A Numerical Study of Viscoelastic Flow Through an Array of ... · overview of viscoelastic constitutive models is presented, including stresses caused by these types of fluids. Finally,

ANumericalStudyofViscoelasticFlowThroughanArrayofCylinders

by

YuenPhilipHoang

AthesissubmittedinconformitywiththerequirementsforthedegreeofMasterofAppliedScience

DepartmentofMechanicalandIndustrialEngineeringUniversityofToronto

©CopyrightbyYuenPhilipHoang2018

ii

ANumericalStudyofViscoelasticFlowThroughanArrayofCylinders

YuenPhilipHoang

MasterofAppliedScience

DepartmentofMechanicalandIndustrialEngineeringUniversityofToronto

2018

Abstract

Thisthesisisastudyoncreepingflowofanidealviscoelasticfluidthroughsquarearraysof

cylinderstopredictthepressuredrop.Numericalsimulationswerecompletedforarraysof

threedifferentsolidvolumefractions:2.5%,5%,and10%.Substantialamountsofelastic

stresseswerefoundbeyondacriticalflowrate,uptosixtimesthatofthehighestNewtonian

stresses.Anincreaseinpressuredropcausedbyelasticitywasfound,incontrasttomanyother

numericalstudieswhichfindadecreaseornochange.Thispressuredropwas,however,

considerablysmallerthanwhatwasfoundexperimentallybyJames,Yip,&Currie(2012).The

absenceofelasticextensionalstressesdownstreamofthecylinderforthe10%arraysupports

theargumentthattheincreaseinpressuredropiscausedbyelasticstressesduetoshear.

iii

TableofContentsTableofContents..........................................................................................................................iii

ListofTables...................................................................................................................................v

ListofFigures................................................................................................................................vi

ListofAppendices.......................................................................................................................viii

Chapter1 Introduction............................................................................................................1

Chapter2 Background.............................................................................................................4

2.1 WhatIsViscoelasticity................................................................................................................4

2.2 PolymerSolutions.......................................................................................................................5

2.3 TheMaxwellModel....................................................................................................................6

2.4 TheOldroyd-BModel..................................................................................................................8

2.5 TheOldroyd-BModelinShearFlow.........................................................................................10

2.6 TheOldroyd-BModelinExtensionalFlow................................................................................12

2.7 TheDeborahNumberandWeissenbergNumber....................................................................13

2.8 ExperimentalLiteratureReview................................................................................................14

2.9 NumericalLiteratureReview....................................................................................................17

2.10 Objectives.................................................................................................................................19

Chapter3 Methodology........................................................................................................20

3.1 Geometry..................................................................................................................................20

3.2 MeshSetup...............................................................................................................................21

3.3 BoundaryConditions.................................................................................................................23

3.4 NewtonianSimulation..............................................................................................................24

3.5 SimulatingElasticStresses........................................................................................................26

3.6 ElasticProperties......................................................................................................................27

3.7 SummaryofAssumptions.........................................................................................................29

Chapter4 ResultsandDiscussion..........................................................................................30

4.1 NewtonianFlowPlots...............................................................................................................30

4.2 ElasticStressPlots.....................................................................................................................33

4.3 ElasticStressesNeartheCylinder.............................................................................................39

iv

4.4 MeshConvergenceAnalysis.....................................................................................................40

4.5 ForceActingontheCylinderandPressureDrop......................................................................43

4.6 ComparisonswithOtherStudies..............................................................................................45

4.7 Discussion.................................................................................................................................47

Chapter5 Conclusion............................................................................................................49

References....................................................................................................................................50

v

ListofTablesTable1.Permeabilityandflowresistanceofaperiodicarrayofcylindersbasedonthesolution

bySangani&Acrivos(1982).................................................................................................25

vi

ListofFiguresFigure1.1.Periodicsquarearrayofcylinders(𝜙=5%)...................................................................2

Figure2.1.Polymerdeformationinshearandextension..............................................................5

Figure2.2.The1-dimensionalupper-convectedMaxwellmodel..................................................6

Figure2.3.The1-dimensionalOldroyd-Bmodel............................................................................9

Figure2.4.NormalizedflowresistancereproducedfromJamesetal.(2012).............................16

Figure3.1.Unitcelltorepresentaperiodicarrayofcylinders....................................................20

Figure3.2.Meshingregions.RegionAusesamappedfacemesh.RegionBusesaquadrilateral

dominantmesh.RegionB*isthesameasregionB,exceptforthe2.5%solidvolume

fractiongeometrywhereitislocallyrefined.......................................................................21

Figure3.3.5%arraycoarsestmesh(23,664elements)...............................................................22

Figure3.4.5%arrayclose-upofcoarsestmesh(23,664elements)aroundthetoppole............23

Figure3.5.Predictedsimulationflowresistancevs.ReynoldsnumberoftheNewtonianflow

field......................................................................................................................................26

Figure3.6.FirstnormalstresscoefficientofB2fluidadaptedfromYip(2011)...........................28

Figure4.1.NormalizedvelocitymagnitudesandpressurecontoursoftheNewtonianflowfield.

Blacklinesonthenormalizedvelocitycontoursrepresentstreamlines,whiletheright-hand

sideofthepressurecontoursareuniformly0.....................................................................31

Figure4.2.Newtonianstressesinaflowthroughasquarearrayofcylinders.Thesestresses

werenormalizedusingthemaximumstressintheflowfield,whichistheshearstressat

thetopandbottompolesofthecylinder............................................................................32

Figure4.3.𝛥𝜏𝑥𝑥stressesnormalizedbythemaximumNewtonianshearstress.Magnitudes

higherthan1.0aretruncated..............................................................................................34

Figure4.4.𝛥𝜏𝑥𝑥stressesnormalizedbythemaximumNewtonianshearstress........................35

Figure4.5.𝛥𝜏𝑥𝑦stressesnormalizedbythemaximumshearstress..........................................37

Figure4.6.𝛥𝜏𝑦𝑦stressesnormalizedbythemaximumshearstress..........................................38

Figure4.7.Additionalelasticstressesincylindricalcoordinatesatthecylindersurface.Notethat

𝛥𝜏𝑟𝜃overlapswith𝛥𝜏𝑟𝑟......................................................................................................39

Figure4.8.Stressesactingonasurfaceelementincylindricalcoordinates...............................40

vii

Figure4.9.Meshconvergenceanalysisof2.5%solidvolumefractionatpoint(0.3L,0).(*)

indicatesthatthedownstreamregionwaslocallyrefinedratherthantheentireflow

domain.................................................................................................................................41

Figure4.10.Meshconvergenceanalysisof5%solidvolumefractionatpoint(0.3L,0)...............42

Figure4.11.Meshconvergenceanalysisof10%solidvolumefractionatpoint(0.3L,0).............42

Figure4.12.Stressesonasurfaceelement..................................................................................43

Figure4.13.Normalizedpredictedflowresistancevs.Deborahnumber....................................45

Figure4.14.ComparisonofflowresistanceresultswithexperimentalresultsbyJamesetal.

(2012)...................................................................................................................................46

Figure4.15.Comparisonofflowresistanceresultstonumericalresultsfrom(Hemingwayetal.,

2018).(*)indicatesthattheDeborahnumberiscalculatedusing𝜆=0.6sforthe

comparison...........................................................................................................................47

FigureC.1:Meshconvergenceanalysisof2.5%solidvolumefractionatthecylinderpole........61

FigureC.2:Meshconvergenceanalysisof5%solidvolumefractionatthecylinderpole...........61

FigureC.3:Meshconvergenceanalysisof10%solidvolumefractionatthecylinderpole.........62

FigureD.1.2.5%arrayvelocityprofilesbetweenparallelcylinders(x/L=0).Particleimage

velocimetryresultsareadaptedfromYip(2011).................................................................63

FigureD.2.2.5%arrayvelocityprofilesattheinletboundary(x/L=-0.5).Particleimage

velocimetryresultsareadaptedfromYip(2011).................................................................64

FigureD.3.5%arrayvelocityprofilesbetweenparallelcylinders(x/L=0).Particleimage

velocimetryresultsareadaptedfromYip(2011).................................................................64

FigureD.4.5%arrayvelocityprofilesattheinletboundary(x/L=-0.5).Particleimage

velocimetryresultsareadaptedfromYip(2011).................................................................65

FigureD.5.10%arrayvelocityprofilesbetweenparallelcylinders(x/L=0).Particleimage

velocimetryresultsareadaptedfromYip(2011).................................................................65

FigureD.6.10%arrayvelocityprofilesattheinletboundary(x/L=-0.5).Particleimage

velocimetryresultsareadaptedfromYip(2011).................................................................66

viii

ListofAppendicesAppendixA:ExpandedUpper-ConvectedMaxwellandOldroyd-BEquations.............................53

AppendixB:FLUENTUserDefinedFunction................................................................................54

AppendixC:MeshConvergenceAnalysisatCylinderPoles.........................................................61

AppendixD:ComparisonofNewtonianVelocityProfilestoParticleImageVelocimetryResults

fromYip(2011).............................................................................................................................63

1

Chapter1 IntroductionFlowsthroughporousmediawereinitiallystudiedbyHenryDarcy,whoinvestigatedtheflowof

waterthroughsand.Fromstudyingtheseflows,hefoundalinearrelationshipbetweenthe

flowrateandthepressuredropforNewtonianfluids,whichbecameDarcy’slaw,

𝑄 =𝐾𝐴𝜂𝛥𝑝𝑙 (1.1)

whereQistheflowrate,Δpisthepressuredrop,Aisthecross-sectionalarea,listhelength

throughtheporousmediumoverwhichthepressuredropistakingplace,𝜂istheviscosityof

thefluid,andKisaproportionalityconstantbasedonthegeometryofthemedium,knownas

theintrinsicpermeability.

Darcy’slawisthefoundationforstudyingaquifersandgroundwaterflows.Thisrelationshipis

validforaReynoldsnumberuptoabout10,andforporousmediaofanyparticles,andnotjust

forgrainsofsand.

Thestudyoffluidsflowingthroughabedoffibresisofinterestinengineeringbecauseof

applicationssuchasresintransfermoldinginmanufacturingandwaterproofingbreathable

fabricsbycoatingthefabricwithpolymericsubstances.Thefluidsusedintheseapplicationsare

polymericliquidswhichexhibitbothelasticandviscousbehaviour,thusthenecessitytostudy

theviscoelasticfluidsthroughfibrousporousmedia.

Thesefibrousbedsofmaterialscanbemodelledbyanarrayofregularlyspacedcylinderscalled

asquarearrayofcylinders(Figure1.1).

2

Figure1.1.Periodicsquarearrayofcylinders(𝝓=5%)

Flowsthroughthisidealizedgeometryhavebeenextensivelystudiedbecausethisgeometryis

well-defined,two-dimensional,andperiodic,allowingforcomparisonsbetweenstudies.The

onlypropertywhichdefinesanarrayisthesolidvolumefraction,𝜙,whichisthefractionof

solidvolumetothetotalvolume.Forasquarearray,thesolidvolumefractionis,

𝜙 =

𝜋4𝐷7

𝐿7 (1.2)

whereDisthediameterofacylinderandListhedistancebetweenneighbouringcylinders,as

illustratedinFigure1.1.

Theflowthroughasquarearrayisknownasamixedflow,meaningthatitisacombinationof

shearandextension,asopposedtopureshearorpureextension.Theflowfieldismainly

dominatedbyshear,particularlyaroundthepolesofthecylinder.Thecylindersurfaceisa

regionofpureshear,whilethereareregionsofpureextensiondirectlydownstreamofthe

cylinderandatalineparalleltothisregionhalfaunitcelllengthupordown.Theseregionswill

stretchthepolymersdifferently,asitwillbeexplainedlaterinbackground.

3

Theviscoelasticflowthroughthisporousmediumhasbeenshownexperimentallytoexhibitan

increaseinpressuredropcausedbytheelasticityofthefluid(Chmielewski,Nichols,&

Jayaraman,1990;Jamesetal.,2012;Khomami&Moreno,1997;Skartsis,Khomami,&Kardos,

1992).Replicatingthisincreasedpressuredropeffectwithnumericalsimulationshassofar

eludedthescientificcommunity.Infact,mostnumericalstudiesactuallypredictamarginal

reductionratherthanadefiniteincreaseinpressuredrop.

Thereasonforthisenhancedpressuredropeffecthasnotbeenestablished,whetheritis

causedbyelasticeffectsduetoshear,toextension,ortoflowinstabilities.Forexample,James

(2016)andYip(2011)havemadeargumentsthattheelasticeffectsareduetoshear,while

Chmielewski&Jayaraman(1993),Khomami&Moreno(1997),andLiu,Wang,&Hwang(2017),

havearguedthattheeffectsarecausedbyextensionortheformationofflowinstabilities.By

numericallysimulatingtheflowofviscoelasticfluidthroughasquarearrayofcylinders,Ihope

toshedsomelightonthisdebate.

