperimentalworkathighconcentrationsofbers.Itisthislackofexperiments,andthelackofconsistentcorroborationbetweenexperimentalworkandtheory,thatmotivatethisproject.Despitetheapparentsimplicityofthissystem,severalphenomenaareobservedthatcontradictexpectations.Adiscrepancyidentiedfrompublishedworksconcernsthemacroscopicrheologyofconcentratedsuspensions.Suspensionsofsemi-concentratedandconcentratedrodsfrequentlyexhibitshear-thinningbehavior,andthisbehaviorismoreprominentathigherconcentrations[ 33 34 ].However,theoreticalanalysesandnumericalsimulationssuggestthatthesteadyvalueoftheviscosityofsuspensionsshouldbeindependentoftheshearratefornon-colloidal,rigidrodssuspendedinNewtonianuids[ 35 36 37 ].Manyhypotheseshavebeenproposedtoexplaintheoriginofthisanomalousshearthinningbehaviorseenintheexperiments.TheseideasrangefromquestioningtheNewtoniannatureofthesuspendinguid[ 38 39 ]totheassertionthattheberswerenotrigidundertheimposedconditions[ 33 40 41 ].In 24

PAGE 25

Figure1-5.Thefourthordermomentoftheorientationdistribution,p2xp2y,fromexperimentalmeasurementsandsimulationsforaspectratiosofA)A16andB)A32.ExperimentsshownarethoseofStoveretal.[ 32 ](A=16:9and31.9,red);thesimulationsarethoseofSalahuddinetal.[ 28 ](blue),Fanetal.[ 29 ](green),Yamaneetal.[ 31 ](purple),andthosefromthesimulationsperformedbySnooketal.[ 30 ](black). 25

PAGE 26

addition,wehaveverylittleinformationregardingtheparticlenormalstressesinthesesuspensions.Furthermore,therehasbeenincreasinginterestinlookingattheconnectionbetweenthemicrostructureandthesolid-transitionstate(i.e.jamming).Thistransitionisrepresentedbyasharpdivergenceinvaluesofmeasuredviscosityandoccursatconcentrationswherethemobilityofthesuspendedparticlesisseverelyrestricted,i.e.thesuspensionjams.Acomprehensiveanalysisofphenomenaandmeasurementsatthisjammingtransitionislackingintheliteratureevenforspheres,andthereislittleornoworkonsuspensionsofnon-sphericalparticles. 1.3.2Shear-InducedMigrationofParticlesShearinducedmigrationisthephenomenawhereinparticlesaredrivenfromaregionofhighsheartothatofalowershear.LeightonandAcrivosrstobservedthisphenomenainexperimentsofparticulatesuspensionsinaCouetteviscometer[ 42 ].Thiswasamajorbreakthroughinunderstandingtherheologyanddynamicsofsuspensionsandledtomanyfuturestudies.Karnisetal.[ 43 ]weretherstgrouptoobservemigrationofparticlesinapipeow.Sincethenseveralexperimentshavebeenconductedbyvariousgroupstoobservemigrationinpressure-drivenowsusingvariousresonanceimagingandvelocimetrytechniques[ 10 44 45 46 47 48 49 ].AtlowReynoldsnumbers,Stokesequationsdictatethattheuidandparticlemotionislinearandreversible.However,athigherconcentrations,variousphenomenaarisefromirreversibledynamics.Shearinducedmigrationisanexampleofthesephenomena.Thereareafewhypothesesregardingtheoriginoftheseirreversibilities.Oneofthehypothesesisthatthereareparticle-particlecontacts,whichcouldexistinconditionswherethereisabreakdownofthelubricationinteractionsbetweentheparticles.Thesecontactscouldarisefromthepresenceofmicroscsopicdeformationsonthesurfaceoftheparticleswhichcausedeviationsinthebehavioroftheparticledynamicsfromidealsmoothsurfacesviz.thebreakdownofthelubricationlayer.These 26

PAGE 27

short-rangedcontactforceshavebeenshowntocausenon-reversibledeviationsinthetrajectoriesofmotion[ 50 51 52 ].Anotherhypothesisisthatthehydrodynamicmultibodyinteractions,thoughformallyreversible,arechaotic[ 53 ].Forspheres,theinitialconditionshadaninsignicanteffectonthechaoticityofthesystem,andhencecouldnotaccountfortheirreversibilitiesinthebulkbehaviorofthesuspension.Insuspensionsofspheres,smallperturbationstotheparticlemotion,whichareinevitablypresentareampliedthroughthenon-linearhydrodynamicinteractionsandgiverisetotheirreversibilitiesdespitethemathematicalreversibilityofthegoverningequations.Howeverpreviousworkdoneonspheresimpliedthatthechaoticinteractionscouldnotexplainthediffusionofspheresinshearedsuspensions[ 54 ].Theseworkshavefocusedonhighvolumefractionssuspensionsandtheresultsreportedwerethemeasurementsofthesteadyfully-developedowproles.OneofthemajorcomplicationsintheviabilityoftheabovementionedexperimentsistheneedforhighstraintoobserveanycollectivedynamicsatStokesowconditions.Toovercomethisdifculty,anoscillatoryowcanbeused.Foralargeamplitudeofoscillation,centerlinemigrationwasobserved[ 44 ].Thesestudieshavebeenlimitedtothedynamicsofsphericalparticles.Mondyetal.[ 10 ]investigatedtheshearinducedmigrationofrods,whichwastherststudyofitskindfornon-sphericalparticles.Thereisadistinctlackofexperimentalworktoobservethecollectiveirreversibledynamicsofnon-sphericalparticlesintheliterature,particularlyforshear-inducedmigration,andthisneedprimarilymotivatedamajorportionofourwork.Chapter 3 providesadetailedexperimentalstudyofshearinducedmigrationofconcentratedsuspensionsofrigidrodsinanoscillatorypressure-drivenow. 27

PAGE 28

CHAPTER2RHEOLOGYOFCONCENTRATEDSUSPENSIONSOFRIGIDFIBERSTherheologicalpropertiesofviscousNewtonianuidscontainingrigidbersremainsrelativelyunexploredascomparedtosuspensionsofsphericalparticles,andaconsensusoneventhequalitativedescriptionoftherheologyisstilllackingforconcentrationsbeyondthedilutelimit.Asoneexample,thesteadyvaluesoftheshearstressesshould,forsuspensionsofbersthatarelargerelativetocolloidalscalesandfreeofexternalbodyforces,followaNewtonianlaw[ 35 ].However,manyexperimentalstudiesndyieldstressesandanonlinearscalingoftheshearstresseswiththerateofshear,wherethesenon-Newtonianeffectsbecomemoreprominentwithincreasingconcentration[ 33 55 ].DifferentexplanationshavebeenproposedtoexplainthedeparturefromaNewtonianresponse.Thisincludesargumentsthattheberswerenotrigidundertheimposedconditions[ 33 40 ],orthatthebersarenotforce-free.Anexampleofthelatteristheassertionthatadhesiveforcescanexistbetweenthebers,eventhoughtheirsizeislargecomparedtotypicalcolloidalscales[ 56 57 58 ].Previousrheologicalstudieshavefocusedonsuspensionsatrelativelysmallvol-umefractions.Identifyingmeasurementsofrheologyforvolumefractions,,above0.1isdifcultforbersofaspectratiosA=L=d>20,whereLanddaretheberlengthanddiameter,respectively.Thelackofdataisattributable,atleastinpart,tothedif-cultyofpreparingandmeasuringtherheologyofsuspensionsathighconcentrationsforlargeaspectratios.Evenformoderateaspectratiosofaround17or18,measurementsareavailableforvolumefractionsofonlyupto=0:15or0.17[ 59 60 ];measurementsashighas=0:23weremadebyBibboetal.[ 60 ]forsmalleraspectratiosofA=9.Asaresult,therheologicalpropertiesofsuspensionsofrigidbersremainstobechar-acterisedinthelimitoflargeconcentrationswheremechanicalcontactsareexpectedtomatter[ 11 61 62 ].Likewise,thevolumefractionatwhichtheshearstressesdiverge, 28

PAGE 29

andtheowofthesuspensionceases(i.e.becomesjammed),hasnotbeendeterminedpreviously.Here,acustom-builtrheometerhasbeenusedtoexploretheshearstressesandnormalforcesinsuspensionsofnon-colloidal,rigidbersforconcentrationsexceeding=0:23.Therheometermeasuresthestressesinbothapressureandvolume-imposedconguration[ 63 64 ].Themeasurementsindicatethepresenceofyieldstressesinthetestedsuspensions,butalsoaviscousscalingwhereinthestressgrowslinearlywiththerateofshear.Theuniquerheometerdesignfacilitatesthestudyofthesehighlyconcentratedsuspensionsofbers,and,forthersttimetoourknowledge,thevolumefractionsatwhichthestressesdivergearemeasured.Thescalingofthestressesnearthisjammingtransitionarefoundtodiffersubstantiallyfromthatofasuspensionofspheres.ThesemeasurementsarereportedinSection 2.2 ,afterpresentingtheexperimentalmaterialsandtechniquesinSection 2.1 ;conclusionsaredrawninSection 2.3 2.1ExperimentsTheobjectivesoftheexperimentalworkaretocharacterizetherheologyofhighlyconcentratedsuspensionsofrigidrodsbythemeasurementofparameterslikevis-cosities,frictioncoefcients,andvolumefractions;tomaketherstmeasurementsofparticlepressuresofsuspensionsofrigidrodsathighconcentration;andtoformconstitutivelawstomodeltherheologynearthejammingtransition.Toenabletheseobjectives,experimentshavebeenperformedinamodiedrheometeroforiginaldesign.Thisrheometersetupconsistsofawide-gapannularshearcell,withamovabletopplate.Varioussupportingdevicesareinstalledfordirectmeasurements,control,andinterfacingwithacomputer. 2.1.1ParticlesandFluidFourbatchesofrod-likeparticleswereusedintheexperiments.Theywereob-tainedbyusingaspecially-designeddevicetocutlongcylindricallamentsofplastic 29

PAGE 30

Figure2-1.Experimentalsetupformeasuringrheologicalparametersnearthejamminglimit,microscopeimageofthebers,andzoomed-inimageoftheporoustop-plate.A)Sketchoftheexperimentalapparatus.SeeAppendix B foranadditionalschematicoftherheometer.B)Microscopicimagesoftheplasticbers.C)Imageofthetopplate(theinsetisablowupoftheimageshowingthenylonmesh). Table2-1.Propertiesofeachbatchofbers.DatashownincludesthemeanvalueandstandarddeviationoftheaspectratioA,berlengthL,andberdiameterd.ValuesofthedimensionlessnumberSp,characterisingtherelativestrengthsoftheviscousandelasticforces,arealsoreported. FiberlabelSymbolAL(mm)d(mm)Sp (I)14:50:85:80:10:400:01510)]TJ /F5 7.9701 Tf 6.586 0 Td[(3(II)46:30:42:50:10:400:012:410)]TJ /F5 7.9701 Tf 6.586 0 Td[(4(III)37:20:45:80:20:810:023:910)]TJ /F5 7.9701 Tf 6.586 0 Td[(4(IV)3:40:32:80:10:810:032:710)]TJ /F5 7.9701 Tf 6.586 0 Td[(5 30

PAGE 31

(PLASTINYL6.6)thatweresuppliedbyPLASTICberS.P.A.(http://www.plasticber.com).ImagesoftypicalbersfromeachbatchareshowninFigure 2-1 (b).Thelengthanddiameterofover100bersweremeasuredwithadigitalimagingsystem.Thedistribu-tionsoflengthsanddiameterswerefoundtobeapproximatelyGaussianforallaspectratios.ThemeanvalueandstandarddeviationoftheberaspectratioA=L=d,lengthL,anddiameterdareshowninTable 2-1 .Notethatbatches(II)and(III)haveverydifferentlengthsanddiameters,butroughlythesameaspectratioofA6)]TJ /F7 11.9552 Tf 11.956 0 Td[(7.TherigidbersweresuspendedinaNewtonianuidthathadamatchingdensityoff=1056kg/m3.Thesuspendinguidwasamixtureofwater(10:72wt%),TritonX-100(75:78wt%),andZincChloride(13:50wt%).Theuidviscosityoff=3Pasandthedensityweremeasuredatthesametemperature(25C)atwhichtheexperimentswereperformed.Thesuspensionswerepreparedbyaddingtheberstotheuid,wherebothquantitieswereweighed,andgentlystirring.Littletonosettlingorcreamingwasobserved.Therheologicalmeasurementswereperformedatamaximumshearrateof_3s)]TJ /F5 7.9701 Tf 6.586 0 Td[(1,ensuringthatamaximumReynoldsnumber(f_L2=f)of0.04wasachieved.Theberscanbeconsiderednon-colloidal,owingtotheirlargesize,andrigidundertheconditionsoftheexperiment.Regardingthelatter,thebucklingcriterionhasbeencharacterisedbyadimensionlessnumber,Sp=128f_A4=EYln(2A),wheretheYoung'smodulus,EY,isapproximately3000MPaforPLASTINYL6.6.ThenumberSp,oftencalledthespermnumber,isaratiooftheviscousandelasticforcesactingontheber[ 65 ].ThevaluesofSp,showninTable 2-1 forourexperiments,weremuchsmallerthanthecriticalSpermnumberof328forthecoil-stretchtransitioninacellularow[ 66 ].AmoredetailedoverviewoftestingrigidityofrodshapedparticlesisdescribedinAppendix A 31

PAGE 32

2.1.2PressureImposedRheometerSetupTheexperimentswereconductedusingacustomrheometerthatwasoriginallyconstructedbyBoyeretal.[ 63 ]andthenmodiedbyDagois-Bohyetal.[ 64 ].Thisrheometer,sketchedingure 2-1 (a),providesmeasurementsofbothshearandnormalstresses.Theshearingcellconsistsof(i)anannularcylinder(ofradiiR1=43:95mmandR2=90:28mm)whichisattachedtoabottomplatethatcanberotatedand(ii)atopcoverplatethatcanbemovedvertically.Thistopplateisporous,enablinguidtoowthroughit,butnotparticles.Theplatewasmanufacturedwithholesofsizes2)]TJ /F7 11.9552 Tf 12.375 0 Td[(5mmandthenwascoveredbya0.2mmnylonmesh(seeFigure 2-1 (c)).Theparallelbottomandtopplateshavealsobeenroughenedbypositioningregularly-spacedstripesofheightandwidth0.5mmontotheirsurfaces.Atransparentsolventtrapcoversthecell,hinderingevaporationofthesuspendinguid.Experimentswerecarriedoutintwomodes,viz.volume-controlledmodeandpressure-controlledmode.Theformerisakintoconventionalrheometry,wherethevolumeofthecellisxedforeachrun,hencethevolumefractionofthesuspensionisdenedforeachrun.Thelatterisanovelwayofperformingthesetypesofexperiments,drawingfromconceptsingranularrheology.Inpressure-imposedrheometry,theparticlepressurePismaintainedatasetvaluethatismeasuredbytheprecisionscale;thevolumefractionandtheshearstressaremeasuredasafunctionoftheshearrate_andpressureP.Involume-imposedrheometry,theheighth,andconsequentlythevolumefraction,aremaintainedataxedvalue,whiletheshearstress,,andparticlepressure,P,aremeasuredasafunctionoftheshearrate,_.SeeAppendix B foradetailedanalysisofthemeasuredquantitiesandcalculatedvalues.Errorsinthemeasurementsof,P,andforthesuspensionsdependuponthecalibrationexperiments,thepreparationofthesuspensionsamples,andtheprecisionoftheheight,torque,andscalemeasurements.Estimates,basedupontestswithindependently 32

