Microscopic modelling of orientation kinematics of non-spherical particles suspended in confined flows using unilateral mechanics

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Adrien SCHEUER, Emmanuelle ABISSET-CHAVANNE, Francisco CHINESTA, Roland

KEUNINGS - Microscopic modelling of orientation kinematics of non-spherical particles

suspended in confined flows using unilateral mechanics - Comptes Rendus - Mecanique - Vol.

346, p.48-56 - 2018

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Microscopic

modelling

of

orientation

kinematics

of

non-spherical

particles

suspended

in

confined

flows

using

unilateral

mechanics

Adrien Scheuer

a

,

b

,

Emmanuelle Abisset-Chavanne

a

,

Francisco Chinesta

c

,

∗

,

Roland Keunings

b

a_{ICI – Institut de calcul intensif & ESI GROUP Chair, École centrale de Nantes, 1, rue de la Noe, 44300 Nantes, France} b_{ICTEAM, Université catholique de Louvain, avenue Georges-Lemaître 4, B-1348 Louvain-la-Neuve, Belgium} c_{PIMM & ESI GROUP Chair, ENSAM ParisTech, 151, boulevard de l’Hôpital, 75013 Paris, France}

a

b

s

t

r

a

c

t

Keywords:

Fibresuspensions Jeffery’sequation Confinement

Thepropertiesofreinforcedpolymersstronglydepend onthemicrostructuralstate, that is, the orientation state of the fibres suspended in the polymeric matrix, induced by the forming process. Understanding flow-induced anisotropy is thus a key element to optimizebothmaterials and process. Despite theimportant progressesaccomplished in the modelling and simulation of suspensions,few works addressed the fact that usual processingflowsevolveinconfined configurations,whereparticlescharacteristiclengths maybegreaterthanthethicknessofthe narrowgapsinwhichthe flowtakesplace.In thosecircumstances,orientation kinematicsmodels proposedfor unconfinedflows must beextendedtotheconfinedcase.Inthisshortcommunication,weproposeanalternative modellingframework basedonthe useofunilateralmechanics,consequently exhibiting aclearanalogywithplasticityandcontactmechanics.Thisframeworkallowsustorevisit themotionofconfinedparticlesinNewtonianandnon-Newtonianmatrices.Wealsoprove thattheconfinedkinematicsprovidedbythismodelare identicaltothosederivedfrom microstructuralapproaches(Perezetal.(2016)[1]).

1. Introduction

Overthelastdecades,compositematerials,madeofasuspendingmatrixandareinforcementcomposedoffibresusedto fortifythematrixintermsofstrengthandstiffness,weresuccessfullyintroducedintheaerospaceandautomotiveindustries and provedto be a lightweight alternativeto producestructural andfunctionalparts. Themechanical propertiesofsuch reinforcedpolymers,however,stronglydependontheorientationstateoftheirmicrostructure,whichisestablishedduring theformingprocess[2].Predictingtheevolutionofthisorientationstateisthusakeyyetcomplextask,sincethemotionof thereinforcingfibresisimpactedbytheflowingmatrixandtheinteractionswiththeneighbouringfibresandcavitywalls. Thus,flow-inducedanisotropymustbeunderstoodandmodelledinordertooptimizebothmaterialsandprocesses.

*

Correspondingauthor.

E-mail addresses:Adrien.Scheuer@ec-nantes.fr,Adrien.Scheuer@uclouvain.be(A. Scheuer),Emmanuelle.Abisset-chavanne@ec-nantes.fr

Thereisalonghistoryofstudiesofthemotionofslenderbodiessuspendedinaviscousfluid,startingfromtheseminal work of Jefferyback in 1922 [3]. A vastliterature dedicatedto fibre and non-spherical particle suspensionsis available, studyingextensivelydifferentmodellingscalesandexploringtheimpactoftheconcentrationregimeandthenatureofthe suspendingmatrix.Schematically,thethreemainmodellingscales involvedwhenaddressingtheorientationkinematicsof suspendedparticlescanbesummarizedasfollows:(i)themicroscopic scale,thescaleofasingleparticle;(ii)themesoscopic

