Effect of viscosity on the avalanche dynamics and flow transition of wet granular matter

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ContentslistsavailableatScienceDirect

Particuology

j o u r n al ho me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / p a r t i c

Effect

of

viscosity

on

the

avalanche

dynamics

and

ﬂow

transition

of

wet

granular

matter

Jens

H. Kasper

a,∗

_,

_Vanessa

_Magnanimo

b

_,

_Sjoerd

_D.M.

_de

_Jong

a

_,

_Arjan

_Beek

a

_,

Ahmed

Jarray

a,∗

a_Multi-Scale_Mechanics_(MSM),_Thermal_and_Fluid_Engineering,_Faculty_of_Engineering_Technology,_University_of_Twente,_P.O._Box_217,₇₅₀₀_AE_Enschede,

TheNetherlands

b_Construction_Management_and_Engineering_(CME),_Faculty_of_Engineering_Technology,_University_of_Twente,_P.O._Box_217,₇₅₀₀_AE_Enschede,_The

Netherlands

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received30July2020

Receivedinrevisedform22October2020

Accepted8December2020 Availableonlinexxx Keywords: Granularavalanche Transition Cohesion Viscosity Rotarydrum DEM

a

b

s

t

r

a

c

t

Thedynamicbehaviourofgranularflowsisimportantingeo-mechanicsandindustrialapplications,yet poorlyunderstood.Westudiedtheeffectsofliquidviscosityandparticlesizeonthedynamicsofwet granularmaterialflowinginaslowlyrotatingdrum,inordertodetectthetransitionfromthe avalanch-ingtothecontinuousflowregime.Adiscreteelementmethod(DEM)model,inwhichcontactforces andcohesiveforceswereconsidered,wasemployedtosimulatethisflowbehaviour.Themodelwas validatedexperimentally,usingglassbeadsinawoodendrumandwater–glycerolmixturestotunethe liquidviscosity.TheDEMsimulationsshowedcomparableresultstotheexperimentsintermsofaverage slopeangleandavalancheamplitude.Weobservedthattheavalancheamplitude,flowlayervelocity andgranulartemperaturedecreaseastheliquidviscosityincreases.Thiseffectismorepronouncedfor smallersizedparticles.Theincreaseinviscousforcescausestheflowingparticlestobehaveasabulk, pushingthefreesurfacetowardsaconvexshape.Inaddition,avalanchesbecomelesspronouncedand thegranularflowtransitionsfromtheavalanchingregimetothecontinuousregime.Theavalanching flowregimeismarkedbyintermittentrigidbodymovementoftheparticulatebedandnear-zerodrops inthegranulartemperature,whilenorigidbodymovementofthebedoccursinthecontinuousflow regime.Weidentifiedtheavalanching-continuousflowtransitionregionasafunctionofadimensionless granularGalileonumber.

©2020ChineseSocietyofParticuologyandInstituteofProcessEngineering,ChineseAcademyof Sciences.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http:// creativecommons.org/licenses/by/4.0/).

Introduction

Granularﬂowshavebeenextensivelystudiedbyresearchers overthepastyears(Duran,2012;Jarray,Magnanimo,Ramaioli,& Luding,2017;Taghizadeh,Hashemabadi,Yazdani,&Akbari,2018;

Xu,Xu,Zhou,Du,&Hu,2010).Understandingthedynamicsof

ﬂow-inggranularmatterisimportantinbothindustrialapplicationsand themanagementofgeo-physicalphenomena,suchaslandslides andavalanches(Pudasaini&Hutter,2007;Wu,2015).Agranular avalancheoccurswhentheslopeofapileofgrainsexceedsits max-imumangleofstabilitym,whereafterthesurfaceangledecreases

untiltheangleofreposerisreached.Rotarydrumsﬁlledwith

∗ Correspondingauthors.

E-mailaddresses:j.h.kasper@student.utwente.nl(J.H.Kasper),

a.jarray@utwente.nl(A.Jarray).

particlesarepracticalgeometrieswhichhavebeenadopted exten-sivelytomimiccontinuousgranularavalanches(CourrechDuPont,

Fischer,Gondret, Perrin,&Rabaud,2005;Fischer,Gondret,

Per-rin,&Rabaud,2008;González,Windows-Yule,Luding,Parker,&

Thornton,2015;Jarrayetal.,2017;Jarray,Shi,Scheper,Habibi,&

Luding,2019;Liu,Yang,&Yu,2013;Li,Yang,Zheng,&Sun,2018;

Taghizadehetal.,2018;Vo,Nezamabadi,Mutabaruka,Delenne,&

Radjai,2020;Weinhart,Tunuguntla,Jarray,&Roy,2017;Xuetal.,

2010).Ata low rotational speed,avalanches occurperiodically, whileathighrotationalspeedsacontinuousﬂowisobserved.

