Simulating flexible fiber suspensions using a scalable

SIMULATING FLEXIBLE FIBER SUSPENSIONS USING A SCALABLEIMMERSED BOUNDARY ALGORITHM ∗

JEFFREY K. WIENS† AND JOHN M. STOCKIE†

Abstract. We present an approach for numerically simulating the dynamics of flexible fibers in a three-dimensional shear flow using a scalable immersed boundary (IB) algorithm based on Guermond and Minev’s pseudo-compressible fluid solver. The fibers are treated as one-dimensional Kirchhoff rods that resist stretching, bending,and twisting, within the generalized IB framework. We perform a careful numerical comparison against experimentson single fibers performed by S. G. Mason and co-workers, who categorized the fiber dynamics into several distinctorbit classes. We show that the orbit class may be determined using a single dimensionless parameter for lowReynolds flows. Lastly, we simulate dilute suspensions containing up to hundreds of fibers using a distributed-memory computer cluster. These simulations serve as a stepping stone for studying more complex suspensiondynamics including non-dilute suspensions and aggregation of fibers (also known as flocculation).

Key words. flexible fibers, immersed boundary method, fluid-structure interaction, Kirchhoff rod theory,pseudo-compressibility method, parallel algorithm

AMS subject classifications. 74F10, 76D05, 76M12, 65Y05

1. Introduction. The behaviour of long, flexible fibers in a suspension plays an impor-tant role in many applications, including pulp and paper manufacture, polymer melts, and fiber-reinforced composite materials [

21,

46].Thedynamicsofsuchsuspensionsdependheavilyontheshapeandflexibilityoftheindividualfibersaswellastheinteractionsbetweenfibers.Becauseofthecomplexityofthefibermotioninsuspensions,manyresearchershavedevelopednumericalmethodsthataffordvaluableinsightintobothindividualfiberdynamicsandtheresultingaggre-gatesuspensionrheology[

20,

37,

46].Thesesimulationscancomplementphysicalexperimentsbyprovidinginformationthatisnoteasilyobtainedthroughdirectmeasurement.Inthispaper,wedevelopanapproachforsimulatingasuspensionofflexiblefibersthatisbasedontheimmersedboundary(IB)method[

36],whichisamathematicalframeworkoriginallydevelopedbyPeskin[

35]tocapturethetwo-wayinteractionbetweenafluidandanimmerseddeformablestructure.Here,thefluiddeformstheelasticstructurewhilethestructureexertsforcesontothefluid.TheIBmethodhasbeenusedtostudyawidevarietyofbiologicalandengineeringapplicationsincludingbloodflowthroughheartvalves[

13,

35],cellgrowthanddeformation[

38],jellyfishlocomotion[

17],evolutionofdryfoams[

24]andparachuteaerodynamics[

23].Wetreattheflexiblefibersasone-dimensionalKirchhoffrods[

7]describedusingthegeneralizedIBframeworkdevelopedbyLimetal.[

28].Inthisapproach,thefibersarerepresentedas1DspacecurvesusingamovingLagrangiancoordinate,whereinateachLagrangianpointanorthonormaltriadofvectorsdescribestheorientationand“twiststate”oftherod.Thispermitsthefibertogeneratenotonlyaforcebutalsoatorquethatisappliedtothesurroundingfluid.Theprimaryobjectiveofthispaperistodevelopanefficientmethodologyforsimulatingsuspensionscontainingalargenumberofflexiblefibers.Sincesolvingthefullfluid-structurein-teractionproblemcomesattheexpenseofadditionalcomputationalwork,theunderlyingparallelalgorithmispurposelydesignedtoscaleefficientlyondistributed-memorycomputerclusters.Thispermitsnon-dilutesuspensionstobesimulatedefficientlybyspreadingtheworkovermultiplepro-cessors.ThenumericalalgorithmisbasedontheworkofWiensandStockie[

53]whoimplementedapseudo-compressiblefluidsolverdevelopedbyGuermondandMinev[

15,

16]intheIBframe-work.WeextendthisoriginalalgorithmtousetheEulerian–LagrangiandiscretizationemployedbyGriffithandLim[

12]whichemploysapredictor-correctorproceduretoevolvetheimmersedboundary.Here,twoseparateforcespreadingandvelocityinterpolationstepsareappliedateachtimestepwhichimprovesthespatialconvergencerateofthemethod.WebegininSection

2byreviewingtheoreticalandexperimentalresultsintheliteraturepertainingtothehydrodynamicsofsuspensionscontainingflexiblefibers,aswellasdiscussing

∗WeacknowledgesupportfromtheNaturalSciencesandEngineeringResearchCouncilofCanada(NSERC)throughaPostgraduateScholarship(JKW)andaDiscoveryGrant(JMS).ThenumericalsimulationsinthispaperwereperformedusingcomputingresourcesprovidedbyWestGridandComputeCanada.†DepartmentofMathematics,SimonFraserUniversity,Burnaby,BC,Canada,V5A1S6(

[email protected],

[email protected]).

1

mailto:[email protected]



2 J. K. WIENS AND J. M. STOCKIE

several prominent computational approaches. In Sections

3and

4,westatethegoverningequationsunderlyingourIBmodelforfluid-fiberinteraction,aswellasthenumericalalgorithmusedtoapproximatetheseequations.InSection

5,wepresentsimulationsoffiberdynamicsinbothsingle-andmulti-fibersystems,andcomparetheseresultstopreviouslypublishedexperimentalwork.

2.Background:PulpFibers.

2.1.TheoryandExperiments.TheoreticalinvestigationsofthedynamicsoffibersinashearflowdatebacktoJefferyinthe1920s[

19],whoderivedananalyticalsolutionforthemotionofasinglerigid,neutrally-buoyantellipsoidalparticleimmersedinanincompressibleNewtonianfluid(specifically,inaStokesflow).Jefferyfoundthatsuchafiberrotateswithawell-definedperiodicorbithavingconstantperiodbutnon-uniformangularvelocity.ItwaslatershownbyBretherton[

3]thatJeffery’sanalyticalsolutioncouldbeextendedtomoregeneralaxisymmetricparticleswithnon-ellipticalcross-sectionsbyreplacingtheellipsoidalaspectratioarbyaneffectiveaspectratioa∗r.Althoughthetheoryforrigidfiberdynamicsisrelativelywell-developed,farlessisknownaboutfibersthatexperiencesignificantbending.Forthisreason,experimentalobservationsareofcriticalimportanceinunderstandingthedynamicsandrheologyofsuspensionscontainingflexiblefibers.Unlikerigidfibers,flexiblefibersundergoamuchwiderandricherrangeofmotionwhensubjectedtoabackgroundlinearshearflowgivenwithvelocityfieldu=(Gy,0,0).ThisproblemwasstudiedinthepioneeringworkofMasonandco-workers[

1,

10,

11]whocategorizedthefiberdynamicsintoseveraldistinctorbitclasses.Whenmotionsareconfinedtothexy-plane,fiberdynamicsfallintooneoffourorbitclasses–rigid,springy,flexible,andcomplexrotations–whichareillustratedinTable

1.TheexperimentsofMasonetal.involvedprimarilysyntheticfibers(madeofrayonanddacron)immersedinhighlyviscousfluids(suchascornsyrup)althoughtheiroriginalmotivationwastheapplicationtonaturalwoodpulpfibersuspensions.