4

Chapter2 BackgroundThischapterfirstgivesabriefintroductiontoviscoelasticityandpolymersolutions.Thenan

overviewofviscoelasticconstitutivemodelsispresented,includingstressescausedbythese

typesoffluids.Finally,areviewofrelatedexperimentalandnumericalresearchonviscoelastic

flowthroughporousmediaispresentedlaterinthischapter,proceedingtotheprimary

researchobjectives.

2.1 WhatIsViscoelasticityNewtonianfluidsexhibitalinearrelationshipbetweenshearstressandshearrate,orinother

words,aviscosityindependentofshearrate.Mosttextbooks,therefore,definenon-Newtonian

fluidsasfluidsthathaveashearratedependentviscosity,fluidsthatareshear-thinning,shear

thickeningfluids,orhaveayieldbehaviour(Cengel&Cimbala,2013;Potter&Wigger,2010;

White,1994).TheshearstressinaNewtonianfluidunderpureshearconditionsiscalculatedas

follows,where𝜏9:istheshearstress,𝜂isviscosity,anduisthevelocityinthexdirection.

𝜏9: = 𝜂𝑑𝑢𝑑𝑦

(2.1)

Thereis,however,anothercategoryofnon-Newtonianfluidsareviscoelasticfluids.Although

theseviscoelasticfluidstendtobeshear-thinning,theyarealsoelastic,whichischaracterized

bytheirstringiness,wherebytheymayformlongfilamentswhichpersist.Examplesofthese

fluidsincludepolymermelts,eggwhites,andsaliva.

Viscoelasticfluidscanexhibitsomepeculiarphenomenanotobservedbyothertypesoffluids

suchastheopenchannelsiphon(James,1966)androdclimbing(Barnes,Hutton,&Walters,

1989).Theopensiphoneffectisaphenomenonwhereavesselcontainingaviscoelasticfluidis

tippedsothatfluidstartsflowoveritsedge.Ifthevesselisthenstraightened,thefluid

continuesclimbingupthesideandoutofthevessellikeasiphonwithoutatube.Therod

5

climbingeffectisexhibitedwhenarotatingrodisinsertedinapoolofviscoelasticfluid.Instead

ofthefluidbeingpushedoutwardfromcentrifugalforces,thefluidclimbsuptherod.

Thefluidswhichexhibitthesephenomenaaresolutionsormeltsoflong-chainedpolymer

molecules.Theselongchainscanstretchandbecomeentangledwitheachother.Inthecaseof

theopensiphoneffect,theentangledpolymerspulleachotherupandoutofthevessel.Inrod

climbing,thepolymerchainsstretchlikeanelasticband,tighteningaroundtherod.This

createsaregionofhigherpressurearoundtherod,whichpushesthefluiduptherod.

2.2 PolymerSolutionsThepolymermolecules,attheirreststate,arerandomlycoiledwithoutapreferreddirection.

Underanapplieddeformationorstrain,however,thesecoilsstarttounravelandaligninthe

directionofthestrain,whichcanbecausedbyeithershearorextension(Figure2.1).

Figure2.1.Polymerdeformationinshearandextension

Asthesepolymercoilsstretch,theBrownianmotionofsurroundingmoleculesinthesolution

randomlybombardthepolymer,causingittoreturntoitsunalignedreststate.Thisrecoil

makesthepolymermoleculesactlikemicroscopicspringsinthesolution.Thecombinedeffect

ofmillionsofthesemicroscopicspringsgivethesepolymersolutionstheircharacteristic

“stringiness”orelasticity.

6

Mostpolymersolutions,however,areshear-thinning,makingitdifficulttoisolatetheeffectsof

elasticityfromthosecausedbyshear-thinning.Fortunately,aclassoffluidscalledBogerfluids

exhibithighelasticitywithlittleshear-thinning,makingthemidealforstudyingtheeffectsof

elasticity.ABogerfluidisadilutesolutionofhighmolecularweightpolymergenerallydissolved

inahighlyviscoussolvent.Bycomparingtheresultsofaviscoelasticflowwiththoseofa

NewtonianflowfieldofthesameReynoldsnumber,itisthenpossibletoseparateviscousand

elasticeffects,allowingpurelyelasticeffectscanbeidentifiedfromthedifference.

2.3 TheMaxwellModelPolymermoleculescreateextrastresseswhenthefluidisbeingdeformedorstretchedunder

shearorextension.Thisbehaviourcanbeunderstoodusingaspringanddampersystemcalled,

theMaxwellmodel(Figure2.2).

Figure2.2.The1-dimensionalupper-convectedMaxwellmodel

TheMaxwellmodelisthesimplestviscoelasticmodel,inwhichmostothermodelsarebased

on.Thestressactingonthespringanddampersystem,𝜏,canbederivedasfollowstoobtain

thefollowingdifferentialequation.Thederivationstartsbydividingwithstrainrateofthe

system,𝛾,intoanelasticcomponent,𝛾>,andaviscouscomponent,𝛾?,

𝛾 = 𝛾> + 𝛾?. (2.2)

Thenrelatingthesestrainratestothestress,𝜏,usingthedamperconstant,𝜂,andthespring

constant,G,

𝛾 =𝜏𝐺+𝜏𝜂, (2.3)

BeforefinallyrearrangingtheequationtoobtaintheequationfortheMaxwellmodel

7

𝜏 + 𝜆𝜏 = 𝜂𝛾,𝜆 =𝜂𝐺. (2.4)

TohelpillustratethebehaviouroftheMaxwellmodel,thestresscalculatedwhenasudden

increaseinshearrate,𝛾,isappliedfromrestis,

𝜏 = 𝜂𝛾 1 − 𝑒GHI . (2.5)

Equation2.5showsthatthestressincreasesasymptoticallytoitssteadystatevalues.Thestress

ofafluidelementdependsnotonlyonthecurrentstrainrate,butonpaststrainratesaswell.

Inotherwords,thefluidhasa‘memory’ofpaststressesandstrains.Therateatwhichthe

stressreachesitssteady-statevaluedependsonthevalueknownastherelaxationtime,𝜆,

whichisameasureoftheelasticityofthefluid.Incontrast,aNewtonianfluidwould

instantaneousreacttothesuddenincreaseinshearrate.

Iftheviscoelasticfluidissubjectedtoasmallamplitudeoscillatoryshearingoftheform,

𝛾 = 𝛾J cos 𝜔𝑡 , (2.6)

where𝛾istheappliedstrain,𝜔istheoscillatoryrate,𝛾Jisthestrainamplitude,andtistime

sincestartingthetest,theresponseoftheMaxwellmodelfromthisoscillatoryshearinputis

𝜏 =𝜂𝜔𝛾J

1 + 𝜔7 𝜆 7 𝜔𝜆 cos 𝜔𝑡 − sin 𝜔𝑡 . (2.7)

Thisresponsecanbeseparatedintoanin-phasecomponentcalledthestoragemodulus,G’,

andanout-of-phasecomponentcalledthelossmodulus,G”,asfollows,

𝜏/𝛾J = 𝐺′ cos 𝜔𝑡 − 𝐺"𝑠𝑖𝑛(𝜔𝑡),

where 𝐺′ = 𝜂𝜆𝜔7

1 + 𝜔7𝜆7

(2.8)

and 𝐺" =𝜂𝜔

1 + 𝜔7𝜆7. (2.9)

ThestoragemodulusisrelatedtotheelasticityofthefluidwhileG”isrelatedtotheviscous

responseofthematerial.Purelyelasticmaterials,suchasmostmetals,havealossmodulusof

zeroandpurelyviscousmaterialswouldhaveazerostoragemodulus.

8

Inafluidflow,thesespringsanddumbbellsarerotatedandstretched.Mathematically,thisis

handledusingtheupper-convectedtimederivative.Thismodelisknownastheupper-

convectedMaxwell(UCM)model:

𝝉 + 𝜆[𝝉∇= 𝜂J𝜸. (2.10)

where𝝉representsthestresstensor,𝜸representsthestrainratetensor,𝜂representsthe

viscosity,andthe𝛻accentrepresentstheupper-convectedtimederivative,whichisasfollows,

𝝉∇=𝐷𝝉𝐷𝑡

− 𝛻𝑢 _ ⋅ 𝝉 + 𝝉 ⋅ 𝛻𝑢 =𝜕𝝉𝜕𝑡+ 𝑢 ⋅ 𝛻𝝉 − 𝝉 ⋅ 𝛻𝑢 _ − 𝝉 ⋅ 𝛻𝑢 , (2.11)

wheretheTsuperscriptisthetransposeofthetensor,𝑢isthevelocityvector,and bbcisthe

materialtimederivative.

AlthoughtherearenofluidsthatbehaveexactlylikeaMaxwellfluid,thetheMaxwellmodelis

thefoundationforothermoresophisticatedmodelssuchastheOldroyd-Bmodel.

2.4 TheOldroyd-BModelTheOldroyd-Bmodel,sometimesalsoknownastheJeffreysmodel,isanextensiontothe

upper-convectedMaxwellmodel.ThismodelcombinesaNewtoniansolventwiththeupper-

convectedMaxwellmodelrepresentingthepolymer.TheOldroyd-Bmodelisthesimplest

modelthatcansimulatethebehaviourtheclassofviscoelasticfluidcalledBogerfluids

introducedearlier(James,2009;Prilutskietal.,1983).Forthisreason,theOldroyd-Bmodelwas

chosenastheconstitutivemodeltofindthestressesinthefluidinthistheflowofapolymer

solutionthroughasquarearrayofcylinders.

9

Figure2.3.The1-dimensionalOldroyd-Bmodel

Theone-dimensionalOldroyd-Bmodelcanberepresentedasastheupper-convectedMaxwell

modelwithanadditionalparalleldamper,shownasthemechanicalsysteminFigure2.3.Inthe

figure,thisparalleldamper,𝜂d,representsthestresscontributedbytheNewtoniansolvent

whilethespring,G,anddamper,𝜂e,combinationinseriesrepresentsthecontributionfrom

thepolymer.Thestressofthecombinedsystemcanbecalculatedasfollowstogivethe

Oldroyd-Bequationbysummingupthesolvent(𝜏d)andpolymer(𝜏e)stressestogether:

𝜏 = 𝜏e + 𝜏d. (2.12)

Thestressesinthesystemcanberelatedwiththestrainrateofthesystem.

𝛾 =

𝜏e𝐺+𝜏e𝜂e=𝜏d𝜂d (2.13)

ThenrearrangingtheaboveequationtosolvetheUCMpolymerstresscomponent,𝜏e,andthe

Newtoniansolventstresscomponent,𝜏d,

𝜏e = 𝜂e𝛾 −𝜂e𝜏e𝐺

(2.14)

𝜏d = 𝜂d𝛾 (2.15)

𝜏eisintermsof𝜏e,sotorelate𝜏ewiththetotalstressofthesystem,𝜏,itcanbeshownthat

𝜏e = 𝜏 − 𝜏d = 𝜏 − 𝜂d𝛾.

Nowsubstituting𝜏ewith𝜏 − 𝜂d𝛾inEquation2.14,

𝜏e = 𝜂e𝛾 −𝜂e𝜏𝐺+𝜂e𝜂d𝛾𝐺

. (2.16)

10

CombiningEquations2.12,2.15,and2.16andsolvingtheequationtobeintermsof𝜏and𝛾

yields,

𝜏 +𝜂e𝜏𝐺

= 𝜂e𝛾 +𝜂e𝜂d𝛾𝐺

+ 𝜂d𝛾. (2.17)

Simplifyingtheaboveequationwiththefollowingconstants,

𝜆 =𝜂e𝐺,𝜂 = 𝜂d + 𝜂e, 𝜆7 =

𝜂d𝜂𝜆, (2.18)

producestheone-dimensionalformoftheOldroyd-Bequation

𝜏 + 𝜆𝜏 = 𝜂(𝛾 + 𝜆7𝛾). (2.19)

Again,theupper-convectedtimederivativecanbeusedtomodelhowthestressesare

convectedthroughoutaflowfield:

𝝉 + 𝜆𝝉∇= 𝜂 𝜸 + 𝜆7𝜸

∇. (2.20)

Expandedequationsoftheupper-convectedMaxwellandOldroyd-BmodelsinCartesian

coordinatesareprovidedinAppendixA.

TounderstandthebehaviouroftheOldroyd-Bmodel,itisinstructivetoobservehowthemodel

actsunderpureshearandpureextensional,forwhichtherearesimpleanalyticalsolutions.The

followingsectionsoutlinethesesolutions.

2.5 TheOldroyd-BModelinShearFlowUnderaconstantshearflow,thepolymersarestretchedintheshearingdirection,x,asshown

inFigure2.1.Thesepolymersstretchinthex-direction,creatingintwo-dimensions𝜏99and𝜏::

stressesthatwouldotherwisenotexistinNewtonianfluids.Thesestressescannotbemeasured

individually,sotheelasticstressesarenormallyrepresentedbythefirstnormalstress

difference(𝑁[),definedas𝜏99 − 𝜏::.FortheOldroyd-Bmodelinsteadyshear,thefirstnormal

stressdifferenceisfoundtobe,

𝑁[ = 2𝜂e𝜆𝛾7, (2.21)

where𝛾istheshearrate.N1isoftentimesusedtorefertotheelasticstressesduetoshear,and

willbereferredtoassuchthroughoutthepaper.

