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JournalofNon-NewtonianFluidMechanics294(2021)104580
ContentslistsavailableatScienceDirect
JournalofNon-NewtonianFluidMechanics
journalhomepage:
BuildingonOldroyd’sviscoplasticlegacy:Perspectivesandnew
developments
,,,∗
aDepartmentofMathematics,UniversityofBritishColumbia,Vancouver,BC,V6T1Z2,Canada
bDepartmentsofMathematics&MechanicalEngineering,ImperialCollege,LondonSW72AZ,UK
cDepartmentofMathematics,UniversityCollege,LondonWC1H0AY,UK
ABSTRACT
Thedecadefollowingthesecondworldwarheraldedthepublicationofacollectionofimportantpapersonnon-Newtonianfluidmechanics;Oldroyd’swork
,butOldroyd’sstyleandmethodssetthesceneforsubsequentwork
inthearea,exploitingmathematicalanalysistoformulateproblems,’snamewillforeverbelinked
withthestudyofelasticfluids,thepurposeofthepresentpaperistoofferamodernperspectiveonanumberofOldroyd’spapersonviscoplasticfluidsfrom
1947–1951[1–8].Alongtheway,wesprinkleinabriefreviewofsomeofthesubsequentdevelopmentsstemmingfromOldroyd’sadvances,togetherwithafew
’soriginalpapers,,
wecomplementthisdiscussionwithalubricationanalysis,extending,clarifyingandcorrectingtheimportantoriginalanalysisofWaltonandBittleston(1991)
[9];althoughlubricationtheorywasnotdirectlyutilizedbyOldroyd,themethodologyalignswithhisphilosophyofusingasymptoticandanalyticalapproaches.
’seightviscoplasticpapersLast,beingtheonlypermittedtensor-invariantformthatisindependent
ofthepressure,OldroydappliesthevonMisescriterionforyield,
.‘Arationalformulation’[1]√∑
1≡=,(2)
2Y
ThefirstofOldroyd’sforaysintoviscoplasticityresultedinhis,
seminalpaperona‘rationalformulationoftheequationsofplasticforsomepositiveconstant‘yieldstress’
flowforaBinghamsolid’[1].Inthispaper,followingthetensorial
formulationofcontinuummechanics,OldroydtakesBingham’sconcept=Y,(3)
ofamaterialthatcan‘supportfinitestresselasticallywithoutflowand
whichflows...whenthestressesaresufficientlygreat’anddeterminesandwearriveattheBinghamlaw,
thenowwell-knownthree-()
Y
paperhasbeenwidelycitedandprovidesadefinitiveformulation=2+̇̇if⩾Y,(4)
ofthistensoriallaw,whichformsthebasisofanymodernstudyof1∑
=,,Oldroydcomplemented
2
Oldroydbeginsarmedonlywiththeassumptionsthattheyield(4)withalinearlyelasticrheologicallaw,accountingforsolid-like
conditiondependsonthedeviatoriccomponentsofthestressalone,-
plasticviscosityaboveyieldisconstant,,andcompletetheconstitutive
Hethuswritesdownageneralrheologicallawforthedeviatoricstressmodelbydemandinġ=0when<Y.
aboveyield,ArmedwiththetensorialBinghamlaw,Oldroydcontinuesonto
discusstheenergeticsofayield-stressmaterial,establishingaminimum
=+2̇,(1)[10]also
intermsofatracelessyield-stresstensorand(deviatoric)strainratederivesthisprinciple,whilstalsoestablishingthemaximumprincipal
=
himtoarguethattheprincipaldirectionsof,anḋallcoincide,theory:together[1,10]laykeyfoundationsforvariationalformulations
.
∗Correspondingauthor.
E-mailaddress:d.******@().
/
Received9February2021;Receivedinrevisedform20May2021;Accepted22May2021
Availableonline1June2021
0377-0257/©:.
-NewtonianFluidMechanics294(2021)104580
.(a)AsketchofOldroyd’(b)and(c)weshownumericalsolutionsforBi≫1(hereBi=2048)adaptedfrom[11]withdifferentjetwidths(=1∕2,1∕8
respectively).
Oldroyd’spaperarrivedduringtheperiodthatplasticitytheorywas
beingshapedintotheformthatwecurrentlyrecognize,withthenow-
classicaltextsbyHill[12]andPrager&Hodge[13]emergingroughly
,Pragerandco-workersestablishedthe
three-dimensionalformulationoftheBinghamlawsomewhatearlier
thanOldroyd,takingaperspectivemuchclosertosolidmechanicsand
plasticitytheory[14,15].Nonetheless,theapproachofOldroyd[1],
anditsrheologicalsetting,marksthestartoffluidviscoplasticityaswe
materialsas‘Binghamsolids’–thatis,elasticsolidsthatcanundergo
plasticflowforlargeenoughstress–ratherthanthemorestandard
modernviewof‘Binghamfluids’thatflowlikeaviscousfluidforlarge
’s
explicitallowanceofthematerialtodeformelasticallybelowyieldil-
lustratedthisviewpoint,whichhasresurfacedonanumberofoccasions
(.[16])buthasonlyrelativelyrecentlyregainedtractioninmodern
viscoplasticmodelling[17].