Table1Two-dimensionalorbitclassesforflexiblefiberswhoseunstressedstateisintrinsicallystraight.AdaptedfromForgacsetal.[

11].

OrbitClass

IRigidrotation

IISpringyrotation

IIIaLooporSturn

IIIbSnaketurn

IVComplexrotation

Theseexperimentsonfibersuspensionsdemonstratethatvaryingeitherthehydrodynamicdragforceorthefiberflexibilitygovernsthetransitionbetweenthevariousplanarorbitclasses.InclassIorbits,thefiberremainsrigidandrotatesaspredictedbyJeffery’sequation.Whenasmall

IB SIMULATIONS OF FLEXIBLE FIBER SUSPENSIONS 3

flexibility is introduced into the fiber, it undergoes a springy rotation (class II) in which it bendsinto a shallow arc as it rotates outside the horizontal plane of shear. When the fiber flexibility isincreased, it experiences significant deformations that take the form of S turns (class IIIa) or snaketurns (class IIIb). Note that S turns require a high degree of initial symmetry so that snake turnsare actually far more prevalent in actual suspensions [

1,

10].Whenthefiberflexibilityisincreasedevenfurther,thefibermayneverstraightenoutasitreturnstothehorizontal,inwhichcasetheorbitisclassifiedasacomplexrotation(classIV).Forthelargestvaluesofflexibilityencounteredinthread-likesyntheticfibers,thefibercantransitionbeyondtheclassofcomplexrotationsandundergoconvolutedself-intersectionsasobservedbyForgacsandMason[

10]inexperiments.Inmanycases,thefiberrotationisnotconstrainedtothexy-planebutinsteadundergoesagenuinelythree-dimensionalorbitthatprotrudesor“buckles”outalongthez-direction,althoughthexy-projectionofthefibermaystillbelongtooneoftheplanarorbitclassesI–IVdescribedabove.Notethatrealsuspensionssuchaswoodpulpalsocontainirregularly-shapedfibersthatareeitherintrinsicallycurvedorcontainkinksorothernon-uniformities;consequently,fiberorbitaldynamicsinsuchsuspensionsarenotnecessarilyconfinedtotheseidealizedorbitclasses.Indeed,theexperimentsofArlovetal.[

1]wereusedtoclassifyamuchbroaderclassofgenuinelythree-dimensionalorbitsforwoodpulpfibershavinganintrinsiccurvature.Weclosethisdiscussionbydefiningadimensionlessparameterthatcanbeusedtoconvenientlyclassifyandpredicttheorbitclasstowhichaspecificfiberbelongs.ForlowReynoldsnumberflows(withRe/1),thehydrodynamicdragforceexperiencedbyafiberisproportionalto

Fd=µGD,(2.1)

whereµisthefluidviscosity,Gistheshearrate,andDisthediameterofthefiber[

51].Bybal-ancingthisdragforcewiththecorrespondingfiberbendingforce,asingledimensionlessparametercanbederivedthatcapturesthefiberflexibility[

42]

χ=µDGL3

EI,(2.2)

whereListhefiberlength,EisYoung’smodulusofthematerial,andIismomentofareaintheplaneofbending.Theparameterχmayalsobeinterpretedasaratiooffiberdeflectiontofiberlength.Inaseriesof2Dnumericalsimulations[

44],theparameterχwasshowntoprovideausefulmeasureoffiberflexibilitythatcharacterizeseachorbitclassoverawiderangeoffluidandfiberparameters.ThisdimensionlessflexibilityparameterhasalsoappearedinthecomputationalstudiesofRossandKlingenberg[

39](wheretheyreferredtoitasadimensionlessshearrate)andWherrettetal.[

50](whereχ−1iscalledabendingnumber).

2.2.OverviewofComputationalApproaches.Apopularclassofnumericalmethodsforsimulatingflexiblefibersistheso-calledbeadmodelsinwhichaflexiblefiberistreatedasastringofrigidbeadsthatarelinkedtogetherbyflexibleconnectors.ThisapproachoriginatedwiththeworkofYamamotoandMatsuoka[

56]whotreatedfibersaschainsofbondedspheresthatarefreetostretch,bendandtwistrelativetoeachother.TheirapproachwasextendedbyRossandKlingenberg[

39]whomodelledfibersaschainsofrigidprolatespheroidsconnectedbyballandsocketjoints.ThedynamicsofthebeadnetworkaregovernedbyNewton’slawsthroughabalanceoflinearandangularmomentumthatincorporatesthehydrodynamicandinterparticleforcesactingoneachbead.Morerecently,Klingenberg’sgrouphasvalidatedtheirmodelresultsagainstexperimentsforsinglefiberdynamics[

41]aswellasdevelopingamulti-fiberextensionthathasbeenusedtosimulateflocculation[

45].AsignificantshortcomingofKlingenberg’smodelandrelatedvariants[

48,

50,

56]isthattheyfailtocapturethefullfluid-structureinteractioninfibersuspensions.Althoughtheirapproachdoesincludethehydrodynamicforceexertedbythefluidonthefiber,thefiberdoesnotitselfexertanyforcebackontothefluid;therefore,thefluidisapassivemediumthatobviouslyneglectsanyofthecomplexfluiddynamicsthatmustoccurintheregionimmediatelyadjacenttoadynamicallydeformingfiber.Severalrecentbead-typemodelshaveattemptedtoaddressthislimitation,forexampleWuandAidunwhoproposedamodelforrigid[

55]andflexible[

54]fibersthatincorpo-ratesthefullfluid-structureinteractionusingaLatticeBoltzmannapproach.Similarly,Lindstr¨om


and Uesaka proposed an alternative model for rigid [

31]andflexible[

29,

30]fibersthatusestheincompressibleNavier–Stokesequationstomodelthefluid.

Acompletelydifferentapproachforcapturingflexiblefiberdynamicsisbasedontheslenderbodytheory[

2]whichexploitsapproximationstothegoverningequationsbasedonasmallfiberaspectratio.ThisistheapproachtakenbyTornbergandShelley[

47]whostudiedflexiblefilamentsinaStokesflowbyderivingasystemofone-dimensionalintegralequations.Theysolvedtheseintegralequationsnumericallyusingasecond-ordermethodthatalsocapturesinteractionsbetweenmultiplefibers.ThisapproachhasbeenfurtherextendedbyLietal.[

26]whousedasimilarmethodologytoinvestigatetheproblemofsedimentation(orsettling)offlexiblefibers.Unlikethebeadmodelsdescribedearlier,thisslender-bodyapproachcleanlyseparatesthefibermodelfromitsnumericaltreatment,whichmakesthemodelmoreamenabletomathematicalanalysisandalsopermitsthenumericaldiscretizationtobeindependentlytestedthroughconvergencestudies.Furthermore,becausethefluidhasbeensimplifiedbyassumingaStokesflowregime,theseslender-bodydiscretizationsdonotrequireafluidgridbecauseoftheavailabilityofnumericalmethodsbasedonGreen’s-functionsolutionsthatgreatlyreducethecomputationalcomplexity.Theonlysignificantdisadvantageofthisapproach,besidetheStokesflowrestriction,isthatthereareasyetnoresultsthatincorporateanyeffectsoffibertwist[

34].