11

UnderanoscillatoryshearflowwiththeOldroyd-Bmodel,G’andG”arefoundasfollowsusing

ananalysissimilartothatinsection2.3:

𝐺′ =

𝜂e𝜆𝜔7

1 + 𝜔7𝜆7 (2.22)

𝐺" =𝜂e𝜔

1 + 𝜔7𝜆7+ 𝜂d𝜔. (2.23)

Oneoftheusefulresultsoftheoscillatorysheartestistheabilitytoobtaintheviscosity

contributionofthesolvent,𝜂d,andthusthepolymerviscosity,𝜂e.Thesolventviscositymaybe

obtainedbymeasuringhighfrequencyG”plateauandthencalculatingthesolventviscosityas

follows(Prilutskietal.,1983):

𝜂d = limj→l

𝐺"𝜔 (2.24)

Thepolymerviscositycanthenbecalculatedbysubtractingthesolventviscosityfromthetotal

viscosity:

𝜂e = 𝜂 − 𝜂d (2.25)

TherelaxationtimecanthenbecalculatedusingthelowfrequencyG’plateauasfollowed:

𝜆 = lim

j→J

𝐺′𝜔7𝜂e

(2.26)

Additionally,insteadyshearflow,theOldroyd-Bmodel(2.20)predictsthat

𝜏9: = 𝜂d + 𝜂e 𝛾9:, (2.27)

thusconfirmingthattheviscosityofthefluidissimplythesumofthesolventandpolymer

viscosities.ThisalsomeansthattheOldroyd-Bmodelassumesnegligibleshear-thinningeffects,

areasonableassumptionforBogerfluids.Additionally,ifsolventviscosityisnegligible,thenthe

Oldroyd-Bmodelturnsbackintotheupper-convectedMaxwellmodel.However,inaBoger

fluid,𝜂e < 𝜂d,oreven𝜂e ≪ 𝜂d.

12

2.6 TheOldroyd-BModelinExtensionalFlowItmaybefurtherinstructivetoexaminewhathappenswiththeOldroyd-Bmodelinextensional

flow.Insuchaflow,thereisanelongationbutnoshearing.Inasteadyplanarextensionalflow

fieldwithaconstantextensionalrateinthex-direction,𝜖,

𝜖 =𝜕𝑢𝜕𝑥

(2.28)

andfromcontinuity,

𝜖 = −𝜕𝑣𝜕𝑦

(2.29)

Theonlywaytosatisfytheaboveequationsis

𝑢 = 𝜖𝑥, 𝑣 = −𝜖𝑦, (2.30)

whereuisthefluidvelocityinthex-directionandvisthefluidvelocityinthey-direction.

Usingthisextensionalrate,theOldroyd-Bmodel(2.20)givesforstretchinginthex-direction

(Bird,Armstrong,&Hassager,1987),

𝜏99 + 𝜆𝑑𝜏99𝑑𝑡

− 2𝜆𝜏99𝜖 = 𝜂 2𝜖 1 − 2𝜆7𝜖 + 2𝜆7𝑑𝜖𝑑𝑥

, (2.31)

yieldingthestress

𝜏99 = 2𝜂𝜖

1 − 2𝜆7𝜖1 − 2𝜆𝜖

+ 2𝜂𝜖 1 −𝜆7𝜆

𝑒Gc[G7qrq

1 − 2𝜆𝜖𝑤ℎ𝑒𝑛𝜆𝜖 ≠

12 (2.32)

and

𝜏99 =𝜂𝜆2𝜖 − 4𝜆7 𝜖 7 𝑡 + 2𝜂d𝜖𝑤ℎ𝑒𝑛𝜆𝜖 =

12 (2.33)

Notably,when𝜆𝜖 ≥ 1/2,𝜏99willcontinuouslyincreasewithnosteady-statevalue.Infact,the

stressgrowsexponentiallywhen𝜆𝜖 > 1/2becauseofthe𝜆𝜖termintheexponent,meaning

thattheelasticstresscanincreaserapidlyandwithoutlimit.

Inextension,thereisalsoa𝜏::stress.Thisstress,however,doesnotexperiencesuch

exponentialgrowth,soitisoftennegligiblecomparedto𝜏99.

13

2.7 TheDeborahNumberandWeissenbergNumberTheDeborahnumber,liketheReynoldsnumber,isadimensionlessquantitythathelpsto

characterizetheflowfield.Itistheratiooftherelaxationtimeofthefluid,𝜆,tothe

characteristictimeofthedeformationprocess,𝑡x (Barnesetal.,1989;Reiner,1964).

𝐷𝑒 = 𝜆𝑡x (2.34)

Thesilicone-basedmaterialcalled“Bouncingputty”maybeusedtoillustratetheimportanceof

theratioofthetwotimescales(Barnesetal.,1989).Whenplacedinacontainer,thematerial

willstarttolevelandtaketheformofthecontainergivensufficienttime,therefore,actinglike

aviscousliquid.When,however,rolledintoaballanddroppedfromaheight,thematerial

bounceslikeanelasticsolid.Thedifferencebetweenthetwobehavioursisduetothedifferent

timescalesofthetwoprocesses:longtimescalesmakethematerialactlikeaviscousliquid,

suchaswhensettlinginacontainer,andshortertimescalesmakethematerialactlikean

elasticsolid,suchastheimpactofthefluidontheground.

IntermsoftheDeborahnumber,avaluemuchlessthanonemeansthatelasticityhaslittleto

noinfluencewhileahighvaluemeansthatelasticitydominatesthestressfield.Athigh

Deborahnumbers,flowprocesseshappentooquicklytoallowthestressesinthematerialto

relax,allowingforstrongelasticeffects.

Anotherdimensionlessnumberusedtocharacterizetheeffectsofelasticityinaflowisthe

Weissenbergnumber,whichisdefinedas

𝑊𝑖 = 𝜆𝛾. (2.35)

Inprinciple,theWeissenbergnumberisusedforflowsdominatedbyshear;however,the

WeissenbergandDeborahnumbersareoftenusedinterchangeably,fortheybothare

measuresoftheelasticeffectsoftheflow.Bothnumbersare,however,proportionaltothe

flowrateandtherelaxationtimeofthefluid,oftenmakingtheminterchangeable.Inthisstudy,

theDeborahnumberwillbeusedtoquantifyflowelasticity.

14

2.8 ExperimentalLiteratureReview

Oneoftheearlieststudiesontheviscoelasticflowsthroughaporousmediawascarriedout

whileattemptingtoincreasetheviscosityofbrinesolutionsinenhancedoilrecover.Pye(1964)

addedsmallamountsofpolyacrylamidepolymertoabrineandmeasuredthepressuredropin

flowsofthroughporoussandstonerocks.Hemeasuredtheviscosityofthebrinesolution,but

noticedthattheincreaseinviscositydidnotcompletelyaccountfortheincreaseinpressure

dropthroughthesandstoneforagivenflowrate.Infact,theymeasuredapressuredrop4.5to

15timesgreaterthanwhatwasexpectedfromapurelyviscousfluid.Theyconcludedthatthe

effectwascausedbythepolymer,buttheydidnotdiscoverthemechanismbywhichthese

polymersincreasethepressuredrop.

Thisenhancedpressuredropeffectwaslaterobservedinvariousexperimentsstudyingthe

flowofviscoelasticfluidsthroughpackedbedsofspheres.Thepressuredropwasfoundtobe

uptotwoordersofmagnitudehigher(Dauben&Menzie,1967;Durst,Haas,&Interthal,1987;

James&McLaren,1975;Marshall&Metzner,1967;Rodriguezetal.,1993;Vorwerk&Brunn,

1991).Atlowflowrates,thepressuredropmatchedwhatwasexpectedfromapurelyviscous

fluid.Dauben&Menzie(1967)notedthatN1stresseswereoftenaslargeorevenlargerthan

theviscousstresses.Laterstudies,however,arguedthattheincreaseinpressuredropwas

causedbyextensionalstressesduetothepolymer(Durstetal.,1987;James&McLaren,1975;

Vorwerk&Brunn,1991).

Skartsis,Khomami,&Kardos(1992)studiedtheflowofviscoelasticfluidsthroughbothasquare

andstaggeredarrayofcylinders,withahighsolidvolumefractionof55%,usingvarioustypes

polymericsolutions,includingtwoBogerfluids.Theyobservedanenhancedpressuredropafter

increasingtheflowratetoaDeborahnumberaboveapproximately0.01forbothgeometries

forallfluids.Theyconcludedthattheonsetofelasticeffectsdoesnotstronglydependonthe

geometryofthebed.

15

Inalaterstudy,KhomamiandMoreno(1997)investigatedviscoelasticflowsthroughsquare

arraysofcylinderswithsolidvolumefractionsof14%and55%.Theyobservedthatwhenthe

Weissenbergnumberincreasedaboveacriticalvalue,theflowresistanceincreased.Theyused

particleimagevelocimetryandstreakphotographytostudythevelocityfieldoftheflow.They

foundthatthecriticalWeissenbergnumbercorrespondedtoachangeinflowregimefroma

steadytwo-dimensionalflowtoasteadythree-dimensionalone,andthen,atevenhigher

numbers,toanunsteadythree-dimensionalone.However,forthe55%array,theflowskipped

thesteadythree-dimensionalflow.Theyattributedtheabruptincreaseinflowresistancetoan

elasticflowinstability.

MuchofthestudybyKhomamiandMorenoconfirmedtheresultsofasimilarstudydone

earlierbyChmielewskiandJayaraman(1992,1993).ChmielewskiandJayaramanfoundan

increaseinpressuredropaboveaDeborahnumberof1usingasquarearraywithasolid

volumefractionof30%.They,however,foundpressurefluctuationsabovethecriticalDeborah

numberandascribedthiseffecttoelasticinstabilities.Theseelasticinstabilitieswerethen

observedusinglaserDopplervelocimetryandstreaklinephotography.Theirstudiessuggested

thattheenhancedpressuredropwascausedbyflowinstabilitiescreatedbyelasticity.

LaterexperimentswerepublishedbyJamesetal.(2012)basedonthethesisbyYip(2011),who

usedsquarearrayswithmuchlowersolidvolumefractions(2.5%,5%,and10%).Theyobserved

asteadyincreaseinflowresistanceaboveacriticalDeborahnumberofapproximately0.5,as

showninFigure2.4.Theincreaseinflowresistancewassimilarforallthreearraysuptoa

Deborahnumberof1.5,abovewhichtheflowresistanceofthe10%arraydivergedfromthe

othertwo.

16

Figure2.4.NormalizedflowresistancereproducedfromJamesetal.(2012).

Jamesetal.(2012)andYip(2011)usedparticleimagevelocimetrytomonitortheflowfield,

andcontrarytothestudybyKhomami&Moreno(1997)andChmielewski&Jayaraman(1993),

theyfoundthattheflowfieldwassteadyandtwo-dimensionalwellabovethecriticalDeborah

number.SomeoftheirparticleimagevelocimetryresultsshowninAppendixD.Furthermore,

nosuddenflowfieldchangeswerefoundafterthecriticalDeborahnumber.Thisabsenceofa

suddenchangeinflowfieldisindirectcontrasttowhatwasfoundbyKhomamiandMoreno

(1997),whoobservedthattheenhancedpressuredropeffectwasassociatedwithathree-

dimensionalflowregimetransition.Themaindifferencebetweenthetwostudies,however,

werethemuchhighersolidvolumefractionsusedinKhomamiandMoreno(1997).Thisthesis

willmainlybebasedontheworkcompletedbyJamesetal.(2012)andYip(2011)becausetheir

flowsweresteady,andthusamenabletonumericalsimulation.

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Normalized

Flow

Resistance,fRe

/fRe

New

t.

DeborahNumber,De

φ=2.5%

φ=5%

φ=10%

17

James,Shiau,&Aldridge(2015)studiedtheviscoelasticflowaroundanisolatedcylinderand

foundamonotonicincreaseinpressuredropaboveanonsetDeborahnumberof0.6.The

pressuredropincreased50%duetoelasticityataDeborahnumberof3.Notably,theyusedthe

samebatchofBogerfluidsasJames,Yip,&Currie(2012).Particleimagevelocimetryresults

showthattheflowfieldremainedsteadyandchangedgraduallyafteronset.Thoughthe

particleimagevelocimetryresultsgouptoDe=10.28,substantialchangesintheflowfield

happenaboveaDeborahnumberof1.3.