scale,thescaleofapopulationofparticle,whoseconformationisusuallyrepresentedbyaprobabilitydensityfunction(pdf) oforientation,givinganunambiguousandcompletedescriptionoftheorientationstate;and(iii)themacroscopic scale,the scaleof thepart,whoseconformationisoften givenbythe firstmoments oftheaforementionedpdf, providinga coarse yet concisedescriptionof theorientation state in thepart.Depending on thelevel ofdetail andaccuracyrequired fora givenapplication,aspecific scale,oracombinationofthemmightbe chosen.Forfurtherdetailonthatsubject,including theso-calledmultiscaleapproach,wereferthereadertothereview[4]andthemonograph[5],aswellastothereferences therein.Onlyasuccinctoverviewofthemicroscopicmodellingisproposedthereafter.

In [3], Jeffery derived the expression of the hydrodynamic torque exerted on an ellipsoidalparticle immersed in an unboundedcreepingflowofaNewtonianfluid.Hethenobtainedanequationofmotionbyassumingthattheparticlerotates soastoachieveinstantaneouszerotorque,resultingintheso-calledJefferyequation.Consideringaspheroid(axisymetric ellipsoid)anddefiningtheorientationofaparticlebytheunitvectorp alongitsprincipalaxis,Jeffery’sequationreads

˙

pJ

=  ·

p

+

κ

(

D

·

p

− (

D

: (

p

⊗

p

)

p

))

(1)

where



andD are,respectively,theskew-symmetricandsymmetricpartsoftheunperturbedvelocitygradient

∇

v ofthe flow,and

κ

istheshapefactorofthespheroid,givenby

κ

=

r2₋₁

r2₊₁ withr theaspectratiooftheparticle.Slenderbodieslike fibresandrodscanbeassimilatedasinfiniteaspectratioellipsoids(

κ

≈

1).

Jeffery’sequation wasexperimentally verifiedby Taylor[6]andMason[7].Bretherton[8]showedthat theequationis alsovalidforanyaxisymmetric particleprovided thatan effectiveaspect ratioisdetermined.HinchandLeal [9–12] also studiedJeffery’smodel,addressing theimpactofBrownianmotionandproposingconstitutiveequationsforthebehaviour ofsuspensions.However, onlya fewworks, eitherexperimental [13,14] ornumericalandtheoretical [15,16],addressthe fact that flows of industrial interest take place in narrow gaps, whose thickness is of the same order of magnitude or smallerthanthelengthofthereinforcingfibres,thusconstrainingthespaceofpossibleorientation,andasaconsequence thekinematics.

In[1],weproposedamultiscalemodeltodescribetheorientationdevelopmentofadilutesuspensionoffibresconfined in anarrow gap.The microscopic model was basedon a dumbbellrepresentation [17] of theconfined rod, with hydro-dynamic andcontactforces(normal tothe gapwall)actingon it.Thisconfinementmodel was laterextended in[18] to includeunilateralcontacts andnon-uniformstrain ratesatthescale oftherod. In anycase,the resultingkinematics are a combinationof the unconfined Jeffery kinematics anda correction term that preventsthe fibre from leavingthe flow domain,thatis,

˙

p

= ˙

pJ

+ ˙

pC (2)

The equation of motion of such a confined rod, derived in [1] and summarized in Eq. (2), presents thus significant similaritieswithequationsofelastoplaticity.Indeed,wecoulddrawaparallel between,ontheonehand,theclassical un-confinedkinematicsandtheelasticdeformation,and,ontheotherhand,theconfinedmotionoftheparticleandelastoplatic deformation.

Hence,thepurposeofthisshortcommunicationisto explorethealternativemodellingframeworkbasedonunilateral mechanicstorevisitthemotionofconfinedparticlesinNewtonianandnon-Newtonianmatrices.

Thepaperisorganisedasfollows.InSection 2,wederivethemodelfortheconfinedkinematicsofsuspendedparticles using unilateral mechanics. Then, in Section 3, we discuss how this generalframework allows us to build the confined kinematics of fibres and spheroids immersed in a Newtonian (based on Jeffery’s model [3]) or second-order (based on Brunn’smodel[19])viscoelasticfluid.Finally,inSection4,wedrawthemainconclusionsandpresentsomeperspectivesof thisapproach.

