Citation
The Dynamics of the Microstructure and the Rheology in Suspensions of Rigid Particles.

Material Information

Title:
The Dynamics of the Microstructure and the Rheology in Suspensions of Rigid Particles.
Creator:
Snook, Braden J
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (156 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:
LADD,ANTHONY J
Committee Members:
CURTIS,JENNIFER S
MEI,RENWEI
GUAZZELLI,ELISABETH
ZOUESHTIAGH,FARZAM
POULIQUEN,OLIVIER
LEMAIRE,ELISABETH
Graduation Date:
5/2/2015

Subjects

Subjects / Keywords:
Aspect ratio ( jstor )
Experimental results ( jstor )
Hydrodynamics ( jstor )
Material concentration ( jstor )
Rheology ( jstor )
Shear flow ( jstor )
Shear stress ( jstor )
Simulations ( jstor )
Velocity ( jstor )
Vorticity ( jstor )
Chemical Engineering -- Dissertations, Academic -- UF
fibers -- multiphase-flow -- spheres -- suspensions
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:
Numerical and experimental methods have been used to reveal the complex relationship between the macroscopic properties and the microstructure, or relative arrangement of particles, of a suspension. Both spheres and fibers were considered. For fibers, a numerical model was developed to predict particle motion which considers the hydrodynamic drag experienced by a particle due to the applied flow and a short range repulsive force to maintain the excluded volume. The model was used to simulate steady and oscillatory shear flow with fully periodic and confined boundary conditions. Comparison with previous calculations and experiments for steady shear indicates that long and short range hydrodynamic interactions are not required to accurately predict the fiber motion at the concentrations simulated. The results aid in explaining current discrepancies in the literature between numerical predictions and experimental observations of the microstructure while also improving the understanding of the effect that fiber aspect ratio and concentration have on particle dynamics. This same model was used to interpret recent experimental observations of fiber alignment perpendicular to the flow direction when subjected to oscillatory shear flow. Numerical predictions agree with experimental observations only when fiber motion is restricted in the gradient direction by bounding walls; fully periodic simulations do not reproduce the experimental measurements. It is also shown that bounding walls can affect the microstructure in steady shear flow for low aspect ratio fibers. In addition to these numerical efforts, experiments were performed for fibers in steady shear flow with the purpose of measuring the first and second normal stress differences. The measurements are in quantitative agreement with numerical predictions, finding that the first normal stress difference is positive and approximately twice the magnitude of the second normal stress difference, which is negative. The predictions show that the contact contribution to the stress tensor must be included in the calculation to accurately predict the normal stress differences. As the contact force acts perpendicular to the fiber, the size and sign of the normal stress differences are owing to the fibers primarily aligning with the flow direction and slightly aligning with the gradient direction. Due to discrepancies with previous work in the literature, numerical simulations were performed for the same level of confinement as used in previous experimental measurements with a rheometer. By confining the fibers, some of the experimental results could be replicated, suggesting that a geometrically influenced measurement was made. Numerical results show the bounding walls act to increase the contact number density in the bulk suspension. Experiments were also performed with spheres to study the dynamics of shear-induced migration in oscillatory parabolic flow. The spatial and temporal dependence of the particle volume fraction and velocity were compared to previous experimental and numerical works, including numerical predictions from the Suspension Balance Model (SBM). The experimental measurements were made at a higher resolution than done previously and confirm the existence of a near-wall particle layer that has been predicted from discrete particle simulations. Comparison with the SBM predictions indicates a need to reconsider several parameters in order to accurately predict both the dynamics and steady state particle volume fraction distribution, including the concentration dependence on the constitutive relationship. ( 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, 2015.
Local:
Adviser: BUTLER,JASON E.
Local:
Co-adviser: LADD,ANTHONY J.
Statement of Responsibility:
by Braden J Snook.

Record Information

Source Institution:
UFRGP
Rights Management:
Copyright Snook, Braden J. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Classification:
LD1780 2015 ( lcc )

Downloads

This item has the following downloads:


Full Text

PAGE 1

THEDYNAMICSOFTHEMICROSTRUCTUREANDTHERHEOLOGYIN SUSPENSIONSOFRIGIDPARTICLES By BRADENJ.SNOOK ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2015

PAGE 2

c ! 2015BradenJ.Snook 2

PAGE 3

ToShannon,youaremyeverything 3

PAGE 4

ACKNOWLEDGMENTS Ihavehadthegoodfortunetobesurroundedbymanywonderfulpeopleduringmy pursuitofadualPhDinGainesvilleandMarseille.Ithasbeenapleasuretospendtime withallofyou. IamingreatdebttomyPhDadvisors,Dr.JasonE.ButlerandDr. « Elisabeth Guazzelli.ItisthroughtheirleadershipandstrictadherencetoexcellencethatIhave developedtheskillsnecessarytosucceedasaresearcher.Theyhaveinstilledinme acommitmenttoworldclassresearchandtoneversacriÞcethequalityofmywork.It isbecauseofthisIamconÞdentinmyfutureandlookforwardtowhatisnext.Icannot thankthemenoughforeverythingtheyhavedone. IowealargethankyoutomyFrenchandAmericancommittees.Dr.AnthonyJ.C. LaddspentmuchofhistimesigniÞcantlyaidinginmytechnicalwritingskills.Although Istillstruggle,IknowIhavegreatlyimprovedbecauseofhim.Dr.RenweiMeiandDr. « ElisabethLemairewerekindtobethereportersonthecommittee,providingexcellent reviewsandideasthatimprovedmydissertation.Greatinsightwasprovidedfrom Dr.JenniferCurtis,sparkingmyinterestinmultiphaseturbulentßowandaidinginmy understandingofthesubject.Inadditiontobeingacommitteemember,Dr.Olivier Pouliquenwasanexcellentresourcefordiscussionandquestionsduringmytimespent inFrance.Dr.FarzamZoueshtiaghwasgeneroustobethepresidentofmycommittee, providingathoroughandwellwrittenreviewofmydefense. IhaveadditionallygainedgreatinsightfromfacultymembersDr.RangaNarayanan andDr.LewisE.Johns.Withtheirdifferingperspectivesandinquisitivenature,I developedstrongerproblemsolvingandscientiÞccommunicationskills.Theyhave shownmetheimportanceofquestioningeveryaspectofone'sownwork. MytimespentinFrancewouldnothavebeennearasenjoyableifitwasn'tfor theaidofandtimespentwithmycollegesandfriendsthere.Ifitweren'tforthetime spentbyJean-FrancüoisLouf,MathieuSouzy,MathieuColombani,andFrancüoisGuillard 4

PAGE 5

tohelpme,IwouldlikelystillberentinganapartmentinFrancewithmultiplecell phonesandinternetsubscriptions.ItisalsothankstothemthatIhavehopeofoneday speakingFrench. Mydevelopmentasagraduatestudentwasstronglyaidedbythementoringof CaseyLeMarcheandMarcoPocci.InadditiontobeneÞtingfromtheirlifeandresearch experiences,theirsupportandoptimismmotivatedmethroughthemostchallenging momentsofgraduateschoolandIamextremelygratefultothem.Ialsoamthankfulfor themanylifeandresearchdiscussionsspentwithViratUpadhyay,BradMessmer,and ThomasLem « ee. Tomyfriendsandfamily,yourkindness,support,andunderstandinghasnotbeen overlookedthepastÞveyears.Thankyoutomyfather,KevinSnook,whotaughtmeif ajobisworthdoing,itisworthdoingright.Thankyoutomymother,SheilaSnook,who taughtmetheimportanceofeducationatanearlyage.Lastly,thankyoutomyÞancee, ShannonBlazek,yourloveandpatiencehasbeenunwavering.Icannotwaitforour futuretogether. 5

PAGE 6

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 2NUMERICALMODELFORUMULATIONFORSUSPENSIONSOFFIBERS .. 19 2.1ModelOverview ................................ 19 2.2FiberMotion .................................. 19 2.3ModelOptimization ............................... 21 3THEMICROSTRUCTUREINSUSPENSIONSOFFIBERSINSTEADYSHEAR FLOWWITHOUTCONFINEMENT ......................... 25 3.1ReviewofPreviousExperimentalandNumericalWorks .......... 25 3.2NumericalPredictionsandComparisonstoPreviousResults ....... 28 3.3Summary .................................... 34 4THEMICROSTRUCTUREINSUSPENSIONSOFFIBERSINSTEADYSHEAR FLOWWITHCONFINEMENT ........................... 36 4.1BackgroundofConÞnementEffects ..................... 36 4.2NumericalPredictionsComparingConÞnedandPeriodicResults ..... 36 4.3Summary .................................... 41 5THEMICROSTRUCTUREINSUSPENSIONSOFFIBERSINUNSTEADY SHEARFLOWWITHANDWITHOUTCONFINEMENT ............. 43 5.1OverviewofParticleDynamicsinOscillatoryShearFlow .......... 43 5.2ResultsandDiscussion ............................ 44 5.3Summary .................................... 53 6DERIVATIONOFTHESTRESSINSUSPENSIONSOFRIGIDFIBERS .... 55 6.1RelevancyofDerivation ............................ 55 6.2StressCalculation ............................... 55 7THERHEOLOGYINSUSPENSIONSOFFIBERSWITHOUTCONFINEMENT 58 7.1OverviewofRheologyinSuspensionsofParticles ............. 58 6

PAGE 7

7.2Experiments .................................. 61 7.2.1ParticlesandFluids ........................... 61 7.2.2Rotating-RodandTilted-TroughFlows ................ 63 7.2.3DataAnalysisforTilted-TroughFlows ................. 66 7.2.4NumericalPredictions ......................... 70 7.3ResultsandDiscussion ............................ 70 7.3.1ComparisonofExperimentalandNumericalResults ........ 70 7.3.2TimeDependence ........................... 72 7.3.3OriginsoftheNormalStressDifferences ............... 75 7.3.4ComparisonswithPreviousWorks .................. 78 7.4Summary .................................... 80 8THERHEOLOGYINSUSPENSIONSOFFIBERSWITHCONFINEMENT .. 82 8.1MotivationforStudyingRheologyofConÞnedSuspensions ........ 82 8.2Results ..................................... 82 8.3Summary .................................... 89 9DYNAMICSOFSHEAR-INDUCEDMIGRATIONOFSPHERICALPARTICLES INPIPEFLOW .................................... 90 9.1BackgroundofShear-InducedMigration ................... 90 9.2Experiments .................................. 93 9.2.1ParticlesandFluids ........................... 93 9.2.2ExperimentalApparatus ........................ 93 9.2.3ExperimentalProcedure ........................ 95 9.2.4DataAnalysis .............................. 97 9.3ResultsandComparisons ........................... 100 9.3.1SteadyState .............................. 100 9.3.2Dynamics ................................ 107 9.3.3MigrationDependenceonConcentrationandStrain ........ 115 9.4Discussions ................................... 118 10CONCLUSIONS ................................... 122 APPENDIX ATHEFREE-SURFACEPROFILEOFASUSPENSIONFLOWINGINASEMICIRCULARTROUGH ................................ 126 BTHESUSPENSIONBALANCEMODEL ...................... 130 B.1TheSBMforPipeFlow ............................ 132 B.2SteadyStateSolution ............................. 135 CTABULATEDRESULTS ............................... 137 REFERENCES ....................................... 150 7

PAGE 8

BIOGRAPHICALSKETCH ................................ 156 8

PAGE 9

LISTOFTABLES Table page C-1Numericalresultsfortheamountofaccumulatedstrain, ú ! t toreachsteady state. ......................................... 138 C-2Numericalresultsfortheorbitconstant, " C b # ,atsteadystate. .......... 139 C-3Numericalresultsforthemoment ! p 2 x p 2 y " atsteadystate. ............. 140 C-4Numericalresultsforthemoment " p 2 x # atsteadystate. .............. 141 C-5Numericalresultsforthemoment ! p 2 y " atsteadystate. .............. 142 C-6Numericalresultsforthemoment " p 2 z # atsteadystate. .............. 143 C-7CharacteristicsforeachsetofÞbers. ........................ 144 C-8Experimentalresultsfor " 2 withthestandarderror. ................ 145 C-9Numericalresultsfor " 2 . ............................... 146 C-10Numericalresultsfor " 1 ............................... 147 C-11Numericalresultsforthesuspensionshearstress, # /µ ú ! . ............. 148 C-12Thenumberofoscillationsandaccumulatedstraintoreachsteadystate. ... 149 9

PAGE 10

LISTOFFIGURES Figure page 2-1DrawingsofcontactingÞbersandgeometricparameters. ............ 20 2-2Theeffectsduetochangingsimulationparameters. ............... 22 3-1PreviousnumericalandexperimentalresultsarerepresentedbyÞlledand opensymbolsfortheorbitconstant, " C B # ,andthefourthordermomentofthe orientationdistribution, ! p 2 x p 2 y " . ........................... 26 3-2Numericalandexperimentalresultsfortheorbitconstant. ............ 29 3-3Thefourthordermomentoftheorientationdistributioncontributingtotheshear stress, ! p 2 x p 2 y " athighandlowaspectratios. .................... 30 3-4Theaccumulatedstrainrequiredtoreachsteadystateandthetemporalevolutionof " C b # . ..................................... 31 3-5Thetemporalevolutionofthesecondordermomentoftheorientationforalignmentwiththeßow,thegradient,andvorticitydirections. ............. 33 3-6Thesecondordermomentoftheorientationforalignmentwiththeßow,the vorticity,andthegradientdirections. ........................ 35 4-1TheeffectsofconÞnementon " p 2 x # atlowandhighaspectratios. ........ 37 4-2TheeffectsofconÞnementon ! p 2 y " atlowandhighaspectratios. ........ 38 4-3TheeffectsofconÞnementon " p 2 z # atlowandhighaspectratios. ........ 39 4-4TheeffectsofconÞnementon " C B # atlowandhighaspectratios. ........ 40 5-1RelevantparametersforÞbersinoscillatoryshearßow. ............. 45 5-2NumericalandexperimentalresultsforÞberalignmentwiththevorticityaxis. . 46 5-3Temporalevolutionof S ! andthenumberofcontacts. .............. 48 5-4Changesin S ! duetobounding,strainamplitude,andconcentration. ...... 50 5-5Theprobabilitydistributionsoftheangle " . .................... 52 7-1MicroscopicimagesofpolyamideÞbers. ...................... 62 7-2Illustrationsoftherotating-rodandthetilted-troughßows. ............ 64 7-3Imagesofthelinesformedbyprojectingninelasersheetsontothefree-surface ofasuspensionßowinginthetroughatdifferinganglesofinclination. ..... 66 7-4ExperimentallymeasuredsurfaceproÞles. ..................... 67 10

PAGE 11

7-5Effectsofmixingon $ S . ............................... 69 7-6Experimentalandnumericalresultsfor " 2 . ..................... 71 7-7Experimentalandnumericalresultsfor " 1 and " 2 + " 1 / 2 . ............ 73 7-8Temporalevolutionof " C b # and " 2 . ......................... 74 7-9Detailedpredictionsofthemicrostructureandcontactforces. .......... 77 8-1ChangesintheÞrstnormalstressdifferenceduetoconÞnementathighand lowaspectratios. ................................... 83 8-2ChangesinthesecondnormalstressdifferenceduetoconÞnementathigh andlowaspectratios. ................................ 84 8-3Changesin ( N 1 $ N 2 ) /µ ú ! duetoconÞnementathighandlowaspectratios. .. 85 8-4ChangesincontactnumberdensityduetoconÞnementat A =11 and A =20 . 86 8-5ChangesincontactnumberdensityduetoconÞnementat A =30 and A =50 . 87 8-6ThenumberdensityofcontactsoccurringinthebulkforconÞnedsimulations normalizedbythenumberdensityofcontactsinfullyperiodicsimulations. ... 88 9-1Theexperimentalapparatusandschematicofparticlevisualizationmethod. .. 94 9-2Examplesoftheinitialparticledistributions. .................... 96 9-3Imageprocessingsteps. ............................... 97 9-4Theevolutionofthevolumefractionforagivenbin. ................ 98 9-5Steadystateresultsforthevolumefractionandtheparticlevelocityat R / a = 8.21 . .......................................... 101 9-6Steadystateresultsforthevolumefractionandtheparticlevelocityat R / a = 15.71 . ......................................... 102 9-7Steadystateresultsofthepresentworkcomparedtopreviousexperimental andnumericalworkatsimilar R / a . ......................... 104 9-8Additionalsteadystateresultsofthepresentworkcomparedtopreviousexperimentalandnumericalworkatsimilar R / a . ................... 105 9-9Steadystateresultsfortheparticlevelocityofthepresentworkcomparedto previousexperimentalwork. ............................. 106 9-10Steadystateresultsfor R / a =8.21 comparedtothenumericalpredictions fromthesteadystateSBM. ............................. 108 11

PAGE 12

9-11Steadystateresultsfor R / a =15.71 comparedtothenumericalpredictions fromtheSBM. .................................... 109 9-12Experimentalresultsforthetemporalevolutionoftheparticlevolumefraction distributionsat R / a =8.21 . ............................. 110 9-13Experimentalresultsforthetemporalevolutionoftheparticlevolumefraction distributionsat R / a =15.71 . ............................. 111 9-14SBMpredictionsandexperimentalresultsforthetemporalevolutionofthelocalparticlevolumefractionat R / a =8.21 . ..................... 112 9-15SBMpredictionsandexperimentalresultsforthetemporalevolutionofthelocalparticlevolumefractionat R / a =15.71 . .................... 113 9-16Changesinthetemporalevolutionofthelocalparticlevolumefractiondueto ! . 116 9-17Theexperimentallyobservedvolumefractiondistributionatthebeginningof theÞnaloscillation. .................................. 116 9-18Meansquaredisplacementoftheparticles. .................... 117 A-1Cut-awayviewofthesemi-circulartroughofradius R . .............. 127 B-1Evolutionofvolumefractionacrossthepipefortherheologyof Boyer etal. ( 2011 a ). ........................................ 134 B-2Evolutionofvolumefractionacrossthepipeusingtherheologyof Boyer etal. ( 2011 a )withthesteadystatesolution. ....................... 136 12

PAGE 13

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulÞllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THEDYNAMICSOFTHEMICROSTRUCTUREANDTHERHEOLOGYIN SUSPENSIONSOFRIGIDPARTICLES By BradenJ.Snook May2015 Chair:JasonE.Butler Major:ChemicalEngineering Numericalandexperimentalmethodshavebeenusedtorevealthecomplex relationshipbetweenthemacroscopicpropertiesandthemicrostructure,orrelative arrangementofparticles,ofasuspension.BothspheresandÞberswereconsidered. ForÞbers,anumericalmodelwasdevelopedtopredictparticlemotionwhichconsiders thehydrodynamicdragexperiencedbyaparticleduetotheappliedßowandashort rangerepulsiveforcetomaintaintheexcludedvolume.Themodelwasusedtosimulate steadyandoscillatoryshearßowwithfullyperiodicandconÞnedboundaryconditions. Comparisonwithpreviouscalculationsandexperimentsforsteadyshearindicates thatlongandshortrangehydrodynamicinteractionsarenotrequiredtoaccurately predicttheÞbermotionattheconcentrationssimulated.Theresultsaidinexplaining currentdiscrepanciesintheliteraturebetweennumericalpredictionsandexperimental observationsofthemicrostructurewhilealsoimprovingtheunderstandingoftheeffect thatÞberaspectratioandconcentrationhaveonparticledynamics.Thissamemodel wasusedtointerpretrecentexperimentalobservationsofÞberalignmentperpendicular totheßowdirectionwhensubjectedtooscillatoryshearßow.Numericalpredictions agreewithexperimentalobservationsonlywhenÞbermotionisrestrictedinthegradient directionbyboundingwalls;fullyperiodicsimulationsdonotreproducetheexperimental measurements.Itisalsoshownthatboundingwallscanaffectthemicrostructurein steadyshearßowforlowaspectratioÞbers. 13

PAGE 14

Inadditiontothesenumericalefforts,experimentswereperformedforÞbersin steadyshearßowwiththepurposeofmeasuringtheÞrstandsecondnormalstress differences.Themeasurementsareinquantitativeagreementwithnumericalpredictions,ÞndingthattheÞrstnormalstressdifferenceispositiveandapproximatelytwice themagnitudeofthesecondnormalstressdifference,whichisnegative.Thepredictionsshowthatthecontactcontributiontothestresstensormustbeincludedinthe calculationtoaccuratelypredictthenormalstressdifferences.Asthecontactforce actsperpendiculartotheÞber,thesizeandsignofthenormalstressdifferencesare owingtotheÞbersprimarilyaligningwiththeßowdirectionandslightlyaligningwiththe gradientdirection.Duetodiscrepancieswithpreviousworkintheliterature,numericalsimulationswereperformedforthesamelevelofconÞnementasusedinprevious experimentalmeasurementswitharheometer.ByconÞningtheÞbers,someofthe experimentalresultscouldbereplicated,suggestingthatageometricallyinßuenced measurementwasmade.Numericalresultsshowtheboundingwallsacttoincreasethe contactnumberdensityinthebulksuspension. Experimentswerealsoperformedwithspherestostudythedynamicsofshearinducedmigrationinoscillatoryparabolicßow.Thespatialandtemporaldependenceof theparticlevolumefractionandvelocitywerecomparedtopreviousexperimentaland numericalworks,includingnumericalpredictionsfromtheSuspensionBalanceModel (SBM).Theexperimentalmeasurementsweremadeatahigherresolutionthandone previouslyandconÞrmtheexistenceofanear-wallparticlelayerthathasbeenpredicted fromdiscreteparticlesimulations.ComparisonwiththeSBMpredictionsindicatesa needtoreconsiderseveralparametersinordertoaccuratelypredictboththedynamics andsteadystateparticlevolumefractiondistribution,includingtheconcentration dependenceontheconstitutiverelationship. 14

PAGE 15

CHAPTER1 INTRODUCTION Thepervasivenessofsuspensionshasmadetheirstudyrelevanttoamultitude ofapplications,whiletheircomplexityhasinhibitedtheabilitytomodelandpredict theirdynamics.Manyuniquephysicalpropertiesarestillnotaccuratelypredictedby currenttheory,requiringcontinuingin-depthexaminations.Accuratelypredictingthe microstructureandrheologyisoftenofmostinterest.Forexample,measuredstresses areusedinthedesignofprocessingequipment,butintruththemeasurementsare frequentlymisinterpretedandoftenincorrect.Rheologicaltestingforthepurpose ofcorrelatingsuspensionperformancetoparameterssuchasviscositywouldbe aidedbyimprovedcapabilitiesregardingtherelationshipbetweenthesuspension's composition(particlesize,polydispersity,particleshape)andthemeasuredproperties. Accordingly,alucidcomprehensionoftheirinterdependenceisgreatlydesired.The presentworkendeavorstocreateimproved,accuratemodels,andtheoriesforpredicting theinterdependenceforsuspensionsofÞbersandspheres. SeveraltheoriesexistforrelatingthemicrostructureofÞberstotherheology,however,forconcentratedsuspensionsthesetheoriesfail( Petrich etal. , 2000 b ; Salahuddin etal. , 2013 ).Accordingly,thisconcentrationregimeisofinteresthereandisdeÞnedas nL 2 d ! 1 ( Doi,M.andEdwards,S.F. , 1986 ),where n isthenumberdensityofÞbers and L and d aretheÞberlengthanddiameter.Therehavebeenseveralexplanations fortheinabilitytoaccuratelypredicttherheologyfromthemicrostructureattheseconcentrations.Oneexplanationisthatexperimentsprimarilymeasurethemicrostructure andrheologyofdifferentsetsofÞbersandsuspendingßuid.Thisposesaproblemas therecanbevariancesinÞberdensity,lengthordiameterdistributions,andmaterial properties,suchasßexibility,whichmaymanifestthemselvesinthemeasurementsof themicrostructureorrheology.Bymeasuringdifferentsuspensions,differentsourcesof 15

PAGE 16

experimentalvariancescouldaffecttheresults,makingitdifÞculttounderstandthediscrepanciesbetweentheexperimentallymeasuredrheologyandtheoreticalpredictions fortherheologyusinganexperimentallymeasuredmicrostructure.Tomitigatethese additionalcomplications, Petrich etal. ( 2000 b )performedexperimentsusingthesame suspensionformeasurementsofthemicrostructureandrheology.Despitethis,there wasstillpooragreementbetweenthesemeasurementsandcurrenttheoriesthatpredict therheologyfromthemicrostructure.Recently, Salahuddin etal. ( 2013 )performednumericalsimulationswheretherheologyandthemicrostructurewereaccessiblewithout theneedoftheoriesrelatingtherheologytothemicrostructure.Agreementwasshown withexperimentalobservationsofthemicrostructureandtheshearstress.Theinability topredicttheshearstressfromthemicrostructurewasthenattributedtonotincluding contactsinthecurrenttheory.Duetotheirsimulationtechnique,thecontactcontribution totherheologycouldnotbeeasilydetermined.Unfortunately,theagreementbetween thenumericalpredictionsandthemicrostructureisonlytruefor ! p 2 x p 2 y " ,where p x and p y aretheorientationsoftheÞberwiththeßowandgradientdirections,respectively. Comparisonofthenumericallypredictedorbitconstant, " C b # ,whichisanothermeasure ofÞberorientation,withpreviousexperimentsexhibitsqualitativedisagreement.Without rectifyingthesediscrepancies,formingaconstitutiverelationshipisanimpossibletask. RectifyingthesediscrepanciesrequiresaclearidentiÞcationoftheprincipal physicsinthesuspension,whichisgreatlyaidedbydirectcomparisonofnumerical andexperimentalresults.Chapter 2 describesanumericalmodelwhichignores longandshortrangehydrodynamicinteractionstodecreasethecomputationalcost ofconcentratedsystemswhileincludingtherelativeinßuencecontactshaveonthe microstructure.Previousnumericsandexperimentsarecomparedandarereported inChapter 3 toestablishtheaccuracyofthemodelandtheimportanceofcontacts. Manyofthecurrentdiscrepanciesinexperimentalobservationsofthemicrostructure areresolved.Additionally,thesensitivityoftheorbitconstanttosmallchangesinÞber 16

PAGE 17

alignmentwiththevorticityisdiscussed.Chapter 4 movestoexaminetheeffectsof conÞnementonthemicrostructureinsteadyshear. ThesimulationmodelfromChapter 2 isalteredtosimulateoscillatoryshearßow inChapter 5 .Resultsarecomparedtorecentexperimentalmeasurementsofthe microstructureatthesameconditions.TheimportanceofconÞnementisshownaswell astheeffectofamplitudeandparticleconcentrationonthemicrostructure. Inordertocalculatethestressfromthepreviousnumericalpredictions,Chapter 6 derivesthestressinasuspension,includingthecontributionduetoparticlecontacts. TheresultingequationisusedinChapter 7 tocomparethenumericalpredictionsto experimentalmeasurements.Experimentsmeasuredthedeßectioninthefree-surface ofatilted-troughandWeissenberggeometries,allowingforcalculationoftheÞrstand secondnormalstressdifferences, N 1 and N 2 .Theimportanceofcontactsonnormal stressdifferencesisshownaswellastheiraffectontheshearstress.Resultsfor normalstressdifferencesfromconÞnedsuspensionsarecomparedtoperiodicresults andpreviousexperimentsusingarheometerinChapter 8 .TheroleofconÞnementis discussedandthegeometricaffectswithrheometricmeasurements. Chapter 9 movestoastudyofshearinducedmigrationinsuspensionsofspheres. LikesuspensionsofÞbers,theconstitutiveequationforthestressisstillbeingdeveloped.Ofinterestisifexistingtheoriescanaccuratelypredictthedynamicsofmigration inparabolicßowforvaryingparticlevolumefractionsandgeometries.Thedynamics areaccessedexperimentallybyusinghighamplitudeoscillatoryparabolicßow.Results arecomparedtonumericalpredictionsusingtheSuspensionBalanceModel(SBM). MultiplerheologieswereusedfortheconstitutiveequationforthestressintheSBM, theiraccuracyisdiscussedalongwithSBM'sabilitytopredictthedynamicsandsteady stateresults. 17

PAGE 18

Chapter 10 providesasummarizedsetofconclusionsaswellasabroaderperspectiveonthenetimpactoftheworkinitsentirety.Aspartofthisdiscussion,continuing researchanddirectionsareenumerated. 18

PAGE 19

CHAPTER2 NUMERICALMODELFORUMULATIONFORSUSPENSIONSOFFIBERS 2.1ModelOverview AuniquemodelhasbeendevelopedtosimulateconcentratedsuspensionsofrigidnonBrownianÞbersinavarietyofßowconditions.Themodelincludestheparticlemotion duetotheappliedßowandashortrangerepulsiveforcetomaintaintheexcluded volume.Thecomputationallyexpensivelongandshortrangehydrodynamicinteractions areignored.Themodelutilizeseither Lees&Edwards ( 1972 )periodicorconÞned boundaryconditions.Itisadditionallycapableofsimulatingsteadyaswellasoscillatory shear. 2.2FiberMotion ForagivenÞber " ,Figure 2-1 AdeÞnesthecenterofmass x ! ,theorientation p ! , whichisaunitvectorparalleltoaÞber'smajoraxis,thelength L ,andthediameter d . Thevelocityofthecenterofmass, ú x ! = u ( x ! )+ $ ! 1 ( I + p ! p ! ) á F ! , (2Ð1) andtherotationalvelocity, ú p ! = ! á p ! + B ( I $ p ! p ! ) á E á p ! + 12 $ ! 1 L 2 L ! % p ! , (2Ð2) aregivenbyforceandtorquebalancesintheabsenceofinertia( Snook etal. , 2012 ; Sundararajakumar&Koch , 1997 ).TheEquations 2Ð1 and 2Ð2 accountforthemotion owingtotheappliedßowÞeldandtorepulsiveforcesandtorquesbetweenÞbers. Theimposedvelocitywaseitherasteadyoranoscillatoryshearßowgivenby u =ú ! y e x .Foroscillatoryshearßow,theshearratewasatimedependent ú ! ( t ) square waveofmagnitudeoneandaperioddeterminedbythestrainamplitude.Equation( 2Ð1 ) showsthatif F ! =0 ,aÞbertranslateswiththeimposedvelocityevaluatedatitscenter ofmass x ! .If L ! =0 ,theÞberrotateswiththerateofrotationoftheshearingßow, 19