2.9 NumericalLiteratureReviewNumericallysimulatingviscoelasticflowsthroughanarrayofcylindersisnoteasy.Talwarand

Khomami(1992)attemptedtonumericallysimulatetheviscoelasticflowthrougharraysofhigh

solidvolumefractions(55%and35%),usingboththeupper-convectedMaxwellmodelandthe

Oldroyd-Bmodel.Despitefindingsubstantialelasticstressesusingbothmodels,theypredicted

asmallmonotonicdecreaseinflowresistancewithincreasingWeissenbergnumber,contraryto

whatwasobservedintheexperimentby(Skartsisetal.,1992)forthe55%array.However,they

werelimitedbythecomputationalpoweroftheday,sotheirmaximumnumberofdegreesof

freedomwaslessthan7,000(lessthan10,000degreesoffreedomisconsideredsmall).

Later,TalwarandKhomamiattemptedthesamesimulationwith12,000degreesoffreedom

(Khomami,Talwar,&Ganpule,1994).Thistime,theyusedadifferentfiniteelementtechnique,

buttheystillpredictedasmalldecreaseinflowresistance.

Talwar&Khomami(1995)againattemptedtosimulatetheflowthrougha45%arrayusingtwo

newmodels:thePhan-Thien-TannerandtheGiesekusmodels.Again,theyfoundsubstantialN1

andextensionalstresses,butthesestressesdidlittletoaffectthepressuredrop.Theyfound

onlyasmall10%decreaseinflowresistancetoaminimumbeforeitincreasedbacktoits

originalvalue.Theysuggestedthatatemporalinstabilityornon-linearflowtransitionmayhave

causedtheincreaseinflowresistanceobservedinexperiments.

18

Othernumericalstudieshavebeensimilarlyunsuccessfulinpredictinganenhancedpressure

drop.SouvaliotisandBeris(1992)andHua&Schieber(1998)studiedtheflowthrougharraysof

cylinderswithasolidvolumefractionsrangingfrom12.6%to55%.Bothstudiespredictedsmall

pressuredropdecreases,incontrasttothepressuredropincreasesfoundexperimentally.Both

studiesspeculatedthatthediscrepancybetweennumericalandexperimentalresultswas

causedbyflowfieldchangescausingthree-dimensionaleffects,time-dependenteffects,orthe

formationofflowinstabilities.

Liu,Wang,&Hwang(2017)wereabletopredictasharpflowresistanceincreaseusing

hexagonally-packedandrandomly-packedarrays,butwereunsuccessfulinpredictingthesame

effectforasquarearrayofcylinders.Theyassociatedtheflowresistanceincreasewiththe

extensionalregiondownstreamofthecylinders.Therefore,theyarguedthatpolymerstretch

waslimitedinasquarearrayofcylindersandthusthepressuredropwaslimited.

Totheauthor’sknowledge,onlythenumericalsolutionbyHemingway,Clarke,Pearson,&

Fielding(2018)hasshownapressuredropincreaseforasquarearray.Theirsimulationsshowa

smalldipbeforeasharpincreaseinflowresistanceupto5%.Thisdecreasetoaminimum,

however,hasnotbeenobservedexperimentally,although,itwouldbedifficulttomeasurea

dipofonly8%fortheirhighestsolidvolumefractionof38.4%.Unfortunately,theirresultsare

cutoffsoonafterthesharpincreaseinflowresistancelikelybecauseofproblemswith

convergence.

Insummary,alloftheexperimentsstudyingthetheflowofviscoelasticfluidthroughporous

mediashowadefiniteincreaseinflowresistanceaboveacriticalDeborahnumber.Incontrast,

thenumericalsimulationshavegenerallypredictedasmalldecreasetoaplateauorasmall

decreasetoaminimumbeforereturningbacktotheoriginalvalue.Onlyonenumericalstudy

completedbyHemingway,Clarke,Pearson,&Fielding(2018)predictedanincrease,thoughthe

amountsofincreaseweresmall.Thecauseofthesediscrepanciesiscurrentlyunknown,

19

althoughtheyhavebeenspeculatedtobecausedbyflowfieldchangeseithertoathree-

dimensionalflowfield,atime-dependentflowfield,orfromtheformationofflowinstabilities.

Toinvestigatethisinconsistency,anumericalstudyofflowusingtheOldroyd-Bmodelwas

carriedout.Differentfrompriorattempts,theflowfieldwasassumedtobeexactlythesameas

itsNewtoniancounterpartwithadditionalelasticstresses.Thisassumptionwasmadebecause

particleimagevelocimetryresultsbyYip(2011)showthattheflowfieldchangesonlygradually

beyondtheonsetDeborahnumberandtosimplifythesimulationsbecausesimulationsof

viscoelasticflowsgenerallytakeconsiderablecomputationalresourcesandaredifficultto

converge.

2.10 Objectives

Asindicatedabove,theobjectiveofthisthesisresearchastonumericallysimulatetheflow

throughaperiodicarrayofcylinders.Thesolidvolumefractions(2.5%,5%,and10%)werethe

sameasthatusedinJamesetal.(2012)andYip(2011).

Consequently,theprimaryobjectivesofthisstudyareasfollows:

• Findtheelasticstressesoftheflowfieldandexaminehowtheyvarywithincreasing

Deborahnumber

• Usingtheseelasticstresses,attempttopredictthepressuredropalongasquarearray

ofcylinders

• TopredictthecriticalDeborahnumberbeforewhichtheflowfieldhasonlymarginal

elasticeffects

• Toshedlightonthecauseoftheobservedincreaseinpressuredrop

• Todeterminewhethertheincreaseiscausedbyelasticstressesinshearorextension

20

Chapter3 MethodologyThecommerciallyavailablesoftwarecalledANSYSFLUENT™wasusedtosetupthedomainand

tosimulatetheNewtonianflowfield.ANSYSFLUENT™isasolverthatusesthefinitevolume

methodwithacell-centredformulation.Auser-definedfunctionwasusedwiththesoftwareto

calculatetheupper-convectedMaxwellstressesforthepolymercontributionofthefluid.The

Oldroyd-BstresseswerethenobtainedbyaddingtheNewtoniansolventcontribution;thus,

findingtheelasticstressesoftheflowfield.

3.1 Geometry

Figure3.1.Unitcelltorepresentaperiodicarrayofcylinders

Theperiodicarraywastreatedusingaunitcell,aspresentedinFigure3.1.Thecellhasasingle

cylinderwithadiameterDatthecentreofasquareandthecelllengthisL.Threedifferentsolid

volumefractionswereusedinthisstudy,2.5%,5%,and10%,representingthelowsolidvolume

fractionsusedinJamesetal.(2012)andYip(2011),enablingpossiblecomparisonswith

experimentaldata.

21

3.2 MeshSetupBeforeanysimulationcanstart,thedomainhastobedividedintosmallercellsorelementsin

whichtheequationsaresolved.

Figure3.2.Meshingregions.RegionAusesamappedfacemesh.RegionBusesaquadrilateraldominantmesh.RegionB*isthesameasregionB,exceptforthe2.5%solid

volumefractiongeometrywhereitislocallyrefined.

Thedomainofthesquarecellwasdividedintotwoorthreedistinctregions.RegionA,shownin

Figure3.2,usesameshofregularquadrilateralelementsmappedtotheconcentriccircles

enclosingtheregion.Themeshofthisinnerregionhadafivefoldbiastothecylinder,sofiner

elementswereusednearthecylinder.

ForregionsBandB*,theautomaticmeshingalgorithmprovidedbyANSYSMeshingcreated

mainlyquadrilateralelementsofsimilarsize.TheseouterregionsweremeshedusingANSYS’s

quadrilateral-dominantmeshalgorithm.Theleftandrighthandsidesoftheunitcellwere

madeidentical,sothattheycouldbematchedperfectlytosatisfytheperiodicboundary

condition.RegionB*wasseparatefromregionBonlyforthe2.5%arraytolocallyrefinethat

region.Otherwise,thenumberofelementswouldhavebecometoonumerous,making

simulationsunnecessarilycomputationallyexpensive.

22

Themeshescomprisedapproximately350,000elements,exceptforthe2.5%array,whichhad

1,200,000elements.Thenumberofelementswasconsideredsufficientusingamesh

convergenceanalysispresentedinsection4.4.

Toshowtheshapeandtherelativesizeoftheelements,Figures3.3-3.4showthecoarsest

meshusedinthe5%array.Thefinestmeshesusedinthisworkhadmeshelementsatthepoles

ofthecylinderwithwidthslessthan0.1%ofthecylinderdiameter.

Figure3.3.5%arraycoarsestmesh(23,664elements)

23

Figure3.4.5%arrayclose-upofcoarsestmesh(23,664elements)aroundthetoppole

3.3 BoundaryConditionsThetreatmentattheboundariesofthedomainmustbedefinedattheoutset.Inour

simulations,threeboundariesrequireddifferenttreatment:theleftandrightboundaries,the

topandbottomboundaries,andthesurfaceofthecylinder.

Attheleftandrighthandsidesoftheunitcell,aperiodicboundaryconditionwasspecified.A

periodicboundaryconditionimposestherestrictionthatthetwoboundariesmusthavethe

samedistributionsofvelocity,stress,andpressuregradient.Simulationsusinguptosix

cylinderslinedupintheflowdirectionconfirmthevalidityofusingaperiodicboundary

condition.Thetopandbottomboundarieswererequiredtohavezeromassfluxthroughthe

boundaryandzeroshearrate.Finally,theboundaryconditionsatthecylindersurfacewerethe

noslipconditionandthattherewaszeromassfluxpassingthroughthesurface.

24

Formally,theseboundaryconditionsare:

𝑢 −𝐿2, 𝑦 = 𝑢

𝐿2, 𝑦 ; 𝝉 −

𝐿2, 𝑦 = 𝝉

𝐿2, 𝑦 ;

𝜕𝑝𝜕𝑥

−𝐿2, 𝑦 =

𝜕𝑝𝜕𝑥

𝐿2, 𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡;

𝑣 𝑥,−𝐿2

= 𝑣 𝑥,𝐿2

= 0;𝜕𝑢𝜕𝑦

𝑥,−𝐿2

=𝜕𝑢𝜕𝑦

𝑥,𝐿2

= 0;

𝑢 ± 𝑥7 + 𝑦7 =𝐷2

= 0,

(3.1)

wherepisthelocalpressure,𝑢isthevelocityvector,and𝝉isthestresstensor,asinsection

2.3.

3.4 NewtonianSimulationDarcy’slaw(Equation1.1)appliestoNewtonianfluidsforflowsthroughporousmedia.Thelaw

isgenerallyvalidforwhichtheReynoldsnumberslessthan1(Marsily,1986)andthehighest

Reynoldsnumberusedinthisthesisis0.0063.LowReynoldsnumbersensuresthatinertial

effectsarenegligible,yieldingStokesflow.Inthisregime,themomentumequationislinear,the

streamlinesaresymmetricupstreamanddownstreamofthecylinder,andthepressuredropis

directlyproportionaltothemassflowratethroughtheflowfield.

Darcy’slawmaybeusedtocomparetherelevantflowparametersandrelatethemtotheir

respectiveNewtoniancounterparts.

Inporousmedia,theReynoldsnumberisdefinedas(Marsily,1986)

𝑅𝑒 = 𝜌𝑈𝐷𝜂, (3.2)

whereDisthediameterofthecylinder,𝜌isthedensityofthefluid,𝜂isthefluidviscosity,and

Uisthebulkvelocitywhichis,

𝑈 =𝑞𝐿, (3.3)

whereqistheflowrateperunitlength.

25

Theflowresistancecanberepresentedthefrictionfactoroftheflowgeometrycanbedefined

as,

𝑓 = −𝛥𝑝𝐷𝐿𝜌𝑈7

. (3.4)

Darcy’slawcanalsobeexpressedbymultiplyingthetwogroupstoyield,

𝑓𝑅𝑒 = −

𝛥𝑝𝐷7

𝜂𝐿𝑈=𝐷7

𝐾. (3.5)

BecauseEquation3.5isindependentofthedensity,viscosity,andflowrate,itisapreferred

waytopresentresults.

Therearevariousanalyticalsolutionsforpredictingtheresistancetoflow.Theoneusedinthis

paperwillbethatbySangani&Acrivos(1982),whousedaFourierseriesmethodtosolvethe

Stokesequation.Accordingtotheirtechnique,thepermeabilityoftheporousmedium,𝐾,is

𝐾 =

𝐷7

16𝜙ln

1𝜙

− 0.738 + 𝜙 − 0.887𝜙7 + 2.038𝜙� + 𝑂 𝜙� , (3.6)

where𝜙isstillthesolidvolumefractionofthearray.

Theflowresistanceistherefore,

𝑓𝑅𝑒 =

𝐷7

𝐾=

16𝜙−0.5 ln 𝜙 − 0.738 + 𝜙 − 0.887𝜙7 + 2.038𝜙� + 𝑂(𝜙�)

. (3.7)

Thisformulawasusedtoensuretheaccuracyofthenumericalresults.Thevaluesfor𝐾and

𝑓𝑅𝑒forthethreearraysaregiveninTable1.