PAGE 33

createdsamplesofsuspension,suggesterrorsof6Pa,5Pa,and0:005for,P,and,respectively.Inatypicalexperiment,theannularcellwaslledwithsuspensionandtheporousplatewasloweredintotheuidtoapositionh.Thisheight,measuredindependentlybyapositionsensor(NovotechnikT-50),providestheinformationnecessarytocalculatethebervolumefraction,.Thebottomannuluswasrotatedataratebyanasyn-chronousmotor(ParvaluxSD18)regulatedbyafrequencycontroller(OMRONMX20.4kW),whilethetorqueexertedonthetopplatewasmeasuredbyatorquetransducer(TEICFF401).Theshearstresswasdeducedfromthesetorquemeasurementsaftercalibrationwithapureuidtosubtractundesiredcontributionsresultingfromthefrictionatthecentralaxisandtheshearinthethingapbetweenthetopplateandthecellwalls.Aprecisionscale(Mettler-ToledoXS6002S)measuredtheapparentweightofthetopplateand,aftercorrectingforbuoyancyandnormalisationbythearea,providedthedeterminationofthenormalstressactinginthegradientdirection.Forsimplicity,thisisreferredtoastheparticlepressure,P.ThescalewasplacedonaverticaltranslationstagedrivenbyaLabVIEWcode.Figure 2-2 showssampledatasetsforthesuspendinguidintheabsenceofparti-cles,suspensioninvolume-controlledandpressure-controlledexperimentsrespectively.Steadystatevalueswereobtainedusingthesignalsobtainedfromthetransducers.Correctionsaremadetothevaluesobtainedbymeansofcalibrationexperimentscar-riedoutusingonlythesuspendinguids.Thesecalibrationsarecarriedouteverytimethesuspendinguidischanged.Thesevalueswereusedtocalculateandplotvariouscharacteristiccurves.AdetailedexplanationforthecalibrationandthedataanalysisisprovidedinAppendixB.ShearandnormalviscositieswerecalculatedusingEquation C-5 33

PAGE 34

Figure2-2.Rawdatasuppliedbytheshearcellforbersofaspectratio,A=11.3.A)puresuspendinguidwhichisruninvolume-controlmode,B)volume-controlmodedataforthesuspension,andC)pressure-controlmodedataforthesuspension. 2.2ResultsandDiscussion 2.2.1ShearandNormalViscosityTypicalrheologicaldatafortheapparentrelativeshearandnormalviscosities,=f_andP=f_,areplottedagainstvolumefraction,,inFigure 2-3 (a)and(b).Thedatawascollectedforbersofbatch(II)usingpressure-imposedandvolume-imposedmeasurements.Asexpected,bothquantitiesincreasewithincreasing.However,multiplevaluesoftheapparentviscositiesaremeasuredforanygiven.Plottingtheshearstress,,andtheparticlepressure,P,againsttheshearratefordifferentvaluesofdemonstratesthatandParelinearin_,buthaveanon-zerovalueat_=0,seeFigure 2-3 (a)and(b).Thisseemstosuggestthatayield-stressexistsforboththeshearstressandtheparticlepressure,0andP0,respectively.Theirvaluescanbedeterminedusingalineartofthestressandpressuredataasafunctionof_,asindicatedbythelinesinFigure 2-4 (a)and(b).Bothyield-stresses,0andP0,increase 34

PAGE 35

Figure2-3.Shearandnormalrheologyanalyses.A)Shear(=f_))andB)normal(P=f_)viscositiesasafunctionofvolumefraction,.PanelsC)andD)showshear[()]TJ /F3 11.9552 Tf 11.955 0 Td[(0)=f_]andnormalviscosities[(P)]TJ /F3 11.9552 Tf 11.955 0 Td[(P0)=f_],asafunctionofvolumefraction,aftersubtractionoftheyieldstresses.Experimentswereconductedinbothpressure-imposed(N)andvolume-imposed(4)modes. withincreasingasshowninFigures 2-4 (c)and(d)forallfourbatchesofbers.Thegrowthin0andP0withrespecttoaremorepronouncedforlargeraspectratiosA.ThedataofFigure 2-4 (a)and(b)demonstratethatthestressesscalelinearlywiththerateofshear,asexpected.Furthermore,theslopesofandPwith_increasewith,whichisevidenceoftheincreaseoftheshearandnormalviscositieswith.Theseshearandnormalviscositiescanbecollapsedintoasinglefunctionofbyremovingtheyieldstresses.Figures 2-4 (c)and(d)showtheresultsof()]TJ /F3 11.9552 Tf 12.043 0 Td[(0)=f_and(P)]TJ /F3 11.9552 Tf 12.559 0 Td[(P0)=f_asafunctionof.Inallofthefollowinganalysis,theyieldstressesaresubtractedsystematicallyfromtherawdata. 35

PAGE 36

Figure2-4.Shearstressandparticlepressuredata.A)Shearstress()andB)particlepressure(P)versusshearrate,_,forthebersuspensionofbatch(II)atdifferentvaluesof0.26(lightestgreyshade),0.30,0.35,0.38,and0.41(black).Thelinesrepresentthelineartforeachdifferentvalue.Yield-stressC)fortheshearstress(0)andD)particlepressure(P0)versusforbersofbatch(I),(II),(III),and(IV)shownusingthesymbols,4,3,and,respectively(seetable 2-1 ).Theinsetsofgraphs(c)and(d)arelog-logplotsversus=mwheremisthemaximumowablevolumefractiongiveningure 2-6 (a). 2.2.2AnalysisofNear-JammingLimitRheologyFigures 2-5 (a)and(b)showtherelativeshear(s=()]TJ /F3 11.9552 Tf 12.762 0 Td[(0)=f_)andnormal(n=(P)]TJ /F3 11.9552 Tf 11.21 0 Td[(P0)=f_)viscositiesforalloftheberbatches.BothquantitiesincreasewithandseemtodivergeatamaximumvolumefractionthatdependsontheaspectratioA.Theinuenceoftheaspectratioisalsoseenontherheologicalfunctionsass()andn()shifttowardlowervaluesofwithincreasingA.Aninterestingobservationisthatthedataforbatches(II)and(III),correspondingtosimilarvaluesofAbutdifferentsizes, 36

PAGE 37

Figure2-5.Comparisonsofshearviscosity(s)vs.volumefraction()andnormalviscosity(n)todimensionlessshearrateJ.A)s=()]TJ /F3 11.9552 Tf 11.955 0 Td[(0)=f_andB)n=(P)]TJ /F3 11.9552 Tf 11.955 0 Td[(P0)=f_versusaswellasC)=s=nandD)versusJ=f_=(P)]TJ /F3 11.9552 Tf 11.955 0 Td[(P0),forberbatches(I),(II),(III),and(IV)asrepresentedbythesymbols,4,3,and,respectively(seeTable 2-1 ).TheinsetsofgraphsC)andD)arelogarithmicplots. collapseontothesamecurve.Thisindicatesthatnitesizeeffectsarenotsignicant.Also,thedecreaseofnismuchstrongerthanthatofsfor.0:35.Analternativerepresentationoftherheologicaldataplotsthefrictioncoefcient=s=nandthevolumefractionasafunctionofthedimensionlessshearrate,J=f_=(P)]TJ /F3 11.9552 Tf 13.019 0 Td[(P0)[ 63 ].Therheologyisthendescribedbythetwofunctions(J)and(J)asshowninFigure 2-5 (c)and(d)forthesamedataasinFigure 2-5 (a)and(b)(referAppendix C ).Astrikingresultisthatacompletecollapseofallthedataisobservedfor(J),indicatingthatthefrictioncoefcientisindependentoftheaspect 37

PAGE 38

Figure2-6.Criticalvaluesofvolumefractionandfrictionnearthejamminglimit.A)m()andB)s()atthejammingpointversusberaspectratio,A,togetherwiththedata( F )obtainedbyBoyeretal.[ 63 ]forsuspensionsofspheres(A=1).ComparisonswithexperimentaldatafromRahlietal.[ 67 ]( )onthedrypackingofrigidbersandthesimulationsofWilliams&Phillipse[ 68 ]( N )forthemaximumrandompackingofspherocylindersaregivenongraphA). ratioA.ThevolumefractionisadecreasingfunctionofthedimensionlessnumberJ.Thereisaclearshiftof(J)towardthelowervaluesofwhenAisincreased.Thedataforbatches(II)and(III),havingsimilaraspectratios,againcollapseontothesamecurve.Thisfrictionalapproachisparticularlywellsuitedtostudythejammingtransition,asitcircumventsthedivergenceoftheviscosities.Fromthelogarithmicplotof(J),shownintheinsetofgure 2-5 (d),thecritical(ormaximumowable)volumefractionmcanbedeterminedfromthelimitingvalueofasJgoestozero.Similarly,thelogarithmicplotof(J)intheinsetofgure 2-5 (c)showsthatthefrictioncoefcienttendstoanitevaluesatthejammingpoint.ThecriticalvaluesmandsareplottedagainsttheberaspectratioAingures 2-6 (a)and(b),respectively.Again,thesimilarresultsforbatches(II)and(III)indicatethatconnementisnotinuencingthemeasurements,andthevaluesobtainedby[ 63 ]forsuspensionsofspheresarealsoplottedonthesegraphs(forA=1,althoughstrictlyspeakingasphereisnotacylinderofaspectratioone).Clearly,mdecreases 38

PAGE 39

Figure2-7.Rescaledrheologicaldata:A)s=()]TJ /F3 11.9552 Tf 11.955 0 Td[(0)=f_,B)n=(P)]TJ /F3 11.9552 Tf 11.955 0 Td[(P0)=f_andC)=s=nversus=maswellasD)=mversusJ=f_=(P)]TJ /F3 11.9552 Tf 11.955 0 Td[(P0),forallthedataofthedifferentbatches(I),(II),(III),and(IV)shownusingthesymbols,4,3,and,respectively(seetable??).Theinsetsofgraphs(a),(b),and(d)arelog-logplots.Theredsolidcurvescorrespondtotherheologicallawsgivenbyequations( 2 ),( 2 ),and( 2 ). 39

PAGE 40

withincreasingA.Thisfollowsthegeneraltrendsofadecreaseinvolumefractionwiththeaspectratioforprocessessuchasdrypacking,asshowninFigure 2-6 (a).AcomparisonisalsomadeinFigure 2-6 (a)betweenthevaluesofmandestimatesfromsimulations[ 68 ]ofthemaximumconcentrationatwhichtheorientationdistributionremainsrandom.ThecriticalfrictionsdoesnotvarysignicantlywithAintheexploredrangeanditsvalue(0:47)islargerthanthatobtainedforspheres(0:32)[ 63 ].Figure 2-7 displaysthesamedataasgure 2-5 ,butwithscaledbym.Thissim-plerescalingleadstoagoodcollapseofthedataforalloftheberbatches,indicatingthattheaspectratioprincipallyimpactsthemaximumvolumefraction,m.Anotherre-markableresultisthattherelativeshearandnormalviscosities,sandn,divergenearthejammingtransitionwithascalingcloseto(m)]TJ /F3 11.9552 Tf 12.497 0 Td[())]TJ /F5 7.9701 Tf 6.586 0 Td[(1,asclearlyevidencedbytheinsetsofgures 2-7 (a)and(b).Thisstarklycontrastswiththedivergenceof(m)]TJ /F3 11.9552 Tf 12.076 0 Td[())]TJ /F5 7.9701 Tf 6.586 0 Td[(2observedforsuspensionsofspheres[ 63 ].Aconstitutivelawforcanbegeneratedbyttingthedatatoalinearcombinationofpowersof(m)]TJ /F3 11.9552 Tf 11.956 0 Td[()=, ()=s+m)]TJ /F3 11.9552 Tf 11.955 0 Td[( +m)]TJ /F3 11.9552 Tf 11.955 0 Td[( 2;(2)wasdoneforspheres[ 64 ].Theredcurveingure 2-7 (c)showstheresult,withs=0:47,=2:44,and=10:20.Asnotedpreviously,thevalueforsislargerthanthatobtainedforsuspensionsofspheres(s=0:3).Thevaluesforandalsodifferfromthoseobtainedforsuspensionsofspheres(=4:6and=6).Thebesttforswasfoundtobe s()=14:51m)]TJ /F3 11.9552 Tf 11.955 0 Td[( m)]TJ /F5 7.9701 Tf 6.586 0 Td[(0:90;(2)asseeningure 2-7 (a).Notethatthebest-texponentis)]TJ /F7 11.9552 Tf 9.298 0 Td[(0:9,ratherthan)]TJ /F7 11.9552 Tf 9.299 0 Td[(1.Therheologicallawfornisthenjustgivenby n()=s()=();(2) 40

PAGE 41

whichisrepresentedbytheredcurveingure 2-7 (b).ThevariationofwithJcanbededucedfromthislastlawsinceJ=1=n();thisresultisshowninFigure 2-7 (d). 2.2.3ObservationofYieldStressesThesuspensionsexhibityield-stresseswhichincreasewithincreasingvolumefraction,,andaremorepronouncedforlargeraspectratios.Yield-stresseshavebeenreportedpreviouslyforrigidberssuspendedinNewtonianuids,andtheyieldstresseshavebeenattributedtoadhesivecontacts[ 56 57 ]despitetherelativelylargesizeofthebers.Arecentmodel[ 58 ],whichconsideredattractiveinteractionsbetweenbersinthediluteregime,predictedsimpleBinghamlawsforboththeshearstressandtherstnormalstressdifference,withtheapparentshearandnormalyieldstressesproportionalto2and3,respectively.ThepresentdataalsofollowsBinghamlaws,buttheyieldstress,0,andpressure,P0,increasewithhigherpowerlawsinthanpredicted.Thiscanbeseenintheinsetsofgures 2-3 (c)and(d),whereitisalsodemonstratedthatthedataforallaspectratioscollapsesontosinglecurvesbyrescalingbym.Itisunclearwhether,forthelargebersusedhere,colloidalforcesarerespon-siblefortheyield-stresses.Finite-sizeeffectsclosetothejammingpointcanalsobeadvocated,particularlysincelubricationforcesareinefcientatpreventingmechanicalcontactsbetweenelongatedparticles[ 61 ].Closetojamming,sincethesystemhasanitesize,percolatingjammingnetworkofparticlescanexist.Whileitistransientphe-nomenon,itmayimpacttheaveragedrheologicalmeasurementswhichconsequentlymayexhibitapparentyieldstresses.Clearly,moreworkisnecessarytoelucidatetheoriginoftheyieldstresses. 2.3ConclusionsUsingacustomrheometerwehaveperformedpressureandvolume-imposedmeasurementsoftherheologyofnon-colloidalrigidberssuspendedinaNewtonianuid.Measurementsfortheshearstressandparticlepressurehavebeenobtainedin 41