PAGE 20

! ! !"#$%& " !"#$%& ! " ! ! !" " !" # !" ! # !" # $ % & ' $ ( ! ) ' ( # $ Figure2-1. DrawingsofcontactingÞbersandgeometricparameters.A)Thecenterof mass x ! andorientation p ! describetheconÞgurationofanyrigidÞber " . Therepulsiveforce f !" betweenÞbers " and % ,separatedbyadistance | h !" | ,actsatthepoint x ! + s !" p ! onÞber " andthepoint x " + s "! p " onÞber % .B)ThealignmentoftheÞberwiththevorticitydirection( z -axis)is measuredbytheangle & .Theangle ' isthatbetweenthegradientdirection ( y -axis)andtheprojectionoftheÞberontotheßow-gradient( x $ y )plane. ! = 1 2 # ( & u ) $ ( & u ) T $ andafraction B = % A 2 e $ 1 & / % A 2 e +1 & oftherateofextension oftheßow, E = 1 2 # ( & u )+( & u ) T $ .Theparameter B containstheÞberaspectratio A , whichhasbeenreplacedbyaneffectiveaspectratio, A e ' 0.8 A .Thiseffectiveaspect ratioalterstheoriginalcalculationof Jeffery ( 1922 ),whichwasperformedforaspheroid, toaccountforthemotionofaspherocylinder( Bretherton , 1962 ; Mason&Manley , 1956 ; Trevelyan&Mason , 1951 ).Moreover,thepresentmodelignorestheinßuenceofvelocity disturbancescausedbythemotionofotherÞbersinthesuspension. TheresponseofaÞberduetoanappliedforce F ! ortorque L ! isderivedfrom slenderbodytheory( Batchelor , 1970 ; Cox , 1970 ).Thecenterofmassmobilityis $ ! 1 ( I + p ! p ! ) ,where $ ! 1 =ln(2 A ) / 4 ( µ L and µ istheviscosityofthesuspendingßuid. TheforcesconsideredhereareduetoÞbercollisions,whicharemodeledbyashort rangerepulsion, f !" = ' ( ) ( * 0 if | h !" | > ) , ± f 0 h !" / | h !" | if | h !" | ( ) , (2Ð3) 20

PAGE 21

where | h !" | istheminimumseparationdistancebetweentwoÞbers " and % asshown inFigure 2-1 A.Acontactoccurswhen | h !" | ( ) ,where ) =0.1 d .Thecontactforceis f 0 = | ú ! L $ / 2 | andthesigninEquation( 2Ð3 )ischosensothattheÞbersrepeleachother. Thenetforceontheßuidiszero,astheforceonÞber " isequalandoppositetothe forceonÞber % .ThetotalforceandtorqueonaÞber " isfoundbysummingoverthe repulsiveforcecontributionsfromallotherÞbers % , F ! = N f + " =1 f !" (2Ð4) L ! = N f + " =1 s !" p ! % f !" , (2Ð5) where s !" isthepointalongÞber " atwhichthecollisionoccurswithÞber % (Figure 2-1 A)and N f isthenumberofÞbers. BothperiodicandconÞnedgeometriesweresimulated.Periodicsimulationsused Lees&Edwards ( 1972 )boundaryconditionstoproperlyaccountforgradientsinvelocity. ConÞnedgeometriesweremodeledbyprohibitingÞbersfrompassingthroughthe boundariesinthegradientdirection.Thiswasachievedbyusingarepulsiveforceofthe sameformusedtomaintaintheexcludedvolume.Althoughsimulationswerecarried outatvariouslevelsofconÞnement,thevorticityandßowdirectionsalwaysremained periodic,withalengthof 5.25 L . Equations( 2Ð1 )and( 2Ð2 )areintegratedintimeusinganEulermethodtogive thepositionsandorientations.Thetimestepischosentogiveamaximumstrainstep of 2 % 10 ! 4 ,ensuringtheaccurateresolutionofcontactsbetweenparticles.Boththe methodandthechoiceoftimestepweretestedsuccessfullytoverifythattheresultsare convergent. 2.3ModelOptimization Wehavedevelopedamodelinwhichshortrangeparticleinteractionsareassumed tobedominate.Severalnumericalconcernswereaddressedpriortosimulatingthe 21

PAGE 22

! !"# $ $"# % %"# !" % # !"% !"& !"' !"( ! $ % " )*+,-./01,2"%# )*+,-./01,#"%# )*+,-./01,'"%# 3 ! !"# $ $"# % !" % # !"# !"## !"& !"&# !"' ! $ % " ()*+,-+,./0123 ()*+,-+,.(45066 7 (0366-2-3+5 Figure2-2. Theeffectsduetochangingsimulationparameters.For A =11 ,A)the simulationboxsizewasaltereduntilitnolongeraffectedthemicrostructure. Aboxsizeof 5.25 L wassufÞcienttoremovetheaffectsofperiodicityonthe microstructure.B)Therepulsiveforceandcutoffdistancewerevariedto determinetheiraffectsonthesimulationresults.Theexactsizeof f 0 didnot altertheresultsaslongasitwaslargeenoughtomaintaintheexcluded volume.However,slightchangesoccurwithchangesin ) .Thesechanges correspondtodifferentexcludedvolumesbeingmaintained. 22

PAGE 23

parameterspaceofinterest.First,thesimulationboxsizewasaltereduntilperiodicity nolongeraffectedtheresults.Assessmentofeffectsduetoperiodicitywasdoneby analyzingthemicrostructure.TheorientationoftheÞbersiscommonlyquantiÞedbythe orbitconstant,givenby C = 1 A e tan & ( A 2 e cos 2 ' +sin 2 ' ) 1 / 2 , (2Ð6) wheretheangles ' and & correspondtotheÞber'salignmentintheßowgradientplane anditsalignmentwiththevorticityaxis,asshowninFigure 2-1 B.Thevalueof C ranges fromzerotoinÞnity,accordinglytheorbitconstantisusuallyrecastas C b = C / ( C +1) , whichiswhatisusedhere.Figure 2-2 Ashowsaboxsizeof 5.25 L issufÞcienttoremove theseeffects.Next,thetwoadjustableparametersinourmodel,thecutoffdistance, ) , andtherepulsiveforce, f 0 ,werealteredtotesttheiraffectsontheresults.Figure 2-2 B showsresultsfortheorbitconstantwhereeither f 0 or ) weremultipliedbyacoefÞcient designatedbythex-axis.Forexample,whenthecoefÞcientwas 1 fortheparameter f 0 ,thisparameterhadavalueaspreviouslydeÞned, | ú ! L $ / 2 | .When f 0 wassufÞciently sizedtomaintaintheexcludedvolume,itsvariationhadnoaffectontheresults.This wastruesolongasthetimestepwasadjustedappropriatelytoensureconvergence. Thecutoffdistancefortherepulsiveforcewasalsoaltered,showingsomechangesin themicrostructurewithchangesinitsvalue,asseeninFigure 2-2 B.Whenthecutoff distance ) wasincreased,theorbitconstantincreasedaswell.Aswillbeshownin Chapter 3 ,theorbitconstantincreaseswithdecreasingaspectratio.Accordingly,the cutoffdistancehadtobesetwithaphysicalsigniÞcance,asincreasingordecreasing itssizeyieldsthesameresultsaschangingtheaspectratio.Theexperimentalworkin Chapter 7 shows10%varianceinÞberdiameter,resultinginchoosing ) =0.1 d . Althoughonlycontactsweremodeled,computationtimeswerestillhighasthe accurateresolutionofcontactsforupto 21,705 Þberswasrequired.Todecreasethe computationtime,Verletneighborlists,generatedfromacelllinked-list,wereusedfor 23

PAGE 24

contactdetection.ThiswasdonebyÞrsttreatingagivenÞberasalineofoverlapping spheres.Next,eachspherewasassignedtoacellinsideofthesimulationbox.This celllinked-list,orlistofspheresinagivencell,wasusedtogeneratetheVerletneighbor list.TheVerletneighborlistwascomposedofthespheresinagivencellwhichwere withinasetdistanceofeachother.Thesetdistancewaschosensothatmultiplesteps intimecouldbetakenbeforeitwouldbenecessarytogeneratethecelllinked-listand Verletneighborlistagaintoensureparticlesdidnotoverlap.Theseparationdistances betweenthespheresontheVerletneighborlistwerethencalculatedtodetermineif anycontactswereoccurring.Thisresultedinamuchsmallernumberofparticlesto consider,greatlydecreasingthecomputationcost. 24

PAGE 25

CHAPTER3 THEMICROSTRUCTUREINSUSPENSIONSOFFIBERSINSTEADYSHEARFLOW WITHOUTCONFINEMENT 3.1ReviewofPreviousExperimentalandNumericalWorks Concentratedsuspensionsofelongatedparticles,speciÞcallyÞbers,arecommonly encounteredinindustry.Predictingthemacroscopicpropertiesofthesesuspensions, suchastheshearstress,offersthepossibilitiesofimprovedproblemsolving,new processcapabilities,andincreasedefÞciency.Asthemacroscopicpropertiesdepend onthemicrostructure,ortheconÞgurationoftheparticlessuspendedintheßuid, understandingtheprincipalphysicsunderlyingthedynamicsofthesuspendedparticles isofgreatvalue.However,duetothehighnumberofparticlesandthecomplexityof theirinteractions,asatisfactoryunderstandinghasnotbeenrealized.Numericalstudies offerthevaluableabilitytoalterthephysicsinthemodel,aidingineffortstoresolvethe inßuenceofthecrucialforcesexperiencedbyaparticleinthemicrostructure. ThedynamicsofthemicrostructuredependupontheÞbergeometryandconcentration.ThegeometryofaÞberisshowninFigure 2-1 Aandisdescribedbythelength L ,thediameter d ,andtheaspectratio A = L / d .Thecenterofmassandorientationof aÞber " isgivenby x ! and p ! ,asshowninFigure 2-1 A.TheorientationoftheÞbersis commonlyquantiÞedbytheorbitconstant,givenby C = 1 A e tan & ( A 2 e cos 2 ' +sin 2 ' ) 1 / 2 , (3Ð1) wheretheangles ' and & correspondtotheÞber'salignmentintheßowgradientplane anditsalignmentwiththevorticityaxis,asshowninFigure 2-1 B.Thevalueof C ranges fromzerotoinÞnity,accordinglytheorbitconstantisusuallyrecastas C b = C / ( C +1) , whichisusedhere.TheorientationoftheÞbersisalsostudiedusingvariousmoments of p ! .Themomentsusedherewillbe " p 2 x # , ! p 2 y " , " p 2 z # ,and ! p 2 x p 2 y " .Theßow,gradient, andvorticitydirectionsforthesteadyshearßowaredesignatedby x , y ,and z and thebrackets, " á # ,designateanensembleaveraging.Thisdesignationforensemble 25

PAGE 26

0123 nL 2 d 0.3 0.4 0.5 0.6 0.7 0.8 ! C b " A =30 Sundararajakumar & Koch 1997 A =32 Wu & Aidun 2010 A =31.9 Stover et al. 1992 A =50 Sundararajakumar & Koch 1997 A =50 Petrich et al. 2000 A ! !"# $ $"# % %"# & !" % # ! !"!$ !"!% !"!& ! $ % & $ ' & " ( '$()*+,+-.//01) )*+,-. ( '$("2)*34567) )*+,-. $22% ( '&%)*+,+-.//01) )*+,-. ( '&$"2)*34567) )*+,-. $22% ( '#%)*+,+-.//01) )*+,-. ( '#!)863709-) )*+,-. %!!! : Figure3-1. PreviousnumericalandexperimentalresultsarerepresentedbyÞlledand opensymbolsfortheorbitconstant, " C B # ,andthefourthordermomentof theorientationdistribution, ! p 2 x p 2 y " .TheorbitconstantA)showsanincrease withincreasingconcentrationfornumericalpredictions,whereas experimentsshowadecreasewithincreasingconcentration.However,the fourthordermomentoftheorientationdistributioncontributingtotheshear stressB)doesnotshowageneraltrendinthenumericalandexperimental results. 26

PAGE 27

averagingwillbeusedfortheorbitconstantaswell.Thesecondordermomentsof theorientationquantifythedegreeofalignmentoftheÞberwiththeßow,gradient,and vorticitydirections.Thefourthordermomentisrequiredtocalculatetheshearstressin asuspensionofÞbersinshearßow. Concentrationisdividedintodilute,semi-dilute,semi-concentrated,andconcentratedregimes( Doi&Edwards , 1978 ).Eachregimeischaracterizedbyaqualitative changeintheoverlapofÞbers.Forexample,inthediluteregimeÞbersarewidelyseparatedandthesphericalvolumessweptoutbyrotatingtheÞbersabouttheircentersdo notintersect.Thetransitionfromdilutetosemi-diluteoccursatthepointwherethese sphericalvolumesbegintooverlap.Thesemi-concentratedtoconcentratedregimesare studiedhere.Thesemi-concentratedregimeisdeÞnedas nL 2 d ! O (1) ,where n isthe numberdensityofÞbers.Inthisregime,particlecontactsinßuencetheÞberdynamics. Therelativeimportanceofcontactscomparedtohydrodynamicinteractionsisstudiedin Chapter 3 .Additionally,theeffectofincreasingconcentration, nL 2 d ,andaspectratio, A , onthemicrostructurethroughthisregimeareexplored. Sundararajakumar&Koch ( 1997 )simulatedÞbersuspensionsincludingonlythe motionduetothehydrodynamicdragfromtheappliedsteadyshearßowanddueto Þbercontacts. Wu&Aidun ( 2010 )performedLattice-Boltzmannsimulations,including longandshortrangehydrodynamicinteractionsaswellasÞbercontactsinsteady shearßow.Numericalpredictionsfrom Sundararajakumar&Koch ( 1997 )and Wu& Aidun ( 2010 )areplottedinFigure 3-1 Afortheorbitconstant " C b # .Thereisstrongdisagreementinthepredictionsofthesetwomodels.Sinceincludinglongandshortrange hydrodynamicinteractionsresultsinquantitativelydifferentpredictions,resultsstrongly suggesttheseinteractionsmustbeincludedinthemodel.However,thisconclusionmay notbeentirelycorrectasthecontactmodelof Sundararajakumar&Koch ( 1997 )did notmaintaintheexcludedvolume,butonlyensuredthatthecenterlinesoftheÞbers didnotcross.Additionally,comparisonoftheexperimentalobservationsof Stover etal. 27

PAGE 28

( 1992 )and Petrich etal. ( 2000 b )onFigure 3-1 AshowssigniÞcantdifferenceswhen comparedtothenumericalpredictionsof Wu&Aidun ( 2010 ).Thereisevenqualitative disagreementwithexperimentsasnumericspredictanincreasein " C b # withincreasing concentrationandexperimentsobserveadecreasewithincreasingconcentration.Interestingly,numericalresultsof Salahuddin etal. ( 2013 ),whoutilizedthesamemodel as Wu&Aidun ( 2010 ),areinagreementwithexperimentalobservationsof Stover etal. ( 1992 )and Petrich etal. ( 2000 b )for ! p 2 x p 2 y " ,asshowninFigure 3-1 B.Theresultsof Sundararajakumar&Koch ( 1997 )arestillindisagreement. Theinconsistenciesbetweenthenumericalandexperimentalresultsremainsto beunderstood.Additionally,thedegreeofimportanceofhydrodynamicsandcontacts ontheevolutionandsteadystatemicrostructureinthesemi-concentratedtoconcentratedregimeisunclear.Thisworkthoroughlytestsasimplemodel,developed inChapter 2 ,whichincludesÞbermotionduetotheappliedsteadyshearßowanda shortrangerepulsiveforcetorigorouslymaintainthefullexcludedvolume.Theresults offerexplanationsforthecurrentinconsistenciesanddeterminetheroleofcontactson themicrostructure.Allresultsareforsteadyshearßowsimulationsbeginningfroma randomlygeneratedpositionandorientationdistributions. 3.2NumericalPredictionsandComparisonstoPreviousResults Previousnumericalandexperimentalmeasurementsoftheorbitconstant " C b # are comparedtothenumericalresultsfromthecurrentmodelinFigure 3-2 .Experimental observationsandnumericalpredictionsareplottedusingopenandÞlledsymbols, respectively.Thereisexcellentagreementbetweenthenumericalworkof Wu&Aidun ( 2010 )andthecalculationsfromthemodelhere.Thisstronglysuggestscontactsare dominantattheseconcentrations.Resultsof Sundararajakumar&Koch ( 1997 )arein pooragreementwiththecurrentworkaswellaswith Wu&Aidun ( 2010 ).Accordingly, contactsarerequiredtoaccuratelypredict " C b # ,andtheymustmaintainthefullexcluded volume.Itisimportanttonote,althoughnotseenintheparameterspacestudiedhere, 28

PAGE 29

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.3 0.4 0.5 0.6 0.7 0.8 ! C b " A =30 Present Work A =30 Sundararajakumar & Koch 1997 A =32 Wu & Aidun 2010 A =31.9 Stover et al. 1992 A =50 Present Work A =50 Petrich et al. 2000 A 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.2 0.3 0.4 0.5 0.6 ! C b " A =11 Present Work A =10 Sundararajakumar & Koch 1997 A =15 Present Work A =16 Wu & Aidun 2010 A =16.9 Stover et al. 1992 B Figure3-2. Numericalandexperimentalresultsfortheorbitconstant.A)HighandB) lowaspectratiosareshownwherenumericalandexperimentalresultsare givenbyÞlledandopensymbolsrespectively.Ournumericalresultsarein agreementwith Wu&Aidun ( 2010 ),butindisagreementwithexperimental results. 29

PAGE 30

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 nL 2 d 0 0.01 0.02 0.03 0.04 0.05 0.06 ! p x 2 p y 2 " A =15 Present Work A =16 Salahuddin et al. 2013 A =16.9 Stover et al. 1992 A =30 Present Work A =32 Salahuddin et al. 2013 A =31.9 Stover et al. 1992 A =50 Present Work A =52 Salahuddin et al. 2013 A =50 Petrich et al. 2000 Figure3-3. Thefourthordermomentoftheorientationdistributioncontributingtothe shearstress, ! p 2 x p 2 y " athighandlowaspectratios.Theavailablenumerics andexperimentscomparedareinagreementandshownoincreaseof ! p 2 x p 2 y " withincreasingconcentration. theimportanceofmaintainingtheexcludedvolumewilllikelydecayathighenough aspectratio.ThisisbecauseatsufÞcientlylargeaspectratiostheÞberswillbethin enoughtobewellapproximatedbyaline. ComparisonsofthepresentnumericswiththepreviousexperimentsonFigure 3-2 areprimarilyindisagreement,evenqualitatively.Athighaspectratios,Figure 3-2 A showslittlechangeinexperimentalobservationsof " C b # withincreasingconcentration. Atloweraspectratios,Figure 3-2 Bshowsadecreasein " C b # withincreasingconcentration.Numericalresultspredictthattheorbitconstantincreaseswithincreasing concentrationforhighandlowaspectratios.UnderstandingthisqualitativedisagreementismademoredifÞcultbytheagreementbetweenexperimentalandnumerical measurementsof ! p 2 x p 2 y " ,shownonFigure 3-3 . 30

PAGE 31

0.5 1 1.5 2 2.5 3 3.5 nL 2 d 10 100 1000 10000 ! A =11 A =15 A =30 A =50 A 020004000 6000 ! 0.3 0.35 0.4 0.45 0.5 0.55 " C b # nL 2 d =1 nL 2 d =1.5 nL 2 d =2 nL 2 d =2.5 nL 2 d =3 B Figure3-4. Theaccumulatedstrainrequiredtoreachsteadystateandthetemporal evolutionof " C b # .Theaccumulatedstrainrequiredtoreachsteadystateis shownA)forvaryingaspectratiosandconcentrations.Thetemporal evolutionof " C b # isshownB)for A =30 atvaryingconcentrations. 31

PAGE 32

Tounderstandthesourceofthediscrepancybetweennumericalandexperimental measurementsofthemicrostructure,theeasilyaccessibletemporaldependenceof theorbitconstantisstudied.Figure 3-4 plotstheaccumulatedstrainrequiredforthe presentnumericstoreachsteadystatealongwithanexampleofthetemporalevolution of " C b # for A =30 .Theaccumulatedstrainistheshearratemultipliedbythetime spentshearing.Whentheremainingaverageorbitconstantvaluesremainedwithinone standarddeviationofeachother,thesystemwasdeterminedtobeatsteadystate.It isimportanttonotethattheaccumulatedstrainislimitedinprecision.Onaverage,the simulationstoresthedetailsofthemicrostructureeverytenstrains,meaningthesystem mayreachsteadystateslightlybeforethevaluesreported.Theaccumulatedstrainto steadystatedecreaseswithincreasingconcentration.Notingthelogarithmicy-axis inFigure 3-4 A,thetimetosteadystategreatlyincreaseswithincreasingaspectratio. ThesetrendsofferalikelyexplanationfortheexperimentalresultsinFigure 3-2 .For example,intheexperimentsof Petrich etal. ( 2000 b )thesuspensionwasshearedfor 20 minutesat 0.5 s ! 1 beforemakingmorethan 1500 measurementsatintervalsof 270 / nL 3 toremovecorrelations,assuggestedby Stover etal. ( 1992 ).At A =50 and nL 2 d =3 , afterthe 1500 th measurementtheexperimentreachedanaccumulatedstrainof 3300 . For A =50 , nL 2 d =1 ,afterthe 1500 th measurementtheexperimenthadreached astrainof 8700 .AccordingtoFigure 3-4 A,theexperimenthadlikelyincludedmany observationsofastilltransientsystem.Unfortunately,comparingsimulationpredictions atthesamestraintoexperimentalresultsmaynotbepossibleasexperimentsalmost certainlyhadadifferentinitialdistribution.Thenumericsusedarandomlygenerated uniformdistributionfortheÞberpositionandorientation,whereastheexperimentsstirred thesuspensionat 120 RPMforatleast5hours.Thisresultsinapotentiallydifferent evolutionofthemicrostructuretosteadystate. Althoughthelongtimetosteadystatecanexplainthediscrepanciesbetween numericalandexperimentalresultsfor " C b # ,itisunclearwhyresultsfor ! p 2 x p 2 y " wouldbe 32

PAGE 33

0 500 1000 1500 2000 ! 0.9 0.91 0.92 0.93 0.94 " p x 2 # A 0 500 1000 1500 2000 ! 0.01 0.02 0.03 0.04 0.05 0.06 0.07 " p y 2 # " p z 2 # B Figure3-5. Thetemporalevolutionofthesecondordermomentoftheorientationfor alignmentwiththeßow,thegradient,andvorticitydirections.Numerical predictionsforthesecondordermomentoftheorientationforA)alignment withtheßowdirectiongivenby " p 2 x # ,andB)alignmentwiththegradientand vorticitydirectionsgivenby ! p 2 y " and " p 2 z # for A =30 at nL 2 d =3 areshown. 33

PAGE 34

inagreement.Tofurtherexaminethedynamicsofthemicrostructure,Figure 3-5 plots theevolutionof " p 2 x # , ! p 2 y " ,and " p 2 z # for A =30 and nL 2 d =3 .Themoments " p 2 x # and " p 2 z # requirethelargestamountofaccumulatedstraintoachievesteadystate.Thefavorable comparisonsfor ! p 2 x p 2 y " butnotfor " C b # arelikelydueto p z stillchangingintimeandthe orbitconstantissensitivetothesechanges.Additionally,theexperimentalprecisionmay notbehighenoughtoaccuratelymeasuretheorbitconstant.Duetothehighsensitivity of " C b # on p z ,itmaybeadvisabletostudytheindividualsecondordermomentsofthe orientationdistributionfornumericalandexperimentalcomparisons. Steadystatenumericalresultsforthesecondordermomentoftheorientation areplottedinFigure 3-6 .Fibersmoststronglyalignwiththeßowdirection,wherethis alignmentincreaseswithincreasingconcentrationandaspectratios.Alignmentwiththe gradientdirectiononlyslightlychangeswithincreasingconcentrationandincreaseswith decreasingaspectratio.Thealignmentwiththevorticityaxisprimarilycompensatesfor thechangingalignmentintheßowdirection. 3.3Summary NumericalsimulationsofrigidÞbersinsteadyshearßowwereperformed.Long andshortrangehydrodynamicinteractionswereignoredinthesimulationswhileashort rangerepulsiveforcemaintainedthefullexcludedvolume.Comparisonwithprevious numericalworkatconcentrations nL 2 d =1 to 3 foraspectratios A =11 to 50 strongly suggestscontactinteractionsaredominantinpredictingthemicrostructure.Additionally, thesecontactsmustmaintaintheexcludedvolumeoftheÞbersoverthisrangeof concentrationsandaspectratios.Experimentalandnumericaldiscrepanciesinresults fortheorbitconstantarelikelyduetoitssensitivitytochangesinÞberorientationand thelargeamountofaccumulatedstrainrequiredtoreachsteadystate.Thelongtimeto steadystateisaresultofcontinualchangesofÞberalignmentwiththeßowandvorticity directions.Theaccumulatedstrainincreaseswithincreasingaspectratioanddecreases withincreasingconcentration, nL 2 d . 34

PAGE 35

0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.6 0.7 0.8 0.9 1 ! p x 2 " A =11 A =15 A =30 A =50 A 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0 0.01 0.02 0.03 0.04 0.05 ! p y 2 " A =11 A =15 A =30 A =50 B 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0 0.05 0.1 0.15 0.2 0.25 ! p z 2 " A =11 A =15 A =30 A =50 C Figure3-6. Thesecondordermomentoftheorientationforalignmentwiththeßow,the vorticity,andthegradientdirections.ResultsareshownforA)theßow direction,B)thevorticityaxisandC)thegradientdirection. 35

PAGE 36

CHAPTER4 THEMICROSTRUCTUREINSUSPENSIONSOFFIBERSINSTEADYSHEARFLOW WITHCONFINEMENT 4.1BackgroundofConÞnementEffects Whendesigningaßowexperiment,theeffectoftheboundingwalls,orconÞnement, ontheresultsisanimportantconsideration.Inexperimentswithashearßow,the boundariesinthegradientdirectionareoftenofthesmallestscale;theboundariesdo notexistoraremuchfartherapartintheotherdirections.Forexample,experimental measurementsofthemicrostructurearecommonlydoneusingaCouetteshearcellwith aclearoutercylinder.Adyedtracerparticleisthenvisualizedfromthesideandfrom abovethecell,enablingmeasurementsoftheorientationdistributions.Thegapinthese experimentsneedstobesmalltoensuretheexperimentisapproximatingashearßow, whilestillbeinglargeenoughtonotinßuencetheparticledynamics.Byunderstanding theeffecttheboundingwallshaveonthemicrostructure,experimentscanbedesigned appropriately.Additionally,ifachangeinorientationoftheÞberduetoconÞnement occurs,itmayhaveimportantbeneÞtstoindustrialprocessesthatrequireaunique alignmentoftheÞber.Tounderstandtheroleoftheseboundariesonthemicrostructure, simulationsofsteadyshearßowwereperformedwhereÞbermotionwasrestrictedinthe gradientdirection;Þberswerekeptfrompassingthroughthetopandbottomboundaries forheightsorseparationdistancesofthreeÞberlengths, 3 L ,andÞveÞberlengths, 5 L . ThenumericalmodelusedwasdevelopedinChapter 2 .Inbrief,itignoreslongand shortrangehydrodynamicinteractions,butincludesparticlecontacts.Themotioninthe gradientdirectionisrestrictedbyarepulsiveforceusedtopreventaÞberfromcrossing theboundary. 4.2NumericalPredictionsComparingConÞnedandPeriodicResults Thepresentedresultsareaveragedatsteadystate.Steadystatewascalculated usingthesamecriteriaasinChapter 3 ,whereoncetheorbitconstantremainswithin 36

PAGE 37

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.6 0.7 0.8 0.9 ! p x 2 " A =11 A =11 3 L A =11 5 L A =20 A =20 3 L A =20 5 L A 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.7 0.8 0.9 1 ! p x 2 " A =30 A =30 3 L A =30 5 L A =50 A =50 3 L A =50 5 L B Figure4-1. TheeffectsofconÞnementon " p 2 x # atlowandhighaspectratios. Comparisonsaregivenforsimulationswithfully-periodicboundariesaswell assimulationsconÞnedatdistancesof 3 L and 5 L .Themoment " p 2 x # is shownforvariousconcentrationsatA)lowandB)highaspectratios. 37

PAGE 38

0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 ! p y 2 " A =11 A =11 3 L A =11 5 L A =20 A =20 3 L A =20 5 L A 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0 0.01 0.02 0.03 ! p y 2 " A =30 A =30 3 L A =30 5 L A =50 A =50 3 L A =50 5 L B Figure4-2. TheeffectsofconÞnementon ! p 2 y " atlowandhighaspectratios.Little changeinalignmentwiththegradientdirectionisshownbythemoment ! p 2 y " forA)lowandB)highaspectratiosatvariousconcentrationsandlevelsof conÞnement. 38