Table1.PermeabilityandflowresistanceofaperiodicarrayofcylindersbasedonthesolutionbySangani&Acrivos(1982)

SolidVolumeFraction Permeability,K[m2] FlowResistanceParameter,fRe

2.5% 2.85e-5 0.355

5% 1.02e-5 0.991

10% 3.19e-4 3.17

26

TheinitialvaluesoftheNewtonianvelocityfieldweresettobethebulkvelocity.The

simulationwasthenallowedtoiterateuntiltheflowratethroughthedomainstabilizedand

matchedwiththevaluesinTable1.ThedifferencebetweentheFLUENTflowrateandthat

predictedbySanganiandAcrivoswaslessthan0.1%,asshowninFigure3.5.

Figure3.5.Predictedsimulationflowresistancevs.ReynoldsnumberoftheNewtonianflowfield.

3.5 SimulatingElasticStressesAftertheNewtonianflowfieldconverged,threeuser-definedscalarswerecreatedtorepresent

thethreeUCMstresses:𝜏99,𝜏9:,and𝜏::.Thesestresseswerecalculatedtoidentifythe

magnitudeofthepolymerstresscontributionstothestressfield.CalculatingthethreeUCM

stressesrequiredsolvingthreecoupledequationsfoundinAppendixA.

0

0.5

1

1.5

2

2.5

3

3.5

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Flow

Resistance,fRe

ReynoldsNumber,Re

ϕ=2.5%

ϕ=5%

ϕ=10%

AnalyticalSolutionfromS&A(1982)

27

Tosolvethesethreecoupledequations,auser-definedfunctionwascreatedtodefinetheUCM

equationinFLUENT.Theseuser-definedfunctionsalongwithsetuponhowtheyweresetup

canbefoundinAppendixB.

Asdescribedearlier,thevelocityfieldwastakentobethesameastheNewtonianvelocityfield.

TheelasticstresseswereinitializedastheNewtonianstresses,asfoundpreviously.The

simulationiteratedandwasconsideredconvergedwhenthestressesstabilizedbetween

iterations.Thestressesonthecylinderandatthedomain’sinletwerealsomonitoredtoensure

convergence.

ItwasfoundthatatthehigherDeborahnumbers,itwasnecessarytointroduceadiffusivity

constantof5e-12mtopreventthepolymerstressesfromdiverging.Theconstantwasdoubled

invaluetoseewhetheritaffectstheresultsbycheckingthemaximumUCM𝜏99stresses

locatedatthetopandbottomofthecylinder,whicharetheregionswiththehigheststress

gradients,andthustheregionsmostaffectedbydiffusivity.Itwasobservedthatdoublingthe

diffusivityconstantdecreasedthe𝜏99stressatthesepointslessthan1%.Becauseofthissmall

magnitude,theintroductionofdiffusivitywasconsiderednegligibleotherthantoensurethe

convergenceofthepolymerstresses.

ThestressesofaNewtoniansolventwerethenaddedtothepolymerstressestoobtainthe

Oldroyd-Bstresses:

𝝉 = 𝝉𝒑 + 𝝉𝒔 (3.8)

where𝝉𝒑isthepolymerstresstensorcalculatedusingtheUCMmodeland𝝉𝒔isthestress

tensorcreatedbytheNewtoniansolvent.

3.6 ElasticPropertiesThepropertiesofthefluidusedinthesimulationwerethesameonesmeasuredbyJamesetal.

(2012).Hemeasuredthesolventviscosityandtotalviscosityandfoundtheratiobetweenthe

twovaluestobe0.525.HealsoprovidedG’andN1thatcouldbeusedtofindtherelaxation

timeofthefluid.

28

Jamesetal.(2012)usedtherelaxationtimemeasuredbythelowoscillatoryshearresults

calculatedfromG’describedinEquation2.26,whichrequiresfindingalowfrequencyplateau

whenmeasuring𝐺′/𝜔7.Whilehedidattempttofindthisplateau,therewasstillsome

variationbetweenthelowestfrequencypoints.Additionally,thesemeasurementswerelikely

unreliableduetothelowshearstressesobservedatthelowestoscillatoryfrequencies.His

relaxationtimefromthesheartestcalculatedfromN1was,therefore,consideredmore

appropriateduetothebetterreliabilityofthemeasurementandtheimportanceofshearin

thisflowfield.He,however,usedtwofluids,sotheresultsforB2fluidwereusedbecauseit

showedlessvarianceinitsN1measurements.

TherelaxationtimemeasuredusingaconstantshearratetestintherheometerwithN1results

showninFigure3.6.Theseresultsareplottedusingthenormalstresscoefficient,𝛹[,

𝛹[ =𝑁[𝛾7

= 2𝜂e𝜆. (3.9)

Figure3.6.FirstnormalstresscoefficientofB2fluidadaptedfromYip(2011)

TheDeborahNumberisdefinedheretomatchthatofJamesetal.(2012),

1.5

1.8

2.1

2.4

2.7

3

0 1 2 3 4 5 6

NormalStressC

oefficien

t,𝛹

1[Pas²]

ShearRate[s⁻¹]

29

𝐷𝑒 = 𝜆𝑈

(1 − 𝜙)𝐿. (3.10)

LatersectionswilluseaDeborahnumberbasedontherelaxationtimefromtheG’results

(3.9s)ratherthantheN1relaxationtimeoftheB2fluid(0.6s)usedinthesimulationsinthis

worktomaintainconsistencywith(Jamesetal.,2012).TheonsetDeborahnumberwill,

therefore,remainatDe=0.5.ThesoleexceptiontothisisFigure4.15,wherebytheDeborah

numbersinthex-axisarescaledtousetheN1relaxationtime(0.6s)tomakeacomparisonwith

anothernumericalstudy.

3.7 SummaryofAssumptionsInsimulations,someassumptionswerenecessary,andtheywere:

• Theflowfieldistwo-dimensionalandsteady,i.e.notvaryingwithtime.

• ThevelocityfieldfortheOldroyd-BfluidisexactlythesameasthatoftheNewtonian

fluid.

• TheOldroyd-BequationisagoodmodelfortheBogerfluidsusedintheexperimentby

Jamesetal.(2012).

• Inertialeffectsarenegligiblesothatasimplifiedmomentumequationmaybeused.

30

Chapter4 ResultsandDiscussion

4.1 NewtonianFlowPlotsAsthefoundationfortheelasticstresscalculations,thevelocityfield,pressurefield,

streamlines,andstressesareshownfortheflowofaNewtonianfluid,whicharedisplayed

usingtwo-dimensionalcontourplots,showninFigures4.1and4.2.

InFigure4.1,thevelocitymagnitudesandpressurecontourplotsareshown.Thevelocity

magnitudeplotsinFigure4.1arenormalizedusingthebulkvelocitydefinedearliertheseare

overlaidwithstreamlines,whilethepressurefieldwasnormalizedusingthepressuredrop

acrosstheunitcell.ConsideringthattheNewtonianflowfieldscaleslinearlyintheStokesflow

regime,onlyonenormalizedplotforeacharraywasnecessarytorepresenteachflowfield.

31

Figure4.1.NormalizedvelocitymagnitudesandpressurecontoursoftheNewtonianflowfield.Blacklinesonthenormalizedvelocitycontoursrepresentstreamlines,whiletheright-

handsideofthepressurecontoursareuniformly0.

32

AsshowninthesetofcontourplotsinFigure4.1,theflowfieldiscompletelysymmetricdueto

theinsignificanceofthenon-linearinertialterm.Asforthepressure,themaximumand

minimumpressureswithintheunitcellarefoundatthefrontandrearstagnationpoints,

respectively.Additionally,theleftandrighthandsidesarelinesofconstantpressurealongwith

theverticalcentreline.

ShownbelowaretheaccompanyingNewtonianstressesforallthreearraysasabackgroundfor

theforthcomingnon-Newtonianstresses.Thesestresseswerenormalizedusingthemaximum

shearstressfoundatthepolesofthecylinder.Becausethesestressesareproportionaltotheir

strainrate,theseplotsalsoshowtheregionsofhighshearandextension.

Figure4.2.Newtonianstressesinaflowthroughasquarearrayofcylinders.Thesestresseswerenormalizedusingthemaximumstressintheflowfield,whichistheshearstressatthe

topandbottompolesofthecylinder

33

InFigure4.2,theNewtonianstressfieldisdominatedbytheshearstressesaroundthecylinder,

thehigheststressesbeingtheshearstressesatthetopandbottompolesofthecylinder.

Smallamountsofstressduetoextensionforeandaftofthecylinderareslightlyoffsetfromthe

stagnationpoints,representedby𝜏99and𝜏::.Notingwhere𝜏99and𝜏::arepositiveand

negative,theregionforeofthecylinderexperiencesextensioninthey-direction,whileaftof

thecylinderexperiencesextensioninthex-direction.Additionally,extensionhaslessinfluence

forthe10%arraythantheothertwoarrays,shownbythelackofcontoursupstreamand

downstreamofthecylinderinthe𝜏99and𝜏::plots.

4.2 ElasticStressPlotsAfterobtainingtheNewtonianstresses,elasticstresseswerecalculatedfromtheOldroyd-B

modelusingtheNewtonianflowfield.TheOldroyd-Bmodel,however,includesbothelasticand

viscousstresses;therefore,thestressesexpectedfromaNewtonianflowfieldusingafluidof

thesameviscosity,𝝉𝑵𝒆𝒘𝒕.,werethensubtractedouttoobtaintheadditionalelasticstresses,

𝛥𝝉,whicharethestressesofinterest,

𝛥𝝉 = 𝝉 − 𝝉𝑵𝒆𝒘𝒕.. (4.1)

ThesestresseswerenormalizedbythehighestNewtonianstresscreatedbyaNewtonianfluid

ofthesameviscosity,i.e.,bytheNewtonianshearstressatthepolesofthecylinder.Theywere

normalizedinthiswaytosimplifycomparisonsbetweenplots.ThehighestNewtonianstress

waschosentofacilitatethenormalizationtoallowforcomparisonswiththeNewtonianstress

field.

𝛥𝝉hasthreecomponents,𝛥𝜏99,𝛥𝜏9:,and𝛥𝜏::,whichareshownusingsetsofninecontour

plotsforDeborahnumbersof0.3,1.0,and2.1andforthreesolidvolumefractionsof2.5%,5%,

and10%(Figures4.3-4.6).ThelowestDeborahnumberislowerthanthecriticalDeborah

numberof0.5,soelasticeffectsareexpectedtobelow.ThetwohigherDeborahnumbersare

twoandfourtimeshigherthanthecriticalDeborahnumber,sothatelasticeffectsareexpected

tobesubstantial.

34

The𝛥𝜏99stressesarepresentedinFigures4.3and4.4.Plottedinbothfiguresaretheexact

samestresses,butthescalesofthecontourlevelsaredifferent,asshownbytheirlegends.In

Figure4.3,valueshigherthanitsmaximumcontourlevelof1.0aretruncatedtobetterreveal

regionsdominatedbyelasticity.Becauseeachplotisshownwiththesamecontourlevels,

Figure4.3alsoallowsforcomparisonsbetweeneachplot.Ontheotherhand,Figure4.4shows

the𝛥𝜏99withadifferentsetofcontourlevelstoshowthemagnitudeoftheelasticstresses.It,

however,failstoshowtheextentthat𝜏99affectstheflowfield.

Figure4.3.𝜟𝝉𝒙𝒙stressesnormalizedbythemaximumNewtonianshearstress.Magnitudeshigherthan1.0aretruncated.

35

Figure4.4.𝜟𝝉𝒙𝒙stressesnormalizedbythemaximumNewtonianshearstress.

ThecontoursinFigures4.3and4.4showthattherearetwomainelasticregionsinthese

contours:nearthepolesofthecylinderanddownstreamofthecylinder.Theregionsaround

thepolesaredominatedbyshear,i.e.byN1,whilethedownstreamregionisdominatedby

extension.AstheDeborahnumberincreases,thestressesincreaseandthesetworegions

increaseinsize,allowingelasticstressesgreaterinfluenceovertheflowfield.Thesetwo

regionscontainthehighestelasticstresses,makingthe𝛥𝜏99stresscomponentofgreatest

interest.

TherearesignificantelasticstressesfoundnearthecylinderpolesevenatthelowestDeborah

number.TheelasticstressesinthisregionareN1stressescausedbyshear.AtDe=0.3,

however,thesestressesareextremelylocalandwouldonlyaffectasmallpartoftheflowfield

aroundthetopandbottom,andtheyactsymmetrically.AsforthelargerDeborahnumbers,N1

36

stressesaffectlargeportionsoftheflowfieldaroundthepolescylinder;thesestressesare

convecteddownstreamslightly,creatingasymmetriesintheflowfield.

Theextensionalstressesdownstreamofthecylinderdecreasewithincreasingsolidvolume

fraction,asshowninFigures4.3and4.4.Infact,therearehighextensionalstressesonlyforthe

2.5%and5%arraysatDe=2.1,i.e.,theextensionalstressesinthe10%solidvolumefraction

geometryaresmallrelativetotheNewtonianstresses,despitebeingsubstantiallyabovethe

criticalDeborahnumber.Thisresultiscausedbythelowerextensionalratesintheregionand

thesmallerdistancesthatallowthepolymerstostretchbeforethenextcylinder.Thisfacthelps

toconfirmtheobservationsmadebyYip(2011)thattheextensionalstrainsforthe10%solid

volumefractiongeometryareinsufficienttocauselargestresses,aswellassupportingthe

argumentthatN1stressesarethemaincauseoftheenhanceddrageffect.