AnalternativeapproachthatpermitssimulatingflexiblefibersimmersedinhigherReynoldsflowsistheimmersedboundarymethod.ThisistheapproachtakenbyStockieandGreen[

44]whosimulatedasingleflexiblefiberintwodimensionsusingasimplerepresentationofthefiberintermsofspring-likeforcesthatresiststretchingandbending.Stockie[

43]laterextendedtheseresultstoasingle3Dwoodpulpfiberusingamuchmoredetailedandrealisticmodelthatexplicitlycapturestheinterwovenmulti-layernetworkofcellulosefibrilsmakingupthewoodcellwall.Morerecently,NguyenandFaucistudieddiatomchainsusingtheIBmethodwithasimilarlydetailedfibermodel[

33].TheIBmethodproperlycapturesthefullinteractionbetweenthefluidandimmersedstructurebyincludingtheappropriateno-slipboundaryconditionalongthefiber,althoughitdoescomeatanadditionalcost.Firstofall,incomparisonwithslender-bodymodels,thefluidsolverportionoftheIBalgorithmcanbesignificantlymoreexpensivebecauseitsolvestheNavier-Stokesequationsonafinitedifferencegrid.Secondly,becausetheIBmethodaimstocapturethedetailedfluidflowaroundthefiber,thefluidgridneedstobeadequatelyrefinedinordertoresolvedetailsontheorderofthefiberdiameter,whichinturnplacespracticallimitationsonthefiberaspectratiothatcanbecomputed.Thirdly,adetailedcharacterizationofthestructureofathree-dimensionalfibersuchasin[

33,

43]typicallyrequiresthousandsofIBpointstoresolveandisthereforecomputationallyimpracticalforsimulatingsemi-dilutesuspensionsofmultiplefibers.

Inthispaper,weapplytheIBapproachtosimulateflexiblefibers,andwehavechosentotreateachfiberinsteadasaone-dimensionalKirchhoffrodthatresistsstretching,bendingandtwisting,asdescribedinthegeneralizedIBmethodofLimetal.[

28].Additionally,weemployahighlyscalableimplementationofthegeneralizedIBalgorithm[

53]thatspreadsthecomputationalworkoveralargenumberofprocessors,therebypermittingustosimulatehydrodynamicinteractionsinsuspensionscontaininglargenumbersofflexiblefibers.

3.GoverningEquations.ConsideraNewtonian,incompressiblefluidthatfillsarectangulardomainΩhavingdimensionsHx×Hy×HzandwhosestateisspecifiedusingEuleriancoordinatesx=(x,y,z).Immersedwithinthefluidisaneutrally-buoyantelasticfiberoflengthL.Thefiberisdescribedbyaone-dimensionalspacecurveΓ⊂Ω,parameterizedbytheLagrangiancoordinates∈[0,L].Thespatialconfigurationoftherodattimetisgiveninparametricformasx=X(s,t)anditsorientationand“twiststate”aredefinedintermsoftheorthonormaltriadofvectorsD1(s,t),D2(s,t),D3(s,t),wherethethirdtriadvectorD3remainstangenttothespacecurveX.Notethatbecauseofnumericalconsiderations(describedshortly),D3(s,t)isnotexactlytangenttothespacecurveXbutisratherpenalizedinawaythatitisonlyapproximatelyinthetangentialdirection.

Thefluidvelocityu(x,t)andpressurep(x,t)atlocationxandtimetaregovernedbythe


incompressible Navier–Stokes equations

ρ

(∂u

∂t+ u · ∇u

)+∇p = µ∇2u + f +

12∇× n,(3.1)

∇ · u = 0,(3.2)

where ρ is the fluid density and µ is the dynamic viscosity (both constants). The Eulerian forceand torque densities, f and n, are written as

f(x, t) =∫Γ

F (s, t) Φw(x−X(s, t)) ds and(3.3)

n(x, t) =∫Γ

N(s, t) Φw(x−X(s, t)) ds,(3.4)

wherein the integrals spread the Lagrangian force and torque densities, F (s, t) and N(s, t), ontopoints in the fluid. The interaction between Eulerian and Lagrangian quantities is mediated usingthe smooth kernel function

Φw(x) =1

w3φ

(x1

w

)φ

(x2

w

)φ

(x3

w

),(3.5)

where

φ(r) =

18 (3− 2|r|+

√1 + 4|r| − 4r2) if 0 ≤ |r| < 1,

18 (5− 2|r| −

√−7 + 12|r| − 4r2) if 1 ≤ |r| < 2,

0 if 2 ≤ |r|.(3.6)

Here, w represents an effective thickness of the rod which is set to some multiple of the fluid meshwidth h; that is, w = Ch for some integer multiple C ∈ Z+. Note that if w = h, the kernel Φw(x) isidentical to the discrete delta function employed in many immersed boundary methods [

14,

25,

32].TherodismodeledasaKirchhoffrod[

7]usingthegeneralizedimmersedboundaryframeworkofLim[

28].BalancinglinearandangularmomentumyieldstheLagrangianforceandtorquedensities

F=∂Frod

∂s,(3.7)

N=∂Nrod

∂s+∂X

∂s×Frod,(3.8)

intermsoftheinternalforceFrod(s,t)andmomentNrod(s,t)transmittedacrossasegmentoftherod.InternalquantitiesareexpandedinthebasisD1,D2,D3as

Frod=F1D1+F2D2+F3D3,(3.9)

Nrod=N1D1+N2D2+N3D3,(3.10)

wherethecoefficientfunctionsaredefinedbytheconstitutiverelations

N1=a1

(∂D2

∂s·D3−κ1),N2=a2

(∂D3

∂s·D1−κ2),N3=a3

(∂D1

∂s·D2−τ),(3.11)

F1=b1

(D1·∂X∂s

),F2=b2

(D2·∂X∂s

),F3=b3

(D3·∂X∂s−1).(3.12)

Equations(

3.11)incorporatetheresistanceoftherodtobendingandtwisting,witha1anda2beingthebendingmoduli(aboutaxesD1andD2respectively)whilea3isthetwistingmodulus.Theconstants(κ1,κ2,τ)definetheintrinsictwistvectoroftherodwhereκ:=√κ21+κ22istheintrinsiccurvatureandτistheintrinsictwistinthestress-freeconfiguration.Theremainingforceterms(

3.12)acttokeepthetriadvectorD3approximatelyalignedwiththetangentcurve∂X/∂s


and also penalize any stretching of the rod from its equilibrium configuration. Accordingly, thegeneralized IB method can be viewed as a type of penalty method in which the rod is only approx-imately inextensible and approximately aligned with the orthonormal triad, and the constants b1,b2 and b3 play the role of penalty parameters.