Remark.Inthispaper,weconsiderthefollowingtensorproducts,assumingEinstein’ssummationconvention:

– ifa andb arefirst-ordertensors,thenthesinglecontraction“

·

”reads

(

a

·

b

)

=

ajbj; – ifa andb arefirst-ordertensors,thenthedyadicproduct“

⊗

”reads

(

a

⊗

b

)

jk

=

aj bk;

– ifa andb arerespectivelysecondandfirst-ordertensors,thenthesinglecontraction“

·

”reads

(

a

·

b

)

j

=

ajkbk; – ifa andb aresecond-ordertensors,thenthedoublecontraction“

:

”reads

(

a

:

b

)

=

ajkbkj.

2. Confinedorientationkinematicsusingunilateralmechanics

Inthissection,wederivestepbystepthekinematicsofaconfinedsuspendedparticleusingtheframeworkofunilateral mechanics.

(i) Additivedecomposition. We assume that the particle kinematics(particle rotary velocity)can be decomposedinto an unconfined(U)andconfined(C)contribution,accordingto

˙

p

= ˙

pU

+ ˙

pC (3)

Bydefinition,theorientationvectorissubjecttothenormalizationcondition

p

·

p

=

1 (4)

(ii) Unconfinedcontribution. The unconfined equation of motion p

˙

U is either givenby a model from the literature, e.g., Jeffery’s equation [3]forellipsoidal particles immersedina Newtonian fluid or Brunn’smodel [19] inthe caseof a second-order (non-Newtonian) matrix,orcould be estimatedfromexperimental observations.In orderto satisfy the normalizationconditionEq.(4),theunconfinedkinematicsshouldverifyp

˙

U

·

p

=

0.

(iii) Alloweddomainandconfinementcondition. We define a function f

:

S → R

calledthe confinement condition (the equivalentoftheyieldconditioninplasticitymechanics)andconstraintheadmissibleorientationstatesp

∈

S

(with

S

theunitsphere)tolieintheflowdomain

D

,ensuringthegapwalls’impenetrability,definedas

D

=



p

∈

S

|

f

(

p

)

=

p

·

n

−

H L

≤

0



(5) whereL isthesemi-lengthoftheparticle,H thegapsemi-width,andn denotesaunitvectornormaltothegapwall. Weassume,withoutlossofgenerality,thatp pointstowardstheupperhemisphere.

Werefertotheinteriorof

D

,denotedbyint

(D)

anddefinedasint

D) = {

p

∈

S |

f

(

p

) <

0

}

,astheunconfineddomain, whereastheboundary

∂

D

,givenby

∂

D = {

p

∈

S |

f

(

p

)

=

0

}

,iscalledtheconfinementsurface.

(iv) Confinedcontributionandconsistencyrequirement. Theconfinedkinematicsp

˙

C _{isobtainedfromthegradientofthe} con-finementconditionintroducedabove,i.e.

˙

pC

= −

γ

d f

dp (6)

where

γ

istheconsistencyparameter.Thederivativeof f

(

p

)

withrespecttop,enforcingthenormalizationcondition, reads

d f

dp

=

n

− (

p

·

n

)

p (7)

whichisobtainedby subtracting fromthe derivativeof f

(

p

)

withrespectto p itsprojection ontothep direction in ordertoensurethatp

˙

C

·

p

=

0.

Theconsistencyparameter

γ

isassumed,ontheonehand,toobeytheKuhn–Tuckercomplementaryconditions

γ

≥

0

,

f

(

p

)

≤

0

,

and

γ

f

(

p

)

=

0 (8)

and,ontheotherhand,tosatisfytheconsistencyrequirement

γ

˙

f

(

p

)

=

0 (9)

Toobtainthederivativeof f withrespecttotime,weproceedasfollows:

˙

f

=

d f

dp

· ˙

p (10)

= (

n

− (

p

·

n

)

p

)

· (˙

pU

−

γ

(

n

− (

p

·

n

)

p

))

(11)

= ˙

pU

·

n

−

γ

(

1

− (

p

·

n

)

2

)

(12)