PAGE 42

thedenseregimeandforaspectratiosbetween3and15,andthevolumefractionsatwhichtherheologydivergeshasbeencharacterisedasafunctionoftheaspectratio.Subtractingtheapparentyield-stressesrevealsaviscousscalingforboththeshearstressesandparticlepressures,whereinbothgrowlinearlywiththerateofshear.Theaspectratioofthebersdoesnotaffectthefrictioncoefcient,,butdoesimpactthemaximumowablevolumefraction,m.Rescalingthevolumefraction,,bythismaximumvolumefraction,m,leadstoanexcellentcollapseofallthedataonmastercurvesfortheshearandnormalviscosities.Hence,wearguethattheaspectratioprincipallyaffectsthemaximumvolumefractionatwhichthesuspensionscanbesheared.Usingthedatapresentedhere,constitutivelawsintheformofexpansionsin(m)]TJ /F3 11.9552 Tf 12.653 0 Td[()havebeengeneratedfortherheologyofdensesuspensionsofrigidbers.Animportantproductofthepresentstudyistheexaminationoftherheologyclosetothejammingtransition.Atjammingthefrictioncoefcientisfoundtobeconstantandtobelargerthanthatfoundforsuspensionsofspheres.Bothshearandnormalviscositiespresentasimilaralgebraicdivergencein(m)]TJ /F3 11.9552 Tf 12.51 0 Td[())]TJ /F5 7.9701 Tf 6.586 0 Td[(1instarkcontrasttothatin(m)]TJ /F3 11.9552 Tf 12.576 0 Td[())]TJ /F5 7.9701 Tf 6.586 0 Td[(2observedforsuspensionsofspheresnearthejammingpoint.Themaximumvolumefractionmisseentodecreasewithincreasingaspectratio,similartothedrypackingofrigidbersfoundinexperiments[ 67 ],seegure 2-6 (a).However,noinferencesaboutthegeneralstructureofthesuspensionatjammingispossible,ascomparisonswithestimatesofmaximumrandompackingdonotclearlyindicatethattheorientationdistributionhasorganized[ 68 ].Directobservations,orsimulations,ofthestructuresneedtobedevelopedinfutureworktoresolvethisquestion. 42

PAGE 43

CHAPTER3DYNAMICSOFCONCENTRATEDSUSPENSIONSOFRIGIDFIBERSAnindividualberthatisrigidandfreeofanyexternalforcesexhibitsreversiblemotionswhensuspendedinaviscousuid,solongastherateofowremainslow.Forexample,suchaberowingthroughacylindricaltubeinresponsetoanoscillat-ingpressuregradientwillreturntoitsinitialpositionandorientationaftereachcycle.However,measurementsdescribedinthispaperdemonstratethatthespatialandori-entationaldistributionofaconcentratedsuspensionofrigidandforcefreebersisnotreversibleduringowthroughatube.Rather,theresultsindicatethatparticlespreferen-tiallymigratetowardthecenterofthetube,withtheextentofmigrationdependingupontheamplitudeoftheoscillatorydisplacementinthetubeandtheconcentrationofthebers.Themigrationoftherigidbersduringtubeowisanexpected,evenifpreviouslyundemonstrated,result,astheidenticalphenomenonforsuspensionsofsphereshasbeenstudiedextensively.Thegeneralobservation,rstmadebyLeighton&Acrivos[ 6 ],isthatspheresmigratefromregionsofhighsheartolowshearoccursuntilbalancedbythetendencyofthespherestomigratefromregionsofhighconcentrationtolowerconcentration.Furthermore,theratesofmigrationscalewiththerateofshearandareindependentoftheviscosityofthesuspendinguidiftheReynoldsnumberremainslow.Thisshear-inducedmigrationhasbeenobservedinCouettegeometries[ 6 69 70 71 ],inpressure-drivenows[ 43 45 46 47 48 ],andforowbetweeenrotatingeccentriccylinders[ 72 ].Inthemorespeciccaseofoscillatoryowforsuspensionsofspheresinatube[ 44 73 ],thedetailedmigrationresultswerefoundtodependuponthestrainamplitudeandconcentration.Theshear-inducedmigrationofparticlessignifcantlyimpactstheoperationofowprocesses.ThemigrationofspheresinaCouettegeometryaffectstherheologicalmeasurementsofsuspensions,complicatingtheevaluationoftheeffectiveviscosity[ 5 ]. 43

PAGE 44

Inthisgeometry,spheresmigrateawayfromtheregionofhighshearinthegapbetweenthecupandbobtowardtheregionbelowthebobwheretheshearrategoestozero.Consequently,thetorquemeasuredbytherheometerchangesintimeandthesteady,long-timemeasurementisnotrepresentativeoftheviscosityofthesuspensionatthedesiredconcentration.Asanotherexample,themigrationofparticlestothecenterofthepipeinapressure-drivenowresultinapressuredropthatisnolongerlinearalongthelengthofthepipeandthatislowerthanexpectedowingtothelowereffectiveviscosityneartheboundingwalls.Theoriginofparticlemigrationrepresentsafundamental,unresolvedproblem,asthegoverningequationsfortheuid(Stokesequations)andparticlemotionareformallyreversibleandmigrationshouldnotoccur.Twoideas,bothofwhichcouldbeoperatingsimultaneously,havebeenadvancedtoresolvethisquestion.Therstattributestheirreversiblemotiontothechaoticityofthehydrodynamicinteractionsbetweentheparticles[ 53 ].Inthisscenario,thesmallperturbationstotheparticlemotion,whichareinevitablypresentinrealsuspensions,areampliedthroughthenonlinearinteractionsandgiverisetothemigrationdespitethemathematicalreversibilityofthegoverningequations.Thesecondideaisthatparticlecontact-collisionsdrivetheirreversiblemigration[ 6 ].Thisintroducesanirreversiblecomponentintothegoverningequations,butrequiresrelaxationofthetraditionalassumptionmadeinStokesowthatlubricationforcespreventparticlesurfacesfromtouching.Experimentalevidencesupportstheideathatparticle-particlecontactsalterthetheextentofirreversibilityinconcentratedsuspensions[ 74 75 ].Experimentshaveyettobeperformed,however,thatdirectlycorrelateparticlemigrationwiththeparticleroughness.Numericalevidencefromsimulationssupporttheideathatchaotichydrodynamicinteractionsdriveirreversibleparticledistributionsduringsedimentation[ 76 ],butnotduringtheshearowofforcefreeparticles[ 77 78 ]wheretheinteractionsareweaker. 44

PAGE 45

Alongwithexperimentalevidenceofshearinducedmigrationinconcentratedsuspensionsofspheres,numericalsimulationsalsopredictsimilarcollectivebehaviorofparticles.Monolayersimulationsofspheresinapressure-drivenowbetweentwoplanewallsusingtheStokesianDynamicsmethod[ 9 ]agreedqualitativelywiththeexperimentalobservationsandalsoveriedmanyassumptionsregardingshear-inducedmigration,includingthefactthatthephenomenonisnotduetoinertialeffects.Morerecentstudieshaveexpandedtothree-dimensionalsimulationsofconcentratedsuspensionsofthousandsofparticlesofmonodispersenon-colloidalparticlesintubeow[ 79 ].Continuummodelshavebeendevelopedtopredictparticlemigrationaswell.Earlymodelsofshear-inducedmigrationpositedaphenomenologicalequationforpredictingtheparticledistribution.Inthesediffusionmodels[ 6 8 ],particlemigrationismodeledasadiffusiveux.Phillipsetal.[ 8 ]denedtwouxesinuencingparticlemigration.Therstuxisbasedonthecollisionfrequencyofparticlesanddescribesauxfromhightolowshearrates(rstterm),andacounteruxgeneratedbyanincreasedconcentrationofparticles(secondterm).Thisuxdependsontheparticleradius,shearrate,andtheconcentrationofparticles.Asecondux,reectsamotionofparticlesfromlowtohighshearratezonesduetotheincreasedviscositycausedbyincreasesinparticleconcentration.TheoweldmustbedeterminedfromtheStokesequation,wheretheviscosityvariesspatiallyaccordingtotheconcentrationofparticles.Modelsofsuspensionviscosity,suchastheKrieger-Doughertymodel[ 80 ],canbeusedforthispurpose.Thediffusionmodelsuccessfullypredictstheexistenceofmigrationinwide-gapCouetteandpressure-drivenPoiseuilleows,andthepredictionsareinagreementwithmeasurementsthatindicatethattheconcentrationproleisindependentoftheappliedshearrateandindependentoftheviscosityofthesuspendingmedium.However,thediffusionmodelisapplicablestrictlyforunidirectionalowsandthemodelfailstopredicttheabsenceofanetmigrationincurvilineartorsionalows[ 81 ]. 45

PAGE 46

Amorerecentmodelrelatesthemigrationuxandrheologyofthesuspension[ 9 82 ].Thissuspensionbalancemodelisatwo-phasemodelwhichprovidesacontinuumdescriptionofthebulksuspensionmotion,aswellastherelativevelocityoftheparticlephaseanduidphase.Themodelissystematicallyderivedbyperformingaphaseaverageofthegoverningmomentumandcontinuityequations.Accordingtothismodel,themigrationuxisdrivenbythedivergenceinthenormalstressoftheparticlephase,whichhasrecentlybeenarguedtoincludethecontactorinterparticlecontributionsaswellashydrodynamiccontributionscomingfromthenon-dragportionoftheinterphaseforce[ 83 84 ].Thisequationmustbesolvedinconjunctionwiththeoverallmomentumequationforthesuspension.Thesuspensionbalancemodelhasanumberofpotentialadvantagesoverthediffusionmodel.Likethediffusionmodel,itpredictsthemajorfeaturesobservedinunidirectionalows,andtheparticlephasestressesneededinthesuspensionbalancemodelcan,inprinciple,bedeterminedfromindependentrheologicalexperiments,thoughthesemeasurementsarenoteasilyperformed.Themodelhasbeenusedtosuccessfullypredictmigrationinarangeofowelds.Here,thepurposeistoprovidedataregardingthemigrationofnon-sphericalparticlesthatcanbecomparedwithmodelsandsimulationsinthefuture.Onlyafewstudiescurrentlyexistforthecollectivedynamicsofconcentratedsuspensionsofbers.Mondyetal.[ 10 ]usednuclearmagneticresonanceimagingtomeasurethespatialdistributionofrigidrodsinaCouetteow.Theyreportedshearinducedmigrationforrod-shapedparticleswithaspectratiosrangingfrom2to18andforvolumefractionsof0.3to0.4inCouetteow,wheretherodmigratedfromregionsofhighershearratenearthewallsofthecelltowardstheregionoflowshearrateclosertotheinnerrotatingcylinderofthecell.Overall,itwasconcludedthattheaspectratioofthebersplayedlittleornoroleintheextentofthemigrationandwasalsothesameasthatofspheres.Alongwiththecenter-of-masspositions,theorientationofbersarealsoexpectedtobeirreversible.Therehavebeennomeasurementsoftheeffectofmigrationonthe 46

PAGE 47

orientationdistribution,butirreversiblechangesinorientationdistributionshavebeenmeasuredinshearingows.StudiesbyPineetal.[ 85 ]indicatedthat,foranoscillatoryshear,theorientationdistributioncouldbecontrolledbyvaryingtheamplitudeofthestrain.Forlargeamplitudes,thebersalignwiththedirectionoftheow,whichresem-bledresultsinasteadyshear[ 32 ];forsmallamplitudes,theorientationdistributiondidnotvarysignicantlyfromitsinitialstate.However,forintermediatestrainamplitudes,theberspreferredtoaligninthevorticitydirection,perpendiculartotheow-gradientplane.SimulationsperformedbySnooketal.[ 11 ]addressedtheoriginofthispreferredvorticityalignmentandattributedittoshort-rangeinteractionsbetweentheparticles.Thisstudyalsoshowedthatthealignmentdependedstronglyontheconnementofthecellinthegradientdirectionbetweentheboundingwalls.Hereweuseopticalimagingtechniquestoprovideobservationsofdemixingofbersinpressure-drivenows.Resultspresentedinthischapteralsorelatetheorientationoftheberstotheircenter-of-masspositions.Tothebestofourknowledgeatthetimeofthiswork,nomeasurementshavebeenmadeforshearinducedmigration,ortheorientationdistributions,ofrod-shapedparticlesinpipe-ows.Themethodsaredescribedinthenextsection,whichisfollowedbytheresults. 3.1ExperimentsTheobjectiveoftheexperimentspresentedhereistoinvestigatethedynamicsofshearinducedmigrationofrodshapedparticlesinsuspension.Experimentsareconductedinanoscillatorypressure-drivenowandparticlemigrationcanbequantiedusingimageanalyses.Theseexperimentsenablethecharacterizationofthemicrostruc-tureofthesuspensionasafunctionofthestrainbyquantifyingnotonlythedistributionofthecenterofmasspositionofthebers,butalsotheorientationdistribution,whichisanareathathasbeenlargelyunresolved.Thetime-dependentdynamicsofthisphenomenonisalsoinvestigated.Thesubsectionsbelowbrieydescribetheparticle 47

PAGE 48

Figure3-1.SamplephotographforPMMAcoreberopticbers,strippedusingdimethylsulfoxide(DMSO)andcutusingacustom-builtguillotinecuttingdeviceatAix-MarseilleUniversitetothedesiredlengths.Thecutedgesintroducedsurfaceimperfectionsleadingtodiffractionofthelasersheetduringexperimentruns. anduidsystemused,theexperimentalapparatus,experimentalprocedure,andimageanalysisprocesses.VariousresultshavebeendescribedanddiscussedinSection 3.3 3.1.1ParticlesandFluidFiberopticcableswithaPoly(methylmethacrylate)(PMMA)corewerechemicallystrippedoftheirouteruorocarboncoatingbysoakingberopticlamentsinDimethylsulfoxide(DMSO)andmechanicallywipingoffthecoating.Thestrippedcablesweremechanicallycutintorodsofthedesiredlengths.AnexamplephotographofthestrippedandcutbersisshowninFigure 3-1 .Thetwoaspectratiosforourexperimentswereselectedbyusingthesamelength,L=5:20:2mm,andusingtwodiameters,d=0:460:06and0:230:02mm.TheaspectratiousedfortheseexperimentswereA=11:31:6and22:62:3,forthetwodiameters,respectively.TheuidusedfortheseexperimentsissimilartotheoneusedfortherheologyexperimentsdescribedinSection 2.1 .Itisatri-componentmixtureofTritonX-100(73.28%),distilledwater(10.72%),andzincchloride,ZnCl2(16%).TheTritonandwatercontroltheviscosityandtherefractiveindex,andthesaltcontrolsthedensityoftheuid.Theweightfractionsofthecomponentsofthesuspendinguidarechosento 48