The final equations required to close the system are evolution equations for the rod configura-tion and triad vectors

∂X

∂t(s, t) = U(s, t),(3.13)

∂Dα

∂t(s, t) = W (s, t)×Dα(s, t),(3.14)

where α = 1, 2, 3, and U(s, t) and W (s, t) are the linear and angular velocities along the axis of therod respectively. These equations require that the rod translate and rotate according to the localaverage linear and angular velocity of the fluid, and are interpolated in the standard IB fashion as

U(s, t) =∫Ω

u(x, t) Φw(x−X(s, t)) dx,(3.15)

W (s, t) =12

∫Ω

∇× u(x, t) Φw(x−X(s, t)) dx.(3.16)

By using the same kernel function Φw as in (

3.3)–(

3.4),weensurethatenergyisconservedduringtheEulerian–Lagrangianinteractions[

28].

3.1.ProblemGeometryandInitialConditions.TheproblemgeometryisillustratedinFigure

1,showingafiberΓimmersedinarectangularfluiddomainΩ.Periodicboundaryconditionsareimposedonthefluidinthex-andz-directions,whilethefluidisshearedinthevertical(y)direction.Theshearflowisinducedbyimpartingahorizontalmotiontothetopandbottomboundaries,withthetopwallmovingatspeedUtopandthebottomwallintheoppositedirectionatspeedUbot.Inpractice,weimposeUtop=Ubot:=Uandsettheinitialfluidvelocitytothelinearshearprofileu(x,0)=(G(y−Hy/2),0,0)thatwoulddevelopintheabsenceofthefiber,withshearrateG=2U/Hy.ThefiberoflengthLisplacedatthecenterofthefluiddomain

Fig.1.ProblemgeometryforasinglefiberΓlocatedatthecenterofaperiodic,rectangularchannelΩofdimensionHx×Hy×Hz.Aplanarshearflowisgeneratedbyforcingthetopandbottomwallstomovewithconstantvelocities±Utop.

whichisspecifiedbytheconstantX0,andweconsiderthreedifferentinitialconfigurationsforthefiber:


Configuration 1. The fiber is initially straight and is parameterized by

X(s, 0) = ((ε0 + 1)s, 0, 0) + X0,

D1(s, 0) = (0, 1, 0) ,

D2(s, 0) = (0, 0, 1) ,

D3(s, 0) = (1, 0, 0) ,

where 0 ≤ s < L and ε0 is a perturbation parameter that initially stretches the fiber.Configuration 2. The fiber is curved in the xy-plane with

X(s, 0) = (r0 cos(s/r0 + π), r0 sin(s/r0 + π), 0) + X0,

D1(s, 0) = (0, 0, 1) ,

D2(s, 0) = (cos(s/r0 + π), sin(s/r0 + π), 0) ,

D3(s, 0) = (sin(s/r0), cos(s/r0 + π), 0) ,

where αbr0π ≤ s < αer0π, and αb and αe are constants with 0 ≤ αb < αe ≤ 1. Here,the fiber is a segment of a circle of radius r0 lying in the xy-plane and having lengthL = (αe −αb)πr0. Choosing a sufficiently large radius r0 generates fiber with small initialcurvature.

Configuration 3. Similar to Configuration 2, except that the fiber is curved in the xz-plane with

X(s, 0) = ((ε0 + r0) cos(s/r0), 0, (ε0 + r0) sin(s/r0)) + X0,

D1(s, 0) = (0, − 1, 0) ,

D2(s, 0) = (cos(s/r0), 0, sin(s/r0)) ,

D3(s, 0) = (sin(s/r0 + π), 0, cos(s/r0)) ,

where αbr0π ≤ s < αer0π, and αb and αe are constants satisfying 0 ≤ αb < αe ≤ 1.For all three configurations, the rod has open ends so that boundary conditions are required ats = 0 and L. We assume that the internal force and moment vanish at the endpoints, correspondingto F rod

−1/2 = F rodNs−1/2 = 0 and N rod

−1/2 = N rodNs−1/2 = 0, which are consistent with the boundary

conditions applied by Lim [

27].

4.NumericalMethod.Here,weprovideonlyaverybriefoverviewofthenumericalmethodusedtosolvethegoverningequations,whileadetaileddescriptionofthemethodanditsparallelimplementationcanbefoundin[

52,

53].Whendiscretizingthegoverningequationsweusetwoseparatecomputationalgrids,oneeachfortheEulerianandLagrangianvariables.ThefluiddomainisdividedintoanNx×Ny×Nz,uniform,rectangularmeshwhereeachcellhassidelengthh.Weemployamarker-and-cell(MAC)discretization[

18]whereinthepressureisapproximatedatcellcenterpointsxi,j,kfori,j,k=0,1,...,N−1,whilevelocitycomponentsarelocatedoncellfaces.TheLagrangianvariablesarediscretizedatNsuniformly-spacedpointsdenotedbys`=`∆sfor`=0,1,...,Ns−1with∆s=L/Ns.Sinceourcurrentimplementationisrestrictedtoperiodicfluiddomains,thetopandbottomwallboundaryconditionsareimposedbyslightlyincreasingthesizeofthefluiddomaininthey-directionandintroducingplanesofIBtetherpointsalongy=0andHythatareattachedbyverystiffspringstopointsmovingatthespecifiedvelocitiesUtopandUbot.Wedidthisforconvienceonly,sinceneitherthegoverningequationsnorthefluidsolverisrestrictedtoperiodicdomains.TheIBequationsareapproximatedusingafractional-stepmethoddescribedbyWiensandStockie[

53]inwhichthecalculationoffluidvariablesisdecoupledfromthatoftheimmersedboundary.Forintegratingthefluidequations,weusethepseudo-compressibilitymethoddevel-opedbyGuermondandMinev[

15,

16],whichemploysadirectional-splittingstrategythatreducestoaseriesofone-dimensionaltridiagonalsystems.Theselinearsystemscanbesolvedefficientlyondistributed-memoryclustersbycombiningThomas’salgorithmwithaSchur-complementtech-nique.