Thusthevalueof

γ

isgivenby

γ

=

(

p

˙

U

_·

_n

₎

(

1

− (

p

·

n

)

2

₎

(13)

(v) Summary. Theresultingkinematicscanbesummarizedasfollows:

f

<

0

⇐⇒

p

∈

int

(

D

),

γ

=

0

=⇒ ˙

p

= ˙

pU unconfined f

=

0

⇐⇒

p

∈ ∂

D

,

⎧

⎪

⎨

⎪

⎩

˙

f

=

0 and

γ

>

0

=⇒ ˙

p

= ˙

pU

+ ˙

pC

˙

f

=

0 and

γ

=

0

=⇒ ˙

p

= ˙

pU

˙

f

<

0

=⇒

γ

=

0

=⇒ ˙

p

= ˙

pU confinement force-free contact unconfined detachment (14)

Theforce-freecontactactuallycorrespondstothecasewheretheconfinedparticleistouchingthegapwall,butisnot tryingtoleavetheflowdomain(to“push”onthewall).

Fig. 1. Confined rod suspended in a Newtonian fluid (solid line: confined particle; dotted line: unconfined particle).

Remark2.1.Forthesake ofclarity, we assumedinthepresentsection andthroughouttheremainder ofthisarticle that underconfinement,bothextremitiesofthesuspendedparticleareincontactwiththegapwallsandthecentreofgravity oftheparticleisfixedinthemid-planeoftheflowchannel.Theseassumptionsare,however,relaxedinAppendix A,where weaddress,usingthesameframeworkjustintroduced, thegeneralcasebyintroducingthepositionoftheparticlecentre ofgravityintheconfinementcondition,allowinginthemeantimecontactwithonlyonegapwall.

3. Discussion

Theconfinedkinematicsobtainedintheprevioussectionread

˙

p

= ˙

pU

+ ˙

pC

= ˙

pU

−

(

p

˙

U

_·

_n

₎

(

1

− (

p

·

n

)

2

₎

(

n

− (

p

·

n

)

p

)

(15)

which coincidesexactly withthe expressionobtainedin [1]following amicrostructuralapproach fora confinedrod im-mersedinaNewtonianfluid(p

˙

U_{wasthusgivenbyJeffery’skinematics).}

However,theunilateralmechanicsapproachdevelopedintheprevioussectiondoesnotassumeanythingontheshape ofthe suspendedparticles orthenature ofthematrix fluid.Onlyan expression oftheunconfined kinematics isactually necessary,whichallowsustoextendstraightforwardlyourmodeldescribingthemotionofaconfinedparticletosituations wherethemicrostructuralapproachfrom[1]mightbetedious.

Inthissection,wethusdiscussthreescenarios:(i) a rodimmersedinaNewtonianfluid(developedin[1]),(ii) a spheroid inaNewtonianfluid,and(iii)aspheroidinasecond-orderfluid.Wealsoprovidesomenumericalillustrationsinthecase ofalinearshearflowv

=



γ

˙

z 0 0

T,with

γ

˙

=

1 inachannelofheight H

=

0

.

25L

3.1. ConfinedrodsuspendedinaNewtonianfluid

Inserting Jeffery’s equation forrods p

˙

J_rod

= ∇

v

·

p

− (∇

v

: (

p

⊗

p

)

p

)

asunconfined kinematics in Eq. (15),we recover thekinematics derivedin [1]using amicrostructuralapproach(hydrodynamicandcontactforcesactingonthe dumbbell representationofarod).

Fig. 1depictstheevolutionoftheorientationofarigidfibreimmersedinashearflowofaNewtonian fluid.Thesolid line shows the trajectory of the confined rod andthe dotted line the trajectory of an hypothetical unconfined particle starting from thesame initial orientation. Fig. 1a showsthe evolution ofthe rod orientation asa trajectory on the unit sphere.Startingfromthesameinitialcondition,theconfined(solidred)andunconfined(dottedblue)rodsbothfollowthe sameJefferyorbit.Whenittouchesthewall,theconfinedrodisconstrainedtoslidealongthegapwall,andfinallycatches anunconfinedJefferyorbittangenttothewallandalignsintheflow.Thisabruptchangeinthetrajectoryasittouchesthe gapwallcanalsobeobservedinFig. 1b,wherethecomponentsoftheunitvectoroforientationp arerepresented.