PAGE 49

matchthedensityandrefractiveindexofthesuspendedparticles.AsmallamountofRhodamine6Gdyewasaddedataconcentrationof910)]TJ /F5 7.9701 Tf 6.587 0 Td[(7gm/cm3toenablecontrastimagingwiththelaser-camerasystemasshowninFigure 2-1 .Carewastakenwhilepreparingthesuspensionstopreventtrappingairintheviscousuid.Massesoftheparticlesanduidweremeasuredtoobtainthedesiredvolumefractionforaparticularseriesofexperimentruns.Therequiredmassofberswererstgraduallyaddedtothesurfaceoftheuid.Thisisdone,becauseanyforcedimmersionofparticlestrapsunwantedair.Oncethebersareadded,thesuspensionisthenmixedbygentlyrotatingthebeakeratanangle.Thisfurtherreducestheamountofairtrappedinthesuspension. 3.1.2ExperimentalApparatusTheexperimentalsetup,previouslyusedtostudythecollectivemotionofspheres[ 73 ],wasusedtoobservethechangeinthemicrostructureofrodshapedparticlesuspensions,aswellasobserveshearinducedparticlemigrationinanoscillatorytubeow,andisshowninFigure 3-2 .Thesuspensionwasloadedintoanacrylictubeoflength46.8cmandofcircularcross-sectionwithdiameter2R=1:65cm.ForthebersoflengthLusedinthisstudy,thegeometricratio,R=L3:17.Thetubeisorientedverticallyandameshscreenisplacedatthetopandbottomendsofthetube.Thesmallmeshsizeensuresaconstantparticlevolumefractionwithinthetestingsection,astheparticlescannotpassthrough.Thesuspensionisoscillatedbyasyringepump,microcontroller,syringe,andhosesconnectedtotheinletoftheglasstube.Twomicroswitchtriggersaremountedalongsidethesyringepumptocontrolthestrokelengthofeachoscillation.Thedistancebetweenthetriggerscanbemanuallyadjustedtoimposedifferentstrokelengths.Theoscillatorystrainwasintheformofasquarewaveratherthanasinusoidalone.Therate 49

PAGE 50

Figure3-2.Experimentalsetuptostudymigrationofsuspendedparticlesinpipe-ows.Asyringepumpisusedtocreatealargeamplitudeoscillatoryow.Thelasersheetandcamerasystemarecontrolledviaamicrocontrollerforxedintervalexposures.Imagescapturedusingthissystemwillbeusedtotrackindividualparticles. ofvolumetricdisplacementQ(t),asafunctionoftimet,canbegivenas Q(t)=(0R)(R2)!cos(!t) jcos(!t)j;(3)wherethejjindicatestheabsolutevalue,forafrequency!chosenforvalueswhichmaintainedalowvalueoftheparticleReynoldsnumber,ofO(10)]TJ /F5 7.9701 Tf 6.587 0 Td[(3)andthePecletnumberofO(109).Thestrainamplitude,0,waschosentominimizepossibleendeffectswhilebeingsufcientlylargerwhencomparedtotheberlengths.Thediameterofthesyringeusedinourexperiments,Rswaslargerthanthetuberadius,R.Tocorrectlysetthedistancebetweenthemicro-triggersandhenceobtaintheappropriatestrainamplitude,0,astrokelength,swassetas0=sR2s=2R3.Theglasstubeishousedwithinarectangularplexiglasjacket.Theinterstitialspacebetweenthewallsofthejacketandtheglasstubeislledwithsuspendinguidthatcontainsnodye.Thepurposeofthisjacketlledwiththeuidistoeliminatetheopticaldistortioncausedbythecurvatureofthetube.ACoherentLasirisGreenPowerLine 50

PAGE 51

Table3-1.Concentration,nbulk,expressedvolumefraction,,foraspectratio,A=11.3and22.6. Anbulk(%) 11.30.845.51.6811213.09319.6322.60.842.751.685.5 Laserwithawavelengthof532nmisusedtouorescethedyeduid.ThelaserpassesthroughadarkenedPMMAmaskwithaslitof250mcutintoit.Themaskandtheslitwerepositionedsothelasersheetwasonlyappliedtothecenterofthetube.Sinceitistheuidthatisdyedandnottheparticles,thelaseruorescesthesuspendinguidandtheparticlesappeardark,creatingacontrastwhichcanbeimaged.Aredlongpasslterof590nmwasusedtoenhancethecontrastoftheuorescedsuspension.ThecamerausedforimagingtheseexperimentswasaNikonD300swithanAF-SMicroNikkor60mmf/2.8GEDlens.Topreventphotobleachingcausedbylongexposureofthelaser,ashutterismountedinfrontofthelaser.Theshutterandthecamerawerecontrolledbythemicrocontrollerandthetriggers. 3.1.3ExperimentalProcedureTheuidcontainingtheberswasgentlyremixedandaddedtotheinnercirculartube.Beforethestartofeachexperiment,thesuspensionwasmixedinthetubeusingawirewithimpellersatdifferentheights.Thesuspensionwasthenallowedtostandtoallowanytrappedairtoescapefromthetopofthetube.Thevolumefraction,correspondingtotheseconcentrationsforthetwoaspectratios,A,canbecalculatedas=(nbulk)=4A,andareshowninTable 3-1 .Thestrokelengthwassetbyadjustingthedistancebetweenthemicroswitchtriggerstoobtainthechosenamplitudeofoscillation(either3.5,6.5,or15).Foreachoscillation,40imagesweretakenwithaonesecondinterval.Camerasettingswerechosentoaccountforthenear-zerolightconditionsand 51

PAGE 52

tocaptureclearimagesofthesuspensioninmotion.ThiswasdonebymaximizingtheshutterspeedandreducingtheaperturesizeandISO.Asmallerapertureischosentolimitthedepthofeldwithintheglasstube,toonlyimagetheparticlesinsidethelasersheet.Forthepurposeofstatisticalrigor,atleast2-3runsofeachexperimentwereperformed.Fromtheimagesobtainedfromtheseexperiments,wecanascertainthemicrostructureofthesuspensionasafunctionofimposedowparametersbymeasuringthespatialandorientationdistributionsusingparticletrackingvelocimetryandimageanalysestechniques. 3.2ImageAnalysesAbriefschematicoftheimageanalysesprocessesareshowninFigure 3-3 .Positionsofthecentersofmassandtheorientationscanbecalculatedfromtheimagesobtainedfromtheexperiments.Bymeasuringtheparticlevolumefractionacrossthetube,wecanexaminedemixingofthesuspension.Theexperimentalimageswereprocessed,usingmultiplestepstocleantheimageasshowninFigure 3-3 (a),andanalyzedtodetermineiftherewasamigrationofthebers.AcroppedimageisusedforthedataanalysisandshowninFigure 3-3 (b).WeusetheskimageanalysispackageavailablethroughastandardPythondistribution.Oneoftheprimaryobjectivesistocalculatethearealfractionandthespatialdistributionoftheparticlesacrosstheradialcoordinateofthetube.Theimagequalityisquestionableinthesecondhalfoftheimage;themechanicalcuttingofthebersrefractsthelasersheetandcreatesstreaksgenerallyfoundinthehalfoftheimagefurthestfromthelasersheet.Therefore,onlytheinitialhalfoftheimagesareusedintheimageprocessing.Forthelocaladaptivecontrastenhancement,showninFigure 3-3 (b),equalize adapthistwasused,whichisasubfunctionoftheexposurelibraryintheskimagepackage.Thisalgorithmuseshistogramscomputedoverauser-speciednumberofbinswhichdeterminethecontrastresolutionofthetreatedimage.Thenumberofbinsdeterminesthespacingofthedomainsandhelpincreasethecontrastbetweentheparticlesandthe 52

PAGE 53

Figure3-3.Exampleschematicshowingthevariousimageprocessingtechniquesforasampleimageofbulkconcentration,nbulk=0.84.A)Therawimagecroppedtosize;B)adaptiveequalizationandC)thresholdingwherethereferenceblocksizefortheprocessisuser-dened;D)thecleanedimageaftertheremovalsomeofthenoisypixelstogiveabetterbinaryimagewhichcanbeanalyzedusingparticletrackingtechniques. background.Thecontrastimagewasthenrunthroughanadaptivethresholdfunctionthreshold localwhichispartofthelterslibrary,showninFigure 3-3 (c).ThisstepisalsoknownasadaptiveordynamicthresholdinginolderversionsofPython.Adaptiveorlocalthresholdingisaweightedmeandeterminedbytheneighboringpixelssubtractedbytheconstantoffsetdened.Theoffsetforthisfunctionhasadramaticeffectonthetreatedimageandishenceuser-denedforimagesfromeachexperimentalrun.ThemethodforthresholdingusedwasGaussianwhichisthedefaultmethod.Theresultthresholdedimagewasofenhancedcontrastandeverypixelabovethethresholdvaluewasconsideredtobepartoftheforeground.However,becauseofthenon-sphericalshapeoftheparticles,itwasstilldifculttodistinguishtheparticlesfrom 53

PAGE 54

residualnoisecausedduetodiffractionorshadowscausedbysurfaceimperfections.Toobtainimagesofhigherclarity,thenalstepoftheimageanalysisprocesswastheremove small objectswhichispartofthemorphologylibrary,showninFigure 3-3 (d).Thisfunctionremovesconnectedpixelssmallerthanadenedsize.Theresultoftheimageanalysisprocessesisabinaryimageofdarkpixelswhichindicatetheberonawhitebackgroundwhichisthesuspendinguid.Asatestofthevalidityofthetreatmentprocess,thearealfractionwascalculatedfromthetreatedimageandcomparedtotheknownvolumefractionoftheparticlesinthetubeforanyrun.Ideally,thetreatmentwouldhaveremovedallnoisefromtheimages,leavingonlythebers,andwouldroughlythesameareafractionastheinitialvolumefraction.Thetreatmentparameterswereadjustedtohelpminimizethedifferencebetweenthecalculatedarealfractionoftheimagesandtheknownvolumefractionofthesuspension.Alargesubsetofeachcyclewasaveragedtocreateadistributionoftheareafractions.Aradialweightingwasappliedtondtheaverageanderroroftheareafractionacrossthediameterofthetube.Thisradialweightingcanbegivenas =RR0(r)rdr RR0rdr=PN1(ri))]TJ /F4 7.9701 Tf 6.675 -1.378 Td[(r2i+1=2 2)]TJ /F4 7.9701 Tf 13.151 8.305 Td[(r2i)]TJ /F17 5.9776 Tf 5.756 0 Td[(1=2 2 R2=2;(3)where,(ri)isthevolumefractiontobecalculatedwhichisafunctionoftheradialpositionracrossauser-setcalculationindexi.Thedistancebetweenthecenterlineandthewallofthetube,R,ishalfthesizeoftheframe. 3.3ResultsandDiscussionThegoaloftheexperimentswastoexaminethekeyparametersthatcontrolshearinducedmigrationinbersuspensions.Figure 3-4 showsarepresentativesetofprocessedimagesforanexperimentwithbersofaspectratioA11ataparticleconcentrationofnbulk=0.84inanoscillatorypressuredrivenowatthestrainamplitude0=15.Thisgureshowstheevolutionofthedistributionofthebersinthetubeowtoqualitativelyillustratetheexistenceofshearinducedmigration.Here,theinitialuniform 54

PAGE 55

distributionisdenedwheretheaccumulatedstrain,=0.Itisimportanttonotethatonlyhalfoftheimage,theoneclosertothelaser,wasusedfortheanalyses.Thisisbecausetheimagequalityseverelydeterioratedaswemovedawayfromthelaser.EachpanelinFigure 3-4 isasnapshotofthesuspensionattheendofaoscillationcycle.Forexample,thepanelatanaccumulatedstrain=300at0=15correspondstothestructureofthesuspensionattheendofthetwentiethoscillation.Thedataispresentedasafunctionoftheaccumulatedstrainasitisamoreconvenientwayofcomparingtheextentandrateofmigrationforexperimentsatdifferentamplitudes,whichdifferedintheirnumberofoscillations.Asthesuspensionisoscillated,theisotropicinitialdistributiondemixes,ascanbeseeninintermediateimages,whichinthecaseofFigure 3-4 isat=300and600,respectively.Atasufcientlylargeaccumulatedstrain,forexample,at=900,themicrostructuredoesnotsignicantlychangeasthestrainincreases.Thisstructureshowsadistributionwhichismoreconcentratedatthecenterofthetube,atr=R=0andlessconcentratednearthewallsatr=R=1,showingdirectqualitativeevidenceofshearinducedmigrationofthesuspensionundertheseconditions.ProcessedsetsofimagesliketheoneshowninFigure 3-4 canbeusedtocalculatelocalparticleconcentrations,whichcanbeusedtoquantifythemigrationphenomenonandisdescribedindetailinSubsection 3.3.1 .Additionally,theseprocessedimagescanbeusedtoquantifytheorientationdistributionofthebersintheseexperiments.Ade-tailedstudyoftheorientationdistribution,andtheeffectofmigrationontheorientationdistributionisdescribedinSubsection 3.3.3 3.3.1ArealFractionDistributionIntensitydatawereextractedfromtheprocessedimages,wherethedarkpixelscorrespondtothebersagainstthebrightbackgroundwhichisthesuspendinguid.Togeneratequantitativeinformationregardingthelocalparticleconcentrationasafunctionofradialpositionandstrain,thefractionofpixelslledbyaparticle(i.e.black) 55

PAGE 56

Figure3-4.Setofprocessedimagesthatqualitativelyillustrateshearinducedmigrationinconcentratedbersuspensions.Theberconcentrationisnbulk=0.84,thestrainamplitude0=15,bershaveanaspectratioA11. ineachverticalrowofpixelsintheimageswasrecorded.Theoverallnumberdensityatavolumefraction,(r),isgivenby4A(r)=.Thearealfractiondatawasaveragedoveratleastthreeexperimentalrunstoreduceuctuationsintheruns.Onesuchaveragedarealdistribution,correspondingtothecasedescribedinFigure 3-4 isshowninFigure 3-5 .Itcanbeseenthatastheaccumulatedstrainincreases,thelocalarealfractionincreasesatthecenterofthechannel,atr=R=0,anddecreasesnearthewall.ThisisinaccordancewiththequalitativeobservationshowninFigure 3-4 .Itisimportanttonoteherethatthearealfractiondistributiondescribestheaveragenumberofparticlesinthepixel-widthframe,andhencethebulkconcentrationnbulkisconserved.Theerrorassociatedwiththecalculationoftheaveragelocalarealfractioniswithin10%.Forexample,foraninitialbulkconcentrationnbulk=0.84,thecalculatedbulkconcentrationfromtheprocessedimagesis0.830.04at=0fortherepresentativecasedescribedinFigure 3-5 .This 56

PAGE 57

Figure3-5.Forinitialbulkconcentrationnbulk=0.84,0=15,andA11,thelocalnumberdensity,4A(r)=,isplottedasafunctionoftheradialposition,r=R,ofthetubeofradiusRfordifferentaccumulatedstrains.Shearinducedmigrationcanbeobservedastheaccumulatedstrainincreasestowards=900.IndividualcurvesforthedifferentstrainscorrespondtotheexamplepanelsshowninFigure 3-4 istherstquantitativeclaimofshearinducedmigrationforbersuspensionsattheseconditionsinanoscillatorypipeow.Toidentifytheeffectofstrainamplitudeontheobservationofmigration,experi-mentswereperformedforstrainamplitudes0=3.5and6,inadditiontothecaseof0=15showninFigure 3-5 .Shearinducedmigrationwasobservedforalloftheabove-mentionedconditions.ThedependenceofstrainamplitudeisshowninFigure 3-6 tothesamplecaseshowninFigure 3-5 .Itcanbeseenthatthestrainamplitudehasaneffectonthearealfractiondistribution.Atthesameconcentration,accumulatedstrain,andaspectratio,thearealfractionnearthecenterofthetubeincreasedwithahigherstrainamplitude.Thisdifferenceinresultscanbeattributedtotherearrangementofthemicrostructurewiththeow.Thereversalofowatdifferentpoints,correspondingtothedifferentstrainamplitudesaltersthemicrostructuretodifferentextents.Forexample, 57