Shownnextaretheothertwostresscomponents(𝜏9:and𝜏::),whichareprovidedfor

completion.

37

Figure4.5.𝜟𝝉𝒙𝒚stressesnormalizedbythemaximumshearstress.

The𝛥𝜏9:stressinFigure4.5arelocatednearthesurfaceofthecylinderandaresignificantonly

60°fromthex-axis.ThoughtheyincreasewithincreasingDeborahnumber,thesestresses

affectonlyasmallportionoftheflowfield.

38

Figure4.6.𝜟𝝉𝒚𝒚stressesnormalizedbythemaximumshearstress.

Figure4.6showsthestresscontoursfor𝛥𝜏::.Thissetofcontoursshowthathighelastic

stressesoccurnearthefrontstagnationpoint.Theseextensionalstresses,however,are

substantiallysmallerinmagnitudethantheonesdownstreamofthecylinder.

InFigures4.3,4.4,and4.6,thethreemainelasticregionsareshowninred:aroundthetopand

bottomsurfacesofthecylinder,aroundtheaftstagnationpoint,andaroundtheforward

stagnationpoint.ThesideregionsareduetoN1stressescreatedbyshearingalongthecylinder,

whilethelattertworegionsarecausedbyextension.Thediscussionwhichfollowsfocuseson

thesideregionsandthezonedownstreamofthecylinder.ThestresscontoursfoundinFigure

4.3forthe10%arrayatDe=2.1qualitativelymatchwiththepolymerstraincontoursfoundin

(Liuetal.,2017)fortheir12.6%arrayatWi=4.52.Unfortunately,makingfurthercomparisons

isdifficultbecausetheyplottedcontoursofpolymerstrainratherthanstresses.

39

4.3 ElasticStressesNeartheCylinderStressesonthecylinderaredirectlyrelatedtothepressuredrop,andsoitmaybeinformative

tostudythestressesactingonthecylinder.TheCartesiancoordinatesystem,however,splits

theseelasticstressesintothethreestresscomponents,soitismoresuitabletousestressesin

cylindricalcoordinates.Theconversionofthestressestoacylindricalcoordinatesystemcanbe

calculatedasfollows(Beer,Johnston,DeWolf,&Mazurek,2009),

𝜏�� =𝜏99 + 𝜏::

2+𝜏99 − 𝜏::

2𝑐𝑜𝑠2𝜃 + 𝜏9:𝑠𝑖𝑛2𝜃, (4.2)

𝜏�� = −𝜏99 − 𝜏::

2𝑠𝑖𝑛2𝜃 + 𝜏9:𝑐𝑜𝑠2𝜃, (4.3)

𝜏�� =𝜏99 + 𝜏::

2−𝜏99 − 𝜏::

2𝑐𝑜𝑠2𝜃 − 𝜏9:𝑠𝑖𝑛2𝜃, (4.4)

where𝜃istheangularpositionfromthepositivex-axis,ristheradialposition,𝜏�� istheradial

stresscomponent,𝜏��isthecircumferentialstresscomponent,and𝜏��istheshearstress

component.ThestressesatthecylindersurfaceareplottedinFigure4.7forthe5%arrayata

Deborahnumberof2.1.

Figure4.7.Additionalelasticstressesincylindricalcoordinatesatthecylindersurface.Notethat𝜟𝝉𝒓𝜽overlapswith𝜟𝝉𝒓𝒓

-0.5

0.5

1.5

2.5

3.5

4.5

5.5

6.5

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Normalize

dτ-τ N

ewt.Stresses

x/DPosition

ΔτrrΔτrθΔτθθTheoreticalN1Stress

φ=5%De=2.1

40

Thesurfaceofthecylinderisaregionofpureshear;therefore,theonlyrelevantelasticstresses

isfromN1.TheN1stresses,showninFigure4.7,actonlyinthecircumferentialdirectionas𝜏��.

SincetheonlyelasticstressisfromN1,thecircumferentialstressaroundthecylindershould

reachitssteadystatevalueof2𝜂e𝜆𝛾7accordingtotheOldroyd-Bmodel,whichisshownasthe

reddashedlineinFigure4.7.ThisclosematchbetweenthegreyandreddashedlinesinFigure

4.7showsthatthesimulationsareabletopredicttheN1stressesproperly.

Figure4.8.Stressesactingonasurfaceelementincylindricalcoordinates

Figure4.8showsafluidelementatthesurfaceofacylinderwith𝜏��,representingtheN1

stress,and𝜏��,representingtheviscousstresses.Theviscousstressescreateastressactingon

thesurface,asrepresentedbytheredarrow.TheN1stress,however,onlyacttangentiallyto

surface,creatingnostressonthesurfaceitself.ThisfactmeansthattheN1stressaroundthe

cylindercannotdirectlyactonthecylinder,whichwillberelevantinsection4.5.

4.4 MeshConvergenceAnalysisInordertoensurethatthepresentresultsareaccurateandmeshindependent,amesh

convergenceanalysiswasperformed.Thisanalysiswascompletedbysuccessivelydoublingthe

numberofmeshelementsinthemeshandmonitoring𝜏99attwopoints:oneatapoleofthe

cylinderandonedownstreamatcoordinates(0.3L,0).Thesepointswerechosentomonitorthe

highN1stressatapoleandthehighextensionalstressdownstreamofthecylinder.

ThemeshconvergenceanalysiswasperformedforeachgeometryatDe=2.1,whichwasthe

highestDeborahnumberorflowrateusedinthiswork,generatingthehighestelasticstresses,

andthusthehigheststressgradients.Thesehighstressgradientsmeanthatthesimulations

41

wouldrequirethefinestmesh,andthesamemeshwouldbesufficientforsimulationsatlower

Deborahnumbers.Belowarethemeshconvergenceanalysesatapointdownstreamofthe

cylinder.

Figure4.9.Meshconvergenceanalysisof2.5%solidvolumefractionatpoint(0.3L,0).(*)

indicatesthatthedownstreamregionwaslocallyrefinedratherthantheentireflowdomain.

150

170

190

210

230

250

270

290

310

330

350

τ xxStressat(0.3L,0)(Pa)

NumberofMeshElements

42

Figure4.10.Meshconvergenceanalysisof5%solidvolumefractionatpoint(0.3L,0).

Figure4.11.Meshconvergenceanalysisof10%solidvolumefractionatpoint(0.3L,0).

Figures4.9-4.11showthatthedifferencesinstressinthelasttwopointsarelessthan1%.To

reiterate,successfullyconvergingtheextensionalstressdownstreamofthecylinderrequireda

85

90

95

100

105

110

115

23665 51948 98506 183044 352389

τ xxStressat(0.3L,0)(Pa)

NumberofMeshElements

27

27.2

27.4

27.6

27.8

28

28.2

23216 44297 85553 167713 329164

τ xxStressat(0.3L,0)(Pa)

NumberofMeshElements

43

highlyrefinedmesh,sothemeshinthatregionwasrefinedlocally,asdetailedinsection3.2.

Themeshconvergenceanalysismonitoringthestressatthecylinderpolecanbefoundin

AppendixC.

4.5 ForceActingontheCylinderandPressureDropOneoftheobjectivesofthesesimulationswastopredictthepressuredrop.Usingthestresses

andpressureactingonthecylinder,itispossibletocalculatetheforceactingonthecylinder

andthenthepressuredrop.Theforceactingonthecylinder,𝐹,hastwocomponents:aviscous

componentcausedbyfluidstressesactingdirectingonthecylinder,𝐹�,andapressure

componentcausedbythepressuredistributionaroundthecylinder,𝐹e.

𝐹 = 𝐹� + 𝐹e (4.5)

Theviscouscomponentwascalculatedbeintegratingthestressesaroundthecylindershownin

Figure4.12.

Figure4.12.Stressesonasurfaceelement

𝐹� = − 𝜏99𝑑𝑦 + 𝜏9:𝑑𝑥

d�

d�

d�

d� (4.6)

where𝑠[and𝑠7arethestartandendpointsoftheintegral.Aspreviouslyexplainedinsection

4.3,theelasticstressesaroundthecylinderactonlytangentiallytothesurface;therefore,the

viscousforcecomponentisidenticaltothatofaNewtonianfluid,makingthepressure

componenttheonlycontributortotheenhancedpressuredrop.

Inordertocalculatethepressure,themomentumequationwasused.Theinertialtermswere

neglectedbecauseoftheStokesflowassumption,i.e.,

44

𝜕𝑝𝜕𝑥

=𝜕𝜏99𝜕𝑥

+𝜕𝜏9:𝜕𝑦

(4.7)

𝜕𝑝𝜕𝑦

=𝜕𝜏::𝜕𝑦

+𝜕𝜏9:𝜕𝑥

(4.8)

Thegradientsofthestresses,� ¡¡�9

,� ¡¢�:

,� ¢¢�:

,and� ¡¢�9

,wereextracteddirectlyfromthe

simulations.

Byusingthesegradients,thepressurearoundthecylindercanbecalculated,

𝑝 𝑥, 𝑦 =

𝜕𝑝𝜕𝑥𝜕𝑥 +

9

9£

𝜕𝑝𝜕𝑦𝜕𝑦 + 𝑝J

:

:£. (4.9)

Thispressurewasintegratedaroundthecylindertoobtainthepressurecomponentofthedrag

forceactingonthecylinder.Thetop-bottomsymmetryoftheflowfieldmeansthatonlythex-

componentoftheforceneedstobeconsidered,

𝐹e = 𝑝 𝑥, 𝑦 𝑑𝑦

d�

d�, (4.10)

where𝑠[and𝑠7areagainthestartandendpointsoftheintegral.

Thepressuredropwasthencalculatedbydividingtheforce,F,actingonthecylinder,withthe

lengthLoftheunitcell,

𝛥𝑝 =𝐹𝐿. (4.11)

Toensuretheaccuracyofthesepressurecalculations,thepressuredropusingthismethodwas

firstcalculatedtopredicttheNewtonianflowresistance.Theintegralswerecalculatedusinga

simplemidpointRiemannsumandthepressuredropwasfoundtomatchwiththeanalytical

solutionpredictedbySanganiandAcrivoswithin0.1%error.

Theelasticpressuredropwasthencalculatedusingthismethodandnormalizedwiththe

Newtonianpressuredropforeacharray(Figure4.13).

45

Figure4.13.Normalizedpredictedflowresistancevs.Deborahnumber

Itmayseempeculiarthatthelargeelasticstresses(upto6timesthehighestviscousstress)

aroundthecylinder(Figures4.3-4.6)causesuchasmallchangeinpressuredrop;however,this

differenceisduetothefactthattheelasticstressesonlyacttangentiallytothesurfaceofthe

cylinder,asexplainedinsection4.3.Inotherwords,theseelasticstressesdonotdirectlycause

aforceonthecylinder,onlythepressureproducedbythemdoes.

4.6 ComparisonswithOtherStudiesNowthattheflowresistancehasbeenpredicted,itisimportanttoensurethatthesevalues

haveabasisinrealitybycomparingthemtotheexperimentalresultsobtainedbyJamesetal.

(2012).

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

0 0.5 1 1.5 2 2.5

Normalized

Flow

Resistance,fRe

/fRe

New

t.

DeborahNumber,De

φ=2.5%

φ=5%

φ=10%

46

Figure4.14.ComparisonofflowresistanceresultswithexperimentalresultsbyJamesetal.

(2012).

Thepredictedpressuredropsareconsiderablysmallerthanthosefoundintheexperimentsby

Jamesetal.(2012)(Figure4.14).Forexample,thepredictedpressuredropforthe10%arrayat

aDeborahnumberof2.1isonly10%comparedtothe125%foundexperimentally.Thislarge

discrepancymeansthatthesesimulationswereunsuccessfulinpredictingtheincreasein

pressuredrop,thoughthetrendofamonotonicpressureincreasematcheswhatwasfound

experimentally.Additionally,thetrendsfoundbetweenthethreegeometriesaresimilartothat

foundbyJamesetal.(2012)aswell(Figure4.14):Thethreegeometrieshavesimilarslopes

untilaDeborahnumberofroughly1.7,wherethe10%arrayseparatesfromtheotherarrays.

0.75

1

1.25

1.5

1.75

2

2.25

2.5

0 0.5 1 1.5 2 2.5

Normalized

Flow

Resistance,fRe

/fRe

New

t.

DeborahNumber,De

φ=2.5%φ=5%φ=10%Yip,φ=2.5%Yip,φ=5%Yip,φ=10%

47

Figure4.15.Comparisonofflowresistanceresultstonumericalresultsfrom(Hemingwayet

al.,2018).(*)indicatesthattheDeborahnumberiscalculatedusing𝝀=0.6sforthecomparison.