When integrating the rod position and orthonormal triad vectors forward in time, we use thepredictor-corrector procedure devised by Griffith and Lim [

12].Thisdifferentiatesournumericalmethodfromtheapproachtakenin[

53],whereanAdams–Bashforthextrapolationwasusedtoevolvetheimmersedboundaryintime.Althoughthepredictor-correctorprocedureintroducesadditionalwork,thischangeisnecessaryinordertoobtainsecond-orderconvergenceratesinspace.Lastly,theconstitutiverelations(

3.7)–(

3.12)arediscretizedinthesamemannerasinLimetal.[

28],withthemaindifferencebeinginhowtheorthonormaltriadvectorsareinterpolatedontohalfLagrangianstepss`+12=(`+12)∆s.Here,weusetheRodrigues’rotationformulaasdescribedin[

52]insteadoftakingtheprincipalsquarerootusedbyLimetal.[

28].Ifweassumethatthestatevariablesareallknownattimetn,theIBalgorithmforasingletimestep∆tproceedsasfollows.1.InterpolatethelinearandangularfluidvelocitiesontotherodusingthethedeltakernelΦw(x)toobtainUnandWn.

2.PredicttherodpositionXn+1,∗andorthonormaltriadvectors(Dα)n+1,∗attimetn+1=(n+1)∆ttofirstorderforα=1,2,3.

3.CalculatetheLagrangianforceandtorquedensities,FandN,attimestnandtn+1usingthediscretizationemployedbyLimetal.[

28].

4.SpreadtheLagrangianforceandtorquedensitiesjustcalculatedontofluidgridpoints.ThenapproximatetheEulerianforceandtorquedensity,fn+12andnn+12,attimetn+12=(n+12)∆tusinganarithmeticaverage.

5.IntegratetheincompressibleNavier–Stokesequationstotimetn+1using(fn+12+12∇×nn+12)astheexternalbodyforce.

6.CorrecttherodpositionXn+1andorthonormaltriad(Dα)n+1tosecondorder.Thisrequiresinterpolatingthelinearandangularfluidvelocityattimetn+1ontotherodloca-tion.

5.NumericalResults.

5.1.IntrinsicallyStraightFibers.Webeginbyconsideringthebehaviourofasingleflex-iblefiberimmersedinashearflow,wheretheequilibriumfiberstateisintrinsicallystraight(withnobend,notwist).AsdescribedearlierinSection

2,experimentalobservationsshowthatsuchfibersarecharacterizedbyawell-definedorbitalmotionthatcanbeseparatedintooneofsev-eraldistinctorbitclassesaccordingtoafiberflexibilityparameterχthatcapturestheratiooffiberbendingforcetohydrodynamicdrag.Thissectionaimstoinvestigatethefullrangeofthesetwo-dimensionalorbitalmotions.Inallsimulations,weusethenumericalparameterslistedinTables

2and

3.Sincethefibermotionisconfinedtothexy-plane,wesignificantlyreducetheexecutiontimeofasimulationbyshrinkingthedomaindepthHz,whichallowsustorun100+simulationsinareasonabletimeframe.Notethattheseresultsarevirtuallyidenticaltosimulationsusingalargerdomain(Hz=2),whichweconfirmedthroughnumerouscomputationalexperiments.Inallsimulations,wechoosephysicalparametersthatareconsistentwithnatural(unbeaten)kraftpulpfibers,takingafiberlengthof0.1−0.3cmandflexuralrigidityof0.001−0.07gcm3/s2[

8,

9].Becausefibersinournumericalsimulationshavediameterthatisproportionaltotheeffectivethicknessw,oursimulatedfibersareactuallythickerthananaturalpulpfiber.Forexample,weuseadeltafunctionregularizationcorrespondingtow≈80µm,whereasanaturalpulpfiberhasadiameterbetween20–80µm.Sincetheprecisedependenceofthesimulatedfiberdiameteronwisunknown,weappealtotheworkofBringleyandPeskin[

4]wheretheyobservedthataone-dimensionalarrayofrigidIBpointshasaneffectivenumericalthicknessofD≈2w.Althoughtheseresultsmaynotbestrictlyapplicableinthepresentsetting,thisapproximationissufficientforourpurposes.Anyremainingdiscrepancyinthefiberdiametercanthenbeaccommodatedforbyadjustingthevalueoffiberdragforce(seeFdfromequation(

2.1)).InFigures

2and

3,wedisplaysnapshotsofthedynamicsofafiberwithinitialconfigurationlyinginthexy-planeandforsixvaluesofthedimensionlessflexibilityparameterχbetween0.19and


Table 2Numerical and physical parameter values used in rigid fiber simulations.

Parameter Symbol Value

Size of fluid domain Ω Hx ×Hy ×Hz 2× 12× 16h cm

Number of fluid grid points Nx ×Ny ×Nz 256× 64× 16Fluid mesh width h 1/128 cmFluid density ρ 1.0 g/cm3

Fluid viscosity µ 10.0 g/(cm·s)Speed of moving plates Utop = Ubot 8 cm/sShear rate G 32 s−1

Time step ∆t 1e−5 sFiber length L 0.3 cmFiber mesh width ∆s L/120 cmBending and twisting modulus (EI) a1 = a2 = a3 0.7 dyne · cm2

Shear and stretch modulus b1 = b2 = b3 540 dyne · cm2

Fiber effective thickness w 0.0078125 cmIntrinsic twist vector (κ1, κ2, τ) (0, 0, 0)Fiber length perturbation ε0 0.001Support of delta kernel C 4

Table 3Parameter modifications for the flexible fiber simulations in Figures

2and

3.OnlythoseparametersthathavechangedrelativetovaluesindicatedinTable

2areshownhere.

OrbitClassConfigurationParameters

Springy2r0=0.45,αb=0.4,αe=0.6,EI=2.5e−2,∆s≈1.25e−3,L≈0.282

Sturn1EI=3.0e−3

Snaketurn2r0=0.45,αb=0.4,αe=0.6,EI=3.0e−3,∆s≈1.25e−3,L≈0.282

Complex2r0=0.4,αb=0.4,αe=0.6,µ=15,EI=1.0e−3,∆s≈1.25e−3,L≈0.251

Coiled1G=64,µ=90,EI=1.0e−4,L=0.5

1.125e5.Asexpected,thesimulationsexhibitarangeofdifferentorbitalmotionsthattransitionbetweenthevariousorbitclasses(rigid,springy,flexible,complex,coiled)astheflexibilityincreases.Wealsonotethatwithintheintermediaterangeofχvalues,weobservebothSturnsandsnaketurnsdependingonthesymmetryoftheinitialfiberconfiguration.Despitebeingveryrareinactualfibersuspensions,Sturnsturnouttoberemarkablystableinouridealizedsettingwithaplanarshearflow;indeed,itisonlywhenasymmetryisintroducedinthefiberthrough(forexample)theinitialshapeoralength-dependentstiffnessthatsnaketurnsareobservedinsteadofSturns.TheseresultsareconsistentwiththoseofMasonandco-workers[

1,

10]whoobservedthatSturnsrequiredahighdegreeofsymmetrythatisrarelyachievedinexperiments.Forthelargestvalueofχ=1.125e5inFigure

3

(c)weobserveacoiledorbitwithself-entanglement,andalthoughthistypeofbehaviourisnotpertinenttopulpfibers,ForgacsandMason[

