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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
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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)