PAGE 39

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.1 0.2 0.3 ! p z 2 " A =11 A =11 3 L A =11 5 L A =20 A =20 3 L A =20 5 L A -0.5 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0 0.05 0.1 0.15 0.2 ! p z 2 " A =30 A =30 3 L A =30 5 L A =50 A =50 3 L A =50 5 L B Figure4-3. TheeffectsofconÞnementon " p 2 z # atlowandhighaspectratios.The changeinalignmentwiththevorticitydirectionisshownbytheorientation moment " p 2 z # forA)lowandB)highaspectratiosforvariousconcentrations andlevelsofconÞnement. 39

PAGE 40

0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.4 0.5 ! C b " A =11 A =11 3 L A =11 5 L A =20 A =20 3 L A =20 5 L A 0 0.5 1 1.5 2 2.5 3 3.5 nL 2 d 0.3 0.35 0.4 0.45 0.5 0.55 ! C b " A =30 A =30 3 L A =30 5 L A =50 A =50 3 L A =50 5 L B Figure4-4. TheeffectsofconÞnementon " C B # atlowandhighaspectratios.Theroleof conÞnementisshownforthe " C B # atA)lowandB)highaspectratiosacross varyingconcentrations. 40

PAGE 41

onestandarddeviationofitself,thesystemisconsideredconstant.Theangledbrackets, " á # ,designateanensembleaveraginghere,aswell. Thesecondordermomentoftheorientationwiththeßowdirection, " p 2 x # isplottedin Figure 4-1 .Forallaspectratios, A ,boundaryconditions, 3 L or 5 L ,andconcentrations, nL 2 d ,theÞbersstronglyalignwiththeßowdirection.Figure 4-1 showsastheaspectratioandconcentrationincrease,theeffectofconÞnementdecreases.WhenconÞnement affectsthesystem,itactstodecreasealignmentwiththeßowdirection. Figure 4-2 plotsthesecondordermomentoftheorientationwiththegradient direction, ! p 2 y " .ItisextremelyinterestingthatconÞnementonlyaffects ! p 2 y " for A =11 . ThisseemscounterintuitiveasconÞnementonlyexistsinthegradientdirection.Itis alsointerestingthatforagivenaspectratio,alignmentwiththegradientexhibitslittle changewithincreasingconcentration. Alignmentwiththevorticityaxis,givenbythemoment " p 2 z # ,isshowninFigure 4-3 . Similartothemoment " p 2 x # ,theimportanceofconÞnementdecreaseswithincreasing concentrationandaspectratio.Also,asthealignmentwiththegradientdirection onlyslightlychangesduetoconÞnement,thedecreaseinalignmentwiththeßow directionwhenconÞningthesystemiscompensatedbyanincreaseinalignmentwith thevorticityaxis.TheseeffectsduetoconÞnementalsomanifestthemselvesintheorbit constant " C b # ,asshowninFigure 4-4 .Theorbitconstantrequiresaseparationbetween boundingwallsgreaterthan 5 L for A =11 toremoveconÞnementeffects.For A ) 20 and nL 2 d ( 2 ,aseparationdistancegreaterthan 5 L isrequiredtoremoveconÞnement effects. 4.3Summary ConÞnementcanaffectthemicrostructureandaccordinglymustbecarefullyconsideredinexperimentaldesign.Interestingly,conÞningthesuspensioninthegradient directionprimarilyactstoalterthealignmentwiththeßowandvorticitydirections.The 41

PAGE 42

effectsofconÞnementonalignmentaredependentonaspectratioandconcentration.IncreasingtheaspectratioandconcentrationdecreasestheroleofconÞnement. ForaÞxed nL 2 d ,decreasingtheaspectratioincreasestheparticlevolumefraction, whereÞberswilllikelyexperiencemoreinteractionswiththewall.As nL 2 d increases,it seemslikelythatÞberswillhavesigniÞcantlymoreparticleinteractions,minimizingany disturbancescausedbytheboundingwall.However,furtherstudyisrequiredtofully understandtheeffectsofconÞnementonthemicrostructure. 42

PAGE 43

CHAPTER5 THEMICROSTRUCTUREINSUSPENSIONSOFFIBERSINUNSTEADYSHEAR FLOWWITHANDWITHOUTCONFINEMENT 5.1OverviewofParticleDynamicsinOscillatoryShearFlow Therheologicalandmechanicalpropertiesofsuspensionscomposedofrod-like particlesaresensitivetothespatialandorientationaldistributionoftheparticleswithin thesuspension.Understandingandcontrollingtheßow-inducedalignmentofthese elongatedparticlesisoffundamentalimportanceandhasimplicationsfornumerous materialprocessingapplications,suchasthemanufactureofpaper( Lundell etal. , 2011 ) andÞbercomposites( Papathanasiou&Guell , 1997 ). Recentmeasurements( Franceschini etal. , 2011 )indicatedthattheorientation distributioninsemi-concentratedsuspensionsofnon-colloidalÞberscanbecontrolled byalteringthestrainamplitudewithinanoscillatoryshearßow.TheorientationdistributionchangeslittlefromtheinitialstateatsufÞcientlysmallamplitudesofstrain.At largestrainamplitudes,theÞbersaligninthedirectionoftheßow,similarlytothesteady shearingßowofasuspensionofrigidÞbers( Stover etal. , 1992 ; Sundararajakumar& Koch , 1997 ).Whiletheseresultsmightbeexpected,theexperimentsalsomeasured astronglypreferredalignmentoftheÞbersinthevorticity(transversetoboththeßow andgradientplanes)directionforintermediatestrainamplitudes.Thissurprisingresultremainstobeunderstood.Alignmentwiththevorticitydirectionisobservedwhen oscillatingÞbersaresuspendedinaweaklyelasticßuid( Petrich etal. , 2000 a ).However,theexperiments( Franceschini etal. , 2011 )ofinterestherewereperformedwitha Newtonianßuid.Foranelasticßuid,theorientationofevenasingleÞberspinstoward thevorticitydirection( Iso etal. , 1996 ; Leal , 2006 ),whereastheorientationofasingle ÞberoscillatedinaNewtonianßuidtracesareversiblepathdeÞnedbytheJefferyorbit correspondingtotheinitialorientation( Okagawa etal. , 1978 ). Theirreversibledynamicsofsuspensionsofnon-colloidalparticlessuspendedand shearedinNewtonianßuidshasbeenstudiedextensively;thevastmajorityofworkhas 43

PAGE 44

focusedonsphericalparticles,butearlyworkusedÞbers( Ennis etal. , 1978 ; Okagawa etal. , 1978 ; Okagawa&Mason , 1973 ).ThereversibilityoftheStokesequationsimplies thattheparticlesshouldretracetheirpathsuponreversaloftheßow.However,any smallsourceofirreversibilityinaphysicalsystem(e.g.Brownianmotion,particlesurface roughness,particledeformability,orinertia)causesalossofmemory.Oscillatingthe suspensionsprovidesaparticularlyconvenientformatforstudyingirreversibilityin suspensionsystems( Breedveld etal. , 2001 ; Pine etal. , 2005 ).ForsufÞcientlylarge strainsandconcentrations,particlesdonotreturntotheirstartingconÞgurationsafter oneormorecyclesandtheirdisplacementsexhibitachaotic,randommotion.Theneed forrelativelylargeconcentrationsofparticlesimpliesthatshortrangeinteractionsare theprimaryoriginofthechaoticbehavior. Thisworkaddressestheoriginoftheunexpectedandirreversiblealignment oftheÞbersinthedirectionofvorticity,whichcannotbeattributedtotheelasticity ofthesuspendingßuid.Ratherthevorticityalignment,andotherbehaviorsofthe orientationdistribution,wasattributedby Franceschini etal. ( 2011 )toshortrange interactionsbetweenÞbersinaprocessthatcandrivethesystemintoareversible, or"absorbing",state.Asimplealgorithm,developedinChapter 2 ,demonstratesthat short-rangeinteractionsbetweenparticlescanaccountforalignmentinthevorticity direction,dependingupontheconcentrationandstrainamplitude.Themodelreveals thattheextentofalignmentdependsstronglyonwhetherornotthesuspensionof ÞbersisconÞnedinthegradientdirectionbyclosely-spacedboundingwalls.Onlythose simulationswithawall-spacingmatchingthatusedintheexperimentsaccuratelypredict themeanorientationreportedby Franceschini etal. ( 2011 ) 5.2ResultsandDiscussion TheconcentrationofÞbersandstrainamplitudearevariedwithinthesimulations. Theaspectratioissetto A =11 tocorrespondwiththeexperimentsand A e =8.8 . Forfullyperiodiccalculationsthatutilize Lees&Edwards ( 1972 )boundaryconditions, 44

PAGE 45

! " Figure5-1. RelevantparametersforÞbersinoscillatoryshearßow.A)EachÞberof aspectratio A = L / d isdescribedbyitscenterofmass x i andorientation p i . ArepulsiveforceactsbetweentwoÞbers i and j atthepointsofclosest approachseparatedbythedistance h ij .B)The x -directionisthatoftheßow; theßowvarieslinearlyinthe y -direction,orgradientdirection,andthe vorticityaxisisinthe z -direction.TheÞber'sprojectionontheßow-vorticity planeandtheßowdirectionformstheangle " : cos 2 ( " )= p 2 x / ( p 2 x + p 2 z ) . thesimulationboxiscubicwithalengthof 3.25 L .Forsimulationswithboundaries, ÞbersareconÞnedinthegradientdirectionbyapplyingthesameforcebetweena ÞberandthewallasusedbetweencontactingÞbers.Thewallseparationof 1.45 L usedinthesimulationscorrespondstothatusedintheexperimentsof Franceschini etal. ( 2011 )Periodicboundariesintheßowandvorticitydirectionsareseparatedby 5.25 L .Thenumberofparticlessimulatedrangesfrom 265 to 1,232 ,correspondingto volumefractions ' of 0.05 to 0.20 .Theuseofvolumefractiondeviatesfromthegeneral quantiÞcationofconcentrationinthisdissertation,butisusedforeaseofcomparing topreviousworks.Thevolumefractioncaneasilybeconvertedusingthefollowing equation, nL 2 d = 4 ( ' A . (5Ð1) TwelveinitialconÞgurationswereusedtocalculateeachmeanvalueandreportederrors arethosebetweenthemeanvaluesofeachconÞguration. Primarilytwomeasuresarecalculatedfromthedatageneratedbythesimulations andpresentedhere:onefortheaverageorientationsandanotherthattracksthe 45

PAGE 46

01234 5 ! (Strain Amplitude) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 S " ( N R ) Periodic Simulations Bounded Simulations Experimental Results 01234 5 ! 0 200 400 N R A B Figure5-2. NumericalandexperimentalresultsforÞberalignmentwiththevorticityaxis. A)Comparisonofsimulationresultsfor S ! withtheexperimentalvaluesfor ' =0.20 andatthethelastoscillationreportedby Franceschini etal. ( 2011 ). Theinsetshowstheoscillationnumber( N R )atwhichthedatawasreported. B)Imagesofthemicrostructurefromasimulationwithboundaries, ' =0.20 , and ! =3 priortooscillating( N =0 ),atthelastpointrecordedinthe experiments( N =400 ),andatsteadystate( N =4000 ). 46

PAGE 47

irreversibleactivityofthesuspensionofÞbers.Theorderparameter S ! =1 $ 2 " cos 2 ( " ) # iscalculatedfromthesimulationresultstofacilitatecomparisonwiththeexperimental measurements Franceschini etal. ( 2011 )ofalignmentthatwerereportedsolelyin termsofthisparameter.AsshowninFigure 5-1 ,theangle " isthatbetweentheßow direction( x )andtheprojectionofaÞber'sorientationontotheßow-vorticity( x z ) plane;thebrackets, " á # ,indicateanaverageoverallrunsandÞbers.Notethat S ! is 0 forasuspensionhavingarandomorientationdistributionandiseither $ 1 or 1 fora suspensionofÞbersalignedperfectlywiththeßoworvorticitydirection,respectively. ThenumberofcontactsbetweenÞbersarecountedovereachoscillationandreported as N c .Inthismodel,irreversiblemotionsoftheparticlesaregeneratedonlybycontact forces,thus N c isameasureofirreversibleactivitywithinthesuspension. Figure 5-2 Acomparesexperimentallymeasuredandcalculatedvaluesof S ! as afunctionofstrainamplitude.Simulationswithboundariesseparatedbyadistance equivalenttotheexperimentscapturesthetrendofthealignmentbehavior,including thesurprisingalignmentoftheÞberswiththevorticitydirectioninthevicinityof ! ' 3 . Usingfullyperiodicboundariesdoesnotreproducetheexperimentalresults.Figure 5-2 Bvisualizestheorientationdistributionchangeswiththenumberofoscillations, N , forasimulationat ! =3 withconÞningwalls.Theorientationisrandomfortheinitial conditionat N =0 andprogressivelyalignsinthevorticitydirectionthrough N =400 , correspondingtothepointatwhichthecomparisonismadewithexperimentsinFigure 5-2 A.ThesimulationpredictsanincreasingalignmentoftheÞberswiththevorticity directionuntilreachingasteadystateat N ' 4000 .Figure 5-3 Aprovidesmoredetails abouttheevolutionof S ! with N andshowsthatthevalueof S ! for ! =3 transitions from 0.60 at N =400 to 0.70 at N =4000 ;thesigniÞcantdifferenceintheorientation distributionrepresentedbythisdifferenceof 0.1 inthevalueof S ! isdemonstratedin Figure 5-2 B. 47

PAGE 48

1101001000 N (Number of Oscillations) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 S ! (N) Bounded " =1 Bounded " =3 Bounded " =5 Periodic " =1 Periodic " =3 Periodic " =5 A 1101001000 N (Number of Oscillations) 1000 10000 1e+05 1e+06 1e+07 N c (Number of Contacts) B Figure5-3. Temporalevolutionof S ! andthenumberofcontacts.Resultsforavolume fractionof ' =0.20 .A)Theevolutionof S ! withthenumberofoscillationsfor boththeperiodicandboundedsimulations.B)Thetotalnumberofcollisions, N c ,experiencedbetweentheÞbersduringeachoscillation. 48

PAGE 49

Figure 5-3 AshowsthatsigniÞcantchangesinorientationoccurwhenoscillating with ! > 1 .Thecurvesexhibitaninitialdecreasein S ! ,indicatingamovementtoward ßowalignmentforinitialoscillations,followedbyanincreasethatismorepronounced forsimulationswithboundaries.Inthislattercase, S ! attainspositivevaluesthatreßect avorticityalignmentoftheÞbers.Notethattheinitialdistribution,bothspatialand orientational,isarandomoneinthesimulations,whereassteadyshearwasappliedin theexperimentsbeforebeginningtheoscillations.Thismayexplainthelackofaninitial decreasein S ! asreportedinthethirdFigureof Franceschini etal. ( 2011 ). Figure 5-3 Bshowsthatanorderofmagnitudereductioninthenumberofcollisions betweenÞbersoccursfor ! =3 and 5 forsimulationswithboundingwalls.Asimilar trendisobservedforsimulationswithoutconÞnementforthesamestrainamplitudes, though N c isanorderofmagnitudelarger.Inthesecases, N c reachessteadystateby thesamevalueof N as S ! .Thenumberofcollisionsalsodecreaseswith N for ! =1 , butasteadystateisnotobtainedwithinthesamenumberofcyclesas S ! .Though S ! doesnotchangesigniÞcantly,individualÞberscontinuetochangeorientationdueto contacts. Franceschini etal. ( 2011 )interpretedthelackofachangeintheorderparameter S ! aftertheÞrstfewoscillationsfor ! < 2.2 tothesuspensionattaininganabsorbing stateinwhichindividualparticlesreturntotheirstartingpositionsandorientationsafter eachcycle.Theyalsoassociatedthevorticityalignmentwithattainmentofanabsorbing state,whereasFigure 5-3 indicatesactivityacrossallstrainamplitudesfortheduration ofthesimulations.Thediscrepancybetweenthenumericalpredictionsandexperimental measurementsofactivitycouldbeexplainedbyresolutionlimitsinthemeasurements. Thesimulationspredictorientationchangesanddisplacementsofonlyafewmicrons forindividualparticlesat ! =1 and N > 10 ;suchsmallchangeswouldbedifÞcultto detectevenwithamicroscope.Also,lubricationinteractionsthatexistbetweenÞbers intheviscousßuidcouldprevent,oratleastreduce,theoccurrenceofcontacts.The 49

PAGE 50

01234 5 ! (Strain Amplitude) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 S " Bounded # =0.05 Bounded # =0.15 Bounded # =0.20 Periodic # =0.05 Periodic # =0.15 Periodic # =0.20 Figure5-4. Changesin S ! duetobounding,strainamplitude,andconcentration.Steady valuesoftheorderparameter S ! asafunctionofstrainamplitudefor simulationsperformedwithfullyperiodicboundaries(Periodic)andwith boundingwalls(Bounded). questionoftheexistenceofcontactsinthepresenceoflubricatingßuidsisaquestionof activeinvestigationforsuspensionsofspheres( Arp&Mason , 1976 ; Metzger&Butler , 2012 ),thougharguments( Petrich&Koch , 1998 )thatcontactsaremorelikelybetween Þbersthansphereshavebeenmade.Thecurrentsimulationsdonotincludelubrication withthegoalofgeneratingaminimalmodelthatcanexplaintheunexpectedalignment phenomena. Alldatapresenteduptothispointhasbeenforavolumefractionof 0.20 .Figure 5-4 showsthesteadyvaluesof S ! at ' =0.20 andtwoadditionalvolumefractions.At thelowervolumefractionof 0.05 ,theorientationoftheÞberstendtoalignmorewiththe ßow,ratherthanthevorticity,directionforboththeboundedandunboundedcases. ConÞningtheÞbershasthemostimpactontheorientationdistributionat ' =0.20 and ! =3 ,wheretheresultsofsimulationswithfullperiodicitypredictameanalignment 50

PAGE 51

inthedirectionofßow.Theprobabilitydistributionoftheangle " giveninFigure 5-5 A showsthesmallpreferenceforßowalignmentwhenusingperiodicboundariesandalso quantiÞes,withmoredetail,thestrongalignmentinthevorticitydirectionwhenusing boundingsurfaces. Themaximumextentofalignmentispredictedat ' =0.15 and ! =3 ,where S ! =0.88 ± 0.02 forthesimulationsofconÞnedÞbers.Usingfullyperiodicboundaries forthesameconcentrationandstrainamplitudealsogeneratesapreferredalignmentin thevorticitydirection,where S ! =0.60 ± 0.05 .Comparingthedetailsofthedistributions, asdoneinFigure 5-5 B,demonstratesthesigniÞcantorientationdifferencebetweenthe twotypesofboundaryconditions. ThislatterresultshowsthatboundingwallsarenotrequiredtoachievesomealignmentoftheÞbersinthevorticitydirection,butconÞningtheÞbersclearlyenhancesthe extentoforganizationandalignmentinthevorticitydirection.Thealignmentphenomena,whetherintheßoworvorticitydirection,areacomplexfunctionoftheinteractions betweentheimposedstrainingßowsandtheiramplitude,theconcentrationoftheÞbers, andthegeometryoftheshearcell. ConÞningelongatedparticlesbetweenboundingwallsgenerallypromotesorganizationoftheorientations,asforliquidcrystalswhichtransitionfromanisotropictonematic phasewhensqueezedbetweenplanarwalls( DeGennes&Prost , 1993 ).Likewise, theboundariesimposesomeorderontheparticlesintheimmediatevicinityofthewall (Figure 5-2 B),evenfortheinitialcondition;ßuctuationsinorientationgeneratedbythe oscillatoryshear,ratherthanthermalßuctuationsasforliquidcrystals,providesthe mechanismfortheparticlestosearchforafavorablearrangement.Perhapsthemore interestingpointhereisthepreferenceforvorticity,versusßow,alignment. ThelackofaboundarythatorganizestheorientationsresultsinasigniÞcantly lowerextentof,orevenno,alignmentinthedirectionofvorticityascomparedtothe boundedsimulations.Asimpletestwasperformedtoverifythatthesolidboundaries 51

PAGE 52

010203040 5060 708090 ! 0 0.05 0.1 0.15 0.2 0.25 probability Bounded Periodic A 010203040 5060 708090 ! 0 0.1 0.2 0.3 0.4 probability Bounded Periodic B Figure5-5. Theprobabilitydistributionoftheangle " .When " =0 o theÞberisaligned withtheßowdirectionandwhen " =90 o theÞberisalignedwiththevorticity direction(Figure 5-1 ),atsteadystateforboundedandfullyperiodic conditions.ResultsforavolumefractionofA) ' =0.20 andB) ' =0.15 . Notethattheplotsutilizedifferentscales. 52

PAGE 53

themselves,andneithertheinitialorientationdistributioncreatedbythewallsnor differentgeometryofthesimulationbox,areresponsibleforthevorticityalignment predictedbythesimulations.Theinitialdistributionsweregeneratedwithinthebounded cell,butthenthesimulationwasperformedwithperiodicboundaries.Thesesimulations returnedonlyaslightlylowertendencyfortheÞberstoalignintheßowdirection.For ' =0.20 and ! =3 ,thesimulationpredicts S ! = $ 0.08 ± 0.03 at N =400 ,whichisclose tothevalueof S ! = $ 0.17 ± 0.03 forperiodicboundarieswhenstartingfromarandom particleconÞguration. ThesensitivityoftheÞberalignmentontheinitialdistributionwasexploredfurther byaligningthesuspensioninsteadyshearpriortooscillating,aswasdoneinthe experiments( Franceschini etal. , 2011 ).For ' =20 with ! =3 andboundingwalls, thesimulationswiththepre-shearpredictthatanydifferencesinthemeanalignment disappearby N =400 ,wherethepre-shearedcasesgive S ! =0.63 ± 0.05 andthe initiallyrandom,butconÞned,distributionsgive S ! =0.6 ± 0.1 . 5.3Summary Theminimalmodelemployedhere,whichconsidersonlyself-mobilitiesandexcludedvolumeoftheÞbers,reproducestheexperimentalobservation( Franceschini etal. , 2011 )ofvorticityalignmentoftheorientationdistributionforoscillatoryshear whenboundariesareincluded. Franceschini etal. ( 2011 )attributedthevorticityalignmenttoparticle-particlecollisionssolely,omittingthewallcontributions.Thoughsome alignmentcanoccurwithoutconÞningthesuspension,theconÞnementclearlyplaysa strongroleintheexperimentalobservationswhichutilizedawallspacingof 1.45 Þber lengths.Accordingly,asimpleandcompleteprincipleoforganizationisstilllackingas thealignmentdependssubtlyuponthreefactors:theamplitudeofthestrainingßow,the concentrationoftheÞbers,andtheboundaryconditions.Forthelatterfactor,onlythe twocasesoffullyperiodicboundariesandawallseparationidenticaltothatintheexperimentswereconsidered.Thewallseparationatwhichtheorientationaldynamicsof 53

PAGE 54

thesuspensiontransitionsfromthehighlyconÞnedtotheunboundedbehaviorremains tobeexplored. 54

PAGE 55

CHAPTER6 DERIVATIONOFTHESTRESSINSUSPENSIONSOFRIGIDFIBERS 6.1RelevancyofDerivation ThenumericalmethoddevelopedinChapter 2 enablespredictionofthemicrostructure,butadditionalcalculationsarerequiredtopredicttherheology.Uniquely,the simulationoutputsthenecessaryinformationtodirectlycalculatethecontactcontributiontothestress,determiningtheimportanceofcontactsontherheology.Inclusionof thecontactsinthestresspredictioniscurrentlyuncommon,buthasbeensuggested by Petrich etal. ( 2000 b )and Salahuddin etal. ( 2013 )tobethesourceofdiscrepancies betweencurrenttheoreticalpredictionsandexperimentalobservations.Toreviewthe originofthecontactcontributiontothestresstensor,abriefderivationismade. 6.2StressCalculation Thebulkstress, " " # ,ofthesuspensioniscalculatedbyvolumeaveragingoverthe ßuidandparticlephases.ThestressforasuspensionofÞbersexperiencingamean strainrateof " E # inaNewtonianßuidofviscosity µ is " " # = $ P I +2 µ " E # + n " S # , (6Ð1) where P isthepressure, I istheidentitymatrix, n istheparticlenumberdensity,and " S # isthemeanparticlestresslet.Both $ P I and 2 µ " E # aretheßuidcontributiontothe stresswhilethelasttermrepresentstheparticlecontributiontothestress.Themean stressletforacollectionofrigidslenderbodiescanbecalculatedfromalineintegral overeachÞber i , " S # = 1 N f N f + ! =1 , L / 2 ! L / 2 ( x ! + s p ! ) F ! ( s ) ds . , (6Ð2) where V isthetotalvolumeand F ! ( s ) isthelineforcedensity.Thelineforcedensityof arigidslenderparticleis( Batchelor , 1970 ), F ! ( s )= $ L / I $ 1 2 p ! p ! 0 á ( u ( x ! + s p ! ) $ ( úx ! + s úp ! ) ) . (6Ð3) 55

PAGE 56

UsingEquations( 2Ð1 )and( 2Ð2 )inEquation( 6Ð3 )andsolvingforthestressgives, " " # = $ P I +2 µ " E # + µ f 1 pppp 2 : E $ N V 1 r !" f !" 2 , (6Ð4) where r !" = x ! + s !" p ! $ ( x " + s "! p " ) , N isthetotalnumberofcontacts,andthelast termofEquation( 6Ð4 )isthecontributionofthecontactforcestothestresswhere f !" istherepulsivecontactforceusedinthesimulation,asdeÞnedinEquation( 2Ð3 ).The termcontaining µ f inEquation( 6Ð4 )representsthehydrodynamiccontributionofthe particlestothestress.Directlycomputing µ f fromtheEquations( 6Ð1 ),( 6Ð2 ),( 6Ð3 ),and ( 6Ð4 )gives ( nL 3 µ/ 6ln(2 A ) .However, Batchelor ( 1971 )calculatedacorrectionto µ f toaccountfortheÞnitethicknessoftheÞbers.Others( Mackaplow&Shaqfeh , 1996 ; Shaqfeh&Fredrickson , 1990 )havederivedcorrectionswhichalsoincludetheeffects ofhydrodynamicinteractionsbetweenparticles.Thetheoryof Mackaplow&Shaqfeh ( 1996 )isusedinthecurrentwork,where µ f = 1 6 ( nL 3 µ # ) f ( ) )+0.206 nL 3 ) 3 $ , (6Ð5) f ( ) )= 1+0.64 ) 1 $ 1.5 ) +1.659 ) 2 , (6Ð6) and ) = [ ln(2 A ) ] ! 1 . ThenormalstressdifferencesareespeciallyimportanttoChapters 7 and 8 and arecalculatedfromEquation( 6Ð4 ).Thenormalstressdifferencesdividedbythe shearstressofthesuspensionyieldsthecoefÞcients " 1 = N 1 / # and " 2 = N 2 / # . Thecomponentsofthestressesaregivenbyaveragesoftheorientationsandcontact 56

PAGE 57

interactions, N 1 = " ! xx #$" ! yy # = µ f ú ! #! p 3 x p y " $ ! p x p 3 y "$ $ N V [ " r x f x #$" r y f y # ] , (6Ð7) N 2 = " ! yy #$" ! zz # = µ f ú ! #! p x p 3 y " $ ! p x p y p 2 z "$ $ N V [ " r y f y #$" r z f z # ] , and (6Ð8) # = " ! xy # (6Ð9) = µ ú ! + µ f ú ! ! p 2 x p 2 y " $ N V " r x f y # , (6Ð10) where r x , r y , r z , f x , f y ,and f z arethecomponentsof r ij and f ij . 57

PAGE 58

CHAPTER7 THERHEOLOGYINSUSPENSIONSOFFIBERSWITHOUTCONFINEMENT 7.1OverviewofRheologyinSuspensionsofParticles Therheologyofsuspensionsofparticlesremainsacomplexproblemdespitebeing aclassicalsubjectwithseminalworkby Einstein ( 1905 )and Batchelor&Green ( 1972 ). Evenforthesimplestcaseofrigid,force-free,andnon-colloidalparticlessuspendedin aNewtonianßuid,suspensionscanexhibitnon-Newtonianproperties,includingnormal stressdifferences.Thesteadyshearrheologyofsuchsuspensionshavesomecommon featuresregardingthescalingsofthestresses,regardlessoftheshapeoftheparticle. However,thedetailsoftherheologycandifferconsiderablyforspheresascomparedto Þbers,thesubjectofthepresentwork,orotherparticleshapes. AtlowReynoldsnumber,linearityoftheStokesequationsleadstoaviscous scalingoftheaveragestressforasuspensionundersteadyshearingconditions.This lawappliedtotheshearstress # canbewrittenas # = µ s ú ! ,where µ s istheeffective viscosityofthesuspensionand ú ! istherateofshear.Formonodispersesuspensions ofspheres,theviscosity µ s dependsonlyonparticlevolumefractionandincreaseswith increasingvolumefraction,divergingatmaximumpacking(e.g. Stickel&Powell , 2005 ). ForÞberparticles,thesuspensionviscosityalsoincreaseswithconcentrationandis muchlargerthanthatofasuspensionofspheres,whensuspendedinthesameßuidat thesamevolumefraction( Petrie , 1999 ). Thisquasi-Newtonianbehavioroftheshearstressdoesnotfullydescribethe rheologyofsuspensions,asitdoesnotaccountforthepossibleexistenceofnormal stressdifferences.ThesenormalstressdifferencesalsoscaleviscouslyinStokes ßowsand,sincethenormalstressesdonotdependonthesignoftheshearrate,are proportionaltothemodulusoftheshearstress, | # | .Thus,thenormalstressdifferences canbewrittenas N 1 = " 1 | # | and N 2 = " 2 | # | ,where N 1 and N 2 aretheÞrstandsecond normalstressdifferences. 58