10]didobservesuchcoilingwiththread-likesyntheticfibers.Eventually,thisfiberformsacomplexwrithingbundleasthefiberundergoesself-contact,butbecauseourmodeldoesn’tincorporateanycontact(fiber-on-fiber)forceswemakenoclaimthattheseresultscorrespondtophysicallyaccuratecoilingdynamics.Whentheinitialfiberconfigurationisrotatedintothexz-plane,theresultingdynamicsarenon-planarbutstillfolloworbitsqualitativelysimilartothosederivedbyJeffery[

19].Examplesofthesenon-planarorbitsaregiveninthefirstauthor’sdoctoralthesis[

52],whichshowthattheflexiblefiberundergoesamotionconsistingofarotationsinthexy-planesuperimposedona


0.80.9

1.01.1

1.2x

0.0

0.1

0.2

0.3

0.4

0.5

y

t=0.0000

0.80.9

1.01.1

1.2x

t=0.3500

0.80.9

1.01.1

1.2x

t=0.4400

0.80.9

1.01.1

1.2x

t=0.5000

0.80.9

1.01.1

1.2x

t=0.6500

−160

−120

−80

−40

0 40 80 120

160

(a)

Rig

idO

rbit

(χ=

0.1

9,E

I=

7.0

e−1,L

=0.3

)

0.80.9

1.01.1

1.2x

0.0

0.1

0.2

0.3

0.4

0.5

y

t=0.0000

0.91.0

1.11.2

x

t=0.2500

0.91.0

1.11.2

x

t=0.3500

0.91.0

1.11.2

x

t=0.4000

1.01.1

1.21.3

x

t=0.6000

−100

−80

−60

−40

−20

0 20 40 60 80 100

(b)

Sprin

gy

Orb

it(χ

=4.4

9,E

I=

2.5

e−2,L≈

0.2

82)

0.80.9

1.01.1

1.2x

0.0

0.1

0.2

0.3

0.4

0.5

y

t=0.0000

0.91.0

1.11.2

x

t=0.1500

0.91.0

1.11.2

x

t=0.2500

0.91.0

1.11.2

x

t=0.3500

0.91.0

1.11.2

x

t=0.6000

−100

−80

−60

−40

−20

0 20 40 60 80 100

(c)Snake

Orb

it(χ

=37.3

8,E

I=

3.0

e−3,L≈

0.2

82)

Fig

.2.Snapsh

ots

offiber

positio

nand

fluid

vorticity

inth

exy-p

lane

for

ahalf-ro

tatio

nin

arigid

,sp

ringy

and

snake

orbit.

Para

meter

valu

esare

listedin

Tables

2and 3.


0.80.9

1.01.1

1.2x

0.0

0.1

0.2

0.3

0.4

0.5

y

t=0.0000

0.80.9

1.01.1

1.2x

t=0.2500

0.80.9

1.01.1

1.2x

t=0.3500

0.80.9

1.01.1

1.2x

t=0.4000

0.80.9

1.01.1

1.2x

t=0.6000

−100

−80

−60

−40

−20

0 20 40 60 80 100

(a)

SO

rbit

(χ=

45.0

0,E

I=

3.0

e−3,L

=0.3

)

0.80.9

1.01.1

1.2x

0.0

0.1

0.2

0.3

0.4

0.5

y

t=0.0000

0.80.9

1.01.1

1.2x

t=0.1500

0.80.9

1.01.1

1.2x

t=0.3500

0.80.9

1.01.1

x

t=0.5500

0.70.8

0.91.0

x

t=0.8000

−100

−80

−60

−40

−20

0 20 40 60 80 100

(b)

Com

plex

Orb

it(χ

=119.0

6,E

I=

1.0

e−3,and

L≈

0.2

51)

0.80.9

1.01.1

1.2x

0.0

0.1

0.2

0.3

0.4

0.5

y

t=0.0000

0.80.9

1.01.1

1.2x

t=0.1000

0.80.9

1.01.1

1.2x

t=0.2000

0.80.9

1.01.1

1.2x

t=0.3000

0.80.9

1.01.1

1.2x

t=0.3500

−100

−80

−60

−40

−20

0 20 40 60 80 100

(c)C

oiled

Orb

it(χ

=1.1

25e5

,E

I=

1.0

e−4,and

L=

0.5

)

Fig

.3.Snapsh

ots

offiber

positio

nand

fluid

vorticity

inth

exy-p

lane

for

an

Stu

rn,co

mplex

and

coiled

orbit.

Para

meter

valu

esare

listedin

Tables

2and 3.


100 101 102

Dimensionless Flexibility (χ)

10−3

10−2

10−1

EI

IIIIIIIV

(a)

100 101 102

Dimensionless Flexibility (χ)10−2

10−1

100

101

102

Dra

gR

ate

(Fd)

IIIIIIIV

(b)

Fig. 4. Summary of all simulations showing the relationship between orbit class and different values of thedimensionless flexibility χ, flexural rigidity EI and drag rate Fd. Open markers denote the experimental data shownin Table

4whereE=3GPa.

rockingmotionbackandforthaboutthez-axisinthexz-plane.Wenextexploreinmoredetailthedependenceofthefiberorbitclassonthedimensionlessflexibilityparameterχ.Tothisend,weperformamuchlargerseriesofsimulationswithvaryingfiberlength(L=0.1–0.3cm),diameter(D≈156–312µm),flexuralrigidity(EI=0.001–0.1dyne·cm2),shearrate(G=20–120s−1)andviscosity(µ=0.07–100.0g/(cm·s))correspondingtoReynoldsnumberslyingintherange0.0027–23.9.Foreachsimulation,weassignthefiberdynamicstooneofthefourorbitclassesI–IVbycalculatingthetotalfibercurvature

λ(t)=∫L0

∣∣∣∣∂D3∂s(s,t)∣∣∣∣ds,andusingthemaximumcurvatureoverahalf-rotationt0≤t≤t1toapplythefollowingcriteria:•I:Theorbitisrigidifmaxt0≤t≤t1λ(t)<0.4.

•II:Theorbitisspringyif0.4≤maxt0≤t≤t1λ(t)<3.7.

•III:TheorbitisanSorsnaketurnif3.7≤maxt0≤t≤t1λ(t)andλ(t1)<2.5.

•IV:Theorbitiscomplexif3.7≤maxt0≤t≤t1λ(t)and2.5≥λ(t1).