PAGE 58

Figure3-6.Thedependenceofthestrainamplitudeonmigrationisshownbylookingatthelocalnumberdensity,4A(r)=,forexperimentswiththesameberaspectratioA11,nbulk=0.84,butforadifferentstrainamplitude,0=6ascomparedtoFigure 3-5 where0=15. forowatsteadystate,denedasthestateatwhichthemicrostructurehasbecomesteady,iftheowisreversed,thereisanitestrainrequiredforthesteadymicrostruc-turetodevelopagain.Thisphenomenaisnotsurprisingandithasbeenstudiedforthecaseofspheres[ 86 ],anditwasconcludedthatatleastfourtosixstraincycleswererequiredtoreclaimthesteadymicrostructure,andthisratedependedonthevolumefraction.Hencewewouldnotnecessarilyexpectresultsforourrodexperimentstobethesame,asmanycycles(beyondtherangeofourexperiments)mayberequiredfortherodstoreturntotheirsteadymicrostructure.Incontrast,ahigheraspectratiodidnotshowachangeinthemigrationbehavior,asshowninFigure 3-7 .Notethatthelengthoftheberwaskeptconstantforthetwoaspectratios.ThismeansthatparticleconcentrationnL2d,andnotthevolumefractionistheappropriateparametertoquantifymigrationinourexperiments. 58

PAGE 59

Figure3-7.Forabulkinitialconcentrationnbulk=0.84,0=15,andaspectratioA=22.6,thedependenceoftheberaspectratiobycomparinglocalnumberdensityascomparedtoFigure 3-5 foridenticalconditionsbutforbershavinganaspectratioA=11.3. However,experimentswereconducted,wherenosignicantmigrationwasseenbyobservingtheevolutionofthearealfractiondistributionasshowninFigure 3-8 .Figure 3-8 (a)showsthatatalowconcentration,atnbulk=0.5,foranaspectratioA11,and0=15,thedistributionafteranaccumulatedstrainof720isnotsignicantlydifferentfromtheinitialdistributionindicatingthatthereisnosignicantmigration.Therearetwoproposedreasonsforthelackofobservablemigrationattheseconditions.Firstly,theconcentrationistoolowforasignicantnumberofparticleinteractionstooccur.Alownumberofparticlecontactsleadstoalowparticlenormalstresswhichcouldaccountforthelackofobservablemigration.Secondly,foralowerconcentration,thestrainrequiredforafullydevelopedmicrostructureismuchhigherandmaybebeyondtherangeofstrainthatismeasuredinourexperiments.Migrationdoesnotoccuratlowstrainamplitude0asthedisplacementpercycleisinsufcienttodisturbtheinitialmicrostructure,andhencenomigrationoccurs.Athigherconcentrationsasshown 59

PAGE 60

Figure3-8.ForA11and0=15,nosignicantmigrationwasseenforA)nbulk=0.5andB)nbulk=3. inFigure 3-8 (b),theeffectsofconnementhinderanycollectivemotionofthebers.Although,moreinvestigationintohigherconcentrationsisrequired.Figure 3-10 plotstheextentofmigrationforallconditionstestedtoenableacomprehensivecomparison.Forthepurposeofcalculatingtheextentofmigrationtwo 60

PAGE 61

Figure3-9.Extentofmigrationasafunctionofthebulkparticleconcentration,nbulk,A)Forastrainamplitude0=3.5,B)0=6,andC)0=15,foraspectratio,A=11.3(red),andA=22.6(blue). 61

PAGE 62

binswerechosen,correspondingto1=16ththewidthoftheframe,nearthecenterandthewallofthetube.Theextentofmigrationwascalculatedas(nr=0)]TJ /F3 11.9552 Tf 12.84 0 Td[(nr=1)=nbulk,wherenr=0isthearealfractionatsteadystateatthecenterofthetube(r=R=0),andnr=1isthesteadystatearealfractionatthewallofthetube,wherer=R=1,andnbulkistheaveragedarealfractionacrossthetubeatthelargestavailablestrain.Notethatthevalueofnbulkisradiallyweighted.Figure 3-9 showsthedependenceoftheextentofmigrationwithparticleconcentrationfordifferentstrainamplitudes,forthetwoaspectratios.Animportantresultisthatmaximumextentofmigrationwasobservedatnbulk=0.84,independentofbothamplitudeandaspectratio.AccordingtotheresultspresentedinFigure 3-9 ,thepeakextentofmigrationwasseenforsuspensionsofconcentrationnbulk=0.84,strainamplitude0=15,forberaspectratioA23.Astheconcentrationisincreased,theextentofmigrationdecreases.TheseresultsarecorroboratedbytheresultsshowninFigure 3-5 whichshowedlargeobservedmigration,and 3-8 (a)and(b)whichshowedasignicantlylowerextentforcasesatlow(nbulk=0.5)andhigh(nbulk=3)concentrations,respectively. 3.3.2DynamicsofshearinducedmigrationOneofthemajorquestionstobeanswerediswhetherthemeasurementsshowninFigure 3-9 areatsteadystate.ThedynamicsofthemigrationareevaluatedbyobservingtheextentofmigrationasafunctionoftheaccumulatedstrainasshowninFigure 3-10 .Asseeninthegure,theclaimthatthesystemhasreachedsteadystateisacomplexone.Thisisduetothelargeuctuationsinthemeasurementsoftheextentofmigration,causedduetotheuctuationsinthemotionoftheuidandbersthemselves.Oneoftheproposedmethodstosmooththeharshuctuationsistomoveawayfromthediscretemethodofevaluatingtheextentofmigrationandmovetowardsanintegralapproach.Onesuchapproachistodeneanequivalentradiusre,wherethebulkarealfractionatr
PAGE 63

Figure3-10.EvaluationofsteadystatecanbeperformedbyobservingtheextentofmigrationastheaccumulatedstrainisincreasedforA=11.3(red)and22.6(blue),and0=15,forA)nbulk=0.84andB)nbulk=1.68. 63

PAGE 64

Figure3-11.Forashearingowinthex-directionhavingagradientinthey-direction,theorientationcanbequantiedusingtheanglewhichistheanglemadewithrespecttothegradientdirectionusingtheprojectionoftherodintheow-gradientplane,andzwhichistheanglemadebytheberandthevorticityorthez-direction. nowbeexpressedasameasureofrelowerthanR=2,astherearemorebersnearthecenterthantowardsthewall.Thisapproachforanalyzingtheextentofmigrationhasnotyetbeenperformed. 3.3.3OrientationdistributionDetailsabouttheorientationdistributioncanalsobeextractedusingtheprocessedimages.Forthepurposeofthiswork,theorientationofaberisquantiedbymeasur-ingtheanglez,theanglemadebytherodandthevorticitydirection,and,whichistheanglethattheprojectionoftherodmakeswiththeow-gradientplanewithrespecttothegradientdirection,asshowninFigure 3-11 .Theprobabilitydistribution,P(z)asafunctionoftheanglezisshowninFigure 3-12 .Theprocessedimagewasdividedintothreeequalbins,andtheaverageprobabilitydistributionwasplotted,asshowninFigure 3-12 .AccordingtothedenitionoftheorientationshowninFigure 3-11 ,theberisorientedinthevorticitydirectionasz!0. 64

PAGE 65

Figure3-12.ForA11,atconcentrationnbulk=0.84,and0=15.Theprobabilityofndingarodatanglez,P(z)decreasesaswemovetowardsthecenterofthetube,asseenbyobservingthedistributionatA)0
PAGE 66

Figure3-13.ForA11atbulkinitialconcentrationnbulk=0.84,and0=15,theprobabilityP()ofndingaparticleatanangleshowsapreferentialalignmentfortheowdirection,where=90. appearascirclesinthe2-Dimagestakenduringtheexperimentruns.Thesecirclesandalldarkregionsbelowadenedareathresholdareneglectedduringthecalculationof.Figure 3-13 showstheaverageprobabilityofndingaparticleP()distributionacrosstheentireframeofthereferencecasedescribedinFigure 3-4 ,andFigure 3-14 showsadistributionacrossthethreebins,similartotheanalysisofz.Asexpected,asignicantlyhighP()wasobservedfor=90implyingthatthebersshowpreferentialalignmentintheowdirection.Negligibleprobabilityfor=0showsaninsignicantpropensityforberstoaligninthegradientdirection. 3.4ConclusionsDirectopticalimagingtechniqueswereusedtoinvestigatetheshear-inducedmigrationofconcentratedsuspensionsofrigidrodsinanoscillatoryparabolicow.At 66

PAGE 67

Figure3-14.ForA11,atconcentrationnbulk=0.84,and0=15,theprobabilityP()ofndingaparticleatanangleshowsapreferentialalignmentfortheowdirection,where=90.Thisprobabilityincreasesaswemovefromthecenterofthetube,atA)0
PAGE 68

extentofmigration,andwasusedtoinvestigatetheeffectofstrainamplitudeshowninFigure 3-6 andaspectratioasshowninFigure 3-7 .Asanimportantdivergencefromthebehaviorofspheres[ 86 ],resultspresentedinFigure 3-6 wasnotindependentofthestrainamplitude.Hencetheresults,evenatthehigheststrainamplitude,cannotbeassumedtorepresentthoseexpectedinasteadytubeow.Theaspectratiodidnotplayasignicantroleinaffectingtheextentofmigration.Atlowconcentrations,forexampleatnbulk=0.5,particleinteractionswerein-frequent.HencenosignicantmigrationwasobservedwithinthestrainlimitofourexperimentsshowninFigure 3-8 (a).Inaddition,forlowamplitudes0,thereisalimitunderwhichthemotionofbersuspensionsremainsreversible,whichfurtherconrmedthisobservation.Similarly,connementpreventedthecollectivemotionofthebersathighconcentrations,forexampleatnbulk=3asshowninFigure 3-8 (b).Theextentofmigrationwascalculatedusinganormalizeddifferencebetweentheconcentrationofthebersnearthewallsandthecenteroftheframe.Fromourresults,itwasconcludedthattheextentofmigrationwashighestatnbulk=0.84,anddecreasedwithfurtherincreaseinconcentration.Thislimitwasindependentofstrainamplitudeandberaspectratio.Theextentofmigrationdidnotdependonthevolumefraction,buttheparticleconcentrationnL2d,asthetwoaspectratiosusedhadthesamelengthsbutdifferentdiameters.Toimprovetheanalysis,anintegralmethodforcalculatingtheextentofmigrationwasproposedwhichwouldhelpsmooththeuctuationsobservedinthedata.Thedynamicsofrodmigrationwerealsoinvestigatedbyobservingthetempo-ralevolutionoftheextentofmigration.Reachingsteadystateremainsasignicantchallengetothesemeasurements,asthestrainrequiredislarge,andtherearelargeuctuationsinthemotionoftheberswhichincreasethevariationinthemeasuredarealfractions.Itisexpectedthatsuspensionswithhigherberconcentrationwould 68

PAGE 69

reachafullydevelopedmicrostructurequickerthanlowerconcentrations,andtheeffectofconcentrationwouldbeindependentofaspectratio.Orientationdatawasalsoextractedfromtheprocessedimages.Theberori-entationwasquantiedusingtheanglemadebytheprojectionoftheberintheow-gradientplane,andthezwhichistheanglemadebytheberandthevorticitydirection.Resultsobtainedfromadistributionoftheanglezshowedanegligibleprob-abilityofaberaligninginthevorticitydirectionnearthecenterofthetube,butahighprobabilityofalignmentinthevorticitydirectionnearthewalls.Aprobabilitydistributionofindicatedstrongpreferenceforberstoalignintheowdirectionandnegligibleprobabilityofalignmentwiththegradientdirection. 69

PAGE 70

CHAPTER4CONCLUSIONSConcentratedsuspensionsareseeninnatureandhaveavarietyofapplicationsinindustry.Previousstudieshavefocusedmainlyondilutetosemi-dilutesuspensionsofsphericalparticles.Therehasalsobeensignicantexperimentalworkandnumericalanalysesperformedonconcentratedsuspensionsofspheres.However,thesestudiesarefarfrompracticalindustrialexamples,whichoftenconcernsuspensionsofparticlesthatarepolydisperseinsizeandshape.Forexample,elongatedparticlesareaddedtoconcreteslurriestoincreaseitsmechanicalstrength[ 2 ]androd-shapedparticlesareaddedtodrillinguidstochangetheirrheologicalproperties[ 4 ].Inmanufacturingprocesses,thereisalwaysaneedtotransporthighsolid-contentsuspensions.Furthermore,thereisasignicantneedforprogressinthemeasurementsofrheologicalproperties,modeling,andinterpretationsofthesemeasurementsinconcentratedsuspensions.Theworkpresentedinthisdissertationresolvesquestionsregardingmorepracticalsuspensionsandtheresultsrepresentasignicantimprove-mentinthemodelingofrheologicalrelationsbetweentheparticlevolumefractionandstresses,viscosities,frictioncoefcientsetc.,thatwillhelppredictthedynamicsofconcentratedsuspensionsofnon-colloidalrigidrods.Accuratemeasurementsandinterpretationofrheologicaldataofconcentratedsuspensionshelpinthedesign,performance,andenergyconsumptionofprocessequipment.Experimentswererunusinganovelpressure-imposedrheologytechniquesusingacustom-builtrheometer.Thistechniqueborrowsheavilyfromthefrictionalapproachtorheology,previouslyusedfordrygranularrheologyandnowadaptedtodensesuspensions(explainedinAppendix C ).ResultspresentedinSection 2.2 aretherstrheologicalmeasurementsfornon-colloidalrigidrodsatparticleconcentrationsnearmaximumpackingfractions.Aconspicuousresultwastheappearanceofyieldstressesinmeasuredvaluesofboththeshearandnormalcomponents,whichisanomalousfor 70

PAGE 71

Newtoniansuspensions,asdescribedinFigure 2-5 .Theoriginoftheseyieldstressesremainsunknown.However,severalpossibleexplanationsfortheoriginoftheyieldstresseswereproposedsuchasattractiveinteractionforcesandnite-sizeeffectsclosetothejammingpoint.SubtractingtheseyieldstressesrevealedaviscousscalingforboththeshearstressesaswellastheparticlepressuresasseeninFigure 2-4 .Wealsorelatedthedivergenceoftherheologicalparameterstothemaximumpackingfraction,andproposedrelationstomodelthisdivergence.Thismodelequationwasafunctionofthemaximumpackingfraction,mandisgivenas s()=14:51m)]TJ /F3 11.9552 Tf 11.955 0 Td[( m)]TJ /F5 7.9701 Tf 6.586 0 Td[(0:90:(4)TheexponentgiveninEquation 4 differstothatforspheresasdescribedbyBoyer[ 63 ],andshowninFigure 2-7 .Similarly,aconstitutivelawforthecoefcientoffrictionwasfound, ()=s+m)]TJ /F3 11.9552 Tf 11.955 0 Td[( +m)]TJ /F3 11.9552 Tf 11.955 0 Td[( 2;(4)whereandwerecalculatedfromttingtheexperimentaldata.ThevaluescalculatedfromFigure 2-7 (c)differedsignicantlyfromthatofspherescalculatedbyDagois-Bohy[ 64 ].Inaddition,thecollapseofourmeasurementsontoasinglecurve,alsoshowninFigure 2-7 ,furtherconrmsourhypothesisthat,athighconcentrations,short-rangelubricationinteractionsandparticlecontactsshouldcapturethedynamicsaccurately,andlong-rangehydrodynamicinteractionscouldsafelybeignored.Thisconclusionwouldgreatlyeasesthecomputationalcomplexitywhilerunningsimulationsofthesesystems.Futureworkregardingtherheologyofconcentratedbersuspensionswillmainlyconcerncomparingresultsobtainedfromthepressure-imposedrheometertothoseobtainedfromnumericalsimulations.Therearelimitedresultsfromnumericalsimu-lationsofconcentratedsuspensions,especiallynearthelimitofjamming.Ongoingworkincludessimulatingfeedbackloopononeoftheboundingwallsinresponseto 71