PAGE 59

Forsuspensionsofspheres,experimental( Boyer etal. , 2011 b ; Couturier etal. , 2011 ; Dai etal. , 2013 ; Dbouk etal. , 2013 ; Singh&Nott , 2003 ; Zarraga etal. , 2000 ) andnumerical( Sierou&Brady , 2002 ; Singh&Nott , 2000 )investigationsofthenonNewtoniannormalstressdifferenceshavebeenperformed.Thesecondnormalstress difference, N 2 ,isnegativeandincreasesastheparticlevolumefractionincreases, growingespeciallyquicklyforparticlevolumefractionsabove 20 percent.Theproperties oftheÞrstnormalstressdifferencearemoreelusive.Themagnitudeof N 1 iscertainly muchsmallerthanthatof N 2 ,butassessingthesignisdifÞcult.SomeexperimentsÞnd that N 1 isquitesmallandnegative( Dai etal. , 2013 ; Singh&Nott , 2003 ; Zarraga etal. , 2000 ).Othersreportpositivevalues( Dbouk etal. , 2013 )andyetothershaveasserted thatthevalueistooclosetozerotodeterminewhetheritisnegative,positive,ornull ( Couturier etal. , 2011 ). Informationregardingnormalstressdifferencesinsuspensionscomposedofnonsphericalparticlesisscarcecomparedtosuspensionsofspheres,buttheoryprovides someimportantinsightsintoqualitativeaspectsofthestressesforÞbersascompared tospheres.Anisolated,sphericalparticlethatisfreelysuspendedinashearingßow doesnotgenerateanormalcomponentofbulkstress;rathernormalstressdifferences arisefromparticleinteractionsinsuspensionsofspheres.Incontrast,anisolated Þberinshearßowcancontributetoanon-zeronormalstressdifference,wherethe valuedependsupontheinstantaneousalignmentoftheÞberwithrespecttotheßow ( Batchelor , 1971 ). IncreasingtheconcentrationofÞbersbeyondthelimitofisolatedÞbersresultsin additionalcontributionstothenormalstressdifferencesduetointeractionsbetween particles.Theoriesthataccountforthevelocitydisturbancesonparticlesduetothe presenceofotherparticleshavebeendeveloped.Thesepredictionsforthestresses dependuponknowledgeofthemicrostructureofthesuspensions,whicharedetermined aspartofthesolutionprocedureinsomecases( Dinh , 1984 )andinothercases 59

PAGE 60

( Mackaplow&Shaqfeh , 1996 ; Shaqfeh&Fredrickson , 1990 )mustbeeitherassumed ordeterminedfromsimulations.Directvisualizationalsohasbeenusedtodetermine themicrostructureandtheresultsthenused,togetherwiththetheoriesforstress,to estimatethenormalstressdifferences( Petrich etal. , 2000 a ; Stover etal. , 1992 ). Direct,experimentalmeasurementsofthenormalstressdifferencesforÞber suspensionshavereliedprimarilyonparallelplategeometries,whichmeasure N 1 $ N 2 . Results( Goto etal. , 1986 ; Keshtkar etal. , 2009 ; Petrich etal. , 2000 a ; Sepehr etal. , 2004 )arelargelyinagreementthat N 1 $ N 2 ispositiveandincreasesastheparticle volumefractiondoes;someoftheseworkssuggestthat N 1 $ N 2 increasesastheaspect ratio,orÞberlengthtodiameter,increases.Manyoftheseauthorsreport N 1 ,rather than N 1 $ N 2 ,undertheassumptionthattherelativemagnitudeofthesecondnormal stressdifferenceismuchsmallerthantheÞrst.Alternatively,cone-and-plategeometries canmeasure N 1 directly,asdoneby Kitano&Kataoka ( 1981 ),thoughÞberßexibility playedaroleintheirresults.Thelackofadditionalreportsfor N 1 fromcone-and-plate geometriescouldbeduetothedifÞcultyinmakingmeasurements,whicharefrustrated bytheinabilitytoÞndparticlesthatarerigidandnon-colloidal,butstillsmallwithrespect tothegapsizeintheshearingcells. Thepresentpaperreportsonmeasurementsofnormalstressdifferencesfor concentratedsuspensionsofrigid,neutrallybuoyant,andnon-BrownianÞbers.Measurementsandanalysisofthefree-surfacedeßectioninatilted-troughexperiment resultsinvaluesforthesecondnormalstressdifferenceasafunctionofconcentrationandaspectratio.Likewise,observationsofthedeßectionsofthefree-surfacein aWeissenberggeometryprovidealinearcombinationoftheÞrstandsecondnormal stressdifferences.Whenusedinconjunction,thesemethodsenableassessmentof both N 1 and N 2 asdescribedinChapter 7.2 .Theseexperimentallymeasuredvaluesof thenormalstressdifferencesarecomparedtovaluescalculatedfromsimulationsofthe 60

PAGE 61

Þbersuspension,whicharedescribedinSection 2 .Theobservedandpredictednormal stressdifferencesarediscussedinSection 7.3 . 7.2Experiments NormalstressdifferencesofÞbersuspensionsarecalculatedfrommeasurements ofthedeßectioninthefree-surfaceinaWeissenberg,orrotating-rod,geometryandin atilted-trough.ThisapproachhassomesigniÞcantadvantagesoverusingastandard rheometerandtooling.Firstly,issueswithregardtoÞberconÞnementorboundary effects(e.g.parallelplategapdistance)canbereduced.Secondly,rheometerstypically haveasensitivityclosetothevalueofthenormalforcesproducedinsomesuspensions. Thedeßectionofthefree-surfaceintherotating-rodgeometryisrelatedtothe combination " 2 + " 1 / 2 ( Beavers&Joseph , 1975 ).Thismethodiswell-knownfor examiningnormalstressdifferencesinpolymericßuids,wheretheßuidclimbstherodin aphenomenonreferredtoastheWeissenbergeffect.Forsuspensionsofspheres,the free-surfacedipsneartherotating-rod( Boyer etal. , 2011 b ; Zarraga etal. , 2000 ). Lesswell-knownisthatthesecondnormalstressdifferencecoefÞcient,or " 2 , distortstheßuid-airinterfaceforßowsinopenchannelsortroughs.Thismethod wasÞrstdescribedby Wineman&Pipkin ( 1966 )andwasusedby Tanner ( 1970 )for measuringnormalstressdifferencesinvisco-elasticßuids.Morerecently,opentrough ßowswereusedformeasuring " 2 insuspensionsofnon-Brownianspheres( Couturier etal. , 2011 ; Dai etal. , 2013 ). 7.2.1ParticlesandFluids Fourbatchesofpolyamiderod-likeparticles(PolyamideMIMATsuppliedbySoci « et « e NouvelleLeFlockagewithadensityof 1.156 g á cm ! 3 )wereusedintheexperiments. ImagesofÞbersofaspectratio 32 , 13 ,and 12 areshowninFigure 7-1 ,wheretheaspect ratiois A = L / d , L istheÞberlength,and d thediameter.Thelengthanddiameterof over 100 Þbersweremeasuredwithadigitalimagingsystem.Thedistributionsof L and 61

PAGE 62

! ! " !" #$%&& ' () Figure7-1. MicroscopicimagesofpolyamideÞbers.ImagesareshownforÞberswithan aspectratioofA) A =32 ,B) A =13 ,andC) A =12 .Allthreeimagesare displayedatanidenticalscaleforwhichthescalebarof 0.5 mmis appropriate.Multipleimageswerecollectedandanalyzedforallaspect ratiostocalculatethelengthanddiameterdistributions;theresultsare reportedinTable C-7 .Photographscourtesyoftheauthor. d werefoundtobeapproximatelyGaussianforallaspectratios;theaveragelength, diameter,andaspectratioareshowninTable C-7 . Forthecurrentstudy,theÞberscanbeconsideredrigid.Thecriticalstress, ! crit , atwhichtheÞbersbegintobendduetoabucklinginstability,isgivenby( Forgacs& Mason , 1959 ) ! crit * = E b (ln2 A $ 1.75) 2 A 4 , (7Ð1) where E b isthebendingmodulus,whichisapproximatelytwicetheYoung'smodulus oftheÞber.Themaximumstress, ! max ,actingontheÞbersintheexperimentwas estimatedbycalculatingtheprojectionofthegravityforcesoverthesurfaceareaof thetrough.ThesevaluesarecomparedtothecriticalstressinTable C-7 .Evenforthe highestaspectratioof 32 ,thestressesintheexperimentwouldneedtobeatleast 30 timeslargerforßexibilitytobecomeaconcern. 62

PAGE 63

TherigidÞbersweresuspendedinaNewtonianßuidthathadamatchingdensity. Thesuspendingßuidwasamixtureofwater( 10.2 wt % ),TritonX100 ( 78.0 wt % ),and ZincChloride( 11.8 wt % ).Theviscositywas * f =1.84 ± 0.03 Pa á sat 25 " C.The suspensionswerepreparedbyaddingtheparticlestotheßuid(bothquantitiesbeing weighted)andgentlystirringbyhand.Themixturewasthenplacedinavacuumand leftovernighttoremoveanyairbubbles;littletonosettlingorcreamingwasobserved duringthistime.Thesuspensionwasthenpouredintotheexperimentaldevice(the rotating-rodcontainerorthesemi-circulartrough).Notethatthesuspensionhasto completelyÞllthedeviceinthecaseofthesemi-circulartrough.TheÞbervolume fractionwasvariedwithvaluesof nL 2 d between 1.5 and 3 ,where n isthenumberof particlesperunitvolume.Thesurfacetension ! ofthesuspensionwasmeasuredusing theDuNo ¬ uyring-method.Thevalueof ! =30 ± 3 dyne á cm ! 1 doesnotvarysigniÞcantly withvolumefraction. Toprobethesurface-proÞlometrytechniqueintherotating-rodandtilted-trough ßows,amixtureofspheresandßuidsimilartothatusedby Boyer etal. ( 2011 b )and Couturier etal. ( 2011 )wastested.Theparticleswerepolystyrenebeads(Dynoseeds suppliedbyMicrobeads)havingaradiusof 82 ± 4 µ m,adensityof 1.049 ± 0.003 g á cm ! 3 , andavolumefractionof 40% .Theparticlesweresuspendedinaßuid[poly(ethylene glycol-ran-propyleneglycol)monobutylether]withamatchingdensityandaNewtonian rheologywithashearviscosityof 2.15 Pa á sat 25 " C. 7.2.2Rotating-RodandTilted-TroughFlows Theexperimentsconsistedofmeasuring,byopticalproÞlometry,thefree-surface deßectioninducedbytheanisotropicstressesinarotating-rodßowandaßowdowna tiltedsemi-circulartroughasillustratedinFigure 7-2 .ThemeasuredproÞlesarethen usedtodeterminethenormalstressdifferences. Therotating-rodapparatuswasthesameasthatusedby Boyer etal. ( 2011 b ), wheremoredetailscanbefound.Acylindricalmetalcontainerwitha 5 cmradiuswas 63

PAGE 64

!"#$%&#'(%)$ *'+"+,-.&%'/ A ! ! !" # $ !" #$%&'(%)&&*($+,(-.+&(*'$/.+0( *)&(/1+*12'(13(*)&(,&31'4&,( .+*&'3$/& 5&4.6/.'/2-$'(*'120)( *.-*&,(7.*)($+($+0-&( ! #$%&'(%12'/& B Figure7-2. Illustrationsoftherotating-rodandthetilted-troughßows.IllustrationsofA) therotating-rodandB)tilted-troughßowsareshown.Lasersheets, projectedatlowanglesontothesurfacesoftheßowingsuspensions,enable quantiÞcationofthesurfacedeformations.Forthetilted-trough,nineparallel lasersheetsareutilized;exampleimagesoftheresultinglinesaregivenin Figure 7-3 .ThecoordinatesystemusedinAppendix A foranalysisofthe trough,tiltedatanangle % withrespecttothehorizontal,isalsoshown. 64

PAGE 65

Þlledwiththesuspensionandarodwitha 1.25 cmradiuswasimmersedatthecenter ofthecontainer[Figure 7-2 A].Therodwasrotated,byanAntonPaarRheolabQC rheometer,ataÞxedangularvelocitywhilealaserdiodeprojectedalasersheetontothe free-surfaceofthesuspensionatalowandÞxedangle.Thissheetformedavisibleline onthefree-surface,thedeviationsofwhichareproportionaltotheverticaldisplacement ofthefree-surface.Thesurfacedeformationisrelatedtothequantity " 2 + " 1 / 2 ,as discussedindetailby Boyer etal. ( 2011 b ).Repeatingtheirexperimentswithspheres, arod-dipping,ornegativerod-climbing,phenomenon,consistentwiththeirresultswas observed.However,nodeformationoftheinterfacewasobservedforthesuspensions ofÞberswithintheexperimentaldetection,implyingthat | " 2 + " 1 / 2 | ( 0.02 aswillbe discussedinSection 7.3 . Asemi-circulartroughofradius R =1.29 ± 0.02 cmwasusedfordetermining " 2 [Figure 7-2 B]. Couturier etal. ( 2011 )utilizedadeep,parallel-sidedtroughfortheir studyofsuspensionsofspheres;usingthesemi-circulartroughreducesthequantity ofsuspensionrequiredfortheexperiments,butdoesnecessitatethereexamination giveninAppendix A oftherelationshipbetweentheinterfaceandthenormalstress differences.Thetroughlengthof 140 cmwaschosentobesufÞcientlylargesothe surfacedeßectionproÞlecouldreachasteadystate.Toinduceßow,thetroughwas rapidlytiltedfromitshorizontalpositiontoanangleofinclination % ,asseeninFigure 7-2 B.Theangleofinclinationwasvariedfrom 0 " to 40 " byincrementsof 10 " ,whereall anglesareaccuratetowithin 1 " .Atleastthreeexperimentswereperformedforeach angleofinclination % .Twodifferentmethodswereusedtotreatthesuspensionbetween runs.IntheÞrstmethod,thesuspensionwasremixedwithinthetroughbeforeeach trial.Inthesecondmethod,thesuspensionwasnotmixedbetweentrials.Thefreesurfacedeßectionwasmeasuredbyimagingnineparallellasersheetsprojectedata lowangleontothesurfaceatthemid-lengthofthetrough.Theimageanalysisanddata 65

PAGE 66

! ! Figure7-3. Imagesofthelinesformedbyprojectingninelasersheetsontothe free-surfaceofasuspensionßowinginthetroughatdifferinganglesof inclination.Imagesareshownfor A =13 and nL 2 d =3 atanangleof inclinationofA) % =10 " andB) % =40 " .Thedeformationoftheinterface increaseswiththeangle % .Photographscourtesyoftheauthor. processingtodeterminethecorrespondingnormalstressdifferencesaredescribedin Section 7.2.3 . 7.2.3DataAnalysisforTilted-TroughFlows Figure 7-3 showstypicalimagesofthefree-surfaceofthesuspensionwhentheßow hasbeeninducedatthesmallestinclinationangleof % =10 " tothehorizontalandat thelargestinclinationangleof % =40 " .Thepositionsofthelinesareextractedfromthe imagesandconvertedintoquantitativemeasurementsofthesurfacedeßectionwiththe aidofacalibrationtargethavingaknowngeometry.Oncetheßowbecomesspatially andtemporallyhomogeneous,resultsareaveragedovertimeandtheninelocations; Figure 7-4 showstwoproÞlesmeasuredfromexperiments. Theexperimentallymeasuredinterfaceiscomparedwiththesurfacedeßection predictedbythetheoryinAppendix A toestimatethevalueof " 2 .Inthetheorythe interfacedeformationisdependentonthreedimensionlessparameters(Equation( AÐ9 )): 66

PAGE 67

00.20.40.60.81 r/R -0.04 -0.02 0 0.02 0.04 0.06 h/R Experimental Data Theoretical Fit A 00.20.40.60.81 r/R -0.04 -0.02 0 0.02 0.04 0.06 h/R Experimental Data Theoretical Fit B Figure7-4. ExperimentallymeasuredsurfaceproÞles.SurfaceproÞlesareshownforA) spheresatavolumefractionof 40% andforB)Þberswith A =13 at nL 2 d =3 .Theheight h oftheinterface,relativetothemeanheight,isshown asafunctionoftheradialposition r / R fromthecenterlinetotheedgeofthe trough.Thevaluesof " 2 aredeterminedfromtheexperimentallymeasured proÞlesbycomparingwithsurfaceproÞlespredictedbythetheoryin Appendix A . 67

PAGE 68

thecontactangleat r / R =1 ;theBondnumber, Bo = + gR 2 cos % /! ,where + isthe suspensiondensity, R istheradiusofthetrough;andtheparameter S = " 2 tan % , whichistheproductofthesecondnormalstressdifferencecoefÞcientandthetangent oftheangleofinclination.TheBondnumberisknownandvariesbetween 42.77 and 68.33 inthepresentÞberexperiments.Boththecontactangleand S aredetermined byminimizingthesquareerrorbetweenthemeasuredandcalculatedsurfaceproÞles (usingEquation( AÐ9 )).ExamplesoftheexperimentalandtheÞttedsurfaceproÞles areshowninFigure 7-4 AforspheresandFigure 7-4 BforÞbers.AsurfaceproÞle approachesatriangularshapeasthesurfacetensiongoestozero.Accordingly,Figure 7-4 Bindicatesthatsurfacetensioncannotbeneglectedinthepresentexperiments, sincetheproÞlecontainssigniÞcantcurvatureatthecenterandontheside. Theresulting S fromthepreviousanalysisisshowninFigure 7-5 Afor A =12 , 13 , and 16 with nL 2 d =3 .Thereisalinearrelationshipbetween S and tan % ,conÞrming theanalysisofAppendix A andallowingdeterminationof " 2 byequatingittotheslope inFigure 7-5 A.Forexample,thismethodresultsin " 2 = $ 0.091 ± 0.005 for A =12 and nL 2 d =3 .Alternatively, " 2 canbecalculateddirectlyfrom S / tan % ,whichisplotted against % inFigure 7-5 Bfor A =12 and nL 2 d =3 .Figure 7-5 Bindicates " 2 doesnot varywith % ortheinitialstateofthesuspension(mixedorunmixed)withintheerror, exceptat % =10 " .Averagingover % forthedatainFigure 7-5 Bgives " 2 = $ 0.10 ± 0.04 . Additionally,thecontactangledidnotvarysigniÞcantlyacrosstheentirerangeofaspect ratiosandconcentrationsstudied,havinganaveragevalueof 60 ± 5 o . Duetothesetrends,theremainderofthechapterreports " 2 averagedoverthose valuescollectedatangles % > 10 " fortheexperimentswherethesuspensionwasnot remixedbetweentrials.Thedataat % =10 " isexcludedfromtheaverageduetothe difÞcultyinaccuratelymeasuringthesmalldeformationsinthefree-surface. Thevalidityoftheexperimentalmethodanddataanalysishavebeenassessed withthesuspensionofpolystyrenespheresataconcentrationof 40% forthedifferent 68

PAGE 69

00.20.40.60.8 tan( ! ) -0.025 0 0.025 0.05 0.075 0.1 -S Mixed Not Mixed Total Average: Linear Fit A 010203040 50 ! -0.25 -0.2 -0.15 -0.1 -0.05 0 " 2 Mixed Not Mixed Total Average Total Average: Linear Fit B Figure7-5. Effectsofmixingon $ S .Resultsfor A =12 , 13 ,and 16 with nL 2 d =3 . PlottingA) $ S versus tan % formixingandnotmixingin-between experimentsshowsnodifferencewithintheexperimentalerrorandthat $ S islinearlyrelatedto tan % ,asexpectedfromAppendix A .PlottingB) S / tan % versus % for A =12 showsnosigniÞcantdifferencebetweenmixingandnot mixingin-betweenexperimentsaswell.Additionally,thereisnochangein S / tan % withtheangleofinclination % withintheexperimentalerrorexceptat anangleof 10 " .TheslopeofthelinearÞtinA),whichis " 2 ,isthesameas " 2 averagedoverall % ,asshowninB). 69

PAGE 70

inclinationangles % .ThelinearÞtof S with tan % resultsin " 2 = $ 0.28 ± 0.02 while averagingoverthedifferentangles % resultsin " 2 = $ 0.27 ± 0.03 .Thesemeasurements areingoodagreementwiththevaluesof $ 0.27 ± 0.02 and $ 0.25 ± 0.02 foundby Couturier etal. ( 2011 )forpolystyrenesphereshavingaradiusof 35 µ mand 70 µ m, respectively,andareslightlylargerthanthevalueof $ 0.30 ± 0.02 foundby Dai etal. ( 2013 )forpolystyrenesphereshavingaradiusof 20 µ m. 7.2.4NumericalPredictions Suspensionsofnon-BrownianrigidÞbersinsimpleshearweresimulatedusing thesamemodelasdevelopedinChapter 2 .Fibermotionwasduetothehydrodynamic dragforcesoneachparticleandashort-rangerepulsiveforcetomaintaintheexcluded volume.Thenumericalpredictionsofthismodelwereusedwiththeequationforthe stressderivedinChapter 6 . 7.3ResultsandDiscussion 7.3.1ComparisonofExperimentalandNumericalResults Measurementandsubsequentanalysisofthefree-surfacedeformationinthe tilted-troughgivesthevaluesof " 2 showninFigure 7-6 ;thevaluesalsoaretabulated inTable C-8 .Forthesmalleraspectratiosof A ( 17 , " 2 isnegative,atleastfor thoseconcentrationsatwhichmeasurementswerepossible.Atconcentrationsof nL 2 d =1.5 and 2 foraspectratio 17 ,adeformationoftheinterfacecouldnotbe detected,meaning | " 2 | ( 0.01 .Thesamewastruefor A =32 and nL 2 d ( 3 . Additionally,themeasurementsof " 2 closelyagreeforÞbershavingsimilaraspect ratiosof 12 and 13 ,despitethedifferentlengthsof L =0.545 mmand L =0.301 mm, respectively.Thefactthattheresultsdependonlyontheaspectratio,andnotthelength ordiameter,indicatesthatthemeasurementsarenotinßuencedbytheboundariesof thetroughorbyßexibility. Numericalresultsarecomparedtotheexperimentalresultsfor " 2 inFigure 76 .NumericalresultsaretabulatedinTable C-9 .Withintheexperimentalerror,the 70

PAGE 71

1 1.5 2 2.5 3 3.5 nL 2 d -0.15 -0.1 -0.05 0 ! 2 A =11 A =12 A =13 A 1 1.5 2 2.5 3 3.5 nL 2 d -0.09 -0.06 -0.03 0 0.03 ! 2 A =15 A =17 A =30 A =32 B Figure7-6. Experimentalandnumericalresultsfor " 2 .Resultsfor " 2 asafunctionof concentrationfromthetroughexperiments(opensymbols)andnumerical results(Þlledsymbols)foraspectratiosA) A =11 to 13 andB) A =15 to 32 . Thedashedlineindicatesthedetectionlimitof " 2 =0.01 . 71

PAGE 72

predictedvaluesareinquantitativeagreementwiththemeasuredvalues,withthe exceptionof A =17 at nL 2 d =3 .Thesimulationsassistinclarifyingthetrendsfor " 2 thatareseenintheexperimentalresults.Foragivenaspectratio, " 2 becomes increasinglynegativewithincreasingconcentration.Foragivenconcentration, " 2 becomesincreasinglynegativewithdecreasingaspectratio. Thevalueof " 1 isfoundbyusing " 2 ,asmeasuredfromthetilted-troughexperiments,togetherwiththemeasurementsfromtheWeissenbergdevice,wherethedeßectionofthefree-surfaceisduetothelinearcombinationof " 2 + " 1 / 2 .AsnotedinSection 7.2 ,theinterfaceremainedßatforalloftheconcentrationsandaspectratiosthatwere tested.ThelackofanobservabledeßectionoftheinterfaceintheWeissenbergexperimentsindicatesthat | " 2 + " 1 / 2 | ( 0.02 ,asthesensitivityofthemeasurementsis 0.02 . Figure 7-7 Ashowstheexperimentallyrealizedresultsfor " 1 ,whicharecomputedunder theassumptionthat " 1 '$ 2 " 2 . Valuesof " 1 arecomputeddirectlyfromthesimulationresultsandcompared, inFigure 7-7 A,tothosedeterminedfromtheexperiments;Table C-10 showsthe values.Thecomparisonprovidesevidencesupportingthedeterminationof " 1 from theexperiments.AsamoredirectconÞrmationoftheresultsfromtheWeissenberg geometry,Figure 7-7 Bplotsthesimulationresultsfor " 2 + " 1 / 2 .Theresultingvalues fromthenumericsarewithinthesensitivitylevelofthemeasurements,conÞrmingthe lackofanyobservablesurfacedeßectionintheWeissenbergdevice.Inconclusion, numericalandexperimentalresultsagreethat " 1 ispositiveandapproximatelytwicethe magnitudeof " 2 . 7.3.2TimeDependence ResultsreportedinFigures 7-6 and 7-7 for " 2 and " 1 arethoseatsteadystate. Evaluatingthesurfacedeformationsinthetilted-troughasafunctionoftimewould requireananalysismorecomplexthanthatgiveninAppendix A .However,thetime dependencepredictedbythesimulationsiseasilyaccessible:Figure 7-8 showsthe 72

PAGE 73

1 1.5 2 2.5 3 3.5 nL 2 d -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 ! 1 A =11 A =13 A =15 A =17 A =30 A =32 A 1 1.5 2 2.5 3 3.5 nL 2 d -0.05 -0.025 0 0.025 0.05 ! 2 + ! 1 /2 A =11 A =15 B Figure7-7. Experimentalandnumericalresultsfor " 1 and " 2 + " 1 / 2 .A)Experimental results(opensymbols)for " 1 asevaluatedfromtheWeissenbergdevice togetherwiththeresultson " 2 fromthetilted-trough.Numericalresults(Þlled symbols)for " 1 arealsoshown.B)Thenumericalresultsfor " 2 + " 1 / 2 are comparedtotheestimatedlimitofdetection(dashedlines)of 0.02 . 73

PAGE 74

0 150 300 450 600 750 900 1050 1200 ! t 0.45 0.5 0.55 0.6 0.65 " C b # A =11 nL 2 d =3 A= 15 nL 2 d =3 A =11 nL 2 d =1.5 A =15 nL 2 d =1.5 A . 020040060080010001200 ! t -0.25 -0.2 -0.15 -0.1 -0.05 0 " 2 A =11 nL 2 d =3 A =15 nL 2 d =3 A =11 nL 2 d =1.5 A =15 nL 2 d =1.5 B . Figure7-8. Temporalevolutionof " C b # and " 2 .SimulationresultsforA)therescaled orbitconstant " C b # andB) " 2 indicatethesecondnormalstress-difference reachessteadystatealmostimmediately,whereasthemicrostructure requiresalongertimetoreachsteadystate. 74

PAGE 75

predictedevolutionofboththestructureandthesecondnormalstressdifference coefÞcient, " 2 ,withstrain ú ! t .Therescaledorbitconstant C b ,whenaveragedoverthe suspensionofÞbers,isusedasthemeasureoftheorientationdistribution.Theratio C / ( C +1) deÞnes C b ,where C = 1 A e tan & ( A 2 e cos 2 ' +sin 2 ' ) 1 / 2 (7Ð2) andtheangles ' and & aredeÞnedinFigure 2-1 B.Theaveragerescaledorbitconstant " C b # goestooneifallÞbersarealignedintheßow-gradientplaneandequalszeroifthe Þbersarealignedwiththevorticityaxis. Figure 7-8 Ashowsthatthetimetotransitionfromtheinitiallyrandomorientation distributiontothesteadydistributiondependsstronglyupontheconcentration.However, inallcasesthesecondnormalstressdifferencecoefÞcients(Figure 7-8 B)reachtheir steadyvaluealmostimmediately.TheseresultsaidinexplainingthelackofdifÞculty inmeasuringthesteadyvaluefor " 2 andwhythemeasurementsoftheinitiallymixed orunmixedsuspension(asdescribedinSection 7.2 )givethesameresults.Athigh concentrations,wherethenormalstressdifferencesaresigniÞcant,thesuspensions transitionrapidlyfromtheinitiallyrandomorientationdistributiontothesteadydistribution;anychangesin " 2 occuronanevenfastertimescale.Forthelowerconcentrations, theorientationdistributionrequiresamuchlargerstraintoattainsteadystate,butthe valuesof " 2 aresmallandanychangesaredifÞculttodetect. 7.3.3OriginsoftheNormalStressDifferences Thefavorablecomparisons,showninFigures 7-6 and 7-7 ,thatexistbetween thesimulationandexperimentalresultssuggestthatthesimulationsareproviding anaccuraterepresentationofthestresses.Hence,informationabouttheoriginsand qualitativebehaviorofthenormalstressdifferencescanbeextractedfromthedetailed simulationresultswithsomeconÞdence. 75