3.2. ConfinedellipsoidsuspendedinaNewtonianfluid

Consideringspheroidalparticles(axisymmetricellipsoids),weinsertnowthegeneralJefferyequation[Eq.(1)]inEq.(15). Theequivalencewiththekinematicsobtainedfromamicrostructuralapproachonatri-dumbbellisdetailedinAppendix B.

Fig. 2. Confined spheroid (aspect ratio r=4) suspended in a Newtonian fluid (solid line: confined particle; dotted line: unconfined particle).

Fig. 2depictstheevolutionoftheorientationofaspheroidofaspectratior

=

4 immersedinthesameshearflowofa Newtonian fluid.The unconfinedparticle(dotted blue)undergoes itsclassicalkayaking motion,whereas theconfinedone (solidred)firstslidesalongthewall,andisthenconstrainedonthelargestkayakingorbitpossibleinthenarrowgap,that is,theorbittangenttobothgapwalls.Notethattheorbitpointstangenttothechannelwallsthuscorrespondtoforce-free contacts.

3.3. Confinedellipsoidsuspendedinasecond-orderfluid

Leal[20]andBrunn[19]published someimportanttheoreticalworkstodescribe,respectively,themotionofarodand anellipsoidinasecond-orderfluid,inthelimitoflowWeissenbergnumbers,whichconstitutesthecounterpartofJeffery’s equation in thecaseofa viscoelasticsuspending matrix.Theconstitutive equation forthesecond-order fluid isgivenby Giesekus[21]

σ

= −

pI

+

2

ηD

+

2

η

k(₀11)D

·

D

+

k(₀2)



∂

D

∂

t

+

v

· ∇

D

+  ·

D

−

D

· 



(16) wherek(₀11)andk(₀2)arematerialparameters.Brunn[19]derivedtheequationofevolutionoftheorientationofaparticle, andtheresultingkinematicsreads

˙

pB

=  ·

p

+

κ

(

D

·

p

− (

D

: (

p

⊗

p

)

p

))

− (

I

−

p

⊗

p

)

·

D

· (

H2D

·

p

+

H1

(

D

: (

p

⊗

p

))

p

)

(17) where H1 andH2 dependonthematerialparametersandtheellipsoidaspectratioasgiveninBrunn[19].

In Appendix C,we briefly showhow towrite Brunn’s kinematics[Eq. (17)] inthe formofJeffery’skinematics Eq.(1) withan effectivevelocity gradient

∇

˜

v.Consequently,the microstructuralvalidationdevelopedinAppendix B canalso be usedinthissection.

In thecaseofasecond-order fluid,itis wellknown thatthe kayakingmotionoftheparticledrifts towards theshear planeforoblatespheroids(r

<

1)ortowardsthevorticityaxisforprolatespheroids(r

>

1)[19,22,23].

Fig. 3depictstheevolutionoftheorientationofa spheroidofaspectratior

=

4 immersedinthesameshearflow,but nowthesuspendingmatrixisa second-orderfluid(inthisillustration, wehavek₀(11)

=

0

.

144 andk(₀11)

= −

0

.

09,andthus,

H1

=

0

.

084 and H2

=

0

.

0056). Again, the confined spheroid (solid red) is constrainedto exert its kayaking anddrifting motiontowardsthevorticityaxisoftheflowinthenarrowflowdomain.

4. Conclusionandperspectives

Thisworkproposesanalternativeroute forderiving theconfinedorientation kinematicsofdilutesuspensionsoffibres andellipsoidalparticles.Thisapproach,basedonunilateralmechanics,allowsustoextenddirectlyourprevious microstruc-turalmodeldevelopedforconfinedrodsinaNewtonianfluid[1]toellipsoidalparticlesandviscoelasticmatrices.

The samestrategy mightthen beapplied atthemacroscopic scale,workingonthe so-calledsecond-orderorientation tensora[24],a

=



(

p

⊗

p

)

ψ(

p

)

dp,toderiveamacroscopicmodelforconfinedsuspensions.However,definingtheadequate confinementcondition f

(

a

)

≤

0 inthatcaseisadelicatetaskthatwillbeaddressedinafuturework.