PAGE 72

theimposedloadappliedtothesuspension,whichissimilartothepressure-imposedrheologysetup.Initialworkiscurrentlyfocusedonsuspensionsofhardspheresforeaseofcomputations,butfutureworkwillincluderigidrods.Questionsregardingtheap-pearanceofyieldstressesandtheeffectoftheseyieldstressesonthebulkrheologicalmeasurementscanbeansweredvianumericalsimulations.Inaddition,themicrostruc-tureatjamming-limitconcentrationsislargelyunresolved.Athigherconcentrations,suspensionsofrigidrodsareexpectedtobealignedinthedirectionofshear.Figure 2-6 showedtherangeofaspectratiosandmaximumpackingfractionsrecordedinourexperiments.Furthermore,comparisonofvaluesofmaximumpackingfractionsob-tainedfromourexperimentswithsimulationsperformedbyPhillipseetal.showedthatthemaximumpackingfractionvalueswereclosertothoseforrandomlypacked,ratherthanhighlyordered,microstructures.Numericalsimulationswillhelpanswerquestionsregardingthemicrostructureintheseows.Ithasbeenestablishedthatthemicrostructureofconcentratedsuspensionsisaffectedby,andhasaneffecton,theimposedows.Thereislimitedworkthatinvestigatesthisnon-linearrelationshipforavarietyofows.Toquantifythedynamicsofthemicrostructureinconcentratedbersuspensions,experimentswereperformedusinganoscillatorypipeow.Shear-inducedmigrationisanoticeablemanifestationoftheirreversibledynamicsinthemicrostructureofconcentratedsuspensions.Althoughthisphenomenahasbeenstudiedforspheres[ 73 ],similarstudiesfornon-sphericalparticlesislacking.TheonlyevidenceofparticlemigrationwasshownviaexperimentsperformedbyMondy[ 10 ]inCouetteows.Chapter 3 highlightsquantiableevidenceofshear-inducedmigrationofnon-sphericalparticlesinpipeows.ExperimentsdescribedinSection 3.1 formtherstsetofexperimentstoshownon-sphericalparticlemigrationinmoregeneralows.Theseexperimentsarealsotherstmeasurementsoftheorientationdistributionofbersinpressure-drivenows,usingdirectopticalimagingtechniques(describedinSection 3.2 ).ResultsshowninSection 72

PAGE 73

3.3 showmigrationforsuspensionsatparticlevolumefractionsaslowas5.5%,whichissignicantlylowerthanthethresholdforspheres.Wealsoobservethatthebershaveadistinctpreferentialalignmentintheowdirection(seeSubsection 3.3.3 ),thoughweexpecttoobserveaninhomogenousorientationdistribution,whereinthebersnearthewallswillbepreferentiallyalignedinthevorticitydirection,whilethebersnearthecenterwillbealignedwiththeow.Thisvorticityalignmentisnotasurprisingresultandhasbeenshowntooccurviadirectnumericalsimulationsinshearingows[ 11 ].Similartotherheology,futureworkaboutthedynamicswillpredictthemicrostruc-tureofconcentratedsuspensionsofrigidrodsinpressure-drivenows.Thesesimu-lationswillbeusedtoconrmtheoccurrenceofshear-inducedmigrationaswellaspredicttherangesofparameters,likestrainamplitudeandbulkinitialconcentrationoverwhichthisphenomenaoccurs.Ithasbeenshownthatcollectivemotionlikeparticlemigrationaffectstheorientationdistribution,andnumericalsimulationswillbeusedtoquantifythiseffect.Thesuspensionbalancemodelhasbeenshowntoworkwelltopredictshearinducedmigrationinsuspensionsofspheres,andwillhencebeusedinanattempttomodelthisphenomenainsuspensionsofrigidrods.Furtherquestionsregardingtheeffectofconnementonparticlemigration,aswellasitseffectontheorientationdistribution,canalsobeansweredviathisapproach.Aframeworkforexperimentsanddataanalyseshasbeenprovidedtostudythesesystems,andourresultslookpromising.Thisworkwilltakeusonestepclosertogeneratinganaccuratecontinuummodelforthesesystems. 73

PAGE 74

APPENDIXARIGIDITYTESTSFORNON-SPHERICALPARTICLESWemakeasimplifyingassumptionthatthebersusedintheexperimentsarerigid:theydonotbendnorstretchundertheforcesandstressesimposedbytheow.Inthisappendix,criteriathathavebeendevelopedforestimatingtheoweldsatwhichthisassumptionmayfail.ForgacsandMason[ 87 ]developedatheoryforthecriticalstress,crit,atwhicharodwillbuckleinanaxialcompressionow.Foraberofaspectratio,A=L=d,whereLanddarethelengthanddiameter,thestresswasestimatedas, crit=Eb[ln(2A))]TJ /F7 11.9552 Tf 11.955 0 Td[(1:75] 2A4;(A-1)whereEbisthebendingmodulus,avaluethatisapproximatelytwicetheYoung'smodulusoftheber.Toconrmthetheory,ForgacsandMason[ 87 ]measuredthebendingofpolymerbers,havingarangeofbendingmoduli,asafunctionowstrength.Theyreportedgoodagreementbetweenthequalitativeobservationsandthepredictedvalueofcrit.Generally,anydeformationofaberisarguedtobenegligiblewhenthemaximumstressthatahydrodynamicowexertsismuchsmallerthanthecriticalstressgivenby A-1 .Forashearingow,themaximumstress(max)canbeapproximatedbytheproductoftheuidviscosity,fandmaximumvalueoftherateofshear,_.SwitzerandKlingenberg[ 88 ]simulatedamicromechanicalmodelofexiblebersandcalculatedthedynamicsandrheology.Theydendedarestoringtorque,Yi,thatincludedtheelasticforcesduetobendingaswellastwistingoftheber.Foreachsegmentiofadiscretizedber,therestoringtorquewasdenedas jYij=b(i)]TJ /F3 11.9552 Tf 11.955 0 Td[(eqi)+t(i)]TJ /F3 11.9552 Tf 11.955 0 Td[(eqi);(A-2)wherethebendingangle,i,andtwistingangle,i,ofthesegmentarereferencedtotheirequilibriumvalues,eqiandeqirespectively.Theconstantb=EyI=liisthebending 74

PAGE 75

constantoftheber,whichdependsupontheYoung'smodulusfortheber(Ey),theareamomentoftheber(I=r4=4withrthediameter),andthelengthofthesegmenti(li).Likewise,thetwistingconstant,tiscalculatedfromthePoissonratio.Simulationswereperformedforarangeofvaluesoftheeffectivestiffness,Seff, Seff=EyI 4f_L4;(A-3)adimensionlessparameterthatcomparesthebendingandhydrodynamicforces.UtilizingthedenitionofIandtheaspectratio,theequationcanberewrittenas Seff=Ey 64f_A4;(A-4)whichhasasimilardependenceontheberpropertiesanduidstressascrit=maxasdenedbyForgacsandMason[ 87 ],saveforthelogarithmiccorrection.AsSeff!1,thebersareconsideredrigid.Laugaetal.[ 89 ]andothers[ 90 ]characterizedthebucklinginstabilityusingthespermnumber,Sp,as Sp=8f_A4 Ey;(A-5)whicharisesfromtheratiooftheviscousforces(f_L2=2)totheelasticforces(Eyr4=L2)[ 91 ].ThespermnumberdenitionhasaninversedependenceontheberpropertiesandtheuidowtothevalueofSeff.Forthespeciccaseofacellularow,Wandersmanetal.[ 90 ]quantiedthepointatwhichaberwouldbuckle,andfoundthattheprobabilitygoestozeroasSpdropsbelow120.YoungandShelley[ 92 ]developedacorrectiontotheelasto-viscousnumber(i.e.spermnumber)utilizingslenderbodytheory.Asaresult,theyproducedadimensionlessparameter, Sp=128_A4 Ey[ln(2A))]TJ /F7 11.9552 Tf 11.955 0 Td[(0:5];(A-6) 75

PAGE 76

thatincludedalogarithmicdependenceontheaspectratio,similarlytotheoriginalworkofForgacsandMason[ 87 ].Foraberinacellularow,thecriteriaforrigiditywasfoundtobeSp400,withlowervaluesbeingrigid[ 92 ]. 76

PAGE 77

APPENDIXBCALCULATIONSANDCALIBRATIONSOFPRESSURE-IMPOSEDRHEOLOGYDATAThegoaloftherheologyexperimentsisthedeterminationoftherelationshipbetweenpropertiessuchasshearstress(),particlepressure(Pp),andvolumefraction()fordensesuspensionsofrigidbers.Thesequantitiesarecalculatedusingvaluesoftorque(T),normalforce(Fn),andgapheight(h)measuredfromtheshearcellshowninFigure B-1 .Inthisdevice,theuidislledtoaheighthfinacylindricalannulusofinnerandouterradiiR1=44mmandR2=90mm;themaximumllheighthf=22mm.Aporousplatethatallowsuidtopassthrough,butnotparticles,issubmergedintotheuidtoanadjustableheighth.Togenerateashearingow,thecylindricalannulusisafxedtoaplatethatcanrotateatangularvelocities!between0.1to0.7rad/swhiletheporousplateremainsxed.Hence,thedirectionofowisangularandthegradientdirectionisparallelwithgravity.BoththetorqueTandnormalforceFnaremeasuredbytransducersattachedtothetopplate.Limitsofthetransducersforthemeasurementoftorqueare0.01to1.2Nm,andthelowestaccuratemeasurementofnormalforceis0.8N.Themajoradvantageofperformingrheologicalmeasurementsinthecustom-builtrheometershowninFigure B-1 isthatexperimentscanbeperformedintwodifferentmodes:volume-controlledandpressure-controlledrheology.Involume-controlledexperiments,theheighthoftheporousplateisxedataspeciedvalue,whileinpressure-imposedrheologyexperiments,hisadjustedthroughafeedbackcontrolmechanismduringtheexperimentinordertomaintainthenormalforceFnatasetvalue.Duetothenatureoftheexperimentalsetupandphysicalconstraints,variouscorrectionsaremadetotherawdataobtainedfromtheexperimentaldevicetoderivethedesiredparameters.Thesetreatmentsaredescribedbelow. 77

PAGE 78

B.1Height,VolumeFraction,andNumberDensityThesimplestrelationshipisbetweentheheightofthetop-plateandtheresultantparticlevolumefraction.Asuspension,ofknownvolumefractioni,isaddedtotherheometertoallheightofhf.Notethatallheightsaremeasuredfromthebottomofthecell.Theparticlevolumefraction,,isnowdeterminedfromthesetheighthas =ihf h:(B-1)Incertaincases,thevolumefractionofrod-likeparticlesisexpressedintermsofadimensionlessnumberdensitynL2d,wherethenumberdensityn=N=ViscalculatedusingthenumberofparticlesNperunitvolumeV,wheretheparticleshavealengthLanddiameterd.Therelationbetweenthisdimensionlessnumberdensityandthevolumefractionis =)]TJ /F3 11.9552 Tf 5.48 -9.684 Td[(nL2d 4A;(B-2)wheretheaspectratioA=L=d. B.2TorqueandStressCalculationsThevalueofthetorqueTmeasuredbytherheometerishigherthanthevalueofthetorqueexertedbythesuspensionTb.Correctionsaremadetothemeasuredvaluesoftorquetoaccountforthemechanicallossesexperiencedbythesystem,T0,andthetorqueexertedbytheuidinthenarrowgapbetweenthemovabletopplateoftherheometerandthewallsoftheannularcell,Tg,asshowninFigure B-1 .Toevaluatethesecorrections,therheometeriscalibratedusingthesuspendinguidintheabsenceofparticles.Theequationforthecorrectionstothemeasuredtorquecanbewrittenas T=T0+Tg+Tb:(B-3)Themechanicallossesareassumedtohaveacomponentthatisconstantandacomponentthatisproportionaltotherotationrate!.SincethesuspendinguidisNewtonian,thetorquecontributionsTgandTbareproportionalto!.Astheheightis 78

PAGE 79

FigureB-1.Schematicoftherheometercell.Thetopplateisporous,canbemovedonlyintheverticaldirection,andisalwayskeptsubmergedduringtheexperiment.Thebottomportionoftherheometercellisxedtoabasethatcanberotatedatacontrolledvalue!,whilethetorqueTandnormalforceFnaremeasuredontheshaftattachedtothetopplate.Theschematicalsoshowsthegapbetweentheedgeofthetopplateandtheinnerwalloftheannularcell(notdrawntoscale),whichallowsfortheunhinderedmovementoftheplate.Thegapissufcientlysmallthatparticlescannotpassthroughit. 79

PAGE 80

changed,neitherT0norTgareaffected.However,theshearrate,andhencetorque,isinverselyproportionaltotheheightinthebulkregionoftherheometer.IncludingtheserelationsinEquation B-3 gives T=a+b!+c! h;(B-4)wherea,b,andcareconstants.Thevaluesofaandbaredeterminedbyperformingaleast-squaresttomeasurementsofTforaspanofvaluesof!andh.Ofcoursecisalsodeterminedfromtheanalysis,butthisvalueisnotneededforthefurtherdevelopments.Furthermore,thevaluesofaandbareassumedtonotchangesolongasthesuspendinguidisnotaltered.Particlesareaddedtothesuspendinguidoncethecorrectionsaredeterminedfromtheexperimentsontheuidintheabsenceoftheparticles.Thenthetorqueduetothesuspensionatanyvalueof!andhiscalculatedfromthemeasuredtorqueT(!;h)byusingthecorrections, Tb(!;h)=)]TJ /F7 11.9552 Tf 11.291 0 Td[((a+b!)+T(!;h):(B-5)ThegoalistorelatethetorqueTb(!;h)tothesuspensionstress,whichisafunctionoftherateofshearandvolumefraction,(_;).Thenettorqueexertedbythebulkofthesuspendinguidisgivenbyintegratingtheproductoftheshearstressandradialdistancefromtherotationaxisoverthetopplate, Tb(!;h)=ZS(_;)rdS:(B-6)Itisconvenienttoworkinthecylindricalcoordinatesystemwhere,afterintegratingovertheangularcoordinate, Tb(!;h)=2ZR2r=R1(_;)r2dr:(B-7) 80