PAGE 76

Twomechanismscontributetothenormalstressdifferences.Thehydrodynamic contributionstothestressesarethoseinEquation( 6Ð4 )whichareproportionaltothe appropriatefourthordermomentsoftheorientations, " pppp # .Thecontributiontothe stressesduetothecontactforcesareproportionalto " r ij f ij # .Thoughtheinstantaneous valuesdependonlyonthemicrostructure,boththehydrodynamicandcontactforces inßuencethedevelopmentofthemicrostructure. Thecontactandhydrodynamiccontributionsto " 2 and " 1 arepresentedseparately inTables C-9 and C-10 .Theseparatecomponentsof N 2 and N 1 aredividedbythetotal shearstress, # ,togivethevaluesof " 2 and " 1 ;theshearstress,ascalculatedfromthe simulationsandnormalizedbytheßuidstress( µ ú ! ),isgiveninTable C-11 .Table C-11 demonstratesthatthecontactforcecontributesrelativelylittletotheoverallvalueofthe shearstress.However,Tables C-9 and C-10 indicatetheoppositeregardingtheorigins ofthenormalstressdifferences. Thehydrodynamiccontributionstothenormalstressdifferencesarerelatively smallascomparedtothecontributionsthatarisefromtheshortrangeinterparticle forces,atleastforthoseconcentrationsandaspectratioswheretherearesigniÞcant normalstressdifferences.Thehydrodynamiccontributionsremainsmall,regardless oftheconditions,becausetheÞberscloselyalignwiththedirectionofßowandhence, areacteduponbyonlyasmallfractionofthestrainingßowovertheirlength.Thatthe simulationspredictstrongalignmentwiththedirectionofßowisdemonstratedinFigure 7-9 A,whichshowsthat " p x p x # greatlyexceedsboth ! p y p y " and " p z p z # . Sincetheshort-rangedrepulsionorcontactforce,intendedtopreventtheoverlapof theÞbers,largelygeneratesthenormalstressdifferences,examiningthemicrostructure oftheÞbersandthedetailsofthecontactforcesprovidesinsightsintotheirsigns.The valuesof " r ij f ij # areshowninFigure 7-9 Boverarangeofaspectratiosat nL 2 d =3 .The gradientcomponent " r y f y # hasthelargestmagnitudeofallthreenormalcomponents, indicatingthatthemeanrepulsionforceactsprimarilyinthatdirection.Theshearing 76

PAGE 77

10 15 20 25 30 35 A 0 0.2 0.4 0.6 0.8 ! p x p x " ! p y p y " ! p z p z " ! p x p y " A 010203040 50 A 0 0.0075 0.015 0.0225 0.03 0.0375 ! r x f x " / # ! r y f y " / # ! r z f z " / # 010203040 A 0 2 4 6 8 N / V B Figure7-9. Detailedpredictionsofthemicrostructureandcontactforces.Thesecond ordermomentoftheorientationdistributionA)foraconcentrationof nL 2 d =3 clearlyshowsastrongalignmentwiththeßowdirection.The normalcomponentsofthecontactforcetensorB)forthesame concentrationdecreasewithincreasingaspectratio.Thesametrendis shown(inset)forthecontactnumberdensity,orthenumberofcontacts N dividedbythevolume V .Asthestressduetocontactsistheproductofthe contactnumberdensityandthecontactforcetensor,thecontactcontribution tothestressquicklydecreaseswithincreasingaspectratio.Additionally,the contactforceactsperpendiculartotheÞbersmakingthestrongalignmentin theßowdirectionshowninA)consistentwith " r y f y # havingthestrongest contributiontothenormalstressdifferences,asshowninB). 77

PAGE 78

ßowactstotumbletheÞbers,butcollisionsfrustratethefreerotationasÞbersapproach alignmentwiththeßow-vorticityplaneandgiverisetothelargecomponentofrepulsion actinginthegradientdirection.Examiningtheothercomponentsindicatesthat " r x f x # isverysmall,whereas " r z f z # issimilarinsizeto " r y f y # .Accordingly,repulsiveforcesact morestronglyinthevorticitydirectionthantheßowdirection,butnotasstronglyasin thegradientdirection.ExaminingEquations( 6Ð7 )and( 6Ð8 )alongwiththisinformation revealsthattheÞrstnormalstressdifferencemustbepositiveandthesecondnormal stressdifferencenegative.Additionally,sincethemagnitudeof " r y f y # isapproximately twicethatof " r z f z # ,theÞrstnormalstressdifferencecoefÞcientisgivenby " 1 '$ 2 " 2 . Thedatafromtheexperimentsandsimulationsindicatethatthenormalstress differencesincreaseasconcentrationisincreasedandaspectratioisdecreased.Since therepulsiveforcelargelygeneratesthenormalstressdifferences,theincreasesin themagnitudeofthecoefÞcients " 2 and " 1 withtheconcentrationisexpectedowing tothenumberofincreasingcontacts.Likewise,decreasingtheaspectratioataÞxed valueof nL 2 d increasesthenormalstressdifferencesduetotheincreasednumber ofcontacts,asseenintheinsetofFigure 7-9 B.Theincreasednumberofcontacts occursbecausethereislessspacefortheÞberstorotatefreely:thevolumefractionis inverselyproportionaltotheaspectratio.Theincreaseinnormalstressdifferenceswith decreasingaspectratioisadditionallyduetothedecreasingdiameteroftheÞber,asthe magnitudeof r ij is 1.1 d inthesimulation. 7.3.4ComparisonswithPreviousWorks Comparisonwithotherworksthathavemeasurednormalstressdifferencesin suspensionsofÞbersisdifÞcultsincemostmeasurementshavebeenperformedfor A ) 50 andhavereliedonparallelplates,whichmeasure N 1 $ N 2 .Theexperiments onÞbersat A =35 and nL 2 d ' 2.9 by Keshtkar etal. ( 2009 )areanotableexception thatcanbecomparedtothesomewhatsimilarconditionsreportedhereof A =30 and nL 2 d =3 . Keshtkar etal. ( 2009 )utilizedparallelplatesandhencereportavalueforthe 78

PAGE 79

differenceoftheÞrstandsecondnormalstressesof ( N 1 $ N 2 ) /µ ú ! ' 0.25 .Thisvalueis muchlargerthanthevalueof ( N 1 $ N 2 ) /µ ú ! =0.040 ± 0.006 fromthenumericalresults givenherefor A =30 and nL 2 d =3 . Theverylargediscrepancybetweentheresultsisalarming,butthegeometries inwhichthemeasurementswereperformedareofverydifferentscales.Forthemeasurementsof Keshtkar etal. ( 2009 ),theÞbersareconÞnedbetweentheplateswhich areseparatedbyadistanceofonlythreeÞberlengths.ConÞningÞbersuspensions candrasticallychangethedynamics,aswasfoundfortheorientationdistributionin oscillatoryshearofÞbers( Franceschini etal. , 2011 ; Snook etal. , 2012 ).Inthepresent case,simulationsidenticaltothosedescribedinSection 2 ,butwithaconÞningwall separatedbyadistanceof 3 L ,wereperformedandevaluatingthestressesgives ( N 1 $ N 2 ) /µ ú ! =0.25 ± 0.05 .Accordingly,disparitiesbetweentheresultsgivenhereand thoseavailableintheliteraturearelikelyduetotheseveregeometricconstraintsthat existinparallelplatemeasurements. Manyresearchershaveassumedthat N 2 isnegligiblecomparedto N 1 whenreportingexperimentalresultsfromparallelplatemeasurementsofthenormalstress differences N 1 $ N 2 ( Keshtkar etal. , 2009 ; Petrich etal. , 2000 a ; Sepehr etal. , 2004 ). AttemptstoconÞrmtheassumptioncalculatedthestressesinthesuspensionfrom eitheranexperimentallymeasured( Petrich etal. , 2000 a ; Stover etal. , 1992 )ornumericallypredictedmicrostructure( Sundararajakumar&Koch , 1997 ).Thesestress calculationsignoredcontactinteractionsandindeedtheyÞndthesecondnormalstress differenceismuchsmallerthantheÞrst,whichisinagreementwiththeresultsinTables C-9 and C-10 whencomparingthehydrodynamiccontributionto N 1 and N 2 .However, includingthecontactcontributioniscriticaltoobtainthecorrectqualitativerelationshipof N 1 '$ 2 N 2 ,evenatanaspectratioof 30 .Consequently,assumingthesecondnormal stressdifferenceismuchsmallerthantheÞrstisinaccurate,atleastfortheaspectratios studiedhere. 79

PAGE 80

7.4Summary MeasurementsoftheinterfacialdeformationforsuspensionsofrigidÞbersßowing inanopentroughandinaWeissenberggeometrywereusedtodeterminetheÞrstand secondnormalstressdifferences.Deformationsofthefree-surfaceofthesuspension ßowingthroughthetroughindicatethatthesecondnormalstressdifferencecoefÞcient, " 2 ,isnegative.Themagnitudeof " 2 increasesastheconcentrationisraisedandthe aspectratioislowered.DeformationsoftheinterfaceintheWeissenberggeometry werenotmeasurable,indicatingthat | " 2 + " 1 / 2 | isclosetozero.Hence,theÞrstnormal stressdifferencesarepositiveandapproximatelytwicethemagnitudeofthesecond normalstressdifferences( " 1 '$ 2 " 2 )andincreaseastheconcentrationisraisedand aspectratioisloweredaswell.Thisresultfor " 1 differsgreatlyfromthatofsuspensions ofsphereswheretheÞrstnormalstressdifferenceisverysmallandevenfoundtobe weaklynegativeinsomeexperiments. Simulationsofthesimpleshearingßowalsowereperformedforsuspensions ofrigidÞbersoverthesamerangeofconcentrationsandaspectratiosasusedin theexperiments;stressesthenwerecalculatedfromthedetailedmicrostructureand particlecontacts.ThenormalstressdifferencecoefÞcients " 1 and " 2 predictedbythe simulationsagree,withintheestimatederrorfornearlyallpoints,withtheexperimental values.Thesimulationsrevealthatthecontactinteractions,whichactprimarilyinthe gradientdirection,arelargelyresponsiblefortheobservednormalstressdifferences. Tosummarize,forsuspensionsofnon-colloidalparticles,thenormalstressesscale withtherateofshearanddependupontherelativearrangement,ormicrostructure,of theparticles.Forsuspensionsofspheres,arecentreview( Hinch , 2011 )arguesthat thenormalstressdifferencesarecausedbyrepulsiveinteractions.Theseinteractions arefairlyequalintheßowandgradientdirections,making N 1 verysmall,whilethelack ofsigniÞcantinteractionsinthevorticitydirectionmakes N 2 largeandnegative.Here ournumericalresultsshowthatnormalstressdifferencesforsuspensionsofÞbersare 80

PAGE 81

determinedbyshort-rangedrepulsiveinteractions.DuetothestrongalignmentofÞbers withtheßowdirection,therepulsiveinteractionsactprimarilyinthegradientdirection andweaklyintheßowdirection,making N 1 positive.For N 2 ,repulsiveinteractionsact morestronglyinthevorticitydirectionthantheßowdirection,butnotasstronglyasin thegradientdirection,makingitnegative. Continuingworkwillfocusoneliminatingthedisparitybetweenthemeasurements ofnormalstressdifferencespresentedhereandmeasurementsperformedwithinparallelplaterheometers,whichgivelargervaluesfor N 1 $ N 2 .Also,theimplicationsofthe normalstressdifferences,whicharequalitativelydifferentthanthoseforsuspensionsof spheres,forßowsofÞbersinmoregeneralsituationsshouldbeexplored. 81

PAGE 82

CHAPTER8 THERHEOLOGYINSUSPENSIONSOFFIBERSWITHCONFINEMENT 8.1MotivationforStudyingRheologyofConÞnedSuspensions ThenumericalresultsreportedinChapters 4 and 5 showedconÞnementcan alterthemicrostructureinconcentratedsuspensionsofÞbers.Themodelusedfor thesepredictionswasdevelopedinChapter 2 andonlyincludescontactsforperturbing theÞbermotioninanappliedßow.Accordingly,contactsmustberesponsibleforthe changesinthemicrostructure.InChapter 7 ,experimentalandnumericalresultsshow contactsarethedominantcontributiontothenormalstressdifferences.Sincecontacts arecausingadifferentmicrostructure,differentnormalstressdifferencesareexpected whenthesystemisconÞned.ThisinßuenceofconÞnementontherheologyisespecially importanttostudyasmostexperimentalmeasurementsofthenormalstressdifferences aremadeusingarheometer,wheresuspensionsarecommonlyconÞnedtoadistance ofapproximatelythreeÞberlengths.ThesamesimulationdevelopedinChapter 2 andusedinChapter 4 ,isusedhere.Thegeometriessimulatedarethesameasin Chapter 4 ,whereÞbersarepreventedfromcrossingtheboundariesinthegradient direction.Simulationswereperformedwheretheboundariesinthegradientdirection wereseparatedbyadistanceofeitherthreeÞberlengths, 3 L ,orÞveÞberlengths, 5 L . ThenumericalresultsutilizedtherheologicalequationsderivedinChapter 6 topredict thestressesintheseconÞnedsystems.Resultswillbecomparedtothefullyperiodic simulationresultsofChapter 3 . 8.2Results Figure 8-1 showstheÞrstnormalstressdifference, N 1 ,forarangeofaspectratios, concentrations,andlevelsofconÞnement.Thereisaclearchangein N 1 withthe presenceofthewallsinthegradientdirection.Interestingly,theinßuenceofconÞnement isnotalteredwhenthegapincreasesfrom 3 L to 5 L .Figure 8-2 showsthesecond normalstressdifference, N 2 ,forthesameaspectratios,concentrations,andlevels 82

PAGE 83

01234 nL 2 d 0 0.5 1 1.5 N 1 / µ ! A =11 A =11 3 L A =11 5 L A =20 A =20 3 L A =20 5 L A 01234 nL 2 d 0 0.1 0.2 N 1 / µ ! A =30 A =30 3 L A =30 5 L A =50 A =50 3 L A =50 5 L B Figure8-1. ChangesintheÞrstnormalstressdifferenceduetoconÞnementathighand lowaspectratios.TheÞrstnormalstressdifferencenormalizedbytheßuid stressisplottedforA)lowandforB)highaspectratiosatvaryinglevelsof conÞnementandconcentration. 83

PAGE 84

01234 nL 2 d -1.5 -1 -0.5 0 N 2 / µ ! A =11 A =11 3 L A =11 5 L A =20 A =20 3 L A =20 5 L A 01234 nL 2 d -0.2 -0.15 -0.1 -0.05 0 0.05 N 2 / µ ! A =30 A =30 3 L A =30 5 L A =50 A =50 3 L A =50 5 L B Figure8-2. ChangesinthesecondnormalstressdifferenceduetoconÞnementathigh andlowaspectratios.Thesecondnormalstressdifferencenormalizedby theßuidstressisplottedforA)lowandforB)highaspectratiosatvarying levelsofconÞnementandconcentration. 84

PAGE 85

01234 nL 2 d 0 0.5 1 1.5 2 2.5 3 ( N 1 N 2 )/ µ ! A =11 A =11 3 L A =11 5 L A =20 A =20 3 L A =20 5 L A 01234 nL 2 d -0.1 0 0.1 0.2 0.3 0.4 ( N 1 N 2 )/ µ ! A =30 A =30 3 L A =30 5 L A =50 A =50 3 L A =50 5 L A =35 Keshtkar et al. 2010 B Figure8-3. Changesin ( N 1 $ N 2 ) /µ ú ! duetoconÞnementathighandlowaspectratios. ThedifferenceoftheÞrstandsecondnormalstressdifferences,normalized bytheßuidstress ( N 1 $ N 2 ) /µ ú ! ,areplottedforA)lowandforB)highaspect ratiosatvaryinglevelsofconÞnementandconcentration.Thisisthesame combinationofnormalstressdifferencesasmeasuredwithaparallelplate toolingonarheometer.Experimentalobservationsusingthistoolingarealso plottedwithhighaspectratios. 85

PAGE 86

0 0.5 1 1.5 2 2.5 3 3.5 4 nL 2 d 0 0.25 0.5 0.75 1 1.25 1.5 N / V A =11 Periodic A =11 3 L A =11 5 L A 0 0.5 1 1.5 2 2.5 3 3.5 4 nL 2 d 0 0.15 0.3 0.45 N / V A =20 Periodic A =20 3 L A =20 5 L B Figure8-4. ChangesincontactnumberdensityduetoconÞnementat A =11 and A =20 .ThecontactnumberdensityisshownforperiodicandconÞned boundaryconditionsforA) A =11 andB) A =20 . 86

PAGE 87

0 0.5 1 1.5 2 2.5 3 3.5 4 nL 2 d 0 0.075 0.15 0.225 0.3 N / V A =30 Periodic A =30 3 L A =30 5 L A 0 0.5 1 1.5 2 2.5 3 3.5 4 nL 2 d 0 0.04 0.08 0.12 0.16 N / V A =50 Periodic A =50 3 L A =50 5 L B Figure8-5. ChangesincontactnumberdensityduetoconÞnementat A =30 and A =50 .ThecontactnumberdensityisshownforperiodicandconÞned boundaryconditionsforA) A =30 andB) A =50 . 87

PAGE 88

0.5 1 1.5 2 2.5 3 3.5 4 nL 2 d 0.8 1 1.2 1.4 N Confined Bulk / N Periodic A =11 3 L A =11 5 L A =20 3 L A =20 5 L A =30 3 L A =30 5 L A =50 3 L A =50 5 L Figure8-6. ThenumberdensityofcontactsoccurringinthebulkforconÞnedsimulations isnormalizedbythenumberdensityofcontactsinfullyperiodicsimulations. Thenumberofcontactsoccurringinthebulkisgreaterthanthatofperiodic simulations. ofconÞnementas N 1 inFigure 8-1 .Numericalresultsfor N 2 predictstatisticallythe samevaluewhentheseparationdistanceisincreasedfrom 3 L to 5 L .Both N 1 and N 2 increaseinmagnitudewithincreasingconcentrationanddecreasingaspectratiounder theconÞnedboundaryconditions.Thisisthesameaswhatisseenforperiodicresults. Whenmeasuringnormalstressdifferenceswitharheometer,itiscommontousea parallelplategeometry.ParallelplategeometriesmeasureacombinationoftheÞrstand secondnormalstressdifferences, N 1 $ N 2 .NumericalpredictionsareplottedinFigure 83 alongwiththepreviousexperimentalresultsof Keshtkar etal. ( 2010 )for ( N 1 $ N 2 ) /µ ú ! . Theexperimentsclearlymeasureadifferent ( N 1 $ N 2 ) /µ ú ! thanwhatispredicted fromperiodicsimulations.However,resultsforconÞnedgeometriesshowexcellent agreementbetweenthenumericsandexperimentsat nL 2 d ' 2.9 .Theexperimentswere performedusingaparallelplategeometrywheretheseparationbetweenthetopand bottomplatewasapproximatelythreeÞberlengths.Theexperimentdidnotobservea 88

PAGE 89

noticeablechangeinmeasurementswhentheseparationdistancewasincreasedfrom 3 L to 5 L .Thisisconsistentwiththenumericalpredictionshere. Tounderstandthesourceofthischangeinnormalstressdifferencesdueto conÞnement,thecontactcontributioninthesimulationisexamined.Asshownin Chapter 6 ,thecontacttermcontributingtothestressis $ N V " r !" f !" # .Thenumber densityofcontacts, N V ,isplottedinFigures 8-4 and 8-4 .Athighconcentrationsthe contactnumberdensityisdifferentforperiodicandconÞnedresults.Itispossiblethis increaseinthecontactnumberdensitymaybeduetothenewÞber-wallcontacts occurringattheboundaries.However,omittingthesecontactsfromthecontacttotaland normalizingbythenumberdensityofperiodicsimulationsresultsinratiosgreaterthan 1 ,asshowninFigure 8-6 .Therefore,thewallsacttoincreasethenumberofcontactsin thesuspension. 8.3Summary Likethemicrostructure,conÞnementcancausechangestotherheology.Itis interestingthatthenormalstressdifferencesprimarilychangeduetoconÞnementat highconcentrations.Thisisinterestingsincethemicrostructureprimarilydoesn'tchange duetoconÞnementathighconcentrations,asshowninChapter 4 .Additionally,forall aspectratiosandconcentrations,resultsdonotchangewhentheseparationbetween theboundariesincreasesfrom 3 L to 5 L .Theseresultsareinagreementwiththe experimentsof Keshtkar etal. ( 2010 ).Analysisofthecontactcontributiontothenormal stressdifferenceshowsthattheconÞnementactstoincreasethenumberofcontacts occurringinthebulk,increasingthemagnitudeofthenormalstressdifferences.The affectofconÞnementrequiresfurtherindepthanalysisasithasstrongimplications onpreviousexperimentalwork.Untilthisinßuenceofboundingwallsisunderstood, theexperimentsusedinChapter 7 arerecommendedformeasurementsofthenormal stressdifferences. 89

PAGE 90

CHAPTER9 DYNAMICSOFSHEAR-INDUCEDMIGRATIONOFSPHERICALPARTICLESINPIPE FLOW 9.1BackgroundofShear-InducedMigration Shear-inducedmigrationisaconspicuousexampleoftheeffectsofirreversible dynamicsinshearingßowsofStokesian,non-colloidalsuspensions.Thisirreversibility issomewhatpuzzlingbecausetheßowsareatlowReynoldsnumbersandthevelocityÞeldinthesuspensionisgenerallyassumedtobegovernedbythelinear,and reversible,Stokesequations.Theirreversibilityandresultingmigrationisduetothe combinedeffectsofhydrodynamicforcesbetweentheparticlesandnon-hydrodynamic interactions.Closecontactinteractions,suchassmallroughnessorshort-rangedrepulsiveforces,canindeedleadtodeviationsoftheparticlesfromreversibletrajectories (e.g. Cunha&Hinch , 1996 ; Okagawa&Mason , 1973 ; Pine etal. , 2005 ). Shear-inducedmigrationcandriveparticles,irreversibly,fromthehightothe lowshearrateregionsoftheßow.ThisphenomenonwasÞrst,clearlyidentiÞedfrom experimentsinaCouetteviscometerby Leighton&Acrivos ( 1987 )andprompteda largenumberofexperimentalstudiesduetotheimpactthatsuchmigrationhason thecharacterizationofsuspensionrheology.Forpressure-drivenßowthroughapipe orchannel,theparticlesmigratetowardthecenterline.TheÞrstobservationofsuch inhomogeneitiesinthesuspensionÞelddatesbackto Karnis etal. ( 1966 ).Sincethen, severalexperimentalstudieshavebeenperformedthatmeasuremigrationinpressuredrivenßows( Altobelli , 1991 ; Butler etal. , 1999 ; Hampton etal. , 1997 ; Koh etal. , 1994 ; Lyon&Leal , 1998 ; Norman etal. , 2005 ; Shauly etal. , 1997 )usingmethodssuchas laser-dopplervelocimetryandmagneticresonanceimaging.Mostoftheseexperiments haveexaminedsuspensionsatlargevolumefractions,sothatmigrationcouldbereadily observable,andutilizedneutrallybuoyantparticles.Additionally,mosthavereported measurementsofthesteadyandfully-developedconcentrationandvelocityproÞles. Ratherthansteadilypumpingthesuspension,onecanalsouseoscillatoryßow.When 90

PAGE 91

theoscillationamplitudeislargeenough,migrationtowardthecenterofthepipeis recovered;atsmallamplitudes,theparticlesmigratetowardthewall( Butler etal. , 1999 ; Morris , 2001 ). Earlyeffortsinmodelingshear-inducedmigrationutilizedadiffusionmodel,in whichgradientsintheparticleconcentration,particleinteractionfrequency,andeffective suspensionviscositydriveadiffusiveßuxofparticles( Leighton&Acrivos , 1987 ; Phillips etal. , 1992 ).TheparticleconcentrationÞeldwasderivedbysolvingthediffusion equationinconjunctionwiththemomentumequationforthesuspensionasawhole. Thisdiffusionmodelsuccessfullypredictstheexistenceofmigrationinwide-gapCouette andpressure-drivenPoiseuilleßows,butfailstopredicttheabsenceofmigrationin curvilineartorsionalßows( Bricker&Butler , 2006 ; Chapman , 1990 ; Chow etal. , 1994 ; Krishnan etal. , 1996 ). Amorerecentandrathersuccessfulmodelrelatesthemigrationßuxtotherheologyofthesuspension( Morris&Boulay , 1999 ; Nott&Brady , 1994 ).Thisso-called suspensionbalancemodel(SBM)isatwo-phasemodelwhichprovidesacontinuum descriptionofthebulksuspensionmotionaswellastherelativevelocityoftheparticlephaseandßuidphase.Thismodelpredictsthatparticlemigrationisdrivenbythe divergenceofthenormalcomponentsoftheparticlephasestress( & á " p ),where " p comprisescontactorinterparticlecontributionsaswellashydrodynamiccontributions comingfromthenon-dragportionoftheinterphaseforce( Lhuillier , 2009 ; Nott etal. , 2011 ).TheSBMrequirescorrelationsfortheparticlephasestresswhichcannotbe easilyobtainedexperimentally. Discreteparticlesimulationsareanotheravenuetostudyinhomogeneitiesin suspensionßows,whereinthemotionsofmanydiscreteparticlesarecalculatedunder theassumptionthattheyaresuspendedinaNewtonianßuid.Suchsimulationsofthe pressure-drivenßowinatwo-dimensionalchannelofasuspensionwereconducted usingStokesianDynamics( Nott&Brady , 1994 ).Thesesimulationspredictedmigration 91

PAGE 92

inqualitativeagreementwiththeexperiments,whilealsodemonstratingthatshearinducedmigrationisnotduetoinertialeffectssinceStokesianDynamicscalculates ßowsinthelimitofzeroReynoldsnumber.Capabilitiestosimulatesuspensionßows havebeenexpandingrapidlyandprovidingenhancedinformationaboutmigration. Forexample,three-dimensionalnumericalsimulationsofconcentratedsuspensions of O (1000) monodispersenon-colloidalparticleswereperformedrecentlyinplane Poiseuilleßowsusingtheforcecouplingmethod( Yeo&Maxey , 2011 ). Theobjectiveofthepresentworkistoexaminethedynamicsofshear-induced migrationinapipeßow,inadditiontothesteadyandfullydevelopedconcentrationand velocityproÞleswhichhasbeenthefocusofmostpreviousstudies.Theexperiments subjectthesuspension,consistingofneutrallybuoyantspheres,tolargeoscillating displacementsatlowReynoldsnumberalongtheaxisofthepipeasdescribedinSection 9.2 .Particlevolumefractionandvelocityaremeasuredduringtheoscillationsat highresolutionbymatchingtherefractiveindexoftheßuidtothatoftheparticlesand usingßuorescencetodistinguishtheparticleandßuidphasesforimaging.Measurementsoftheshear-inducedmigrationprocessareanalyzedandcomparedtoprevious experiments( Hampton etal. , 1997 ; Lyon&Leal , 1998 )aswellastothepredictionsof discreteparticlesimulations( Nott&Brady , 1994 ; Yeo&Maxey , 2011 )inSection 9.3 . Theresultsarealsocomparedwithpredictionsofthesuspensionbalancemodelusing realisticrheologicallaws;thismodelisdetailedintheAppendix B .Following Morris& Boulay ( 1999 ),weuseaphenomenologicalformforthenormalstressoftheparticle phasewhichhasaviscousscalingandasimilarconcentrationdependenceinalldirections.Tworheologicalmodelsweretested:thatof Morris&Boulay ( 1999 ),whichwas chosentomatchexperimentalresultsonmigration,andthatof Boyer etal. ( 2011 a ), whichwasderivedfrompressure-imposedrheologicalmeasurements.Adiscussionand conclusionsregardingthevalidityoftherheologicalmodelsaregiveninSection 9.4 . 92

PAGE 93

9.2Experiments 9.2.1ParticlesandFluids TwosetsofspherescomposedofPMMAwithadensityof 1.19 g á cm ! 3 wereusedin theexperimentswithdiameters 2 a =2.01 ± 0.02 mmand 2 a =1.05 ± 0.02 mm.Each diameterwasdeterminedbyopticallyimagingmorethan100particleswiththeresulting measurementshavinganapproximatelyGaussiandistribution.Thesuspendingßuid wasdensityandindexedmatchedtothespheres.Thesuspendingßuidwascomposed ofZnCl 2 ( 17.70 wt%),water( 9.06 wt%),andTritonX-100( 76.24 wt%).Theßuidmixture wasNewtonian,withaviscosityof * f =2.43 Pa á sat 27 " C.Approximately 3 mLofHCl wasaddedforevery 700 mLofsuspendingßuidinordertoinhibitprecipitationofZnCl 2 . Additionally,rhodaminewasaddedtothesuspendingßuid( ' 6 % 10 ! 7 g á cm ! 3 )toaidin particlevisualization.TheparticleReynoldsnumberrangedfrom 8.7 % 10 ! 6 to 3.3 % 10 ! 5 andtheP « ecletnumbersrangedfrom 4.3 % 10 9 to 3.3 % 10 10 .Thesevaluesensured experimentswerecarriedoutintheStokesregimewithoutBrownianmotion. Thesuspensionswerecarefullypreparedtoavoidentrappingairinthehighly viscoussuspendingßuid.Therequiredamountofsphereswereaddedtothesurface ofthesuspendingßuid,bothitemsbeingweightedtoobtainthepropervolumefraction. Afterthesphereswereimmersedintheßuid,thesuspensionwasmixedbyslowly stirringwithametalspatulaandslowlyrotatingthebeakeratanangle.Thiswellmixed suspensionwasthenaddedtotheglasstube.Furthermixinginsidetheglasstubeprior toeachexperimentwasperformedaspartoftheexperimentalprocedure. 9.2.2ExperimentalApparatus TheexperimentalapparatuscanbeseeninFigure 9-1 A.Thesuspensionwasoscillatedinaglasstubebyasyringepumpconnectedtoaprogrammablemicrocontroller. Thetubehadadiameterof 2 R =1.65 cm,givinggeometricratios R / a =8.21 and R / a =15.71 for 2 a =2.01 ± 0.02 mmand 2 a =1.05 ± 0.02 mmrespectively.Twotrigger relaysconnectedtothemicrocontrollersignaledthelimitsofdisplacementthrough 93