NotethatS/snaketurnsandcomplexrotationshavethesamerangeofmaximumcurvature,andthatweusethefibercurvatureλ(t1)attheendofthehalf-rotationtodeterminewhetherornotthefiberhasstraightenedout.SimulationsaredepictedgraphicallyinFigure

4intermsoftwoplotsofflexuralrigidityEIanddragforceFdversusdimensionlessflexibilityχ.Eachpointontheplotcorrespondstoasimulationusingaspecificchoiceofphysicalparameters,andthepointtypeisassignedbasedontheorbitclassificationcriteriaabove.Fromthesetwoplots,itisevidentthatthereisacleardivisionoforbitsintoclassesI,IIandIIIalongverticaldivisionsthatcorrespondtovaluesofχ∼=3.85andχ∼=20.0.TheboundarybetweenclassesIIIandIVisnotassharplydefined,butcanstillbeassignedtoavalueofflexibilityχ≈65.0.Basedontheseobservations,weconcludethatthedimensionlessflexibilityχprovidesausefulmeasureforcharacterizingorbitclassesatthelowerReynoldsnumbersconsideredhere.WeconcludethissectionbyperformingafurthercomparisonofournumericalsimulationswiththeexperimentsofForgacsandMason[

10]ondacronfibersincornsyrup.Firstofall,welisttheparametersandobservedorbitclassforseveraloftheseexperimentsinTable

4.Basedon


values of χ · EI, we see that this rescaled flexibility parameter may be used to classify each orbit,assuming that EI is constant in all experiments. However, we emphasize that since Forgacs andMason did not provide a value for the flexural rigidity (EI), we were unable to determine the valueof χ explicitly.

Table 4Experimental results obtained from Forgacs and Mason [

10]forsyntheticdacronfibers.

OrbitClassχ·EIG(s−1)µ(g/(cm·s))L(cm)D(µm)

Rigid1.96e−43.92111.40.17787.8Rigid1.01e−35.14391.20.14047.8Springy1.43e−34.76311.40.32297.8Springy2.39e−35.96591.20.17787.8Springy4.91e−34.87991.20.24187.8Flexible1.16e−24.82591.20.32297.8

Becausetheseexperimentswereallperformedwithdacronfibers,wenextexplorefurthertheassumptionthatEIisroughlyconstant,andalsowhethertheexperimentalresultsareconsistentwiththedivisionoforbitclassesinoursimulationsinFigure

4.Firstofall,weremarkthatallexperimentaldatapointsareconsistentwithoursimulationsif2.46e−4<EI<3.71e−4(dyne·cm2).Unfortunately,theYoung’smodulusEfordacronisknowntovaryoveranextremelywiderangeof71.5MPa≤E≤22.1GPabetweenvariousmanufacturers[

6].However,themanufacturerofthefibersusedbyForgacsandMasonwasidentifiedasE.I.duPontdeNemoursandCo.,andwewereabletofindapatentfiledbythiscompanyin1969[

5]forseveraldacronblendsthatlistsamuchtighterrangeforYoung’smodulusof2.0GPa<E<3.5GPa.Therefore,thehypotheticalEIofthesesyntheticfiberswouldbebetween3.63e−4<EI<6.36e−4,whichisconsistentwithournumericalresults!Furthermore,mostdatapointsarestillclassifiedcorrectlywhentheEIfallsoutsideourconsistencyrange(2.46e−4<EI<3.71e−4).Toillustrate,wehaveplottedtheexperimentaldatainFigure

4usingopenmarkers,assumingE=3GPa(givinganEI=5.45e−4).Here,weobservethatallexperimentaldataareclassifiedcorrectly,exceptforonetroublesomedatapoint.Therefore,weconcludefromtheseresultsthatoursimulationsareinexcellentagreementwithexperimentaldata.

5.2.IntrinsicallyCurvedFibers.Wenextconsidersingleflexiblefibersthathaveanin-trinsiccurvatureatequilibrium,asituationthatisoftenencounteredfornaturalfiberssuchaswoodpulp.WeusethebaseparametervaluesinTable

2andsimulatetwocasescorrespondingtothemodificationslistedinTable

5.Inbothcases,thefiberisinitializedasacurvedsegmentofacirculararcwithintrinsictwistvector(κ1,κ2,τ)=(1/r0,0,0),whichkeepstheinitialfiberconfigurationatequilibrium(thatis,N1=N2=N3=0att=0).TheresultingorbitsdepictedinFigures

5and

6clearlycorrespondtoS-andsnake-likeor-bits.Theprojectionsofbothfibersinthexy-planebehavelikethecorrespondingplanarorbitsconsideredinSection

5.1,butprotrudeintothexz-plane.ThesesimulationsreproducesimilarorbitaldynamicstothoseobservedinexperimentsofArlovetal.[

1].Thefirstauthor’sthesis[

52]showsadditionalsimulationsforafiberinitiallyorientedalongthez-directionandundergoinganadditionalaxialspin,forwhichthefiberrotatesaroundthez-axisandslightlystraightensoutasitrotatesintotheshearflow.

Table5ParametermodificationsfortheflexiblefibersimulationsinFigure

5and

6.OnlythoseparametersthathavechangedrelativetovaluesinTable

2areshownhere.

OrbitClassConfigurationParameters

Sturn3Hz=2,r0=0.45,αb=0.4,αe=0.6,EI=3.0e−3,ε0=1e−3,∆s≈1.25e−3,L≈0.282

Snaketurn2Hz=2,r0=0.45,αb=0.4,αe=0.6,EI=3.0e−3,Θxz=π/16,∆s≈1.25e−3,L≈0.282


Fig

.5.Snapsh

ots

ofan

Stu

rnorbit

for

an

intrin

sically

curved

fiber

with

para

meters

inTables

2and 5.


Fig

.6.Snapsh

ots

ofsn

ake

turn

for

an

intrin

sically

curved

fiber

with

para

meters

inTables

2and 5.


5.3. Multiple Flexible Fibers. For our last series of simulations, we consider an idealizedrepresentation of a fiber suspension that permits us to employ the domain tiling techniques de-scribed in [

53].Inthesecomputations,wesimulateaPx×1×PzarrayoffibersimmersedinthefluiddomainΩ=[0,PxHx]×[0,Hy]×[0,PzHz]usingtheboundaryconditionsstatedinSection

3.1.ThecoderunsinparallelonaP=Px×Pzarrayofcomputerprocessors(Py=1)andthefluiddomainΩispartitionedalongthex-andz-axessothatoneprocessorlabelledI,KisresponsibleforeachsubdomainΩI,K=[(I−1)Hx,IHx]×[0,Hy]×[(K−1)Hz,KHz],forI=1,2,...,PxandK=1,2,...Pz.Wehaveconstructedthisproblemsothatitcanbeusedasaweakscalabilitytest,whereinthelocalproblemsizeisheldfixedasboththenumberofprocessorsandglobalproblemsizeareincreased.Itisimportanttorecognizethatourmethodisinnowayrestrictedtosuchidealizedarraysoffibers,butratherwehaveemployedthisarrangementhereinordertoclearlyillustratetheparallelscalabilityofouralgorithm.Initially,eachsubdomainΩI,Kcontainsasingleintrinsically-curvedfiberlocatedatitscentroid,witharandomly-chosenorientationangleandwhoseinitialshapeisdefinedinthesamemannerasdescribedearlierforConfiguration3.ThenumericalandphysicalparametersareasinTable