InFigure4.15arenormalizedpressuredropresultscompletedinthisworkarecomparedwith

thenumericalresultsbyHemingwayetal.(2018)withthex-axisscaledforthecomparison.The

resultsbyHemingwayetal.predictasmalldecreaseinpressuredropbeforeincreasingagain,

comparedtothemonotonicincreasefoundinthisstudy.Thisflowresistancedecrease,

however,istoosmalltobemeasuredexperimentally.Additionally,theypredictedpressure

dropincreasesatmuchhigherDeborahnumbers.Theirpredictedpressuredrops,however,are

stillconsiderablysmallerthantheexperimentalresultsbyJamesetal.(2012).

4.7 DiscussionConsideringthatoneofthemainassumptionsofthesesimulationswasthattheflowfieldis

identicaltotheNewtonianflowfield,thediscrepancyinincreasedpressuredropmaybe

causedbyflowfieldchanges.TheparticleimagevelocimetryresultscompletedbyYip(2011)

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized

Flow

Resistance,fRe

/fRe

New

t.

DeborahNumber,De*

φ=2.5%

φ=5%

φ=10%

Hemingwayetal.,φ=3.1%

Hemingwayetal.,φ=7.1%

Hemingwayetal.,φ=12.6%

48

showsomedecreasedflowvelocitiesaroundanddownstreamofthecylindersataDeborah

numberaround2.1foreachsolidvolumefraction(AppendixD);hencethepresentedresults

areexpectedtobelessaccurateatDeborahnumbersmuchhigherthanthecriticalDeborah

number.

Othernumericalstudieswhichmodifiedtheflowfield,however,wereunabletopredictthese

increasesinthepressuredrop;therefore,theabovereasonmaynotbeadequate.

Furthermore,othernumericalstudieshavealreadyusedotherconstitutivemodelssuchasthe

FENEmodels,thePhan-Thien-Tannermodel,andtheGiesekusmodel,buttheyweresimilarly

unsuccessfulinpredictinganincreaseinpressuredrop(Hemingwayetal.,2018;Hua&

Schieber,1998;Talwar&Khomami,1995).

Analternatereasonastowhythesenumericalresultsdonotmatchexperimentalresultsmay

berelatedtotheOldroyd-Bfluidinmodellingthepolymersolution.Forexample,theOldroyd-B

modelpredictsthatthesecondnormalstressdifference(N2),thenormalstressperpendicular

toN1,iszero,whichmaynotbetrue.N2is,however,unlikelytobeasourceofthisdiscrepancy

becauseitisgenerallyanorderofmagnitudelowerthanN1(Barnesetal.,1989).

Also,themodelpredictsinfiniteextensibility,meaningtheelasticstressescanincreasewithout

limit.Finiteextensibility,however,isunlikelytobetheproblembecause𝛥𝜏wouldneedtobe

oforderofmagnitudeofhundredsorthousands(Tirtaatmadja,1993),whichisimplausible

giventhatthesimulationswerelimitedtoflowsnearthecriticalDeborahnumber.Andso,the

causeoftheenhancedpressuredropeffectduetoelasticitystillremainsamystery,despiteall

oftheeffortmadetonumericallysimulateelasticflowfields.

49

Chapter5 Conclusion

Thestresseswithinasquarearrayofcylindersinthreedifferentsolidvolumefractions(2.5%,

5%,and10%)havebeenfoundusingtheOldroyd-Bmodel,withtheassumptionthattheflow

fieldisidenticaltothatofaNewtonianflowfield.Themodelpredictslargenon-Newtonian

elasticstressescausedbyshearandextension,butthesedonotproducepressuredifferences

comparabletotheonesfoundexperimentally.However,thepredictionsarequalitatively

correctwiththepressuredifferenceincreasingwithsolidvolumefraction.Theelasticstresses

duetoextensiondownstreamofthecylinderdecreasewithincreasingsolidvolumefraction,in

constrasttotheincreaseinpressuredropwhichissimilarforallthreearrays.Thisfinding

supportstheargumentmadein(James,2016;Jamesetal.,2012;Yip,2011)thattheelastic

stressesduetoshearcausetheincreaseinpressuredrop.

Thereasonwhythereissuchalargediscrepancybetweenexperimentalandnumericalresults

stillremainsamystery,whichillustratesthedifficultiesinnumericallysimulatingviscoelastic

fluids.

50

References

ANSYSInc.(2015).ANSYS®FLUENTUDFManual(Release16).

Barnes,H.A.,Hutton,J.F.,&Walters,K.(1989).AnIntroductiontoRheology(1sted.).

Amsterdam,TheNetherlands:ElsevierSciencePublishersB.V.

Beer,F.P.,Johnston,E.R.,DeWolf,J.T.,&Mazurek,D.F.(2009).MechanicsofMaterials(5th

ed.).NewYorkCity,NY:McGrawHill.

Bird,R.B.,Armstrong,R.C.,&Hassager,O.(1987).Dynamicsofpolymericliquids.Vol.1,2nd

Ed. :Fluidmechanics.UnitedStates:JohnWileyandSonsInc.,NewYork,NY.

Cengel,Y.,&Cimbala,J.(2013).FluidMechanicsFundamentalsandApplications:ThirdEdition

(3rded.).Boston,Massachusetts:McGraw-HillHigherEducation.

Chmielewski,C.,&Jayaraman,K.(1992).Theeffectofpolymerextensibilityoncrossflowof

polymersolutionsthroughcylinderarrays.JournalofRheology,36(6),1105–1126.

Chmielewski,C.,&Jayaraman,K.(1993).Elasticinstabilityincrossflowofpolymersolutions

throughperiodicarraysofcylinders.JournalofNon-NewtonianFluidMechanics,48(3),

285–301.

Chmielewski,C.,Nichols,K.L.,&Jayaraman,K.(1990).Acomparisonofthedragcoefficientsof

spherestranslatingincorn-syrup-basedandpolybutene-basedbogerfluids.Journalof

Non-NewtonianFluidMechanics,35(1),37–49.

Dauben,D.L.,&Menzie,D.E.(1967).FlowofPolymerSolutionsThroughPorousMedia.Journal

ofPetroleumTechnology,19(8).

Durst,F.,Haas,R.,&Interthal,W.(1987).Thenatureofflowsthroughporousmedia.Journalof

Non-NewtonianFluidMechanics,22(2),169–189.

Hemingway,E.J.,Clarke,A.,Pearson,J.R.A.,&Fielding,S.M.(2018).Thickeningofviscoelastic

flowinaporousmedium.JournalofNon-NewtonianFluidMechanics,251,56–68.

Hua,C.C.,&Schieber,J.D.(1998).Viscoelasticflowthroughfibrousmediausingthe

CONNFFESSITapproach.JournalofRheology,42(3),477.

James,D.F.(1966).OpenChannelSiphonwithViscoelasticFluids.Nature,212(5063),754–756.

James,D.F.(2009).BogerFluids.AnnualReviewofFluidMechanics,41(1),129–142.

51

James,D.F.(2016).N1stressesinextensionalflows.JournalofNon-NewtonianFluid

Mechanics,232,33–42.

James,D.F.,&McLaren,D.R.(1975).Thelaminarflowofdilutepolymersolutionsthrough

porousmedia.JournalofFluidMechanics,70(4),733–752.

James,D.F.,Shiau,T.,&Aldridge,P.M.(2015).FlowofaBogerFluidAroundanIsolated

Cylinder,1–41.

James,D.F.,Yip,R.,&Currie,I.G.(2012).SlowflowofBogerfluidsthroughmodelfibrous

porousmedia.JournalofRheology,56(5),1249–1277.

Khomami,B.,&Moreno,L.D.(1997).Stabilityofviscoelasticflowaroundperiodicarraysof

cylinders.RheologicaActa,36(4),367–383.

Khomami,B.,Talwar,K.K.,&Ganpule,H.K.(1994).Acomparativestudyofhigher-andlower-

orderfiniteelementtechniquesforcomputationofviscoelasticflows.JournalofRheology,

38(2),255–289.

Liu,H.L.,Wang,J.,&Hwang,W.R.(2017).Flowresistanceofviscoelasticflowsinfibrous

porousmedia.JournalofNon-NewtonianFluidMechanics,246,21–30.

Marshall,R.J.,&Metzner,A.B.(1967).FlowofViscoelasticFluidsthroughPorousMedia.

Industrial&EngineeringChemistryFundamentals,6(3),393–400.

Marsily,G.De.(1986).QuantitativeHydrogeology:GroundwaterHydrologyforEngineers.

Orlando,Florida:AcademicPress.

Potter,M.C.,&Wigger,D.C.(2010).MechanicsofFluids(3rdedition).Stamford,CT:Cengage

Learning.

Prilutski,G.,Gupta,R.K.,Sridhar,T.,&Ryan,M.E.(1983).Modelviscoelasticliquids.Journalof

Non-NewtonianFluidMechanics,12(2),233–241.

Pye,D.J.(1964).ImprovedSecondaryRecoverybyControlofWaterMobility.Journalof

PetroleumTechnology,16(8),911–916.

Reiner,M.(1964).TheDeborahNumber.PhysicsToday,17(1),62.

Rodriguez,S.,Romero,C.,Sargenti,M.L.,Müller,A.J.,Sáez,A.E.,&Odell,J.A.(1993).Flowof

polymersolutionsthroughporousmedia.JournalofNon-NewtonianFluidMechanics,

49(1),63–85.

52

Sangani,A.S.,&Acrivos,A.(1982).Slowflowpastperiodicarraysofcylinderswithapplication

toheattransfer.InternationalJournalofMultiphaseFlow,8(3),193–206.

Skartsis,L.,Khomami,B.,&Kardos,J.L.(1992).Polymericflowthroughfibrousmedia.Journal

ofRheology,36(4),589.

Souvaliotis,A.,&Beris,A.N.(1992).Applicationsofdomaindecompositionspectralcollocation

methodsinviscoelasticflowsthroughmodelporousmedia.JournalofRheology,36(7),

1417–1453.

Talwar,K.K.,&Khomami,B.(1992).Applicationofhigherorderfiniteelementmethodsto

viscoelasticflowinporousmedia.JournalofRheology,36(7),1377.

Talwar,K.K.,&Khomami,B.(1995).Flowofviscoelasticfluidspastperiodicsquarearraysof

cylinders:inertialandshearthinningviscosityandelasticityeffects.JournalofNon-

NewtonianFluidMechanics,57(2–3),177–202.

Tirtaatmadja,V.(1993).Afilamentstretchingdeviceformeasurementofextensionalviscosity.

JournalofRheology,37(6),1081.

Vorwerk,J.,&Brunn,P.O.(1991).PorousmediumflowofthefluidA1:effectsofshearand

elongation.JournalofNon-NewtonianFluidMechanics,41(1),119–131.

White,F.M.(1994).FluidMechanics(3rded.).Boston,Massachusetts:McGrawHill.

Yip,R.(2011).SlowFlowofViscoelasticFluidsthroughFibrousPorousMedia.Universityof

Toronto.

53

AppendixA:ExpandedUpper-ConvectedMaxwellandOldroyd-BEquations

Theequationsforthe2-dimensionalupper-convectedMaxwellmodelinCartesiancoordinates

are,

𝜏99 + 𝜆𝜕𝜏99𝜕𝑡

+ 𝑢𝜕𝜏99𝜕𝑥

+ 𝑣𝜕𝜏99𝜕𝑦

− 2𝜏99𝜕𝑢𝜕𝑥

− 2𝜏9:𝜕𝑢𝜕𝑦

= 𝜂 2𝜖99 (A.1)

𝜏9: + 𝜆

𝜕𝜏9:𝜕𝑡

+ 𝑢𝜕𝜏9:𝜕𝑥

+ 𝑣𝜕𝜏9:𝜕𝑦

− 𝜏99𝜕𝑣𝜕𝑥

− 𝜏::𝜕𝑢𝜕𝑦

= 𝜂 𝛾9: (A.2)

𝜏:: + 𝜆

𝜕𝜏::𝜕𝑡

+ 𝑢𝜕𝜏::𝜕𝑥

+ 𝑣𝜕𝜏::𝜕𝑦

− 2𝜏::𝜕𝑣𝜕𝑦

− 2𝜏9:𝜕𝑣𝜕𝑥

= 𝜂 2𝜖:: (A.3)

where,

𝜖99 =𝜕𝑢𝜕𝑥, 𝜖:: =

𝜕𝑣𝜕𝑦. (A.4)

Theequationsforthe2-dimensionalOldroyd-BequationsinCartesiancoordinatesare,

𝜏99 + 𝜆𝜕𝜏99𝜕𝑡

+ 𝑢𝜕𝜏99𝜕𝑥

+ 𝑣𝜕𝜏99𝜕𝑦

− 2𝜏99𝜕𝑢𝜕𝑥

− 2𝜏9:𝜕𝑢𝜕𝑦

= 𝜂 2𝜖99 + 𝜆7 2𝜕𝜖99𝜕𝑡

+ 2𝑢𝜕𝜖99𝜕𝑥

+ 2𝑣𝜕𝜖99𝜕𝑦

− 4𝜖997 − 2𝛾9:𝜕𝑢𝜕𝑦

(A.5)