PAGE 94

Figure9-1. Theexperimentalapparatusandschematicofparticlevisualizationmethod. A)Theexperimentalapparatusisshowninthetemperaturecontrolled environment.B)Thelasersheet,rhodaminedye,andlongpassÞlterenable visualizationofthesphericalparticlesasshowninthediagramwiththeraw experimentalimage.Photographscourtesyoftheauthor. contactwithametalleverattachedtothesyringe.Theamplitudeofoscillation ! ,or onequarterofthetotaldisplacementinthetubedividedby R ,wassetbythedistance betweentheÞxedtriggerrelays.Thetotaldisplacementistheaccumulateddistance traveledbytheßuidinthetubefrompumpingforwardandbackwardforasingleoscillation.Theaveragevelocityisthenthetotaldisplacementdividedbythetimetoachieve thedisplacementforasingleoscillation.Ascreenwasplacedattheentranceandexitof theglasstubetomaintainaconstantvolumefractionastheßuidpumpedintothetube didnotcontainparticles.Theamplitudewassettominimizepossibleendeffectswhile stillbeinglargerelativetothetuberadius.Theamplitudeofoscillationwas ! =5.4 , unlessotherwisenotedinSection 9.3 .Theaccumulatedstrainforoneoscillationwas then 21.7 .Theaccumulatedstrainistheaccumulatedtotaldisplacementdividedby R . Theglasstubewashousedinarectangularplexiglasschamber.Thesuspending ßuidwasaddedin-betweenthechamberandthetube.Thisreducedeffectsdueto 94

PAGE 95

curvatureofthetubeandchangesinrefractiveindex.Alasersheetwasproducedbya CoherentLasirisGreenPowerLineLaser(wavelength 532 nm)andwaspassedthrough amaskattachedtotheapparatus.Themaskhadapreciselycut 1 mmslitandwas positionedsothelasersheetwasonlyappliedtothecenterofthetube.Therhodamine dyeinthesuspendingßuidßuorescedduetotheappliedlaser.ANikonD 300 swithan AF-SMicroNikkor 60 mmf/ 2.8 GEDwasusedforimaginganda 590 nmredlongpass Þlterwasplacedbetweenthetubeandthecameratoenhancethecontrastbetween particlesandthesuspendingßuid.Aschematicoftheset-upandtypicalimageisshown inFigure 9-1 B.Toreducebleachingofthesuspendingßuid,ashutterwasaddedto thelaser.Boththeshutterandthedigitalcamerawereoperatedbyamicrocontroller connectedtothetriggerrelays. 9.2.3ExperimentalProcedure Priortoeachexperiment,thesuspensionwasmixedinsidetheglasstubeusinga longwirewithimpellersattachedatvariousheights.Aftermixing,thesuspensionwas heldstationaryforapproximatelyonehourtoremoveanyair.Examplesoftheinitial volumefractiondistributionareshowninFigures 9-2 Aand 9-2 B,where ' isthelocal particlevolumefraction, ' B isthebulkparticlevolumefraction,and x isthehorizontal positionwithitsoriginatthecenterofthecylinder.Thedataanalysisandaveraging procedureusedtoproducethesedistributionsareexplainedinSection 9.2.4 . Thetriggerrelayswerepositionedtoprovidethedesiredamplitudeofoscillation. Foreachoscillation,fortyimageswererecordedatonesecondintervals.Thecamera wasadjustedtomaximizetheshutterspeedwhileminimizingtheISOandaperture used.ThislimitedthenoisetosignalratiowhilemaintainingashallowdepthofÞeldto ensureparticlesonlyinthelasersheetwereimaged.Directlyfollowingtheexperiment, arulerwasphotographedinordertoscaleproperlytheparticledisplacements.For agivenbulkvolumefractionandpipetoparticleradius,severalexperimentswere performedusingexactlythesameprocedureasdescribedinsection ?? . 95

PAGE 96

-1 -0.5 0 0.5 1 x / R 0 0.2 0.4 0.6 0.8 1 ! ! B =40% ! B =30% ! B =20% ! B =10% A -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 0.6 ! ! B =30% ! B =20% ! B =10% B Figure9-2. Examplesoftheinitialparticledistributions.Examplesoftheinitialparticle distributionsareshownbulkvolumefractionsatA) R / a =8.21 andB) R / a =15.71 .Photographcourtesyoftheauthor. 96

PAGE 97

!"#$%"&'( )&*+", !#-.%/� 1'*# 2134 Figure9-3. Imageprocessingsteps.Examplesofanimageateachstepintheparticle velocityanalysis.FirsttheA)rawimagehasaB)histogramequalizationand thresholdapplied.Next,C)aGaussianblurandlastlyaD)CircularHough Transformtolocatetheparticlepositions.Photographcourtesyoftheauthor. 9.2.4DataAnalysis Imageswereanalyzedtoobtaintheparticlevolumefractionandvelocitydistributions.Theaccuracyinmeasuringthesetwodifferentpropertiesdependsonidentifying twodifferentcharacteristicsintheanalyzedimages:(i)distinguishinganindividual particleor(ii)maintainingtheparticleshape.Consequently,twodifferentanalyseswere required. Calculationoftheparticlevolumefractiondistributionrequiredseveralstepsof imageprocessing.First,anadaptivehistogramequalizationmethod(MATLAB R 2012 a functionadapthisteq)wasused.Thismethodlocallyenhancescontrastbymatchingthe givendomainhistogramtoaspeciÞeddistribution.Thesedomainsaresubsetsofthe totalimageandareconnectedthroughbilinearinterpolation.Thenumberandspacing ofthedomainsandhistogramdistributionsweresettodistinguishtheparticlesfrom 97

PAGE 98

!"!#!$! %&'())*+(,-./01234 ! !5" !5# !5$ !56 !57 !58 !59 :0*4+34.;(< =3-+34.;(:0*4+34.;(!> ! =3-+34.;(! " #! #" $! %&'())*+(,-./01234 ! !5# !5$ !56 !57 !5" !58 !59 !5: !5; <0*4+34.=(= >3-+34.=(Figure9-4. Theevolutionofthevolumefractionforagivenbin.Theevolutionisshown forA) R / a =8.21 at ' B =20% andB) R / a =15.71 at ' B =30% . 98

PAGE 99

thebackground.Next,athresholdwascalculated(usingMATLAB R 2012 afunction graythresh)whichusesOtsumethodofassumingabimodalhistogramandcalculates athresholdlevelbyminimizingthecombinedspreadfromthetwoclasses.Then, theimagewasÞltered(usingMATLAB R 2012 afunctionin2bw)byassigningpixel valuesgreaterthanthecalculatedthresholdwhite,andtherestblack.Examplesof animagebeforeandafterthresholdingcanbeseeninFigures 9-3 Aand 9-3 B.The volumefractionwascalculatedfromtheresultingbinaryimagesbytreatingblackpixels asparticlesandwhitepixelsassuspendingßuid.Toaverageacrossexperiments, columnsofpixelswereassignedtoabinandtheaveragevolumefractioninthebinwas calculated.Bothdynamicandsteadystateresultswereaveragedacrossexperiments wherethenumberofbinsandtheimagesusedforaveragingwillbestatedinthetext. Thepreviousstepswerealsousedtomeasuretheparticlevelocity,butthespeciÞedparameterswereoptimizedtodistinguishindividualparticles.Thisrequirederosion oftheparticles,whichwouldleadtoanincorrectcalculationofthevolumefraction.After usingtheadaptivehistogramequalizationandthresholdingtechniquestodistinguish theparticles,aGaussianblurwasapplied,followedbyacircularHoughtransformto calculatetheparticlepositions.ExamplesoftheresultsareshowninFigures 9-3 Cand 9-3 D.TheHoughtransformusesgradientsinpixelvaluesandaspeciÞedmaximum andminimumradiustolocatetheparticlepositions.Themaximumandminimumparticleradiuswaschosentoapproximatelylimitlocatingparticleswhosecenterofmass wereintheappliedlasersheet.Thisistoensurethatthecalculatedparticlevelocity isinthelasersheet.However,asparticleradiiwereerodedduringtheprocessing,an exactspeciÞcationofmaximumandminimumradiuswasdifÞcult.Theminimumradius wasapproximately 70% ofthemaximumradiusintheprocessedimage.Figure 9-3 D demonstratetheabilityofthisimageprocessingmethodusingthecircularHoughtransformtodetectthecenterofeachparticleintheimage.Finally,particlevelocitieswere calculatedbyusingaparticletrackingalgorithmwhichreliesonthesmalldisplacement 99

PAGE 100

ofthetrackedparticlesbetweentwosequentialimagesbyimposinganupperboundon particledisplacement. Inadditiontothedynamics,steadystateresultsareofinterest.Steadystate wasdeterminedbyanalyzingtheaveragevolumefraction.Thevolumefractionwas calculatedbyaveragingoverimages 10 to 35 ofagivenoscillation.Whenamajorityof thebinswerewithin 1.5 standarddeviationsoftheremainingoscillations,anexperiment wasconsideredtobeatsteadystate.Thedeterminationofsteadystatedidnotexhibit adependenceonthenumberofbins,whichwasvariedfrom 20 to 50 withintervalsof 10 .Figure 9-4 Ashowstheevolutionof ' atvariousbinlocationsforanexperimentwith ' B =20% and R / a =8.21 .Twodifferentlocationsareshown,eachhavingabinwidth of 1 20 ofthediameter.Thecenterbinbeganatthecenterofthetube.Theotherbinis centeredataquarterof 2 R andisaccordinglytermedthequarterbin.Thepositionsof thesebinsareshownintheinset,whichhasnotbeendrawntoscale,onFigure 9-4 A. Thiscalculation,performedontheexperimentsshowninFigure 9-4 (AandB),indicates steadystateafter 16 oscillationsand 19 oscillations,respectively.Resultsshownin Section 9.3.1 wereaveragedacrossallthesteadystateoscillationsaswellasacross thedifferentexperimentalruns(typically 5 ). 9.3ResultsandComparisons 9.3.1SteadyState Theparticlevolumefraction ' observedatsteadystateareshowninFigure 9-5 A forbulkvolumefractions ' B =10% to 40% at R / a =8.21 .Theamplitudeofoscillation was ! =5.4 (or 44.5 particleradii).Themaximumobservedvolumefractionatsteady statedecreaseswithdecreasingbulkvolumefraction.Distinctparticlelayersare observedacrossthetubefor ' B =40% ,likelyduetothehighlevelofconÞnementat thisconcentrationand R / a .Thereisnoobservedmigrationfor ' B =10% ,whichwillbe discussedfurtherinSection 9.3.3 .Figure 9-5 Bshowstheparticlevelocitynormalizedby thecenterlinevelocity U / U center for R / a =8.21 ,wherethecenterlinevelocityisthatofa 100

PAGE 101

-1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ! ! B =40% ! B =30% ! B =20% ! B =10% A -1 -0.5 0 0.5 1 x / R 0 0.2 0.4 0.6 0.8 1 U/U center ! B =40% ! B =30% ! B =20% ! =10% B Figure9-5. Steadystateresultsforthevolumefractionandtheparticlevelocityat R / a =8.21 .Resultsareensembleaveragedoverallexperimentsat R / a =8.21 forA)thevolumefractionandB)particlevelocity. 101

PAGE 102

-1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 0.6 ! ! B =30% ! B =20% ! B =10% A -1 -0.5 0 0.5 1 x / R 0 0.2 0.4 0.6 0.8 1 U/U center ! B =30% ! B =20% ! B =10% B Figure9-6. Steadystateresultsforthevolumefractionandtheparticlevelocityat R / a =15.71 .Resultsareensembleaveragedoverallexperimentsat R / a =15.71 forA)thevolumefractionandB)particlevelocity. Newtonianßuid,withoutparticles,atthesameßowrate.AbluntinginthevelocityproÞle isconsistentwiththeobservedincreaseinvolumefractionatthepipecenter. Observationsoftheparticlevolumefractionandparticlevelocityareshownin Figure 9-6 atbulkvolumefractionsfrom 10% to 30% for R / a =15.71 .Theamplitude ofoscillationwas ! =5.4 (or 85.2 particleradii).Itisimportanttonotethattheimage 102

PAGE 103

analysisforthisgeometryresultedinartiÞciallyhighvolumefractionmeasurements,especiallypastthecenterlineofthecylinderoppositethesideoftheappliedlaser( x / R =0 to 1 ).Thiswasduetotheincreaseinthenumberofinterfacesduetothedecreasein particlesize.Accordingly,falseparticles,whichweredarkimperfectionsintheoriginal image,artiÞciallyincreasedthevolumefraction.Thisislikelywhythemaximumvolume fractionishigherherethanat R / a =8.21 .Additionally,thisissueinhibitedexperiments at ' B =40% .Despitecomplicationsintheimaging,thetrendofdecreasingmaximum volumefractionwithdecreasingbulkvolumefractionisshown.Again,nomigrationat ' B =10% wasobservedaswillbediscussedinSection 9.3.3 .Theparticlevelocity proÞleshowsabluntingconsistentwiththeincreaseinconcentrationatthecenterofthe tube. Figure 9-7 comparesresultsfortheparticlevolumefractionfromthepresentwork tothenumericalworkof Yeo&Maxey ( 2011 )andtheexperimentsof Lyon&Leal ( 1998 ).Althoughthesepreviousresultsareforchannelßow,comparisonsaremade atsimilarvaluesof H / a ,where H ishalfthechannelheight.Thereisagreementinthe numericallypredictedparticlelayerandadecreaseinthemaximumconcentrationwith adecreaseinbulkvolumefraction.Figure 9-8 comparesthepresentobservationsto theexperimentalresultsof Hampton etal. ( 1997 )forpipeßowaswellastonumerical results( Nott&Brady , 1994 ; Yeo&Maxey , 2011 )andtheexperimentalresultsof Lyon &Leal ( 1998 )forchannelßow.Again,theresultshereagreewithnumericalpredictions oftheexistenceofaparticlelayernearthewall.Adecreaseinmaximumconcentration withadecreaseinbulkvolumefractionwasobservedforallexperimentalandnumerical results. ResultsfortheparticlevelocityinFigure 9-9 Ashowalargerbluntinginvelocity proÞlecomparedtoresultsforchannelßow.ThisisexpectedastheÞnitesizedparticle experiencesadifferentßowÞeldinpipeßowthaninchannelßow.Comparisonwiththe 103

PAGE 104

-1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 0.6 ! Present Work R / a =8.21 Yeo & Maxey (2011) H / a =9 Lyon & Leal (1998) H / a =11 A -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! Present Work R / a =8.21 Yeo & Maxey (2011) H / a =9 Lyon & Leal (1998) H / a =11 B Figure9-7. Steadystateresultsofthepresentworkarecomparedtoprevious experimentalandnumericalworkatsimilar R / a .ResultsareshownforA) ' B =40% andB) ' B =30% . 104

PAGE 105

-1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! Present Work R / a =15.71 Hampton et al. (1997) R / a =16 Lyon & Leal (1998) H / a =14 Yeo & Maxey (2011) H / a =20 A -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 ! Present Work R / a =15.71 Yeo & Maxey (2011) H / a =20 Hampton et al. (1997) R / a =16 B Figure9-8. Additionalsteadystateresultsofthepresentworkcomparedtoprevious experimentalandnumericalworkatsimilar R / a .ResultsareshownforA) ' B =30% andB) ' B =20% . 105

PAGE 106

-1 -0.5 0 0.5 1 x / R 0 0.2 0.4 0.6 0.8 1 U/U centre ! B =40% Lyon & Leal (1998) H / a =11 ! B =30% Lyon & Leal (1998) H / a =11 ! B =40% Present Work R / a =8.21 ! B =30% Present Work R / a =8.21 A -1 -0.5 0 0.5 1 x / R 0 0.2 0.4 0.6 0.8 1 U/U centre ! B =30% Hampton et al. (1997) R / a =16 ! B =20% Hampton et al. (1997) R / a =16 ! B =30% Present Work R / a =15.71 ! B =20% Present Work R / a =15.71 B Figure9-9. Steadystateresultsfortheparticlevelocityofthepresentworkcomparedto previousexperimentalwork.ResultsareshownforA) R / a =8.21 andB) R / a =15.71 . 106

PAGE 107

resultsof Hampton etal. ( 1997 )inFigure 9-9 Bshowsthatthepresentobservationsare slightlymoreblunted. ExperimentsarecomparedtothesteadystatepredictionsoftheSBM,givenby Equations( BÐ16 )and BÐ17 oftheAppendix B ,inFigures 9-10 and 9-11 .Thissteady statesolutioncomesfromrequiringthatthemigrationßuxiszeroandassumingthat thereisajammedregionatthecenterofthepipe.Twoslightlydifferentmodelsofthe rheologiesgivenby Morris&Boulay ( 1999 )and Boyer etal. ( 2011 a )wereusedinthe numerics.TheSBMisacontinuummodelandthuscannotpredictthelayeringobserved intheexperiments.Nonethelesstakingintoaccountonlytheaveragemagnitudeofthe experiments,Figure 9-10 AshowsexcellentagreementwiththeSBMpredictionsfor ' B =40% and R / a =8.21 .ResultsinFigure 9-10 Bfor ' B =30% and R / a =8.21 show asmalldiscrepancybetweenthenumericalpredictionsandexperimentalobservationfor themaximumvolumefractionatthecenter.Thediscrepancyincreasesfor ' B =20% asshowninFigure 9-10 C.Altering ' m tobethemaximumconcentrationobserved experimentallyyieldsnumericalpredictionsthatareinmuchbetteragreementwiththe experiments. Numericalpredictionsandexperimentalobservationsforthegeometry R / a =15.71 areshowninFigure 9-11 .For ' B =30% ,experimentalobservationsonFigure 9-11 A for ' m donotquitereachmaximumpacking.Thisislikelyduetotheexperimentalerror artiÞciallyincreasingtheobservedvolumefractionfortheseparticles.ResultsinFigure 9-11 Bshowalargedisagreementbetweennumericalpredictionsandtheexperimental observationsfor ' B =20% .Againaltering ' m tobethemaximumconcentration observedexperimentallyimprovesagreementbetweennumericalandexperimental results. 9.3.2Dynamics ThenumberofoscillationstoachievesteadystateisgiveninTable C-12 ;the methodforcalculatingthesenumbersisgiveninthelastparagraphofSection 9.2.4 . 107

PAGE 108

-1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 0.6 ! SS MB ! m =0.585 SS BGP ! m =0.585 Present Work A -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! SS MB ! m =0.585 SS BGP ! m =0.585 SS MB ! m =0.534 SS BGP ! m =0.534 Present Work B -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! SS MB ! m =0.585 SS BGP ! m =0.585 SS MB ! m =0.439 SS BGP ! m =0.439 Present Work C Figure9-10. Steadystateresultsfor R / a =8.21 comparedtothenumericalpredictions fromthesteadystateSBM.Eithertherheologyof Morris&Boulay ( 1999 ) or Boyer etal. ( 2011 a )wasusedandresultsareshownatA) ' =40% ,B) ' =30% andC) ' =20% . 108

PAGE 109

-1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! SS MB ! m =0.585 SS BGP ! m =0.585 SS MB ! m =0.571 SS BGP ! m =0.571 Present Work A -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! SS MB ! m =0.585 SS BGP ! m =0.585 SS MB ! m =0.379 SS BGP ! m =0.379 Present Work B Figure9-11. Steadystateresultsfor R / a =15.71 comparedtothenumericalpredictions fromtheSBM.Eithertherheologyof Morris&Boulay ( 1999 )or Boyer etal. ( 2011 a )wasusedandisshownatA) ' =30% andB) ' =20% . 109

PAGE 110

-1 -0.5 0 0.5 1 x / R 0 0.2 0.4 0.6 0.8 ! Osc 1 Osc 2 Osc 3 Osc 4 A -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! Osc 1 Osc 2 Osc 3 Osc 5 Osc 7 B -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 ! Osc 1 Osc 5 Osc 10 Osc 15 Osc 20 C Figure9-12. Experimentalresultsforthetemporalevolutionoftheparticlevolume fractiondistributionsat R / a =8.21 .ResultsareshownforA) ' B =40% ,B) ' B =30% andC) ' =20% . 110

PAGE 111

-1 -0.5 0 0.5 1 x / R 0.1 0.2 0.3 0.4 0.5 0.6 ! Osc 1 Osc 5 Osc 10 Osc 15 Osc 20 Osc 25 A -1 -0.5 0 0.5 1 x / R 0.1 0.2 0.3 0.4 0.5 ! Osc 1 Osc 10 Osc 20 Osc 30 Osc 40 Osc 50 Osc 60 B Figure9-13. Experimentalresultsforthetemporalevolutionoftheparticlevolume fractiondistributionsat R / a =15.71 .ResultsareshownforA) ' B =30% andB) ' B =20% . 111

PAGE 112

0100200300 Accumulated Strain 0.3 0.4 0.5 0.6 0.7 ! SBM MB ! m =0.585 SBM BGP ! m =0.585 Present Work: Centre Bin Present Work: Quarter Bin A 0100200300400 500 Accumulated Strain 0.2 0.3 0.4 0.5 ! SBM MB ! m =0.534 SBM BGP ! m =0.534 Present Work: Centre Bin Present Work: Quarter Bin B 0200400 600 800 Accumulated Strain 0.1 0.2 0.3 0.4 0.5 ! SBM MB ! m =0.439 SBM BGP ! m =0.439 Present Work: Centre Bin Present Work: Quarter Bin C Figure9-14. SBMpredictionsandexperimentalresultsforthetemporalevolutionofthe localparticlevolumefractionatthecenterandquarterbinsat R / a =8.21 . ResultsareshownforA) ' B =40% ,B) ' B =30% andC) ' =20% . 112

PAGE 113

0100200300400 500600 Accumulated Strain 0.2 0.3 0.4 0.5 ! SBM MB ! m =0.575 SBM BGP ! m =0.575 Present Work: Centre Bin Present Work: Quarter Bin A 0 500 1000 1500 2000 Accumulated Strain 0.1 0.2 0.3 0.4 0.5 ! SBM MB ! m =0.379 SBM BGP ! m =0.379 Present Work: Centre Bin Present Work: Quarter Bin B Figure9-15. SBMpredictionsandexperimentalresultsforthetemporalevolutionofthe localparticlevolumefractionat R / a =15.71 forvaryingtubelocations. ResultsareshownforA) ' B =30% andB) ' B =20% . 113

PAGE 114

Wereportedtheaverageacrossthedifferentexperimentalrunsaswellasthestandard deviation.Withinthestandarddeviation,thenumberofoscillationstoreachsteadystate isfourtimessmallerforthesmallerparticles,inagreementwiththescalingof a 2 / R 2 that isgivenbytheSBM(Equation BÐ12 oftheAppendix B ). Theaverage ' -distributionsatdifferentoscillationsareshowninFigure 9-12 for ' B =40% to 20% at R / a =8.21 andinFigure 9-13 for ' B =30% to 20% at R / a =15.71 . TheseresultsarecalculatedbyÞrstdividinganimageinto 50 bins.Thenforagivenbin, volumefractionmeasurementsareaveragedacrossimages 20 to 35 .Asmallernumber ofimagesisusedascomparedtotheaveragingprocessforthesteadystateresults. Indeed,thesystemisstilldynamicanditisdesirabletoobservequicklythechanging distribution.Lastly,resultsareaveragedacrosstheexperimentalrunsforagiven R / a and ' B .Particlemigrationisevidencedbytheincreaseinparticlevolumefractionat thecenterofthecylinderanditsdecreaseattheedges.Themigrationisalsoseenin theonlinemovie 1 where,forevery 5 oscillations,images 1 to 40 oftheoscillationare shown. Toexaminefurtherthetemporalevolutionof ' ,dynamicexperimentalresultsare plottedinFigures 9-14 and 9-15 .Thedataareaveragedacrossdifferentexperimental runsforeachgivenbinlocationusingimages 20 to 35 .Thecenterbincorrespondsto 1 20 ofthecylinderwidthbeginningatthecenterofthetube.Thequarterbincorresponds to 1 20 ofthetubewidthcenteredatonequarterofthetotaltubewidth.Numerical predictionsgivenbytheSBM,whichusesthesamemaximumparticlevolumefraction asobservedexperimentallyaswellasthesamebinlocation,arealsoplottedonthese Þgures.Thesepredictionscomefromnumericallysolvingthemigrationequationgiven byEquation( BÐ12 )oftheAppendix B togetherwiththebulkequationfortheshear stressgivenbyEquation( BÐ14 ).AsexplainedintheAppendix B ,weneedtouseanonlocalstresscorrectiontoavoidsingularitiesinthecenterofthepipewheretheshear rateisnull.Wechoseanon-localstress,givenbyEquation( BÐ15 )intheAppendix B , 114

PAGE 115

withanexponentofsix.ThisensuredavolumefractionproÞle,whenmigrationwas completed,thatmatchesthesteadystatenumericalsolutiondiscussedinSection 9.3.1 . Figure 9-14 comparesthenumericalandexperimentalresultsofthevolume fractionatdifferentpointsofaccumulatedstrain(wherethepiperadius R isusedfor normalization)for R / a =8.21 .Theagreementisgoodatalargebulkvolumefractionof ' B =40% .Atasmallerbulkvolumefractionof ' B =30% ,theexperimentalobservations showaslightlyslowerevolutiontoatemporallyconstantsystemwhencomparedto thenumerics.Thediscrepancyincreasesatlower ' B =20% .Thedisagreement betweenthedynamicnumericalpredictionsandexperimentalobservationsiseven greaterfor R / a =15.71 ,asseeninFigure 9-15 .Itshouldbenotedthattheparticles ofsize 2 a =1.05 ± 0.02 exhibitedobservablesedimentingwhenleftovernight.Asthe experimentstookaconsiderableamountoftime,evenupto 8 hoursfor ' B =20% ,it islikelythatsettlingplayedaroleinthedynamics,possiblydelayingthetimetoreach steadystate. 9.3.3MigrationDependenceonConcentrationandStrain Thediscrepancybetweenthedynamicnumericalpredictionsandtheexperimental observationsdiscussedinSection 9.3.2 couldbeattributedtotheexperimentutilizing oscillatoryßowinsteadofsteadyßow.Totestthishypothesis,theamplitudewas increasedto ! =6.75 for R / a =8.21 and ' B =20% .Experimentalmeasurements wereaveragedinthesamewayasinFigure 9-14 and 9-15 andasdescribedinSection 9.3.2 .Figure 9-16 showsthattheexperimentsreachsteadystateatapproximatelythe sameaccumulatedstrain.Itwouldhavebeendesirabletotestanevenlargeramplitude ofoscillation,butthiswouldrequireamuchlongertubetoavoidendeffects.Wehave plottedonthesamegraphexperimentalresultsforasmalleramplitudeofoscillationof ! =2.7 whichindicatesnomigration.Thissuggeststhatthereisanamplitudethreshold formigrationtobeobserved.Theexperimentsshowingmigrationreportedinthepresent workareundertakenabovethisthreshold. 115

PAGE 116

0200400 600 800 Accumulated Strain 0 0.1 0.2 0.3 0.4 0.5 ! " =6.75 Center Bin " =6.75 Quarter Bin " =5.4 Center Bin " =5.4 Quarter Bin " =2.7 Center Bin " =2.7 Quarter Bin Figure9-16. Changesinthetemporalevolutionofthelocalparticlevolumefractiondue to ! .Temporalevolutionofthelocalparticlevolumefractionatthecenter (circle)andquarter(square)binsfor R / a =8.21 at ' B =20% forthree differentnormalizedamplitudesofoscillation ! =2.7,5.4, and 6.75 . -1 -0.5 0 0.5 1 x / R 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ! Present Work R / a =8.21 " =5.4 Present Work R / a =15.71 " =5.4 Present Work R / a =15.71 " =14.3 SBM BGP Figure9-17. Theexperimentallyobservedvolumefractiondistributionatthebeginning oftheÞnaloscillation.Resultsareshownforvarying R / a , ' B ,and ! .There isnoobservablemigrationwhiletheSBMusing Boyer etal. ( 2011 a ) rheology(solidblueline)predictsmigration. 116

PAGE 117

0100200300400 500 Accumulated Strain -3 -2 -1 0 1 2 3 ( ! x) 2 / a 2 R / a =8.21 " =5.4 R / a =15.71 " =5.4 R / a =15.71 " =14.3 A 0 500 Accumulated Strain -10 -8 -6 -4 -2 0 2 4 6 8 10 ( ! z) 2 /a 2 R / a =8.21 " =5.4 R / a =15.71 " =5.4 R / a =15.71 " =14.3 B Figure9-18. Meansquaredisplacementoftheparticles.Resultsforthemeansquare displacementA)transversetotheßowandB)alongtheßowforvarying R / a , ' B ,and ! areshown. 117

PAGE 118

Nomigrationwasobservedat ' B =10% for R / a =8.21 and R / a =15.71 . For R / a =8.21 ,experimentswerestoppedafter 50 oscillationsoranaccumulated strainof 1085 .For R / a =15.71 ,experimentswerestoppedafter 30 oscillationsoran accumulatedstrainof 630 .Tofurthertestthelackofobservablemigration,theamplitude wasincreasedto 14.32 (or 225 particleradii)for R / a =15.71 at ' B =10% .The experimentswerestoppedafter 20 oscillations(oranaccumulatedstrainof 945 )withno observablemigration.Figure 9-17 plotsthelocalvolumefractionatthebeginningofthe Þnaloscillation,demonstratingthelackofanobservablemigrationfortheseconditions. Theseresultssuggestthereisaconcentrationthresholdtohaveobservablemigration. Toexaminefurthertheabsenceofobservedmigrationat ' B =10% ,wehaveplottedin Figure 9-18 themeansquaredisplacementoftheparticlesensembleaveragedacross particlesandexperimentalruns.Clearly,thetransversemeansquaredisplacementis nullwithinerrorbar.Moreover,thesamebehaviorisalsoobservedintheßowdirection, showingthattheparticlemotioncanbeconsideredasquasireversible. 9.4Discussions Thespatialandtemporaldependenceoftheparticlevolumefractionandvelocity forsuspensionsofneutrally-buoyantspheresinoscillatorypipeßowwasexperimentally studiedusingdirectimagingtechniques.Thebulkvolumefraction ' B wasvariedfrom 10% to 40% andthepipetosphereradiuswassetateither R / a =8.21 or R / a =15.71 . NotonlytheÞnalsteadystate,butalsothedynamicsoftheshear-inducedmigration, wasanalyzedandcomparedtopreviousworkaswellastothepredictionsoftheSBM usingrealisticrheologies. Comparisonswithprevioussteadystateresultsobtainedbyexperimentsand discrete-particlesimulationsatsimilarconcentrationsandgeometriesshowexcellent agreementandconÞrmthepresenceofparticlelayeringduetotheconÞnementatthe pipewalls.SteadystatepredictionsoftheSBMquantitativelyagreewiththepresent experimentalresultsatlarge ' B .TheSBMhoweverfailstoshowthedecreasein 118