PAGE 81

Inallthatfollows,itisassumedthattherateofshearatanyradialpositioncanberelatedsimplytotherotationrateandtheheightoftheplatethrough _(r)=!r h:(B-8)Thisassumptionignorestheinuenceofthesidewallsatr=R1andR2ontheowprole.Also,allmeasurementsarerecordedonlyaftertheowproleandassociatedtorquehaveattainedasteadystatevalue,hencethetime-dependentpropertiesoftheshearratearenotrequired.Equation B-7 isill-posed,asareallintegralequations,sincetheunknownappearswithintheintegrand.Solvingrequiresregularizingtheproblembyimposingafunctionalformontheunknownstress.OneapproachistoassumethatthesuspensionfollowsageneralizationoftheNewtonianuidconstitutivelaw.Inpreviousworkusingthisdeviceonconcentratedsuspensionsofspheres,thatapproachwasutilized[ 63 ].However,publishedexperimentalresultshaveconsistentlydemonstratedthatsuspensionsofrigidrodsshear-thin[ 93 ].Consequently,hereweconsiderbothageneralizedNewtonianuidmodelandamoregeneralone. B.2.1GeneralizedNewtonianuidmodelInthegeneralizedNewtonianmodel,thestressofthesuspensionisassumedtoscalelinearlywiththerateofshear.Asaconvenientreferencepoint,allresultsobtainedfromtheexperimentsarereportedbaseduponvaluesofshearrateatthemidpointoftheannulus, _c=!(R1+R2) 2h;(B-9)wherethestressis(_c).Consequently,thestressatanyradialpositioninthecellcanbewrittenas (r)=2(_c)r (R1+R2):(B-10) 81

PAGE 82

AftersubstitutingthisintoEquation B-7 ,integrating,andthensolving,thestresscanberelatedtothetorquemeasuredattherotationrate!, (_c)=Tb(R1+R2) (R42)]TJ /F3 11.9552 Tf 11.955 0 Td[(R41):(B-11)UsingEquation B-11 wecannowcalculatetheshearstressgeneratedbythesus-pensionfordifferentshearratesmeasuredviadifferentrotationrates!andheightsh. B.2.2GeneralStressModelAnalternativeapproachtocalculatingthestressassumesamoregeneralformoftheconstitutivelawusedtoregularizetheintegralequation(Equation B-7 ).Inthismethod,thestressislinearizedaroundthevalueofthestressatthecenterofthecellwheretherateofshearis_c=!(R1+R2)=2h, (_)=(_c)+(_)]TJ /F7 11.9552 Tf 13.69 0 Td[(_c)@ @__c:(B-12)SubstitutingintoEquation B-7 andintegratinggives T(!)=2"A(_c)+(Bj!)]TJ /F7 11.9552 Tf 13.691 0 Td[(_cA)@ @__c#;(B-13)where A=ZR2r=R1r2dr(B-14)and Bj!=ZR2r=R1!r3 hdr:(B-15)Toeliminatethegradientofthestresswithrespecttotherateofshear,asecondmeasurementofthetorqueismadeatarateofrotationthatisdifferentbyadifferentialamount!, T(!+!)=2"A(_c)+)]TJ /F3 11.9552 Tf 6.675 -9.683 Td[(Bj!+!)]TJ /F7 11.9552 Tf 13.69 0 Td[(_cA@ @__c#:(B-16) 82

PAGE 83

Solvingfor@=@_j_c,thendividingbythedifferentialrotationrate!andtakingthelimitasitgoestozerogives @ @__c=1 2@B(!) @!!)]TJ /F5 7.9701 Tf 6.586 0 Td[(1@T(!) @!!:(B-17)Usingthisresult,anexplicitexpressionforthestresscanbecalculatedas (_c)=1 2A"T(!)+(_cA)]TJ /F3 11.9552 Tf 13.15 0 Td[(Bj!)@B(!) @!!)]TJ /F5 7.9701 Tf 6.587 0 Td[(1@T(!) @!!#:(B-18)EvaluatingtheintegralsAandB,aswellasthegradientofB,givestheexplicitexpres-sionforthestress, (_c)=3 2(R32)]TJ /F3 11.9552 Tf 11.956 0 Td[(R31)T(!)+2(R1+R2)(R32)]TJ /F3 11.9552 Tf 11.955 0 Td[(R31))]TJ /F7 11.9552 Tf 11.955 0 Td[(3(R42)]TJ /F3 11.9552 Tf 11.955 0 Td[(R41) 3(R42)]TJ /F3 11.9552 Tf 11.955 0 Td[(R41)!@T(!) @!!:(B-19)ForasuspensionsystemthatfollowsageneralizedNewtonianlaw,theresultgivenbyevaluatingEquation B-19 reducestotheresultderivedbyassumingaNewtonianuidfromtheoutset(Equation B-11 ). B.3NormalForceMeasurementsWhenconcentratedsuspensionsofrigid,non-colloidalparticles,areacteduponbyanimposedow,theyexhibitnon-Newtonianbehaviorathighparticlevolumefractions.Theexistenceofnormalstressesisoneexampleofnon-Newtonianphenomena.Twomechanismspossiblycontributetonormalstressesinsuspensions:hydrodynamiccon-tributionsandcontributionsduetocontactsbetweenparticles.Forsuspensionsofrods,calculationshaveindicatedthatthenormalstressesareweakandareoverwhelminglyduetocontacts[ 11 ].However,hydrodynamicsareanessentialpartoftheproblemstill,sincethecontactinteractionsdependonthemicrostructureofthesuspensionandtheowisresponsibleforalteringthemicrostructure.Inourrheologyexperiments,aconcentratedsuspensionisshearedresultinginthehinderedrotationofrodsduetocontactsbetweennearneighbors.Thecontactforcesthatthismotioncausesresultsinaforcechainbetweentheparticles.Adensenetwork 83

PAGE 84

FigureB-2.Schematicshowingforcesexertedbytherodsonthetopplate.Asmallmeshsizeallowstheuidtomovefreelyacrossthetop-plateandmaintainsaconstantmassofparticlesinthecell.Contactforcechainsarecausedbyhighnumbersofcontactsbetweenparticlesathighparticlevolumefractions. oftheseforcechainsoccurinadensesuspensionathighvolumefractionsduetothelargenumberofparticlecontacts.Theseforcenetworksexertanormalforceonthetopplateoftherheometer,asvisualizedinFigure B-2 .Therheometercanmeasurethisnormalforceusingaforcetransducercontainedinanaccuratebalance.Thenormalforceexertedbytheuidisnotequivalenttotheforcemeasuredbytheinstrument.Inadditiontothecorrectionstothetorquedescribedpreviously,calibrationexperimentsusingonlythesuspendinguidareusedtoobtaincorrectionstothenormalforceaswell.ItisassumedthatsincetheuidisNewtonian,itdoesnotexertanynormalforceonthetopplate.ThevalueobtainedfromtheforcetransducerFmeasuredbythebalanceincludestheforceexertedbyweightofthetop-plateFtandthebuoyancyforceFbexertedbytheuidontheplate.Theweightoftheplateisconstantandcanberemoveddirectlyfromthemeasuredforce.Thecorrectionforthebuoyancyforcedependsdependsontheheighth:thetopplatemust 84

PAGE 85

becompletelysubmergedintheuidfortheentiredurationoftheexperiment,butastheheightchanges,more(orless)ofthebracketsthatsupporttheweightoftheplateFtaresubmerged.Consequently,thebuoyancycorrectionissimplydeterminedfromthecalibrationexperimentandusedtocorrectthemeasureforce, ^Fn=F)]TJ /F3 11.9552 Tf 11.955 0 Td[(Ft+Fb;(B-20)where^Fnisthenormalforceexertedonthetopplateoftherheometer.AnexpressionfortheresultingpressurePpontheplateisgivenby Pp=^Fn (R22)]TJ /F3 11.9552 Tf 11.955 0 Td[(R21):(B-21) 85

PAGE 86

APPENDIXCUNIFYINGRHEOLOGYOFDENSESUSPENSIONSANDGRANULARMEDIAUniversalconstitutivelawshavebeenproposedfordensegranularows[ 94 95 96 ].Boyeretal.demonstratedthatdensesuspensionsofhardspheresofdiameterdanddensitypshearedatarateof_underaconningpressurePpcanbetreatedusingsimilarconstitutiveequationsasgranularmedia.Applyinggranularconceptstodensesuspensionsrequiresmodicationstotheexperimentalmethods.Inthecaseofdensegranularmediaofsphericalparticles,onedimensionlessnumber,canbeusedasacontrolvariable:I=dp p=Pp_.ThisinertialnumberIisaratiobetweentheinertialtimeofrearrangement,tmicro=dp p=Pp,andthetimescaleofthestraintmacro=1=_.Thegranularrheologyisthendescribedusingtwofunctionsoftheinertialnumberforawiderangeofows:theshearstress=(I)Pp,andvolumefraction,=(I).Boyeretal.demonstratedthattheseideascouldbeappliedtodensesuspen-sionsofhardspheres[ 63 ]withafewmodications.Fromaphysicalstandpoint,theexperimentsmustbemodiedbyallowingthesuspendinguidtopassthroughtheboundaryoftheshearingcell.Thisenablescompressionoftheparticlephase.Inthiscaseofviscoussuspensionsofhardspheres,thetheorymustbemodiedbyconsid-eringthedominantforcesthatareviscousratherthaninertialsincetheStokesnumberSt=pd2_=fissmall.Theinternaltimeofrearrangementcanbedenedusingaviscousscalingastmicro=f=Pp.ThissystemcannowbecharacterizedusingadimensionlessviscousnumberJas J=f_ Pp;(C-1)wherefistheviscosityofthesuspendinguid.Macroscopicpropertieslikethecoefcientoffriction,,andthevolumefraction,,shouldnowbefunctionsofJalone, =(J)Ppand=(J):(C-2) 86

PAGE 87

Conventionalrheologyexperimentsareperformedusingacontrolledvolume.How-ever,thesevolume-controlledexperimentscanbereconciledwiththeabove-describedpressure-controlledexperiments.Whenasuspensionisshearedataconstantvolumefraction,shearandnormalstressesscaleviscouslyasf_andcanbeexpressedasfunctionsof =s()f_andPp=n()f_;(C-3)wheres()andn()arethedimensionlessshearandnormalviscosities,respectively[ 97 82 ].RelationscanbedrawnintermsofthedimensionlessviscousnumberJtodeterminetheparticlepressureandtheshearstressas Pp=1 J()f_and=[J()] J()f_:(C-4)Furthermore,thiscanbeusedwithEquation C-3 whichgives s()=[J()] J()andn()=1 J():(C-5) 87

PAGE 88

REFERENCES [1] R.G.Larson.TheStructureandRheologyofComplexFluids.OUPUSA,1999. [2] M.Hassanpoura,P.Shaghb,andH.B.Mahmudb.LightweightaggregateconcreteberreinforcementAreview.Constr.Build.Mater.,37:452,2012. [3] C.H.Bivens,C.Boney,C.Fredd,J.Lassek,P.Sullivan,J.Engels,E.O.Fielder,T.Gorham,T.Judd,A.E.S.Mogollon,L.Tabor,A.V.Munoz,andD.Willberg.Newbersforhydraulicfracturing.OileldReview(publishedbySchlumbergerLtd.),SummerIssue:34,2005. [4] R.Elgadda,R.Ahmed,M.George,andF.Growcock.Settlingbehaviorofsphericalparticlesinber-containingdrillinguids.J.Petr.Sci.Engr.,84:20,2012. [5] AAcrivos.BinghamAwardLectureShear-inducedparticlediffusioninconcentratedsuspensionsofnoncolloidalparticles.J.Rheol.,39:813,1995. [6] D.LeightonandA.Acrivos.Theshear-inducedmigrationofparticlesinconcen-tratedsuspensions.J.FluidMech.,181:415,1987. [7] D.LeightonandA.Acrivos.Measurementofshear-inducedself-diffusioninconcentratedsuspensionsofspheres.J.FluidMech.,177:109,1987. [8] R.J.Phillips,R.C.Armstrong,R.A.Brown,A.L.Graham,andJ.RAbbott.Aconstitutiveequationforconcentratedsuspensionsthataccountsforshear-inducedparticlemigration.Phys.FluidsA:FluidDynamics,4(1):30,1992. [9] P.R.NottandJ.F.Brady.Pressure-drivenowofsuspensions:simulationandtheory.J.FluidMech.,275:157,1994. [10] L.A.Mondy,HBrenner,S.A.Altobelli,J.R.Abbott,andA.L.Graham.Shear-inducedparticlemigrationinsuspensionsofrods.J.Rheol.,38(2):444,1994. [11] B.Snook,L.M.Davidson,J.E.Butler,O.Pouliquen,andE.Guazzelli.Normalstressdifferencesinsuspensionsofrigidbres.J.FluidMech.,758:486,2014. [12] M.Keshtkar,M.C.Heuzey,andP.J.Carreau.Rheologicalbehaviorofber-lledmodelsuspensions:Effectofberexibility.J.Rheol.,53:631,2009. [13] S.Bounoua,P.Kuzhir,andE.Lemaire.Normalstressdifferencesinnon-Brownianbersuspensions.J.Rheol.,60(4):661,2016. [14] J.Park,J.M.Bricker,andJ.E.Butler.Cross-streammigrationindilutesolutionsofrigidpolymersundergoingrectilinearownearawall.Phys.Rev.E,76:040801(R),2007. 88

PAGE 89

[15] J.ParkandJ.E.Butler.InhomogeneousdistributionofarigidbreundergoingrectilinearowbetweenparallelwallsathighPecletnumbers.J.FluidMech.,630:267298,2009. [16] R.HsuandP.Ganatos.Gravitationalandzero-dragmotionofaspheroidadjacenttoaninclinedplaneatlowReynoldsnumber.J.FluidMech.,268:267,1976. [17] G.B.Jeffery.Themotionofellipsoidalparticlesimmersedinaviscousuid.Proc.R.Soc.LondonA,102:161,1922. [18] F.P.Bretherton.ThemotionofrigidparticlesinashearowatlowReynoldsnumber.J.FluidMech.,14:284,1962. [19] B.J.TrevelyanandS.G.Mason.Particlemotionsinshearedsuspensions.I.Rotations.J.ColloidSci.,6:354,1951. [20] S.G.andR.J.Manley.Particlemotionsinshearedsuspensions:Orientationsandinteractionsofrigidrods.Proc.R.Soc.London,Ser.A,238:117,1956. [21] G.K.Batchelor.Slender-bodytheoryforparticlesofarbitrarycross-sectioninStokesow.J.FluidMech.,44:419,1970. [22] R.G.Cox.Themotionoflongslenderbodiesinaviscousuid.Part1.Generaltheory.J.FluidMech.,44:791,1970. [23] M.DoiandS.F.Edwards.TheTheoryofPolymerDynamics.OxfordUniversityPress:NewYork,1986. [24] J.E.Butler.Collectivedynamicsofparticlesinviscousowswithanemphasisonslenderrods.InCollectiveDynamicsofParticles,pages99.Springer,2017. [25] S.KimandS.J.Karrila.Microhydrodynamics:PrinciplesandSelectedApplica-tions.Butterworth-Heineman,2005. [26] G.K.Batchelor.Thestressgeneratedinanon-dilutesuspensionofelongatedparticlesbypurestrainingmotion.J.FluidMech.,46:813,1971. [27] M.B.MackaplowandE.S.G.Shaqfeh.Anumericalstudyoftherheologicalpropertiesofsuspensionsofrigid,non-Brownianbres.J.FluidMech.,329:155186,1996. [28] A.SalahuddinandC.K.Wu,J.andAidun.Studyofsemidilutebresuspensionrheologywithlattice-Boltzmannmethod.RheologicaActa,52(10-12):891,2013. [29] X.Fan,N.Phan-Thien,andR.Zheng.Adirectsimulationofbresuspensions.J.Non-NewtonianFluidMech.,74:113,1998. [30] BradenSnook.Thedynamicsofthemicrostructureandtherheologyinsuspen-sionsofrigidparticles.PhDthesis,UniversityofFlorida,Gainesville,FL,2015. 89