𝜏9: + 𝜆𝜕𝜏9:𝜕𝑡

+ 𝑢𝜕𝜏9:𝜕𝑥

+ 𝑣𝜕𝜏9:𝜕𝑦

− 𝜏99𝜕𝑣𝜕𝑥

− 𝜏::𝜕𝑢𝜕𝑦

= 𝜂 𝛾9: + 𝜆7𝜕𝛾9:𝜕𝑡

+ 𝑢𝜕𝛾9:𝜕𝑥

+ 𝑣𝜕𝛾9:𝜕𝑦

− 2𝜖99𝜕𝑣𝜕𝑥

− 2𝜖::𝜕𝑢𝜕𝑦

(A.6)

𝜏:: + 𝜆𝜕𝜏::𝜕𝑡

+ 𝑢𝜕𝜏::𝜕𝑥

+ 𝑣𝜕𝜏::𝜕𝑦

− 2𝜏::𝜕𝑣𝜕𝑦

− 2𝜏9:𝜕𝑣𝜕𝑥

= 𝜂 2𝜖:: + 𝜆7 2𝜕𝜖::𝜕𝑡

+ 2𝑢𝜕𝜖::𝜕𝑥

+ 2𝑣𝜕𝜖::𝜕𝑦

− 4𝜖::7 − 2𝛾9:𝜕𝑣𝜕𝑥

(A.7)

54

AppendixB:FLUENTUserDefinedFunctionInANSYSFLUENT™,theupper-convectedMaxwellmodelstresseswerefoundintheprogram

weredefinedasuser-definedscalars.Userdefinedscalarshaveanequationtemplatedefined

asfollows(ANSYSInc.,2015),

𝜕𝛼¦𝜕𝑡

+𝜕𝜕𝑥§

𝐹§𝛼¦ − 𝛤¦𝜕𝛼¦𝜕𝑥§

= 𝑆ª«, 𝑤ℎ𝑒𝑟𝑒𝑘 = 1,…𝑁dx®¯®�d

Where𝛼¦arethescalarsorvariables,𝐹§ aretheconvectivecoefficients,𝛤¦arethediffusivity

constants,and𝑆ª« arethesourceterms.

Thecoefficientfortheconvectiveterm,𝐹§,wassetas𝑢𝜆torepresentthe𝑢 ⋅ 𝛻𝝉term,the

unsteadyterm,�°«�c

,wassettozero,andthesourcetermswereprogrammedtomatchwiththe

restoftheupper-convectedMaxwellequationsfoundinAppendixA.Followingistheuser-

definedfunctioncodeusedtoobtainthestressresults.

#include"udf.h"#include"math.h"#definedomain_ID1#defineM_Lambda0.6/*RelaxationTime[s]*/#defineM_Eta_P1.9/*PolymerViscosity[Pas]*/#defineM_Eta_S2.1/*SolventViscosity[Pas]*/enum{ Txx, Txy, Tyy, Txx_OB, Txy_OB, Tyy_OB, Txx_Newt, Txy_Newt, Tyy_Newt, Txx_OB_Dif, Txy_OB_Dif, Tyy_OB_Dif, N_REQUIRED_UDS

55

};DEFINE_ON_DEMAND(calculate_UCM_values){ Domain*d; cell_tc; cell_tc0; Thread*t; Thread*t0; face_tf; d=Get_Domain(1); thread_loop_c(t,d) { begin_c_loop_all(c,t) { C_UDMI(c,t,0)=C_UDSI_G(c,t,Txx_OB)[0]; C_UDMI(c,t,1)=C_UDSI_G(c,t,Txy_OB)[0]; C_UDMI(c,t,2)=C_UDSI_G(c,t,Txy_OB)[1]; C_UDMI(c,t,3)=C_UDSI_G(c,t,Tyy_OB)[1]; C_UDMI(c,t,4)=C_UDSI_G(c,t,Txx_Newt)[0]; C_UDMI(c,t,5)=C_UDSI_G(c,t,Txy_Newt)[0]; C_UDMI(c,t,6)=C_UDSI_G(c,t,Txy_Newt)[1]; C_UDMI(c,t,7)=C_UDSI_G(c,t,Tyy_Newt)[1]; C_UDMI(c,t,8)=C_UDSI_G(c,t,Txx_OB_Dif)[0]; C_UDMI(c,t,9)=C_UDSI_G(c,t,Txy_OB_Dif)[0]; C_UDMI(c,t,10)=C_UDSI_G(c,t,Txy_OB_Dif)[1]; C_UDMI(c,t,11)=C_UDSI_G(c,t,Tyy_OB_Dif)[1]; } end_c_loop_all(c,t) } }DEFINE_ON_DEMAND(initialize_UCM){ Domain*d; cell_tc; Thread*t; d=Get_Domain(1);

56

thread_loop_c(t,d) { begin_c_loop_all(c,t) { C_UDSI(c,t,Txx)=2*M_Eta_S*C_DUDX(c,t); C_UDSI(c,t,Txy)=M_Eta_S*(C_DVDX(c,t)+C_DUDY(c,t)); C_UDSI(c,t,Tyy)=2*M_Eta_S*C_DVDY(c,t); } end_c_loop_all(c,t) } }DEFINE_EXECUTE_ON_LOADING(define_variables,libname){ if(n_uds<N_REQUIRED_UDS) Internal_Error("notenoughuserdefinedscalarsallocated"); offset=Reserve_User_Scalar_Vars(3); Set_User_Scalar_Name(Txx,"txx"); Set_User_Scalar_Name(Txy,"txy"); Set_User_Scalar_Name(Tyy,"tyy"); Set_User_Scalar_Name(Txx_OB,"txxOB"); Set_User_Scalar_Name(Txy_OB,"txyOB"); Set_User_Scalar_Name(Tyy_OB,"tyyOB"); Set_User_Scalar_Name(Txx_Newt,"txxNewt"); Set_User_Scalar_Name(Txy_Newt,"txyNewt"); Set_User_Scalar_Name(Tyy_Newt,"tyyNewt"); Set_User_Scalar_Name(Txx_Newt,"txxOB_Dif"); Set_User_Scalar_Name(Txy_Newt,"txyOB_Dif"); Set_User_Scalar_Name(Tyy_Newt,"tyyOB_Dif");}DEFINE_SOURCE(txx_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=2*M_Eta_P*C_DUDX(c,t)+2*M_Lambda*C_UDSI(c,t,Txy)*C_DUDY(c,t)-C_UDSI(c,t,Txx)+2*M_Lambda*C_UDSI(c,t,Txx)*C_DUDX(c,t); dS[eqn]=2*M_Lambda*C_DUDX(c,t)-1;

57

returnsource;}DEFINE_SOURCE(txy_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=M_Eta_P*(C_DVDX(c,t)+C_DUDY(c,t))+M_Lambda*C_UDSI(c,t,Txx)*C_DVDX(c,t)+M_Lambda*C_UDSI(c,t,Tyy)*C_DUDY(c,t)-C_UDSI(c,t,Txy); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(tyy_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=2*M_Eta_P*C_DVDY(c,t)+2*M_Lambda*C_UDSI(c,t,Txy)*C_DVDX(c,t)-C_UDSI(c,t,Tyy)+2*M_Lambda*C_UDSI(c,t,Tyy)*C_DVDY(c,t); dS[eqn]=2*M_Lambda*C_DVDY(c,t)-1; returnsource;}DEFINE_SOURCE(test_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=C_UDSI(c,t,1)-C_UDSI(c,t,0); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Txx_OB_source,c,t,dS,eqn){ realsource; realx[ND_ND];

58

source=2*M_Eta_S*C_DUDX(c,t)+C_UDSI(c,t,Txx)-C_UDSI(c,t,Txx_OB); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Txy_OB_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=M_Eta_S*(C_DVDX(c,t)+C_DUDY(c,t))+C_UDSI(c,t,Txy)-C_UDSI(c,t,Txy_OB); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Tyy_OB_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=2*M_Eta_S*C_DVDY(c,t)+C_UDSI(c,t,Tyy)-C_UDSI(c,t,Tyy_OB); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Txx_Newt_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=2*(M_Eta_S+M_Eta_P)*C_DUDX(c,t)-C_UDSI(c,t,Txx_Newt); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Txy_Newt_source,c,t,dS,eqn){ realsource; realx[ND_ND];

59

source=(M_Eta_S+M_Eta_P)*(C_DVDX(c,t)+C_DUDY(c,t))-C_UDSI(c,t,Txy_Newt); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Tyy_Newt_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=2*(M_Eta_S+M_Eta_P)*C_DVDY(c,t)-C_UDSI(c,t,Tyy_Newt); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Txx_OB_Dif_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=C_UDSI(c,t,Txx)-2*M_Eta_P*C_DUDX(c,t)-C_UDSI(c,t,Txx_OB_Dif); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Txy_OB_Dif_source,c,t,dS,eqn){ realsource; realx[ND_ND]; source=C_UDSI(c,t,Txy)-(M_Eta_P)*(C_DVDX(c,t)+C_DUDY(c,t))-C_UDSI(c,t,Txy_OB_Dif); dS[eqn]=-1; returnsource;}DEFINE_SOURCE(Tyy_OB_Dif_source,c,t,dS,eqn){ realsource;

60

realx[ND_ND]; source=C_UDSI(c,t,Tyy)-2*(M_Eta_P)*C_DVDY(c,t)-C_UDSI(c,t,Tyy_OB_Dif); dS[eqn]=-1; returnsource;}DEFINE_UDS_FLUX(uds_flux,f,t,i){ realflux=0.0; realrho=C_R(F_C0(f,t),THREAD_T0(t)); flux=F_FLUX(f,t)/rho*M_Lambda; returnflux;}

61

AppendixC:MeshConvergenceAnalysisatCylinderPoles

FigureC.1:Meshconvergenceanalysisof2.5%solidvolumefractionatthecylinderpole.

FigureC.2:Meshconvergenceanalysisof5%solidvolumefractionatthecylinderpole.

200

210

220

230

240

250

260

26711 49629 93918 181266 351883

τ xxStressat(0.5D

,0)(Pa)

NumberofElements

227

228

229

230

231

232

233

234

235

236

23665 51948 98506 183044 352389

τ xxStressat(0.5D

,0)(Pa)

NumberofElements

62

FigureC.3:Meshconvergenceanalysisof10%solidvolumefractionatthecylinderpole.

260

262

264

266

268

270

272

274

276

23216 44297 85553 167713 329164

τ xxStressat(0.5D

,0)(Pa)

NumberofElements

63

AppendixD:ComparisonofNewtonianVelocityProfilestoParticleImageVelocimetryResultsfromYip(2011)

ThefollowinggraphscompareNewtoniancomputationalresultsfromthisworkandparticle

imagevelocimetryresultsfromYip(2011)fortheB2Fluid.

FigureD.1.2.5%arrayvelocityprofilesbetweenparallelcylinders(x/L=0).Particleimage

velocimetryresultsareadaptedfromYip(2011).

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized

Velocity

,u/U

y/LPosition

Newt.(CFD)

Newt.PIV,Yip(2011)De=0.2PIV,Yip(2011)De=1.6PIV,Yip(2011)

64

FigureD.2.2.5%arrayvelocityprofilesattheinletboundary(x/L=-0.5).ParticleimagevelocimetryresultsareadaptedfromYip(2011).

FigureD.3.5%arrayvelocityprofilesbetweenparallelcylinders(x/L=0).ParticleimagevelocimetryresultsareadaptedfromYip(2011).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized

Velocity

,u/U

y/LPosition

Newt.(CFD)

Newt.PIV,Yip(2011)De=0.2PIV,Yip(2011)De=1.6PIV,Yip(2011)

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized

Velocity

,u/U

y/LPosition

Newt.(CFD)

Newt.PIV,Yip(2011)

De=0.3PIV,Yip(2011)

De=2.1PIV,Yip(2011)

65

FigureD.4.5%arrayvelocityprofilesattheinletboundary(x/L=-0.5).Particleimage

velocimetryresultsareadaptedfromYip(2011).

FigureD.5.10%arrayvelocityprofilesbetweenparallelcylinders(x/L=0).Particleimage

velocimetryresultsareadaptedfromYip(2011).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized

Velocity

,u/U

y/LPosition

Newt.(CFD)

Newt.PIV,Yip(2011)De=0.3PIV,Yip(2011)De=2.1PIV,Yip(2011)

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized

Velocity

,u/U

y/LPosition

Newt.(CFD)

Newt.PIV,Yip(2011)

De=0.3PIV,Yip(2011)

De=1.4PIV,Yip(2011)

66

FigureD.6.10%arrayvelocityprofilesattheinletboundary(x/L=-0.5).ParticleimagevelocimetryresultsareadaptedfromYip(2011).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized

Velocity

,u/U

y/LPosition

Newt.(CFD)

Newt.PIV,Yip(2011)De=0.3PIV,Yip(2011)De=1.4PIV,Yip(2011)