PAGE 119

maximumconcentrationatthecenterofthepipewithdecreasing ' B observedintheexperimentsaswellasinthediscrete-particlesimulations.Goodagreementisrecovered whenthemaximumconcentrationisalteredtobethatobservedexperimentally.Thedynamicsoftheshear-inducedmigrationprocesswasonlycomparedtothatgivenbythe SBMsincedatafrompreviousexperimentalordiscrete-particlenumericalworkarenot readilyavailable.DynamicpredictionsfromtheSBMareinqualitativeagreementwhen ' m isthemaximumconcentrationobservedexperimentally.However,thepredictions reachsteadystatesigniÞcantlyfasterthantheexperiments. TheoriginoftheSBM'sinabilitytoquantitativelypredicttheexperimentalresults regardingthefasterdynamicsandthelargerconcentrationinthemiddleofthepipeis unclear.ItcouldbefromafailureofthecontinuummodeltocaptureconÞnementeffects oritcouldbethatfurtherreÞnementisneededintheconstitutiveequationusedforthe particlephasestress.Toaidinresolvingtheseissuesinfuturestudies,severalpoints shouldbenoted.Firstly,thetworheologicallawsusedarestrictlyvalidforlarge ' B and havebeenextrapolatedfor ' B " 40% .Therheologicallawof Morris&Boulay ( 1999 ) wasindeedmeanttomatchexperimentalresultsofmigrationatlarge ' B andthatof Boyer etal. ( 2011 a )wasobtainedfrommeasurementsfor ' B ! 40% .Theserheologies mayhavetobealteredforlowerconcentrations.Secondly,thephenomenologicalform forthenormalstressoftheparticlephasewaschosenforsimplicitytohaveasimilar concentrationdependenceinalldirectionsasproposedby Morris&Boulay ( 1999 ). However,theremaybesomeslowvariationinthedifferentdirectionswhichmayneed tobeincludedinthemodel( Dbouk etal. , 2013 ; Gallier etal. , 2014 ).Finally,normal stressesnearthepipewallswherelayeringoccursduetoconÞnementeffectsmay differfromthoseinthebulkmiddleregionofthepipeasfoundintheforcecoupling simulationsof Yeo&Maxey ( 2011 )andintheÞctitiousdomainsimulationsof Gallier ( 2014 ).ThisisnotcapturedbythepresentSBMwhichusesarheologywhichis homogeneousacrossthepipe. 119

PAGE 120

Thepresentexperimentsshowthatanamplitudeandconcentrationthresholdis requiredtoinducemigration.Theobservedamplitudethresholdisinagreementwith theearlierÞndingsof Butler etal. ( 1999 ).Largeoscillationamplitudeswereusedinthe presentworktoobservemigrationtowardthecenterofthepipe.However,for ' B =10% , nomigrationwasobservedregardlessofthelargeamplitudeofoscillation,upto 225 particleradii.Previousexperimentsinpipeßowof Hampton etal. ( 1997 )didnotsee migrationatthatvolumefractionwhileexperimentsinachannelof Koh etal. ( 1994 )and StokesianDynamicssimulationsof Nott&Brady ( 1994 )observemigration.Throughthe directimagingoftheparticles,thepresentexperimentsprovideclearevidenceofthe absenceofmigrationandofthequasireversibilityofthemotionoftheparticles.This canbediscussedinthelightofexperimentsonthereversibilityofparticlemotionin oscillatoryshearßows( Pine etal. , 2005 )andoscillatorypressuredrivenßows( Guasto etal. , 2010 ).Thethresholdstrainamplitudefortheonsetofirreversibilityhasbeen plottedasafunctionofvolumefraction,seeFigure3of Pine etal. ( 2005 ).Itincreases dramaticallybelow ' B =10% andevenseemstodiverge.Consequentlythereversible behaviorobservedinthepresentexperimentsat ' B =10% maynotbesurprising. Notethatthesituationisfarmorecomplexfornonuniformstrainasthetransitionis pushedtolargerwallstrainamplitudes,seeFigure4of Guasto etal. ( 2010 ),andoccurs simultaneouslyacrossthechannel,evenwherethelocalstrainisnegligible. TheSBMpredictsmigrationat ' B =10% becausetherheologicalmodelsextrapolatedforthislowconcentrationhavesigniÞcantnormalstresses.Therefore,ourpresent experimentsseemtosuggestthatat ' B " 10% normalstressesarenotsigniÞcant enoughtoproducemigration.Atthesesmallvolumefractions,contactstresses,which seemtobeessentialinthemigrationprocess,mayindeedbeinconsequential.Clearly abetterdescriptionoftherheologyoftheparticlephaseisneeded.Inparticular,the relativeimportanceofthecontactandhydrodynamiccontributionsshouldbeassessed. 120

PAGE 121

Thepresentexperimentaldatacanbeusedasabenchmarkforrheologicalmodelsin theSBM,butalsoforparticleinteractionmodelsindiscrete-particlesimulations. 121

PAGE 122

CHAPTER10 CONCLUSIONS Thepresentedworkhasmadecriticalcontributionstothetheoryofsuspensions ofÞbersandspheres.ForÞbers,thiswasachievedthroughathoroughnumericaland experimentalstudyoftherheology,themicrostructure,andtheirinterdependence.This wasdoneforvaryingaspectratios,concentrations,boundaryconditions,andßowÞelds. Resultsclarifyexistingdiscrepancieswhileprovidingin-depthunderstandingusefulfor futureexperimentformulationandtheoreticaldevelopment.Forspheres,anintensive experimentalstudyonthedynamicsofparticlemigrationwascarriedoutforvarying concentrationsandgeometries.Resultsprovidebenchmarkhighresolutiondatafor futurenumericalandexperimentalcomparisons.Additionally,thisdataprovidesthe necessaryinformationforcomparisontotheoreticalpredictionsutilizingconstitutive rheologicalequationsofvaryingformstoaidinclosingthetwo-phaseequations. AnumericalmodelwasdevelopedforconcentratedsuspensionsofÞbers,where longandshortrangehydrodynamicinteractionsareignoredbutparticlecontactsare included.Agreementwiththepreviousnumericalworkof Salahuddin etal. ( 2013 )and Wu&Aidun ( 2010 ),whichincludelongandshortrangehydrodynamicinteractions aswellascontacts,conÞrmsthattheaccurateresolutionofcontactsisessentialin predictingthemicrostructureinthesesystems.ThisallowsforgreatsimpliÞcation inmodelingsoftwareusedinindustry.Numericalresultsenableexplanationforthe disagreementbetweenpreviousnumericalandexperimentalmeasurementsoftheorbit constant " C b # ,butagreementforthemoment ! p 2 x p 2 y " intheliterature.Inthesesystems, theÞberalignmentwiththeßowandvorticitydirectionsrequiresasigniÞcantamount ofaccumulatedstraintoreachsteadystate.Theorbitconstantismoresensitiveto changesofalignmentwiththevorticitythan ! p 2 x p 2 y " istochangesinalignmentwiththe ßowdirection.Accordingly,althoughthedynamicsofthemicrostructuremaystillbe transient, ! p 2 x p 2 y " willlikelyexhibitastatisticallysteadyvalue.However,theorbitconstant 122

PAGE 123

ampliÞesthechangingalignmentwiththevorticity,resultingindisagreementbetween experimentsandnumericsasthemicrostructureisstillchanging.Thisexplainsthe perplexingagreementbetweenexperimentsandnumericsfor ! p 2 x p 2 y " ,butnotfor " C b # , seenintheliterature.Thenumericalmethodalsoenabledseparationofthecontact contributiontothestress.Uniqueexperiments,whichdeterminethenormalstress differencesfrommeasuringthedeßectioninthefree-surfaceofaßowingsuspension, wereperformed.TheseexperimentsdidnotinvolvethehighlevelofconÞnement requiredwhenmeasuringnormalstressdifferencesusingarheometer.Resultswerein quantitativeagreement,indicatingthatcontactsarethedominantcontributionforthese non-Newtonianproperties. Inadditiontothetheseconclusions,itwasfoundconÞnementcanalterboththe particledynamicsandtherheology.Thishasseriousimplicationsonpreviousandfuture experimentalmeasurementsaswellasindustrialapplicationswhichinvolvevarying levelsofconÞnement.Forparticledynamics,thenumericalmodelpredictsthesame alignmentwiththevorticitydirectioninoscillatoryshearßowasobservedpreviouslyin experiments( Franceschini etal. , 2011 ),butonlywiththeproperdegreeofconÞnement. Insteadyshear,thechangeinparticledynamicsislesspronounced.Thenumerics predictonlyaslightchangeinalignmentwiththeßowandgradientdirectionswhen thesystemisconÞned.However,thenumericspredictasubstantialchangeinthe normalstressdifferencesduetoconÞnement.Thecontactnumberdensityincreases whenthesystemisconÞned,increasingthemagnitudeofthenormalstressdifferences whencomparedtotheperiodicresults.Interestingly,thepredictedrheologywasnot statisticallydifferentwhenthedistancebetweenboundingwallsincreasedfromthree Þberlengths, 3 L ,toÞveÞberlengths, 5 L .Thisisinagreementwithexperimental observationswherepreviousexperimentsincreasedthedistancebetweenparallel platesonarheometeruntiltherewasnoobservablechangeinthemeasurednormal stressdifferences( Keshtkar etal. , 2010 ; Petrich etal. , 2000 a ).Thepresentnumerics 123

PAGE 124

predicttheseobservations,butshowthemeasurementsarestilldifferentthanfully periodicpredictions.ThischangeinnormalstressdifferencesduetoconÞnementlikely explainsthevariabilityacrosspreviousexperimentsusingvaryinggeometriesandthe pooragreementofthepresentnumericalworkwiththepreviousexperiments. Thedynamicsofshearinducedmigrationinsuspensionsofsphereswerestudied usinglargeamplitudeoscillatoryparabolicßow.Highresolutionmeasurementsofthe particlevolumefractionandvelocitydistributionsweremade.Steadystatevolume fractiondistributionswereinagreementwithpreviousexperimentalandnumericalworks atsimilarbulkvolumeconcentrations, ' B ,andgeometries, R / a .Thehighresolution resultsofthecurrentexperimentsconÞrmthepresenceofanearwallparticlelayer previouslypredictedbydiscreteparticlesimulations( Yeo&Maxey , 2011 ).Dynamic experimentalresultswerecomparedtotheSuspensionBalanceModel,withmultiple rheologiesutilized.Experimentsandnumericswereonlyinagreementfor ' B =40% . Thecontinuummodelingpredictedafastertemporalevolutionoftheparticlevolume fractiondistributionforallotherbulkvolumefractions.Itislikelytheparametersofthe rheologicalequationrequirefurtherdependenceonthebulkvolumefraction.Moreover, experimentsexhibitedachangeinthemaximumobservedvolumefactionwitha changeinbulkvolumefraction.Additionally,nomigrationwasobservedfor 10% .These observationsarenotpredictedbytheSuspensionBalanceModel. ThepresentworkhassigniÞcantlycontributedtothetheoryofsuspensiondynamics.Thesecontributionsimprovetheunderstandingofsuspensions,enablingimproved interpretationoffutureexperimentalobservations,theoreticaldevelopments,suchas constitutiverheologicalequations,andmodelformulation.Despitethesecontributions, muchworkremains.ExperimentsmeasuringthemicrostructureinsuspensionsofÞbers insteadyshearßow,whichwaituntilafterthepredictedaccumulatedstraintoreach steadystategiveninChapter 3 tomakeobservations,areneeded.Furtherexperiments measuringthemicrostructureforoscillatoryandsteadyshearßowatvaryinglevelsof 124

PAGE 125

conÞnementarealsoneeded.Additionalexperimentsusingarheometeratvarying levelsofconÞnementarenecessitatedtovalidatethepresentednumericalresultsin Chapter 8 .Forsuspensionsofspheres,therheologicalconstitutiveequationsusedin theSuspensionBalanceModelrequirefurtherstudytounderstandthecurrentdiscrepanciesbetweenitspredictionsandthepresentedresults.SpeciÞcally,determiningthe originsforthelackofanobservablemigrationatabulkvolumefactionof 10% anda changeinthemaximumconcentrationwithchangingbulkvolumefraction. 125

PAGE 126

APPENDIXA THEFREE-SURFACEPROFILEOFASUSPENSIONFLOWINGINASEMICIRCULARTROUGH Thecalculationfollowstheworkof Tanner ( 1970 )andof Kuo&Tanner ( 1974 )which includestheeffectofsurfacetension.Itisadaptedfromthatof Couturier etal. ( 2011 )to thecylindricalgeometryofthesemi-circulartrough.Thesemi-circulartroughofradius R isinclinedatanangle % relativetothehorizontaldirectionasshowninFigure 7-2 B. Weuseacylindricalcoordinatesystem( r , & , z )wheretheoriginispositionedatthe middleofthetroughatalevelequaltotheheightofthefree-surfaceofthesuspension averagedacrossthewidthofthetrough.ThesuspensioncompletelyÞllsthesemicirculartroughandthecoordinatesystemisshowninFigure 7-2 BandinFigure A-1 . Theßowinthetroughisassumedtobesteadyandfully-developed,hencetheinterface isafunctionof x onlyandisdesignatedby h ( x ) . Whenthesurfacedeßection h issmall,i.e. h / R + 1 ,thevelocityÞeldisassumed tobepurelyinthe z $ directionandtodependonlyon r .Withinthisassumption,the shearstress # = ! rz onlydependson r ,where " isthebulkstressofthesuspension. Notethatwedropthebrackets, " á # ,usedinsection 2 todesignatethebulkstress.For Stokesiansuspensionsofrigid,non-Brownian,andneutrally-buoyantparticles,theshear stress # islinearinshearrateandsoarethenormalstressdifferences N 1 and N 2 .As theselatterquantitiesshouldbeindependentofthesignoftheshearrate,theyarethus proportionaltothemodulusoftheshearstress.Ofparticularinteresthere,wecanwrite N 2 = ! rr $ ! ## = " 2 | # | . (AÐ1) TorelatethecoefÞcient " 2 tothedeformation h ( x ) ,wesolvethemomentumequations forthesuspension,requiringthatthenormalforcesontheinterfacebalance. Thesuspensionmomentumequationalong z is 1 r , ( r ! rz ) , r = $ + g sin % , 126

PAGE 127

! " ! " ! ! " # ! # #$% " FigureA-1. Cut-awayviewofthesemi-circulartroughofradius R .Thelocationofthe free-surfaceisdesignatedby h ( x ) ,wherethecoordinates x , y , r ,and & are deÞnedinthesketch.Flowisdirectedoutofthepageinthe z -direction;the componentofgravityinthe $ y -directionisgivenby g cos % . where g sin % isthecomponentofgravitythatdrivestheßow.Integratingthisexpression from 0 to r gives # = ! rz = $ 1 2 + gr sin % . (AÐ2) Combiningthislatterequationfortheshearstress( AÐ2 )withtheconstitutiveEquation ( AÐ1 )gives N 2 = " 2 2 + gr sin % . (AÐ3) Thesuspensionmomentumequationalong & isgivenby $ ! !! $# = + gr cos % cos & , which,usingEquation( AÐ1 ),canbewrittenas $ ! rr $# = $ N 2 $# + + gr cos % cos & = + gr cos % cos & , since N 2 onlydependsupon r .Integratingthisequationover & from thefree-surfacelocation y = h ( x ) toalocation( r , & ),whilekeeping r constant,gives ! rr ( r , & )= $ + g cos % [ h ( x ) $ r sin & ]+! S rr | h ( x ) . (AÐ4) Thesuperscript S inEquation( AÐ4 )indicatesthatthestressisevaluatedatthefreesurface. 127

PAGE 128

Thesuspensionmomentumequationalong r iswrittenas $ ! rr $ r = + g cos % sin & $ N 2 r , whichcanbeintegratedover r fromthelocation( r , & )tothelocation( h (0), & ).After integratingandusingEquations( AÐ3 )and( AÐ4 )yields + g cos % [ h ( x ) $ h (0) ] + ! S rr | h (0) $ ! S rr | h ( x ) = " 2 2 + g sin % [ r $ h (0) ] . (AÐ5) Attheinterface,thetangentialandnormalcomponentsofthestressesmust balance.Weassumethatviscouseffectsintheairarenegligible,thatthesurface tensiondoesnotdependuponposition,andthattheslopeofthefree-surfaceissmall, i.e. dh / dx + 1 .Undertheseconditions,thetangentialstressbalancereducesto ! S xy | h ( x ) =0 andthenormalstressbalanceto ! S yy | h ( x ) = $ ! C (theatmosphericpressure takenasreference),where ! isthesurfacetensionofthesuspensionand C '$ d 2 h / dx 2 isthecurvatureoftheair-suspensioninterface.Usingthenormalandtangentialstress balancesgives ! S rr = N S 2 cos 2 & cos 2 & $ sin 2 & + ! S yy = " 2 2 + gr sin % cos 2 & cos 2 & $ sin 2 & + ! d 2 h dx 2 , (AÐ6) wherewehaveusedtheidentities ! S rr = ! S xx cos 2 & + ! S yy sin 2 & , N S 2 =(! S xx $ ! S yy )(cos 2 & $ sin 2 & ). SubstitutingEquation( AÐ6 )intoEquation( AÐ5 )givesarelationshipbetweenthe curvatureoftheinterfaceasafunctionof x andthecoefÞcient " 2 , ! + g cos % 3 d 2 h dx 2 ( x ) $ d 2 h dx 2 (0) 4 = h ( x ) $ h (0) $ " 2 tan % 3 X $ h (0) 2 4 , (AÐ7) with X = 1 2 3 r + r cos 2 & cos 2 & $ sin 2 & 4 = 5 x 2 + h 2 ( x ) 2 x 2 $ h 2 ( x ) 2 x 2 $ 2 h 2 ( x ) , 128

PAGE 129

wherethesecondformoftheequalitywasobtainedusingtherelations x = r cos & and h = r sin & .Theconstant h (0) canbedeterminedbyapplyingthemassconservation equation( 6 R ! R h ( x ) dx , 0 ). ItisconvenienttorecastEquation( AÐ7 )indimensionlessformbydeÞning ö x = x / R and ö h = h / S R aswellas S = " 2 tan % ,togive 1 Bo d 2 ö h d ö x 2 (ö x )= ö h (ö x ) $ ö h 0 $ , 7 ö x 2 + S 2 ö h 2 (ö x ) 2ö x 2 $ S 2 ö h 2 (ö x ) 2ö x 2 $ 2 S 2 ö h 2 (ö x ) $ S ö h (0) 2 . , (AÐ8) wheretheBondnumberis Bo = + gR 2 cos % /! andthedimensionlessconstantis ö h 0 = ö h (0) $ 1 Bo d 2 ö h d ö x 2 (0). Inthisdimensionlessform,wecanseethatthetermsinthesquarebracketsonthe right-handsideofEquation( AÐ8 )areapproximatelyequalto | ö x | when S ö h + 1 and | ö x | is large.When | ö x | approacheszeroandbecomessmallerthan S ö h ,thetermsinthesquare bracketsinEquation( AÐ8 )isapproximatelyequalto S [ ö h (ö x ) $ ö h (0)] / 2 + 1 .Equation ( AÐ8 )canthenbereducedtothesimpliÞedform 1 Bo d 2 ö h d ö x 2 (ö x )= ö h (ö x ) $ ö h 0 $ | ö x | , (AÐ9) whichissimilartoEquation(2.12)usedby Couturier etal. ( 2011 )foradeeptrough. Equations( AÐ8 )and( AÐ9 )canbesolvednumericallybyimposingaslope d ö h / d ö x ofzero at ö x =0 andbyspecifyingthecontactangleat ö x =1 .Itisalsoconvenienttousethe analyticalsolution[Equations(2.13),(2.14),(2.15),and(2.16)]providedby Couturier etal. ( 2011 ). 129

PAGE 130

APPENDIXB THESUSPENSIONBALANCEMODEL TheSuspensionBalanceModel(SBM)isderivedfromatwo-phasemodelingofthe particulatesuspensionwrittenatlowReynoldsnumber( Lhuillier , 2009 ; Morris&Boulay , 1999 ; Nott&Brady , 1994 ; Nott etal. , 2011 ).Theequationsofmotionareusuallywritten forthesuspensionmixtureandtheparticulatephase. Thebalanceequationsforthesuspensionare & á u =0, (BÐ1) & á " + + g =0, (BÐ2) where u = ' u p +(1 $ ' ) u f isthevolumeaveragevelocityofthesuspension( u p and u f arethelocalmeanparticleandßuidvelocities,respectively), + = '+ p +(1 $ ' ) + f isthe densityofthesuspension( + p and + f aretheparticleandßuiddensity,respectively),and " isthebulkaveragesuspensionstress. Theparticlemassconservationequationyields ,' , t + & á ( ' u p )=0, (BÐ3) which,usingtheincompressibilityofthesuspension( BÐ1 ),canberewrittenas ,' , t + u á & ' = $&á ' ( u p $ u ), (BÐ4) toexhibitthemigrationßux ' ( u p $ u ) . Thebalanceequationsfortheparticlephaseare & á " p + n " F H # + ' ( + p $ + f ) g =0, (BÐ5) wheretheparticle-phasestress " p comprisescontact(orinterparticleforce)contributionsaswellashydrodynamiccontributionscomingfromthenondragpartofthe interphaseforce( Lhuillier , 2009 ; Nott etal. , 2011 ).Thehydrodynamicforce n " F H # 130

PAGE 131

isusuallyreducedtothedragforceproportionaltotherelativevelocitybetweenthe phaseswhichcanbewrittenforsphericalparticlesofradius a as F D = $ 9 * f 2 a 2 ' f ( ' ) ( u p $ u ), (BÐ6) whereonecanusetheempiricalhinderedsettlingfunction f ( ' )=(1 $ ' ) n (BÐ7) (with n 5 atlowReynoldsnumbers)proposedby Richardson&Zaki ( 1954 )andwith * f theviscosityofthepureßuid. Forneutrallybuoyantparticles, + = + p = + f ,themomentumEquation( BÐ5 )together withtheexpressionforthedrag( BÐ6 )providethemigrationßux ' ( u p $ u )= 2 a 2 f ( ' ) 9 * f & á " p . (BÐ8) Onlythediagonalcomponentsoftheparticlephasestresstensor " p ,i.e.thenormal stresstensoroftheparticlephase,generateacross-streammigrationßux.Itisnecessarytoassumeaphenomenologicalformwhichforsimplicitycanbechosentohavea viscousscalingandasimilar ' -dependenceinallthedirections( Morris&Boulay , 1999 ) $ * f ú ! ö * n ( ' ) 8 9 9 9 9 : 1 00 0 2 0 00 3 ; < < < < = , (BÐ9) where ú ! istheshearrate,the i ( i =1,2,3 )arescalarparameterswhichareindependentof ' ,and ö * n ( ' ) isadimensionless"normalviscosity"thatcanbedeÞnedas ö * n ( ' )= $ ! p 11 / * f ú ! togive 1 =1 . 131

PAGE 132

B.1TheSBMforPipeFlow Intheaxisymmetricpipegeometry,themigrationequationgivenbyEquations( BÐ4 ) and( BÐ8 )iswrittenusingcylindricalcoordinates( r , & , z ) ,' , t = $ 2 a 2 9 * f , r , r 3 rf ( ' )( , ! p rr , r + ! p rr $ ! p ## r ) 4 , (BÐ10) which,usingthephenomenologicalexpressions( BÐ7 )and( BÐ9 ),becomes ,' , t = 2 a 2 9 , r , r > r (1 $ ' ) n 3 2 , ö * n ú ! , r +( 2 $ 3 ) ö * n ú ! r 4? . (BÐ11) Notethatweassume u á & ' ' 0 intheEquations( ?? )and( BÐ11 )asthecross-stream migrationismuchslowerthanthemeanßowdynamicsandthevolumeaveraged velocityisstillconsideredtobepurelyparallel,i.e.along z ,andnottodependon z ,i.e. u z ( r ) .Theshearrateis ú ! rz = du z / dr . Indimensionlessform(usingthepiperadius R asthelengthscaleandthemean velocity U ofthepipeßowasthevelocityscale),Equation( BÐ11 )iswrittenas ,' , ö t = 2 a 2 9 R 2 , ö r , ö r > ö r (1 $ ' ) n 3 2 , ö * n ö ú ! , ö r +( 2 $ 3 ) ö * n ö ú ! ö r 4? . (BÐ12) Thecross-streammigrationßux(theterminsidethebrackets)hastobenullonthepipe walls( ö r =1 )aswellasonthepipecenter( ö r =0 ). Thedimensionlessshearrate ö ú ! isobtainedbysolvingthe(steadystate)suspension momentumequationacrossthepiperadiuswhichiswrittenindimensionlessformas ö K + 1 ö r , , ö r (ö r ö * s ö ú ! )=0, (BÐ13) wheretheshearstressis # = ! rz = * s ú ! , * s ( ' ) istheviscosityofthesuspension ( ö * s = * s / * f ),and $ K = $ , ! zz / , z isthepressuregradientalongthepipewhichdrives theßow( ö K = KR 2 / * f U ).Integrating( BÐ13 )acrossthepipeshowsthattheshearstress islinearin r ö * s ö ú ! = | $ ö K 2 | ö r . (BÐ14) 132

PAGE 133

Tosolvetheseequations,weneedrheologicallawsfor ö * s and ö * n .Wehavetested twoconstitutivelawsthatcanbefoundintheliterature: (i) Arheologicalmodelproposedby Morris&Boulay ( 1999 ): * s ( ' )=1+(5 / 2) ' (1 $ ' / ' m ) ! 1 + K s ( ' / ' m ) 2 (1 $ ' / ' m ) ! 2 and * n ( ' )= K n ( ' / ' m ) 2 (1 $ ' / ' m ) ! 2 with K s =0.1 and K n =0.75 .TheexponentoftheRichardson-Zakicorrelationis n =4 and 2 =0.8 and 3 =0.5 . (ii) Arheologicalmodelrecentlyproposedby Boyer etal. ( 2011 a ),whichcomes fromrheologicalmeasurementsofdensesuspensionsofneutrally-buoyant spheresusingpressure-imposedrheometry: * s ( ' )=1+(5 / 2) ' (1 $ ' / ' m ) ! 1 + µ c ( ' )( ' / ' m ) 2 (1 $ ' / ' m ) ! 2 and * n ( ' )=( ' / ' m ) 2 (1 $ ' / ' m ) ! 2 with µ c ( ' )= µ 1 +( µ 2 $ µ 1 ) / [1+ I 0 ' 2 ( ' m $ ' ) ! 2 ] and µ 1 =0.32, µ 2 =0.7, I 0 =0.005 .The exponentoftheRichardson-Zakicorrelationis n =5 and 2 =0.95 and 3 =0.6 as deducedfromthenormalstressdifferencemeasurementsof Boyer etal. ( 2011 a ) and Couturier etal. ( 2011 ). Inbothcases,wehavetakenthemaximumpackingfractiontobe ' m =0.585 asfound experimentallyby Boyer etal. ( 2011 a )forsphericalparticlessimilartothoseusedinthe presentexperiment. Insolvingtheseequations,difÞcultiesariseatthecenterlineofthepipewherethe shearrateapproacheszeroandthevolumefractionreachesthemaximumpacking fraction ' m .Thisbehaviorrepresentsabreakdownofthe"local"descriptionofthe stress.Whenthescaleofinterestapproachesthatoftheindividualparticles,the continuumapproximationmaybeexpectedtobreakdown.Toovercomethisissue, usinganonlocalstresscorrectionthataccountsforthesamplingofneighboringßow regionsbytheparticlesovertheirownsizescalehasbeensuggested.Inthiswork,the nonlocalstressmodelaimsatmimickingthisspatialaveraging.Following Miller&Morris ( 2006 ),thenonlocalcorrectionhasbeenchosentodependonthemeanshearrateand theparticletothepipesizeratio ú ! NL = U R @ a R A m , (BÐ15) wheretheexponent m canbeadjusted.Figure B-1 showsthatwithoutthenonlocal correction,thesolutionoftheequationsisblowingupinthemiddleofthepipe. 133

PAGE 134

! 1 ! 0.8 ! 0.6 ! 0.4 ! 0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 r/R ! A ! 1 ! 0.8 ! 0.6 ! 0.4 ! 0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 r/R ! B ! 1 ! 0.8 ! 0.6 ! 0.4 ! 0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 r/R ! C FigureB-1. Evolutionofvolumefractionacrossthepipefortherheologyof Boyer etal. ( 2011 a ).Resultsareshownatdimensionlesstimes ö t =0,10,30,60,90,120,1000 with(red)andwithout(green)thenonlocal correctionforA) m =2 ,B) m =4 ,andC) m =6 andusingtherheologyof Boyer etal. ( 2011 a )( 2 a =0.105 cmand 2 R =1.64 cm). 134