PAGE 90

[31] Y.Yamane,Y.Kaneda,andM.Dio.Numericalsimulationofsemi-dilutesuspen-sionsofrodlikeparticlesinshearow.J.Non-NewtonianFluidMech.,54:405,1994. [32] C.A.Stover,D.L.Koch,andC.Cohen.Observationsofbreorientationinsimpleshearowofsemi-diluesuspensions.J.FluidMech.,238:277,1992. [33] R.L.Powell.Rheologyofsuspensionsofrodlikeparticles.J.Stat.Phys.,62,1991. [34] E.GananiandR.L.Powell.Suspensionsofrodlikeparticles:Literaturereviewanddatacorrelations.J.Comp.Mat.,19:194,1984. [35] S.M.DinhandR.C.Armstrong.Arheologicalequationofstateforsemiconcen-tratedbersuspensions.J.Rheol.,28:207,1984. [36] E.S.G.ShaqfehandH.Fredrickson.Thehydrodynamicstressinasuspensionofrods.Phys.FluidsA,2:7,1990. [37] R.L.SchiekandE.S.G.Shaqfeh.Anonlocaltheoryforstressinbound,Browniansuspensionsofslender,rigidbers.J.FluidMech.,296:271,1995. [38] M.A.Bibbo,S.M.Dinh,andR.C.Armstrong.Shearowpropertiesofsemicon-centratedbersuspensions.J.Rheol.,29:905,1985. [39] R.O.MaschmeyerandC.T.Hill.Rheologyofconcentratedsuspensionsofbersintubeow.II.Anexploratorystudy.Trans.Soc.Rheol.,21:183,1977. [40] M.Sepehr,P.J.Carreau,M.Moan,andG.Ausias.Rheologicalpropertiesofshortbermodelsuspensions.J.Rheol.,48:1023,2004. [41] C.P.J.Bennington,R.J.Kerekes,andJ.R.Grace.Theyieldstressofbresuspensions.Can.J.Chem.Eng.,68:748,1990. [42] D.LeightonandA.Acrivos.Theshear-inducedmigrationofparticlesinconcen-tratedsuspensions.J.FluidMech.,181:415,1987. [43] A.Karnis,H.Goldsmith,andS.Mason.Thekineticsofowingdispersions:I.Concentratedsuspensionsofrigidparticles.J.ColloidInterfaceSci.,22:531,1966. [44] J.E.Butler,P.D.Majors,andR.T.Bonnecaze.Shear-inducedparticlemigrationforoscillatoryowofasuspensionwithinatube.Phys.Fluids,11:2865,1999. [45] R.E.Hampton,A.A.Mammoli,A.L.Graham,N.Tetlow,andS.A.Altobelli.Migrationofparticlesundergoingpressure-drivenowinacircularconduit.J.Rheol.,41(3):621,1997. [46] S.A.Altobelli,R.C.Givler,andE.Fukushima.Velocityandconcentrationmea-surementsofsuspensionsbynuclearmagneticresonanceimaging.J.Rheol.,35:721,1991. 90

PAGE 91

[47] C.J.Koh,P.Hookham,andL.G.Leal.Anexperimentalinvestigationofcon-centratedsuspensionowsinarectangularchannel.J.FluidMech.,266:132,1994. [48] M.K.LyonandL.G.Leal.Anexperimentalstudyofthemotionofconcentratedsuspensionsintwo-dimensionalchannelow.Part1:Monodispersesystems.J.FluidMech.,363:2556,1998. [49] J.T.Norman,H.V.Nayak,andR.T.Bonnecaze.Migrationofbuoyantparticlesinlow-Reynolds-numberpressure-drivenows.J.FluidMech.,523:135,2005. [50] A.Okagawa,R.G.Cox,andS.G.Mason.Thekineticsofowingdispersions.VI.Transientorientationandrheologicalphenomenaofrodsanddiscsinshearow.J.ColloidInterfaceSci.,45(2):303,1973. [51] D.J.Pine,J.P.Gollub,J.F.Brady,andA.M.Leshansky.Chaosandthresholdforirreversibilityinshearedsuspensions.Nature,438:997,2005. [52] F.R.DaCunhaandE.J.Hinch.Shear-induceddispersioninadilutesuspensionofroughspheres.J.FluidMech.,309:211223,1996. [53] G.Drazer,J.Koplik,B.Khusid,andA.Acrivos.Deterministicandstochasticbehaviourofnon-brownianspheresinshearedsuspensions.JournalofFluidMechanics,460:307,2002. [54] B.Metzger,P.Pham,andJ.E.Butler.Irreversibilityandchaos:Roleoflubricationinteractionsinshearedsuspensions.Phys.Rev.E,87:052304,May2013. [55] E.GananiandR.L.Powell.Suspensionsofrodlikeparticles:Literaturereviewanddatacorrelations.J.CompositeMater.,19:194,1985. [56] A.MongruelandM.Cloitre.Shearviscosityofsuspensionsofalignednon-Brownianbres.RheologicaActa,38(5):451,November1999. [57] M.ChaoucheandD.L.Koch.Rheologyofnon-Brownianrigidbersuspensionswithadhesivecontacts.J.Rheol.,45(2):369,March2001. [58] S.Bounoua,E.Lemaire,J.Ferec,G.Ausias,A.Zubarev,andP.Kuzhir.Apparentyieldstressinrigidbresuspensions:Theroleofattractivecolloidalinteractions.J.FluidMech.,802:611,August2016. [59] S.Bounoua,E.Lemaire,J.Ferec,G.Ausias,andP.Kuzhir.Shear-thinningincon-centratedrigidbersuspensions:Aggregationinducedbyadhesiveinteractions.J.Rheol,60(6):1279,2016. [60] M.A.Bibbo.Rheologyofsemiconcentratedbersuspensions.PhDthesis,MassachusettsInstituteofTechnology,1987. 91

PAGE 92

[61] R.SundararajakumarandD.L.Koch.Structureandpropertiesofshearedbersuspensionswithmechanicalcontacts.J.Non-NewtonianFluidMech.,73:205239,1997. [62] M.P.PetrichandD.L.Koch.Interactionsbetweencontactingbers.Phys.Fluids,10(8):2111,1998. [63] F.Boyer,E.Guazzelli,andO.Pouliquen.Unifyingsuspensionandgranularrheology.Phys.Rev.Lett.,107(18):188301,October2011. [64] S.Dagois-Bohy,S.Hormozi,E.Guazzelli,andO.Pouliquen.Rheologyofdensesuspensionsofnon-colloidalspheresinyield-stressuids.J.FluidMech.,776:R2,July2015. [65] L.E.BeckerandM.J.Shelley.Instabilityofelasticlamentsinshearowyieldsrstnormalstressdifferences.Phys.Rev.Lett.,87(19):198301,2001. [66] Y.N.YoungandM.J.Shelley.Stretch-coiltransitionandtransportofbersincellularows.Phys.Rev.Lett.,99(5):58303,2007. [67] O.Rahli,L.Tadrist,andR.Blanc.Experimentalanalysisoftheporosityofran-domlypackedrigidbers.C.R.Acad.Sci.,Ser.IIb:Mec.,Phys.,Chim.,Astron.,327(8):725,October1999. [68] S.R.WilliamsandA.P.Philipse.Randompackingsofspheresandspherocylin-derssimulatedbymechanicalcontraction.Phys.Rev.E,67(5):051301,May2003. [69] J.R.Abbott,N.Tetlow,A.L.Graham,S.A.Altobelli,EiichiFukushima,L.A.Mondy,andT.S.Stephens.Experimentalobservationsofparticlemigrationinconcentratedsuspensions:Couetteow.J.Rheol.,35(5):773,1991. [70] A.W.Chow,S.W.Sinton,J.H.Iwamiya,andT.S.Stephens.Shearinducedparticlemigrationincouetteandparallelplateviscometers:Nmrimagingandstressmeasurements.Phys.Fluids,6(8):2561,1994. [71] N.Tetlow,A.L.Graham,M.S.Ingber,S.R.Subia,L.A.Mondy,andS.A.Altobelli.Particlemigrationinacouetteapparatus:Experimentandmodeling.J..Rheol.,42(2):307,1998. [72] Z.FangandN.Phan-Thien.Numericalsimulationofparticlemigrationinconcen-tratedsuspensionsbyanitevolumemethod.JournalofNon-NewtonianFluidMechanics,58(1):6781,1995. [73] B.Snook,J.E.Butler,andE.Guazzelli.Dynamicsofshear-inducedmigrationofsphericalparticlesinoscillatorypipeow.J.FluidMech.,786:128,2016. 92

PAGE 93

[74] M.S.Ingber,VorobieffP.Mammoli,A.A.,T.McCollam,andA.L.Graham.Exper-imentalandnumericalanalysisofirreversibilitiesamongparticlessuspendedinacouettedevice.J.Rheol.,50(2):99,2006. [75] P.Pham,B.Metzger,andJ.E.Butler.Particledispersioninshearedsuspensions:Crucialroleofsolid-solidcontacts.Phys.Fluids,27:051701,2015. [76] I.M.Janosi,T.Tel,D.E.Wolf,andJ.A.C.Gallas.Chaoticparticledynamicsinviscousows:Thethree-particleStokesletproblem.Phys.Rev.E,65:2858,1997. [77] B.MetzgerandJ.E.Butler.Irreversibilityandchaos:Roleoflongrangehydrody-namicinteractionsinshearedsuspensions.Phys.Rev.E,82:051406,2010. [78] B.Metzger,P.Pham,andJ.E.Butler.Irreversibilityandchaos:Roleoflubricationinteractionsinshearedsuspensions.Phys.Rev.E,87,2013. [79] K.YeoandM.R.Maxey.Numericalsimulationsofconcentratedsuspensionsofmono-disperseparticlesinaPoiseuilleow.J.FluidMech.,682:491,2011. [80] I.M.Krieger.Rheologyofmonodisperselatices.Adv.Coll.InterfaceSci.,3(2):111136,1972. [81] J.M.BrickerandJ.E.Butler.Oscillatoryshearofsuspensionsofnoncolloidalparticles.J.Rheol.,50:711,2006. [82] J.F.MorrisandF.Boulay.Curvilinearowsofnoncolloidalsuspensions:Theroleofnormalstresses.J.Rheol.,43:1213,1999. [83] D.Lhuillier.Migrationofrigidparticlesinnon-brownianviscoussuspensions.Phys.Fluids,21(2):023302,2009. [84] P.R.Nott,E.Guazzelli,andO.Pouliquen.Thesuspensionbalancemodelrevisited.Phys.Fluids,23:043304,2011. [85] A.Franceschini,E.Filippidi,E.Guazzelli,andD.J.Pine.Transversealignmentofbersinaperiodicallyshearedsuspension:Anabsorbingphasetransitionwithaslowlyvaryingcontrolparameter.Phys.Rev.Lett.,107:250603,2011. [86] J.M.BrickerandJ.E.Butler.Correlationbetweenstructuresandmicrostructuresinconcentratedsuspensionsofnon-Browniansphericalparticlessubjecttounsteadyshearows.J.Rheol.,51:735,2007. [87] O.L.ForgacsandS.G.Mason.Particlemotionsinshearedsuspensions:IX.Spinanddeformationofthreadlikeparticles.J.ColloidSci.,14(5):457,1959. [88] L.H.SwitzerandD.J.Klingenberg.Rheologyofshearedexiblebersuspensionsviaber-levelsimulations.J.Rheol.,47(3):759,2003. 93

PAGE 94

[89] E.Lauga.Floppyswimming:Viscouslocomotionofactuatedelastica.Phys.Rev.E,75:041916,Apr2007. [90] E.Wandersman,N.Quennouz,M.Fermigier,A.Lindner,andO.DuRoure.Buckledintranslation.SoftMatter,6:5715,2010. [91] C.H.WigginsandR.E.Goldstein.FlexiveandpropulsivedynamicsofelasticaatlowReynoldsnumber.Phys.Rev.Lett.,80:3879,1998. [92] Y.N.YoungandM.J.Shelley.Stretch-coiltransitionandtransportofbersincellularows.Phys.Rev.Lett.,99(5):058303,2007. [93] S.Bounoua,P.Kuzhir,andE.Lemaire.Normalstressdifferencesinnon-Brownianbersuspensions.J.Rheol.,60(4):661,2016. [94] F.daCruz,S.Emam,M.Prochnow,J.Roux,andF.Chevoir.Rheophysicsofdensegranularmaterials:Discretesimulationofplaneshearows.Phys.Rev.E,72:021309,2005. [95] P.Jop,Y.Forterre,andO.Pouliquen.Aconstitutivelawfordensegranularows.Nature,441(7094):727,062006. [96] Y.ForterreandO.Pouliquen.Flowsofdensegranularmedia.Ann.Rev.FluidMech.,40(1):1,2008. [97] J.J.StickelandR.L.Powell.Fluidmechanicsandrheologyofdensesuspensions.Annu.Rev.FluidMech.,37:129,2005. 94

PAGE 95

BIOGRAPHICALSKETCHSaifShaikhwasborninMumbai,India.Hismotherwasapre-schoolteacherandhisfatherwasalaboratorytechnicianforaninstituteofcardiologyinPune,India.HeattendedSt.Vincent'sHighSchoolforhiselementary,middle,andhighschoolwhereheplayedbasketballandwaspartoftheboyscoutmovement.SaifattendedtheUniversityofPune,wherehereceivedaBachelorofEngineeringdegree,majoringinChemicalEngineering.HisfavoritecoursesattheUniversityofPunewereuidmechanics,andengineeringmaterials.HepursedaresearchinternshipattheNationalChemicalLaboratoryinPune,IndiaunderthesupervisionofDr.SanjeevTambeafterhisundergraduatestudies.Uponthecompletionofhisinternship,SaifwasadmittedtothegraduateprogramattheUniversityofFloridawhereheobtainedaMasterofSciencedegree,majoringinChemicalEngineering.HeparticipatedinaresearchprogramunderDr.JasonE.ButlerwhichledhimtobeacceptedtothePhDprogramupongraduation.HecontinuedhiseducationattheUniversityofFloridainchemicalengineeringandAix-MarseilleUniversityinphysicsofuids.Hewasco-advisedbyDr.JasonE.ButlerattheUniversityofFloridaandDr.ElisabethGuazzelliatAix-MarseilleUniversity.HecarriedoutexperimentalworkatAix-MarseilleUniversityandperformeddataanalysesattheUniversityofFlorida.HereceivedadualPhDinthesedisciplinesin2017andislookingforwardtocontinuingworkinuidmechanicsandrheologyinthepharmaceuticalormanufacturingindustry. 95