Fig. 3. Confined spheroid (aspect ratio r=4) suspended in a second-order fluid (solid line: confined particle – dotted line: unconfined particle).

Acknowledgements

A.ScheuerisaResearchFellowofthe“FondsdelarecherchescientifiquedeBelgique”–F.R.S.–FNRS.

Appendix A. Confinedorientationkinematicsusingunilateralmechanics–interactionwithonlyonegapwall

Inthisappendix,we extendtheframeworkintroducedinSection2insituationswherethesuspendedparticle’scentre ofgravitydoesnotnecessarilyliesinthechannelmid-plane,allowing interactionwithonlyonegapwall.Inthatcase,the confinementconditionnowreads

f

(

p

)

= (

xG

+

pL

)

·

n

≤

H (18)

withxG thepositionoftheparticlecentreofgravityG.Thealloweddomain

D

isthusnowgivenby

D

=



p

∈

S

|

f

(

p

)

=

1 L

(

xG

·

n

)

+ (

p

·

n

)

−

H L

≤

0



(19) Notethatinthiscase,weconsiderthecasewhereGliesintheupper-halfofthechannel.

Usingthesenewdefinitions,thederivationoftheconfinedkinematicsisobtainedfollowingthesamerationaleasthat presented in the step (iv)in Section 2. The derivation of f with respect to p [Eq. (7)] is left unchanged, whereas the derivationof f withrespecttotime[Eq.(10)]nowreads

˙

f

=

∂

f

∂

xG

· ˙

xG

+

∂

f

∂

p

· ˙

p (20)

=

1 L

(

n

· ˙

xG

)

+ (

n

− (

p

·

n

)

p

)

· ˙

p (21)

=

1 L

(

n

· ˙

xG

)

+ (

n

− (

p

·

n

)

p

)

· (˙

p U

₋

_γ

₍

_n

_{− (}

_p

_·

_n

₎

_p

₎₎

₍₂₂₎

=

1 L

(

vG

·

n

)

+ (˙

p U

_·

_n

₎

₋

_γ

₍

₁

_{− (}

_p

_·

_n

₎

2

₎

₍₂₃₎

withvG

= ˙

xGthevelocityoftheparticle’scentreofgravity.Finally,thevalueoftheconsistencyparameter

γ

isgivenby

γ

=

1

L

(

vG

·

n

)

+ (˙

pU

·

n

)

(

1

− (

p

·

n

)

2

₎

(24)

leadingtothefollowingexpressionfortheconfinedkinematics

˙

p

= ˙

pU

+ ˙

pC

= ˙

pU

−

1

L

(

vG

·

n

)

+ (˙

pU

·

n

)

(

1

− (

p

·

n

)

2

₎

(

n

− (

p

·

n

)

p

)

(25)

Thisexpressionisexactlytheonethatwederivedin[18]whenaddressingthekinematicsofaconfinedparticleinteracting withonlyonegapwallusingamicrostructuralapproach(dumbbell).

Fig. 4. Hydrodynamic and contact forces acting on a confined suspended spheroid.

Appendix B. Confinedkinematicsofanellipsoidusingthedumbbellapproach

In order toaddress the confinedkinematics ofan ellipsoid immersedin a Newtonian fluid flow, we considerits cor-responding tri-dumbbellrepresentation.For thesake ofsimplicity, we considerin thisappendix the2D case, that is the bi-dumbbellrepresentedinFig. 4,whichwillprovidetheorientationkinematicsofaspheroid(axisymmetricellipsoid).The orientationofthetwoprincipleaxesaregivenbytheunitvectorsp1 andp2.