.‘Aplasticboundary-layertheory’[2]
Oldroyd’ssecondpaperoutlinesaplasticboundary-layertheoryfor
atwo-dimensionalBinghamfluidmotivatedbytheclassicalBlasius
predictsthattheusualviscousshearstressacrosstheboundarylayer
canbecombinedwithcontributionsfromtheyieldstresstobalance
−1313
boundarylayeremergesthatscalesasBi,where
Y
Bi=(5)
,andaretypicallengthandvelocity
≫
1,.(a)AsketchofOldroyd’(b)and(c)weshowanumerical
[11]withBi=512;(b)showstheflowspeedintheboundary
Oldroyd’sboundary-layerequationisrathermoredauntingthanlayeragainsttheknife,and(c)showsthelogarithmofthestrainrateoverawider
thatinBlasiustheory,(white)streamlinesareincludedinbothpanels.
,
Oldroydwentontoshowthattheequationadmittedaself-similar
solutioncorrespondingtoathickeningboundarylayerthatbridgedordownacircularpipe[6](hisequation(57)),suggestthat,insuch
betweeneithertwoplugs,tworegionsofalmostperfectlyplasticdefor-cases,,it
mation,,afreeviscoplasticshearturnsoutthatboundarylayersinaBinghamfluidagainstarigidwall1
−2[11,18],and
Oldroyd’sanalysisrunsintodifficultieswhentheboundarylayerinvolvearathersimplerbalancebetweenviscousstressesandpressure
buffersawall,beingunabletosatisfyalltheboundaryconditionsandgradients,withplastictermsplayingnorole.
,theillustratehisboundary-layertheory:plasticflowaroundamoving
analysisofexactunidirectionalflowsolutions,suchasOldroyd’sforknifeandaplasticjetemergingfromanorificeinaplanewall.
viscoplasticflowbetweencoaxialcylinders[3](hisequations(7)-(8)),Theseexampleshaveindeedbeenfoundby[11]topossessboundary
2:.
-NewtonianFluidMechanics294(2021)104580
layerswithOldroyd’sself-similarstructure;).Consequently,foraneccentricannulus,theinneryielded
firstfigureshowstheboundary-layerstructuredevelopingattheedgesregionbecomesaxisymmetricaboutthecentreoftheinnercircleonce
’
numericalsimulations(),hisconstructionisnotconstructsolutionswithhigheryieldstress(wheretheouterboundary
thecompletestory:ifthewidthofthejetistoosmall,theboundaryislikelytoplugup)withoutknowingpreciselyforwhatrangeof
layersinteractandamorecomplicatedpatternofnearlyperfectlythesemightarise.
plasticflowdevelopsacrosstheorifice;thejettherebyexpandsbeforeIn[4],Oldroydrepeatstheperturbationexpansionforboundaries
settlingintoOldroyd’spattern(see[11]forfurtherdetails).,thesolutionsarelimitedto
Oldroyd’ssecondexampleistheviscoplasticflowaroundatrans-thefullyyieldedregime,withapluggedouterellipseexpectedathigh
lating,two-dimensionalplate,or‘‘knife’’(),,however,theellipticalgeometryprecludesa
hasmotivatedanumberofexperiments[19–21].However,Oldroyd’,Oldroydinvents
theorydoesnot,infact,applytotheboundarylayersagainstthesideaniterativetechniquetoconvergetothecorrectsolution:heusesthe
oftheknife,aspointedoutbyPiau[18,22];theselayersinsteadhavea11solutionforflowbetweentwowallstoapproximatethatbetweena
Bi−2scalingforBinghamfluid(orBi−+1forHerschel–Bulkleyfluid).,byacleverconstruction,herefinesthis
Nevertheless,theexampleprovidesaconvenientillustrationofthisapproximationtocorrecttheyieldsurfaceposition,inamannerthat
lattertypeofboundarylayer,asillustratedbythenumericalexamplecouldbeappliedrepeatedlyforsuccessiveimprovements.
’spointaboutcloakingiswidelyexploitedinviscoplastic
insteadofthehalf-(.[33,34]).Itappliesinotherconduit
surprises:mostnotably,theboundarylayersagainsttheplatearenotflowswhenaplugintervenesbetweenthetwoboundaries,isolating
,eachshearedzoneisaffectedonly
plasticdeformationalsoarisesatthefrontandbackoftheplateasbytheshapeofadjacentboundary,andindependentoftheotherone.
,thefreeshearlayersthatTheasymmetriesbetweentheboundaryshapesisthereforeentirelyac-
bordertherotatingplugaredescribedbyOldroyd’sboundary-
’stheorydoesthereforeapplytotheknifeproblem,&Hassager[25],
butnotasinOldroyd’-scaledeformationofthisandhowsolutionsforflowincertaingeometriesleadtosolutionsin
formawayfromtheplatehasalsobeenobservedexperimentally[20]otherconduits().
andfoundtheoreticallyforflowsaroundellipseswithhighaspectratioOldroyd’sthirdpaperinthisseriesonconduitflowdigressesfurther
[23].intothedifferentialgeometryunderlyingtheproblem,considering
WerevisitOldroyd’sboundarylayertheoryandcanonicalexamplesmoregeneralboundingsurfaces,assumingonlythatthevelocitycon-
belowinavariationofthetwo-(andthereforewallsandplugs)arecoordinatelinesofsome
Inparticular,
arbitrarycross-
thesubjectofthethirdandfourthseriesofOldroyd’spapers,discussedflowofaBinghamfluidwithinarelativelycomplicatedgeometry(see
inthenextsubsections,).
[6]takesasomewhatdifferenttack,considering
stress,orclosetotheonsetofflow,
thetwotypesofviscoplasticboundarylayer