2withthefollowingmodifications:Hx=0.421875,Hy=12,Hz=0.3125,∆t=5e−5,r0=0.45,αb=0.4,αe=0.6,EI=3.0e−3,∆s≈1.25e−3,L≈0.282,Utop=8.5andUbot=7.5.Anotherdifferencefromourearliersimulationsisthatthetopandbottomboundariesthatinducetheshearflownowmoveatdifferentspeeds(thatis,Utop6=Ubot);consequently,fibersaretransportedacrosssubdomainboundarieswhichprovidesanontrivialtestofouralgorithm’sabilitytohandleinter-processcommunicationaswellaschangestotheIBdatastoredoneachprocessorovertime.Figure

7presentsthreesnapshotsofthedynamicsofa16×16arrayoffibersattheinitialandtwolatertimes.Theimageattimet=0.25emphasizesthefactthatallfibersspendthemajorityoftheirtimealignedhorizontallywiththeshearflow(i.e.,alongthex-axis)andthatonlyasmallproportionofthefibersatanytimeinstantarerotatedoutoftheshearplane.Asthesuspensionevolvesovertime,thefibersarepronetodriftandclustertogether,leadingtodevelopmentofmorecomplexbehaviorsuchasisshownintheimageattimet=1.80.Thislastsnapshotsuggeststhatouralgorithmiscapableofsimulatingatleasttheinitialphasesfiberflocculationinasuspensionwithareasonableconcentrationoffibers.Thenextsetofresultsattemptstoquantifytheimportanceofincludingthefulltwo-wayfluid-structureinteractionbetweenfluidandfibers,relativetoothermorecommonnumericalapproachesthatsimplifyoreliminatethisinteraction.Forthispurpose,wedefineaquantitywecallthelocaldeviationas

Erel(x,t)=|u(x,t)−u(x,0)|maxx(|u(x,0)|),

whichisalocalmeasureoftherelativedifferencebetweenthecomputedfluidvelocityandthecorrespondinglinearshearflowthatwouldariseintheabsenceofanyfibers.Wealsodefinearelatedglobaldeviationfromlinearshearusingeitherthe`∞-norm

‖Erel(x,t)‖∞=maxi,j,k|Erel(xi,j,k,tn)|,

or`1-norm

‖Erel(x,t)‖1=h3

V

∑i,j,k

|Erel(xi,j,k,tn)|,

whereVisthefluidvolume.Fora25-fibersimulationcomputedwith(Px,Py,Pz)=(5,1,5)processors,weprovideplotsinFigure

8ofthelocaldeviationErelattimet=1.80andalongtwodifferenthorizontalslices.ThefigureshavetruncatedthevaluesofErelabovethethreshold0.025sothatsmallerdeviationscanbevisualized.Fromtheseplotsweobservethatthelocaldeviationislargestadjacenttotheindividualfiberswheretheno-slipconditionforcesthefluidtofollowthedeformingandrotatingfibers,butthatthedeviationdecaysrapidlyawayfromthefibers.Nonetheless,therearestillsignificantfluiddisturbancesspreadthroughouttheentirefluiddomainthatinfluencefibermotionandarerelatedtohydrodynamicinteractionsbetweenindividualfibers.Thecorrespondingglobaldeviationvaluesare‖Erel‖1=0.0159and‖Erel‖∞=0.135whichshow


Fig. 7. A suspension of 256 intrinsically-curved fibers (Px = Pz = 16) in Configuration 3. Parameters aredescribed in Section

5.3.

thatrelativedeviationsintheflowareashighas13.5%nearthefibersbutthattheaverageovertheentireflowfieldisonlyabout1.6%.Othersimulationsusingdifferentparametersandinitialconditionsyieldsimilarresults(see[

52])withtheaveragerelativedeviationhoveringaround2%andthemaximumrangingupto40%.Theseresultssuggestthatincorporatingthefullfluid-structureinteractionintomodelsfornon-dilutesuspensionsisimportantintermsofproperlycapturingthedynamicsoftheflexiblefibers.WealsonotethatthesesimulationsareperformedatrelativelowvaluesofReynoldsnumberandfiberconcentration,andthatthedeviationmeasurewillonlygetlargerastheReynoldsnumberandconcentrationincrease.Finally,weclosebyinvestigatingtheparallelperformanceofourIBalgorithmbyconsideringsimulationsofdifferent-sizedsuspensionsoffibersonmultipleprocessors.Basedonourproblemsetup,theexecutiontimewouldideallystayconstantastheglobalproblemsizeandnumberofprocessorsincrease.Indeed,Table

6showsthatasthesizeofthefiberarray(Px,Pz)isincreased,thereisonlyaslightincreaseinexecutiontimeandhenceouralgorithmissaidtobeweaklyscalable.Weremarkthatourcodeisstillnotfullyoptimizedandthatthealgorithmperformancecouldbefurtherimprovedbymakingenhancementssuchasenforcingthetop/bottomwallboundaryconditionsdirectlyinsteadofourapproachoftreatingthewallsusingIBtetherpoints.

6.Conclusions.Inthispaper,wehavepresentedaparallelimmersedboundaryalgorithmforsimulatingsuspensionsofflexiblefibers,whereindividualfibersaremodelledasKirchhoffrods.Thenoveltyofthisworkderivesfromitsapplicationtomulti-fibersuspensionflowswithnon-


Fig. 8. Fluid deviation Erel on two horizontal planes for the 25 fiber simulation computed in Table

6.PlottedvaluesaretruncatedatthethresholdErel=0.025.

Table6Weakscalingresultsshowingtheaverageexecutiontimepertimestep(inseconds)forthemultiplefiberproblem.ThelocalproblemsizeisheldfixedasthenumberofprocessorsP(andglobalproblemsize)isincreased.SimulationsarerunontheBugabooclustermanagedbyWestGrid[

49].

P(Px,Py,Pz)WallTime

25(5,1,5)0.5764(8,1,8)0.58144(12,1,12)0.58225(15,1,15)0.62256(16,1,16)0.61

zeroReynoldsnumberandtheinclusionofthethefulltwo-wayinteractionbetweenthefluidandsuspendedfibers.Inournumericalsimulations,wereproducethefullrangeoforbitaldynamicsobservedexperimentallybyMasonandco-workersforisolatedfibersimmersedinalinearshearflow.Whenextendingtheresultstomulti-fibersuspensions,wedemonstratethroughaweakscalabilitytestthattheparallelscalingofouralgorithmisnearoptimalandhenceshowspromiseforsimulatingmorecomplexscenariossuchassemi-dilutesuspensionsandfiberflocculation.Inthefuture,weplantoimproveontheunderlyingmodel,whichwillallowustosimulatemorerealisticfibersuspensions.First,weplanonincorporatingthecontactforcesbetweenfiberssuchasthefrictionalforcesmodelledbySchmidetal.[

40].Second,wewillincorporatetheeffectofaddedfibermassusingthepenaltyIBmethod[

22].Afterincorporatingtheseextensions,amoreextensivecomparisontoexperimentaldatawouldberequired,comparingquantitiessuchasthespecificviscosityofthesuspension[

37].

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