Avalanchedynamicsareoftencharacterizedbytheevolution oftheslopeangleandparticlevelocity.Theavalancheamplitude =m−risfoundtobeawell-deﬁnedquantity,thatincreases

withparticlesize (Balmforth&McElwaine, 2018;Jaeger, Liu,& Nagel,1989).Courrech Du Pont etal. (2005)observed thatthe velocity proﬁleduringan avalanche decreasesexponentially in piledepthdirection(perpendiculartothepilesurface)andthatno https://doi.org/10.1016/j.partic.2020.12.001

undertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

Pleasecitethisarticleas:Kasper,J.H.,etal,Effectofviscosityontheavalanchedynamicsandﬂowtransitionofwetgranularmatter, Particuology,https://doi.org/10.1016/j.partic.2020.12.001

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Nomenclature ˛ contactangle

ˇ glycerolconcentration(inwater–glycerolmixture) visco-elasticdampingconstant

ı overlapbetweentwoparticles surfaceangle

angularpositionvectorofaparticle m maximumangleofstability

r angleofrepose

˙_rel relativerotationalvelocityvectorbetweentwo par-ticles

avalancheamplitude viscosity

c characteristicviscosity

F coefﬁcientoffriction(CoF)

RF coefﬁcientofrollingfriction(CoRF)

Poisson’sratio particledensity surfacetension unittangentialvector packingfraction

rotationalspeedofthedrum Ca capillarynumber

d piledepth

d0 liquidbridgerupturedistance

E Young’smodulus

e coefﬁcientofrestitution(CoR) F totalforcevector

Fc _contact_force_vector

Fcap _capillary_force_vector

Fl _cohesive_force_vector

Fvis _viscous_force_vector

G shearmodulus

Gag granularGalileonumber

g gravity g gravityvector

h distance-to-contactpointvector h measureofﬂuctuationinsurfaceangle I massmomentofinertiaofaparticle k elasticspringconstant

m particlemass N numberofbins

Na numberofadjacentparticles

Np numberofparticlesinthedrum

n casenumber n unitnormalvector

q torquevectorduetorollingresistance R drumradius

r particleradius

r* effectiveradiusatcontactbetweentwoparticles

Tg granulartemperature

ta totalavalancheduration

ts samplingduration

U ﬂuctuationvelocityvector u particlevelocityvector Vbond liquidbondvolume

v

ﬂowlayervelocity

vc

characteristicvelocityofaparticleintheﬂowlayer w drumwidth

x,y (particle)coordinates x particlepositionvector

steady state develops. Li et al. (2018) found that the variance of theparticlevelocity remains nearly zero when theparticles areatrest,whileitreachesapeakandgraduallydecreasesasan avalancheoccurs.

Theadditionofinterstitialliquidbetweentheparticlesinduces viscous and capillary cohesive forces and alters the dynamics of granular avalanches (Courrech du Pont, Gondret, Perrin, &

Rabaud,2003;Hornbaker,Albert,Albert,Barabási,&Schiffer,1997),

resultingina qualitativelynewbehaviour.Severalstudieshave demonstratedtheimportanceoftheliquids’propertieson gran-ularﬂowandpilestability(Chou,Liao,&Hsiau,2010;Chou,Yang,

&Hsiau,2019;CourrechduPontetal.,2003;Finger&Stannarius,

2007;Kanule,Ng’etich,&Rotich,2019;Liao,2018;Liu,Yang,&Yu,

2011;Louati,Oulahna,&DeRyck,2017;Royetal.,2019;Samadani

&Kudrolli,2001;Shi,Roy,Weinhart,Magnanimo,&Luding,2020).

Chouet al. (2019)examined theeffect of low viscosityliquids

(<10mPas)oncreepinggranularﬂows,andobservedthatthe thicknessoftheﬂowinglayerslightlyincreaseswithliquid

viscos-ity.SamadaniandKudrolli(2001)showedthattheinterstitialliquid

causesparticlestoclumptogether,yieldinganincreaseintheangle ofrepose.Shietal.(2020)demonstratedthatliquid-induced cohe-sioncaneitherdecreaseorincreasethepackingfraction,depending ontheinter-particlefriction.Liquid-inducedcohesionhasalsobeen showntopromotethecollectivemotionofparticles,bykeeping themincontact(Royetal.,2019).Liuetal.(2011)showedthat byincreasingtheliquidsurfacetension,theflowofwetparticles transitions froma steady continuous surfaceflow to a discrete avalanchingflow.Otherstudies byTegzesetal.(1999), Tegzes,

Vicsek,andSchiffer(2003)andTegzes,Vicsek,andSchiffer(2002)

investigatedthetransitionfromintermittentavalanchesto contin-uousﬂowasafunctionoftheliquidcontent,andidentiﬁedthree fundamentalregimesofwetparticles:thegranular,thecorrelated, andtheviscoplasticregime.

Despitethese efforts,it is not yetfullyunderstood howthe interstitialliquidaffectsparticlecontacts,andinturntheﬂowof granularavalanches.Amicromechanicalinterpretationislacking, especiallyforhighlyviscousliquids(>100mPas).Whetherthe viscousforcesenhanceorreducethevolumefractionofwet parti-clesalsoremainsunanswered.Furthermore,theviscosity-induced transitionfromdiscreteavalanching(alsonamedslumping)to con-tinuousﬂowiscurrentlypoorlyunderstood.