PAGE 135

B.2SteadyStateSolution Whentheßowisfullydeveloped,Equation( BÐ12 )becomes(noßux,thus & á " p =0 ) 2 , ö * n ö ú ! , ö r +( 2 $ 3 ) ö * n ö ú ! ö r =0, (BÐ16) which,togetherwiththelinearvariationin r oftheshearstress * s ú ! givenbyEquation ( BÐ14 ),providesthefrictioncoefÞcient( Boyer etal. , 2011 a ) µ ( ' )= * s ( ' ) * n ( ' ) = µ w ö r (2 ! % 3 / % 2 ) . (BÐ17) Here, µ w isthefrictioncoefÞcientatthepipewall( ö r =1 ).Thisfrictionalapproachis similarinessencetothatusedby Lecampion&Garagash ( 2014 ). Thesuspensionßowachievesajammedstatewhenthevolumefraction ' reaches maximumpackingfraction ' m .Thisoccursinacentraljammedregionofthepipe delimitedby ö r jam =[ µ ( ' m ) /µ w ] 1 / (2 ! % 3 / % 2 ) .Outsidethisjammedregion,theconcentration proÞleisgivenbyreversingthefrictionfunction( BÐ17 ).Ensuringthatthetotalvolume fractioninthepipecorrespondstotheinitial(uniform)volumefraction ' 0 providesthe determinationof µ w .Bothrheologicalmodelsof Morris&Boulay ( 1999 )and Boyer etal. ( 2011 a )whichweredescribedintheprecedingsectioncanbetested.Notethat µ ( ' m )= µ 1 inthe Boyer etal. ( 2011 a )rheologywhile µ ( ' m )= K s / K n inthe Morris& Boulay ( 1999 )rheology.Figure B-2 comparesthisobtainedsteadystatesolutionwith thesolutionoftheSBMmodelwithanon-localcorrectiongiveninFigure B-1 .Clearlya non-localcorrectionusingalargeexponent( m ) 4 )satisfactorilyreproducesthesteady statesolution. 135

PAGE 136

! 1 ! 0.8 ! 0.6 ! 0.4 ! 0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 r/R ! A ! 1 ! 0.8 ! 0.6 ! 0.4 ! 0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 r/R ! B ! 1 ! 0.8 ! 0.6 ! 0.4 ! 0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 r/R ! C FigureB-2. Evolutionofvolumefractionacrossthepipeusingtherheologyof Boyer etal. ( 2011 a )withthesteadystatesolution.Resultsareatdimensionless times ö t =0 , 10 , 30 , 60 , 90 , 120 , 1000 with(red)thenonlocalcorrectionwith A) m =2 ,B) m =4 ,andC) m =6 andusingtherheologyof Boyer etal. ( 2011 a ).Thesteadystatesolutionisalsogiveninblue. 136

PAGE 137

APPENDIXC TABULATEDRESULTS 137

PAGE 138

TableC-1. Numericalresultsfortheamountofaccumulatedstrain, ú ! t toreachsteadystate. AnL 2 d =11.522.53 1174030017013050 151150580290190140 3038101940995720530 508130404524251560880 138

PAGE 139

TableC-2. Numericalresultsfortheorbitconstant, " C b # ,atsteadystate. AnL 2 d =1 1.5 2 2.5 3 110.406 ± 0.0020.470 ± 0.0020.508 ± 0.0010.536 ± 0.0010.560 ± 0.001 150.380 ± 0.0030.444 ± 0.0020.481 ± 0.0010.507 ± 0.0010.527 ± 0.001 300.348 ± 0.0020.408 ± 0.0020.431 ± 0.0010.464 ± 0.0010.481 ± 0.001 500.348 ± 0.0010.399 ± 0.0010.431 ± 0.0010.450 ± 0.0010.463 ± 0.0004 139

PAGE 140

TableC-3. Numericalresultsforthemoment ! p 2 x p 2 y " atsteadystate. AnL 2 d =1 1.5 2 2.5 3 110.0170 ± 0.00020.0191 ± 0.00020.0193 ± 0.00020.01864 ± 0.00020.0175 ± 0.0002 150.0129 ± 0.00020.0148 ± 0.00020.0153 ± 0.00020.0153 ± 0.00010.0150 ± 0.0001 300.0070 ± 0.00010.0082 ± 0.00010.0087 ± 0.00010.0089 ± 0.00010.0090 ± 0.0001 500.0046 ± 0.00010.0053 ± 0.00010.0056 ± 0.000050.00575 ± 0.000050.0058 ± 0.00005 140

PAGE 141

TableC-4. Numericalresultsforthemoment " p 2 x # atsteadystate. AnL 2 d =1 1.5 2 2.5 3 110.728 ± 0.0030.788 ± 0.0020.823 ± 0.0010.850 ± 0.0010.873 ± 0.001 150.758 ± 0.0030.813 ± 0.0020.843 ± 0.0010.864 ± 0.0010.8801 ± 0.0008 300.828 ± 0.0020.870 ± 0.0010.8909 ± 0.00070.9036 ± 0.00050.9040 ± 0.0008 500.8779 ± 0.00010.9073 ± 0.00080.9224 ± 0.00060.9312 ± 0.00050.937 ± 0.0003 141

PAGE 142

TableC-5. Numericalresultsforthemoment ! p 2 y " atsteadystate. AnL 2 d =1 1.5 2 2.5 3 110.0403 ± 0.00080.0435 ± 0.00070.0427 ± 0.00060.0401 ± 0.00060.0364 ± 0.0006 150.0295 ± 0.00060.0329 ± 0.00050.0334 ± 0.00050.0327 ± 0.00040.0315 ± 0.0004 300.0152 ± 0.00030.01740 ± 0.00030.018253 ± 0.00020.0185 ± 0.00020.0185 ± 0.0002 500.0097 ± 0.00020.0110 ± 0.00020.0116 ± 0.00010.0118 ± 0.00010.0119 ± 0.0001 142

PAGE 143

TableC-6. Numericalresultsforthemoment " p 2 z # atsteadystate. AnL 2 d =1 1.5 2 2.5 3 110.232 ± 0.0020.169 ± 0.0020.134 ± 0.0010.110 ± 0.0010.091 ± 0.0001 150.212 ± 0.0030.154 ± 0.0010.123 ± 0.0010.103 ± 0.0010.08837 ± 0.0007 300.157 ± 0.0020.113 ± 0.0010.0908 ± 0.00070.0779 ± 0.00040.0775 ± 0.0008 500.112 ± 0.0010.0817 ± 0.00080.0660 ± 0.00050.0570 ± 0.00050.0511 ± 0.0003 143

PAGE 144

TableC-7. CharacteristicsforeachsetofÞbers.Datashownincludesthemeanvalueandstandarddeviationoftheaspect ratio A ,Þberlength L ,andÞberdiameter d .Valuesofthecriticalstressnormalizedbythemaximumapplied stress, ! crit / ! max ,arealsoreported. AL (mm) d (mm) ! crit / ! max 12 ± 10.54 ± 0.020.047 ± 0.0051060 13 ± 20.30 ± 0.040.024 ± 0.003761 17 ± 20.76 ± 0.020.046 ± 0.005320 32 ± 41.50 ± 0.030.047 ± 0.00632 144

PAGE 145

TableC-8. Experimentalresultsfor " 2 withthestandarderror.Adashindicatesadeformationoftheinterfacewas undetectableforthegivenconditions. AnL 2 d =1.5 2 2.5 3 12 $ 0.02 ± 0.02 $ 0.03 ± 0.02 $ 0.10 ± 0.01 $ 0.09 ± 0.01 13 $ 0.02 ± 0.02 $ 0.03 ± 0.02 $ 0.08 ± 0.02 $ 0.13 ± 0.03 17 $ 0.03 ± 0.01 $ 0.07 ± 0.02 32 0.01 ± 0.01 145

PAGE 146

TableC-9. Numericalresultsfor " 2 .Thedataintheuppersectionarethehydrodynamiccontributionto " 2 andthelower sectiondataarethecontactcontributionto " 2 .Figure 7-6 showsthetotalvalueof " 2 . AnL 2 d =1.5 2 2.5 3 11 $ 0.000 ± 0.001 $ 0.001 ± 0.001 $ 0.001 ± 0.002 $ 0.001 ± 0.002 15 $ 0.000 ± 0.001 $ 0.000 ± 0.001 $ 0.000 ± 0.002 $ 0.000 ± 0.002 300.001 ± 0.001 0.001 ± 0.0010.001 ± 0.0010.001 ± 0.002 11 $ 0.023 ± 0.003 $ 0.042 ± 0.004 $ 0.067 ± 0.005 $ 0.092 ± 0.007 15 $ 0.010 ± 0.002 $ 0.017 ± 0.002 $ 0.028 ± 0.003 $ 0.041 ± 0.004 30 $ 0.0014 ± 0.0004 $ 0.0026 ± 0.0007 $ 0.0042 ± 0.0007 $ 0.0061 ± 0.0010 146

PAGE 147

TableC-10. Numericalresultsfor " 1 .Thedataintheuppersectionarethehydrodynamiccontributionto " 1 andthelower sectiondataarethecontactcontributionto " 1 . AnL 2 d =1.5 2 2.5 3 11 $ 0.001 ± 0.0050.001 ± 0.0050.008 ± 0.0060.026 ± 0.006 15 $ 0.002 ± 0.004 $ 0.002 ± 0.0040.003 ± 0.0050.012 ± 0.006 30 $ 0.002 ± 0.0030.000 ± 0.0040.007 ± 0.0050.015 ± 0.004 110.032 ± 0.0030.064 ± 0.0040.115 ± 0.0050.179 ± 0.007 150.015 ± 0.0020.027 ± 0.0020.047 ± 0.0030.074 ± 0.004 300.0024 ± 0.00040.0046 ± 0.00050.0078 ± 0.00070.0119 ± 0.0009 147

PAGE 148

TableC-11. Numericalresultsforthesuspensionshearstress, # /µ ú ! .Thedataintheuppersectionarethehydrodynamic contributionto # /µ ú ! andthelowersectiondataarethecontactcontributionto # /µ ú ! . AnL 2 d =1.5 2 2.5 3 110.153 ± 0.0020.215 ± 0.0020.270 ± 0.0030.315 ± 0.003 150.08 ± 0.040.198 ± 0.0020.260 ± 0.0020.310 ± 0.003 300.116 ± 0.0010.177 ± 0.0020.242 ± 0.0020.313 ± 0.002 110.0002 ± 0.00010.024 ± 0.0020.0008 ± 0.00040.063 ± 0.003 150.0045 ± 0.00050.008 ± 0.0070.018 ± 0.0010.022 ± 0.001 300.0005 ± 0.00010.0009 ± 0.00020.0015 ± 0.00020.0023 ± 0.0003 148

PAGE 149

TableC-12. Thenumberofoscillationsandaccumulatedstraintoreachsteadystate. R / aa 2 / R 2 ' OscillationNumberAccumulatedStrain 8.210.01540% 3 ± 0.6 50 ± 13 8.210.01530% 5 ± 1.4 87 ± 30 8.210.0040520% 13 ± 3.6 260 ± 78 15.710.0040530% 15 ± 3.6 304 ± 78 15.710.0040520% 67 ± 12.6 1400 ± 273 149

PAGE 150

REFERENCES A LTOBELLI ,S.A.1991Velocityandconcentrationmeasurementsofsuspensionsby nuclearmagneticresonanceimaging. J.Rheol. 35 (5),721Ð734. A RP ,A.&M ASON ,S.G.1976ThekineticsofßowingdispersionsIX.Doubletsofrigid spheres(experimental). J.ColloidInterfaceSci. 61 ,44Ð61. B ATCHELOR ,G.K.1970Slender-bodytheoryforparticlesofarbitrarycross-sectionin Stokesßow. J.FluidMech. 44 ,419Ð440. B ATCHELOR ,G.K.1971Thestressgeneratedinanon-dilutesuspensionofelongated particlesbypurestrainingmotion. J.FluidMech. 46 ,813Ð829. B ATCHELOR ,G.K.&G REEN ,J.T.1972Thedeterminationofthebulkstressina suspensionofsphericalparticlestoorderc2. J.FluidMech. 56 (03),401Ð427. B EAVERS ,G.S.&J OSEPH ,D.D.1975Therotatingrodviscometer. J.FluidMech. 69 , 475Ð512. B OYER ,F.,G UAZZELLI , « E.&P OULIQUEN ,O.2011 a UnifyingSuspensionandGranular Rheology. Phys.Rev.Lett. 107 (18),188301. B OYER ,F.,P OULIQUEN ,O.&G UAZZELLI , « E.2011 b Densesuspensionsinrotating-rod ßows:normalstressesandparticlemigration. J.FluidMech. 686 ,5Ð25. B REEDVELD ,V., VANDEN E NDE ,D.,J ONGSCHAAP ,R.&M ELLEMA ,J.2001Shearinduceddiffusionandrheologyofnoncolloidalsuspensions:Timescalesandparticle displacements. J.Chem.Phys. 114 (13),5923. B RETHERTON ,F.P.1962ThemotionofrigidparticlesinashearßowatlowReynolds number. J.FluidMech. 14 ,284Ð304. B RICKER ,J.M.&B UTLER ,J.E.2006Oscillatoryshearofsuspensionsofnoncolloidal particles. J.Rheol. 50 (5),711. B UTLER ,J.E.,M AJORS ,P.D.&B ONNECAZE ,R.T.1999Observationsofshearinducedparticlemigrationforoscillatoryßowofasuspensionwithinatube. Phys. Fluids 11 ,2865Ð2877. C HAPMAN ,B.K.1990 Shear-inducedmigrationphenomenainconcentratedsuspensions .PhDThesis,UniversityofNotreDame. C HOW ,A.W.,S INTON ,S.W.,I WAMIYA ,J.H.&S TEPHENS ,T.S.1994Shear-induced particlemigrationinCouetteandparallel-plateviscometers:NMRimagingandstress measurements. Phys.Fluids 6 ,2561Ð2576. C OUTURIER , « E.,B OYER ,F.,P OULIQUEN ,O.&G UAZZELLI , « E.2011Suspensionsina tiltedtrough:secondnormalstressdifference. J.FluidMech. 686 ,26Ð39. 150

PAGE 151

C OX ,R.G.1970ThemotionoflongslenderbodiesinaviscousßuidPart1.General theory. J.FluidMech. 44 ,791Ð810. C UNHA ,F.R. DA &H INCH ,E.J.1996Shear-induceddispersioninadilutesuspension ofroughspheres. J.FluidMech. 309 ,211Ð223. D AI ,S.,B ERTEVAS ,E.,Q I ,F.&T ANNER ,R.I.2013ViscometricfunctionsfornoncolloidalspheresuspensionswithNewtonianmatrices. J.Rheol. 57 (2),493Ð510. D BOUK ,T.,L OBRY ,L.&L EMAIRE ,E.2013NormalstressesinconcentratednonBrowniansuspensions. J.FluidMech. 715 ,239Ð272. D E G ENNES ,P.G.&P ROST ,J.1993 ThePhysicsofLiquidCrystals .OxfordUniversity Press. D INH ,S.M.1984ARheologicalEquationofStateforSemiconcentratedFiberSuspensions. J.Rheol. 28 (3),207. D OI ,M.&E DWARDS ,S.F.1978Dynamicsofrod-likemacromoleculesinconcentrated solution.Part2. J.Chem.Soc.,FaradayTransactions2 74 ,918. D OI ,M. AND E DWARDS ,S.F.1986 Thetheoryofpolymerdynamics .OxfordUniversity Press. E INSTEIN ,A.1905 ¬ UberdievondermolekularkinetischenTheoriederW ¬ arme geforderteBewegungvoninruhendenFl ¬ ussigkeitensuspendiertenTeilchen. AnnalenderPhysik 322 (8),549Ð560. E NNIS ,G.J.,O KAGAWA ,A.&M ASON ,S.G.1978Memoryimpairmentinßowing suspensions.II.Experimentalresults. Can.J.Chem. 56 . F ORGACS ,O.L.&M ASON ,S.G.1959ParticleMotionsinShearedSuspensions9. Spinanddeformationofthreadlikeparticles. J.ColloidInterfaceSci. 14 ,457Ð472. F RANCESCHINI ,A.,F ILIPPIDI ,E.,G UAZZELLI , « E.&P INE ,D.J.2011TransversealignmentofÞbersinaperiodicallyshearedsuspension:Anabsorbingphasetransition withaslowly-varyingcontrolparameter. Phys.Rev.Letter 107 ,1Ð5. G ALLIER ,S.2014 Simulationnumeriquedessuspensionsfrictionnelles.Applicationaux propergolssolides .PhDUniversit « edeNice. G ALLIER ,S.,L EMAIRE , « E.,P ETERS ,F.&L OBRY ,L.2014Rheologyofshearedsuspensionsofroughfrictionalparticles. J.FluidMech. 757 ,514Ð549. G OTO ,S.,N AGAZONO ,H.&K ATO ,H.1986TheßowbehaviorofÞbersuspensionsin Newtonianßuidsandpolymersolutions.I.Mechanicalproperties. Rheol.Acta 25 , 119Ð129. G UASTO ,J.S.,R OSS ,A.S.&G OLLUB ,J.P.2010Hydrodynamicirreversibilityin particlesuspensionswithnonuniformstrain. Phys.Rev.E 81 (6),061401. 151

PAGE 152

H AMPTON ,R.E.,M AMMAL ,A.A.,G RAHAM ,A.L.&T ETLOW ,N.1997Migration ofparticlesundergoingpressure-drivenßowinacircularconduit. J.Rheol. 41 (3), 621Ð640. H INCH ,E.J.2011Themeasurementofsuspensionrheology. J.FluidMech. 686 ,1Ð4. I SO ,Y.,K OCH ,D.L.&C OHEN ,C.1996OrientationinsimpleshearßowofsemidiluteÞbersuspensions1.Weaklyelasticßuids. J.Non-NewtonianFluidMech. 62 , 115Ð134. J EFFERY ,G.B.1922Themotionofellipsoidalparticlesimmersedinaviscousßuid. Proc.R.Soc.Lord.A 102 (715),161Ð179. K ARNIS ,A.,G OLDSMITH ,L.&M ASON ,S.G.1966TheKineticsofFlowingDispersions. J.ColloidandInterfaceSci. 22 ,531Ð553. K ESHTKAR ,M.,H EUZEY ,M.C.&C ARREAU ,P.J.2009RheologicalbehaviorofÞberÞlledmodelsuspensions:EffectofÞberßexibility. J.Rheol. 53 (3),631. K ESHTKAR ,M.,H EUZEY ,M.C.,C ARREAU ,P.J.,R AJABIAN ,M.&D UBOIS ,C.2010 Rheologicalpropertiesandmicrostructuralevolutionofsemi-ßexibleÞbersuspensions undershearßow. J.Rheol. 54 (2),197. K ITANO ,T.&K ATAOKA ,T.1981TherheologyofsuspensionsofvinylonÞbersin polymerliquids.I.Suspensionsinsiliconeoil. Rheol.Acta 20 ,390Ð402. K OH ,C.J.,H OOKHAM ,P.&L EAL ,L.G.1994Anexperimentalinvestigationofconcentratedsuspensionßowsinarectangularchannel. J.FluidMech. 266 ,1Ð32. K RISHNAN ,G.P.,B EIMFOHR ,S.&L EIGHTON ,D.T.1996Shear-inducedradialsegregationinbidispersesuspensions. J.FluidMech. 321 ,371Ð393. K UO ,Y.&T ANNER ,R.I.1974Ontheuseofopen-channelßowstomeasurethesecond normalstressdifference. Rheol.Acta 13 (3),443Ð456. L EAL ,L.G.2006Theslowmotionofslenderrod-likeparticlesinasecond-orderßuid. J. FluidMech. 69 (02),305. L ECAMPION ,B.&G ARAGASH ,D.I.2014ConÞnedßowofsuspensionsmodelledbya frictionalrheology. J.FluidMech. 759 ,197Ð235. L EES ,A.W.&E DWARDS ,S.F.1972Thecomputerstudyoftransportprocessesunder extremeconditions. J.Phys.C:SolidStatePhys 5 ,1921Ð1929. L EIGHTON ,D.&A CRIVOS ,A.1987Theshear-inducedmigrationofparticlesinconcentratedsuspensions. J.FluidMech. 181 ,415Ð439. L HUILLIER ,D.2009Migrationofrigidparticlesinnon-Brownianviscoussuspensions. Phys.Fluids 21 (2),023302. 152

PAGE 153

L UNDELL ,F.,S ¬ ODERBERG ,L.D.&A LFREDSSON ,P.H.2011FluidMechanicsof Papermaking. Annu.Rev.FluidMech. 43 (1),195Ð217. L YON ,M.K.&L EAL ,L.G.1998Anexperimentalstudyofthemotionofconcentrated suspensionsintwo-dimensionalchannelßow.Part1.Monodispersesystems. J.Fluid Mech. 363 ,25Ð56. M ACKAPLOW ,M.B.&S HAQFEH ,E.S.G.1996Anumericalstudyoftherheological propertiesofsuspensionsofrigid,non-BrownianÞbres. J.FluidMech. 329 ,155Ð186. M ASON ,S.G.&M ANLEY ,R.S.J.1956ParticleMotionsinShearedSuspensions: OrientationsandInteractionsofRigidRods. Proc.R.Soc.Lond.A 238 (1212), 117Ð131. M ETZGER ,B.&B UTLER ,J.E.2012Cloudsofparticlesinaperiodicshearßow. Phys. Fluids 24 (2),021703. M ILLER ,R.M.&M ORRIS ,J.F.2006Normalstress-drivenmigrationandaxialdevelopmentinpressure-drivenßowofconcentratedsuspensions. J.Non-NewtonianFluid Mech. 135 (2-3),149Ð165. M ORRIS ,J.F.2001Anomalousmigrationinsimulatedoscillatorypressure-drivenßowof aconcentratedsuspension. Phys.Fluids 13 (9),2457. M ORRIS ,J.F.&B OULAY ,F.1999Curvilinearßowsofnoncolloidalsuspensions:The roleofnormalstresses. J.Rheol. 43 (5),1213. N ORMAN ,J.T.,N AYAK ,H.V.&B ONNECAZE ,R.T.2005Migrationofbuoyantparticlesin low-Reynolds-numberpressure-drivenßows. J.FluidMech. 523 ,1Ð35. N OTT ,P.R.&B RADY ,J.F.1994Pressure-drivenßowofsuspensions:simulationand theory. J.FluidMech. 275 ,157Ð199. N OTT ,P.R.,G UAZZELLI , « E.&P OULIQUEN ,O.2011Thesuspensionbalancemodel revisited. Phys.Fluids 23 (4),043304. O KAGAWA ,A.,E NNIS ,G.J.&M ASON ,S.G.1978Memoryimpairmentinßowing suspensions.I.Sometheoreticalconsiderations. Can.J.Chem. 56 ,2815Ð2823. O KAGAWA ,A.&M ASON ,S.G.1973Suspensions:FluidswithFadingMemories. Science 181 (4095),159Ð161. P APATHANASIOU ,T.D.&G UELL ,D.C.1997 Flow-InducedAlignmentinComposite Materials .WoodhousePublishingLtd. P ETRICH ,M.P.,C HAOUCHE ,M.,K OCH ,D.L.&C OHEN ,C.2000 a Oscillatoryshear alignmentofanon-BrownianÞberinaweaklyelasticßuid. J.Non-NewtonianFluid Mech. 91 ,1Ð14. 153

PAGE 154

P ETRICH ,M.P.&K OCH ,D.L.1998InteractionsbetweencontactingÞbers. Phys. Fluids 10 (8),2111. P ETRICH ,M.P.,K OCH ,D.L.&C OHEN ,C.2000 b Anexperimentaldeterminationof thestressÐmicrostructurerelationshipinsemi-concentratedÞbersuspensions. J. Non-NewtonianFluidMech. 95 ,101Ð133. P ETRIE ,C.J.S.1999TherheologyofÞbresuspensions. J.Non-NewtonianFluidMech. 87 ,369Ð402. P HILLIPS ,R ONALD J,A RMSTRONG ,R OBERT C,B ROWN ,R OBERT A,G RAHAM ,A LAN L &A BBOTT ,J AMES R1992Aconstitutiveequationforconcentratedsuspensionsthat accountsforshearinducedparticlemigration. Phys.FluidsA 4 (1),30Ð40. P INE ,D.J.,G OLLUB ,J.P.,B RADY ,J.F.&L ESHANSKY ,A.M.2005Chaosand thresholdforirreversibilityinshearedsuspensions. Nature 438 (7070),997Ð1000. R ICHARDSON ,J.F.&Z AKI ,W.N.1954Sedimentationandßuidization:PartI. Trans. Inst.Chem.Eng. 32 ,35Ð53. S ALAHUDDIN ,A.,W U ,J.&A IDUN ,C.K.2013StudyofsemidiluteÞbresuspension rheologywithlattice-Boltzmannmethod. Rheol.Acta 52 (10-12),891Ð902. S EPEHR ,M.,C ARREAU ,P.J.,M OAN ,M.&A USIAS ,G.2004Rheologicalpropertiesof shortÞbermodelsuspensions. J.Rheol. 48 (5),1023Ð1048. S HAQFEH ,E.S.G.&F REDRICKSON ,G.H.1990Thehydrodynamicstressina suspensionofrods. Phys.FluidsA 2 (1),7Ð24. S HAULY ,A.,A VERBAKH ,A.,N IR ,A.&S EMIAT ,R.1997Slowviscousßowsofhighly concentratedsuspensions.2.particlemigration,velocityandconcentrationproÞlesin rectangular. Intl.J.MultiphaseFlow 23 ,613Ð629. S IEROU ,A.&B RADY ,J.F.2002Rheologyandmicrostructureinconcentratednoncolloidalsuspensions. J.Rheol. 46 (5),1031. S INGH ,A.&N OTT ,P.R.2000Normalstressesandmicrostructureinboundedsheared suspensionsviaStokesianDynamicssimulations. J.FluidMech. 412 ,279Ð301. S INGH ,A.&N OTT ,P.R.2003Experimentalmeasurementsofthenormalstressesin shearedStokesiansuspensions. J.FluidMech. 490 ,293Ð320. S NOOK ,B.,G UAZZELLI , « E.&B UTLER ,J.E.2012VorticityalignmentofrigidÞbersinan oscillatoryshearßow:RoleofconÞnement. Phys.Fluids 24 (121702),1Ð7. S TICKEL ,J.J.&P OWELL ,R.L.2005Fluidmechanicsandrheologyofdensesuspensions. Annu.Rev.FluidMech. 37 (1),129Ð149. S TOVER ,C.A.,K OCH ,D.L.&C OHEN ,C.1992ObservationsofÞbreorientationin simpleshear. J.FluidMech. 238 ,277Ð296. 154

PAGE 155

S UNDARARAJAKUMAR ,R.R.&K OCH ,D.L.1997Structureandpropertiesofsheared Þbersuspensionswithmechanicalcontacts. J.Non-NewtonianFluidMech. 73 , 205Ð239. T ANNER ,R.I.1970SomeMethodsforEstimatingtheNormalStressFunctionsin ViscometricFlows. J.Rheol. 14 (4),483. T REVELYAN ,B.J.&M ASON ,S.G.1951Particlemotionsinshearedsuspensions.I. Rotations. J.ColloidInterfaceSci. 6 ,354Ð367. W INEMAN ,A.S.&P IPKIN ,A.C.1966Slowviscoelasticßowintiltedtroughs. Acta Mechanica 2 (1),104Ð115. W U ,J.&A IDUN ,C.K.2010AnumericalstudyoftheeffectofÞbrestiffnessonthe rheologyofshearedßexibleÞbresuspensions. J.FluidMech. 662 ,123Ð133. Y EO ,K.&M AXEY ,M.R.2011Numericalsimulationsofconcentratedsuspensionsof monodisperseparticlesinaPoiseuilleßow. J.FluidMech. 682 ,491Ð518. Z ARRAGA ,I.E.,H ILL ,D.A.&L EIGHTON ,D.T.J R .2000Thecharacterizationofthe totalstressofconcentratedsuspensionofnoncolloidalspheresinNewtonianßuids. J. Rheol. 2 ,185Ð220. 155

PAGE 156

BIOGRAPHICALSKETCH BradenSnookwasborninValparaiso,IN.Hismotherwasanelementaryspecial educationteacherandhisfatherwasaschoolbusdriverandgeneralcontractor.He attendedMorganTownshipHighSchoolwhereheplayedvarsityvolleyball,basketball andtrack.Hegraduatedvaledictorianandpresidentofhisclass. BradenattendedPurdueUniversity,majoringinchemicalengineering.Hisfavorite coursesatPurdueUniversitywererelatedtoßuidmechanics,leadinghimtopursue advancedcoursesintransportphenomenaaswellasdoingresearchwithDr.Stephen BeaudoinandDr.MichaelHarris.BradenalsowasaCo-OpstudentwithKimberly-Clark duringhisundergraduatestudies. Upongraduation,BradencontinuedhiseducationattheUniversityofFlorida inchemicalengineeringandatAix-MarseilleUniversityinphysicsofßuids.Hewas coadvisedbyDr.JasonE.ButlerattheUniversityofFloridaandDr. « ElisabethGuazzelli atAix-MarseilleUniversity.HecarriedoutnumericalworkattheUniversityofFlorida andperformedexperimentsatAix-MarseilleUniversity.HereceivedadualPhDinthese disciplinesin2015andislookingforwardtocontinuingworkinßuidmechanicsintheoil andgasindustry. 156