Oneach beadactsahydrodynamic force (Stokesdrag),scalingwiththedifferenceofvelocity betweenthefluid atthe beadpositionandthebeaditself.Forthebeadlocatedatp1L1,thatforcereads

FH

(

p1L1

)

= ξ(

v0

+ ∇

v

·

p1L1

−

vG

− ˙

p1L1

)

(26) where

ξ

isa frictioncoefficientandvG andv0 denoterespectivelythevelocity ofthecentreofgravityGandthevelocity ofthefluidatG.Withoutanylossofgenerality,weassumethat L1 isthespheroidlongestaxis,onwhichcontactwiththe gapwallwilloccur.Thecontactforcesontheupperandlowerbeadsread

FC

(

p1L1

)

=

μn

and FC

(

−

p1L1

)

= −

μn

(27) withn

=



0 0 1

T (sincethecontactforce isorthogonal tothewall assoonasfrictionisneglected–itcouldbeeasily introduced),anditsintensity

μ

isaprioriunknown.

Whenbothbeadsofthedumbbellalignedalongdirectionp1 areincontactwiththegapwalls(moregeneralsituations wereanalysedin[18]),balanceofforcesyields

2

ξ(

v0

−

vG

)

=

0 (28)

thatis,v0

=

vG,theparticle’scentreofgravityismovingwiththefluidvelocityatthatposition. Balanceoftorquesthusyields

2p1L1

×



∇

v

·

p1L1

− ˙

p1L1

+

μ

ξ

n



+

2p2L2

× (∇

v

·

p2L2

− ˙

p2L2

)

=

0 (29)

Introducingthefactthatp

˙

i

=

ω

×

pi,i

=

1

,

2 (with

ω

theangularvelocityvector),andtakingintoaccountthatpi

×

ω

×

pi

=

ω

,i

=

1

,

2,wehave

ω

=

L1 L2₁

+

L2₂

(

p1

× (∇

v

·

p1L1

+

μ

ξ

n

))

+

L2 L2₁

+

L2₂

(

p2

× (∇

v

·

p2L2

))

(30)

Thus,thespheroidrotaryvelocityisgivenby

˙

p1

=

ω

×

p1

=

L1 L2₁

+

L2₂

((

p1

× (∇

v

·

p1L1

+

μ

ξ

n

))

×

p1

)

+

L2 L2₁

+

L2₂

((

p2

× (∇

v

·

p2L2

))

×

p1

)

(31) Applyingthetriplevectorproductformula

(

a

×

b

)

×

c

= (

a

·

c

)

b

− (

b

·

c

)

a,thepreviousequationreads:

˙

p1

=

L1 L2₁

+

L2₂

((

∇

v

·

p1L1

+

μ

ξ

n

)

− ((∇

v

·

p1L1

)

·

p1

+

μ

ξ

(

n

·

p1

))

p1

)

−

L2 L2₁

+

L2₂

(((

∇

v

·

p2L2

)

·

p1

)

p2

)

(32)

Wenowdevelop thelasttermofthisequationinordertoobtainanexpressionthatonlydependsonp1.First,we decom-posethevelocitygradientaccordingto

∇

v

=

D

+ 

,

((

∇

v

·

p2

)

·

p1

)

p2

= ((

D

·

p2

)

·

p1

)

p2

+ (( ·

p2

)

·

p1

)

p2 (33)

Thenwedevelopeachoftheseterms.UsingthefactthatD issymmetric,wehave

((

D

·

p2

)

·

p1

)

p2

=

p2

(

pT1

·

D

·

p2

)

=

p2

(

pT2

·

D

·

p1

)

= (

p2

⊗

p2

)

·

D

·

p1 (34)

Sincep1 andp2 aremutuallyperpendicular,

(

p1

⊗

p1

)

+ (

p2

⊗

p2

)

=

I (35)

andthusEq.(34)nowreads

((

D

·

p2

)

·

p1

)

p2

= (

I

− (

p1

⊗

p1

))

·

D

·

p1 (36)

Similarly,usingthefactthat



isskew-symmetric,wehave

((

·

p2

)

·

p1

)

p2

=

p2

(

pT1

·  ·

p2

)

= −

p2

(

pT2

·  ·

p1

)

= −(

p2

⊗

p2

)

·  ·

p1 (37) andthus

((

·

p2

)

·

p1

)

p2

= −(

I

− (

p1

⊗

p1

))

·  ·

p1 (38)

Finally,comingbacktoEq.(32),wehave

˙

p1

=

L1 L2₁

+

L2₂

((

∇

v

·

p1L1

+

μ

ξ

n

)

− ((∇

v

·

p1L1

)

·

p1

+

μ

ξ

(

n

·

p1

))

p1

)

−

L22 L2₁

+

L2₂

(

D

·

p1

− (

p1

⊗

p1

)

·

D

·

p1

−  ·

p1

+ (

p1

⊗

p1

)

·  ·

p1

)

(39) or

˙

p1

=  ·

p1

+

L2 1

−

L22 L2₁

+

L2₂D

·

p1

−

L2 1

−

L22 L2₁

+

L2₂

(

p1

⊗

p1

)

·

D

·

p1

+

L1 L2₁

+

L2₂

(

μ

ξ

(

n

− (

n

·

p1

)

p1

))

(40)

The first partof Eq.(40)is actually theclassical Jeffery equation fora spheroid, since L 2 1−L22 L2₁+L2₂

=

r2−1

r2₊₁

=

κ

isthe spheroid shapefactorandthesecondpartisthustheconfinementcontribution,

˙

p1

= ˙

pJ1

+

L1 L2₁

+

L2₂

(

μ

ξ

(

n

− (

n

·

p1

)

p1

))

(41)

Imposingtheimpenetrabilitycondition[1]

(

vG

+ ˙

p1L1

)

·

n

=

0 (42)

allowsustoobtaintheintensity

μ

ofthecontactforce.Sincetheparticlecentreofgravityisinthemid-planeoftheshear flow,vG

=

0 andthus

˙

p1L1

·

n

= ˙

p1JL1

·

n

+

L2 1 L2₁

+

L2₂

(

μ

ξ

(

1

− (

n

·

p1

)

2

₎₎

₌

_{0 (43)}

Thevalueof

μ

isthengivenby

μ

= −

L 2 1

+

L22 L2₁

ξ

L1

(

1

− (

n

·

p1

)

2

)

(

p

˙

1J

·

n

)

(44)

Eventually,theconfinedkinematicsofarigidspheroidaregiven,asexpected,by

˙

p1

= ˙

pJ₁

−

(

p

˙

1J

·

n

)

(

1

− (

n

·

p1

)

2

)

Appendix C. RewritingBrunn’skinematicswithaneffectivevelocitygradient

Brunn’sorientationkinematicsforaspheroidparticleimmersedinasecond-orderfluidreads[19]:

˙

pB

=  ·

p

+

κ

(

D

·

p

− (

D

: (

p

⊗

p

)

p

))

− (

I

−

p

⊗

p

)

·

D

· (

H2D

·

p

+

H1

(

D

: (

p

⊗

p

))

p

)

(46) wherethespheroidshapefactorisgivenby

κ

=

r2₋₁

r2₊₁,withr theaspect-ratiooftheparticle,and H1 and H2 aregivenby Brunn, H1

= (

r2

−

1

)

H2

= −

2



r2₋₁ r2₊₁



2



k(₀2)

+

1₄k(₀11)



.DevelopingEq.(46),wehave

˙

pB

=  ·

p

+

κ

(

D

·

p

− (

D

: (

p

⊗

p

)

p

))

−

H2D2

·

p

+

H2

(

p

⊗

p

)

·

D2

·

p

−

H1

(

D

: (

p

⊗

p

))

D

·

p

+

H1

(

D

: (

p

⊗

p

)) (

p

⊗

p

)

·

D

·

p

(47) Wenowdefineaneffectivevelocitygradient:

˜

∇

v

= ∇

v

−

H2

κ

D

2

₋

H1

κ

(

D

: (

p

⊗

p

))

D (48)

andtheeffectivevorticity andstrainratetensors

˜ =

1₂



˜

∇

v

− ˜

∇

vT



andD

˜

=

1₂



˜

∇

v

+ ˜

∇

vT



.Equipped withthiseffective velocitygradient,Brunn’skinematicsEq.(46)canberewritteninthesameformasJeffery’skinematics[Eq.(1)],thatis,

˙

pB

= ˜ ·

p

+

κ

( ˜

D

·

p

− ( ˜

D

: (

p

⊗

p

)

p

))

(49)

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