Thisworkfocusesontheeffects of liquidviscosityand par-ticlesize onthedynamics and ﬂow behaviourof wetgranular avalanches.Weusedadiscreteelementmethod(DEM)modelto simulateavalanches.Thesimulationsprovidedextrainformation andallowedawiderrangeofresearchconditionsthanwecould achieveexperimentally.Weusedalimitedsetofexperimentsto validateourDEMmodel,whichyieldedcomparableresultsinterms ofaverageslopeangleandavalancheamplitude,withoutany cali-brationofthesimulationparameters.Weexaminedtheevolution oftheslopeangle,particlevelocity,volumefractionandgranular temperature,asfunctionsofliquidviscosity.Thetransitionfrom theavalanchingtothecontinuousﬂowregimewasthen charac-terizedusingdimensionlessquantitiesthatcompareparticleand liquidproperties.Theobtainedresultshaveimportantimplications forgeo-mechanicsandlandslidecontrol.

Materialsandmethods Experimentalmethod Drumsetup

Aschematicrepresentationoftheexperimentalsetupisshown inFig.1. Thesetupconsistsof a drum,that waspartiallyﬁlled withgranular materials and rotated at a speed of =0.5rpm.

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Table1

Propertiesoftheliquidmixtures.

Casenumber,n Glycerolconc.ˇ(%) Mixtureviscosity(mPas) CapillarynumberCa(–) Exp./sim.

0 – – – Exp./sim. 1 0 1.01 0.003 Exp./sim. 2 85 109 0.41 Exp./sim. 3 91 219 0.97 Exp./sim. 4 93 367 1.64 Exp./sim. 5 95 523 2.34 Sim. 6 98 939 4.23 Sim. 7 100 1410 6.39 Sim.

Fig.1.Schematicrepresentationofgranularavalanchesinarotatingdrum.

Thisrotationalspeedprovedlowenoughtoobserveintermittent avalanchingﬂowinthedryandlowviscositycases.Thedrumisan updatedversionoftheoneusedpreviouslybyWindows-Yuleetal.

(2016)andJarray,Shi,etal.(2019).Itiscomposedofacylinder

witharadiusofR=60.5mmandawidthofw=22mm.Thebases ofthedrumconsistoftwoclearcircularplexiglass(PMMA)walls of5mmthicknesstoallowopticalaccess.Thesewerecoatedwith ﬂuorinatedethylenepropylene(FEP)topreventwetglassparticles fromstickingtothewalls.Thesideofthedrumismadeofpoplar wood.

Ineachexperiment,thedrumwasﬁlledtocirca35%ofits vol-umewithmonodisperseborosilicateglassparticles,withadensity of=2500kg/m3_and_a_radius_of_r₌_1.25_mm._We_studied_a_dry_case,

aswellasseveralcaseswheretheparticlesweremixedwith4cm3

ofwater-glycerolmixtures,leadingtoapendularstate(Gabrieli,

Lambert,Cola,&Calvetti,2012;Iveson,Beathe,&Page,2002;Jarray,

Shi,etal.,2019;Urso,Lawrence,&Adams,1999).Here,a small

amountofliquidissharedbyneighbouringparticles,butbetween themismostlyair.Thesurfacetensionandcontactangleofthe mixtures staynear-constantasafunctionofglycerol concentra-tion,whiletheliquidviscositystronglyvaries(GlycerineProducers’

Association,1963;Vicente,André,&Ferreira,2012).Therefore,the

inducedviscousforcesstronglyincreasewithglycerol concentra-tion,whilethecapillaryforcesremainrelativelyconstant.Table1

displaysthepropertiesofthemixtures.Thecasen=0indicatesdry conditions.Asubsetofcaseswerereproducedwithboth experi-mentsandsimulations(seeSection“Numericalmodel”),allowing validation. Forliquidswith>370mPas,particlesstarttostick tothefrontwalloftherotatingdrum,obstructingaccurateimage post-processing,thusonlysimulationswereperformed.

Table1alsoshowsthevaluesofthecapillarynumberCa,deﬁned

as Ca=

vc

cos˛, (1)

withliquidviscosity,surfacetension,contactangle˛and char-acteristicvelocity

vc

.Thecapillarynumberistheratiooftheviscous forcetothecapillaryforce.Sincethevelocityoftheparticlesinthe ﬂowinglayerduringanavalancheismainlyinducedbygravity,the characteristicvelocitywastakenequaltothefree-fallingspeedofa particle,afterfallingadistanceof2r,i.e.

vc

=

4gr,whichyieldsan approximatehigh-endvalueofCa.InTable1,weusedr=1.25mm forthecalculationofCa.

Imagepost-processing

ImagesoftherotatingdrumwererecordedusingaCanonLegria HFG40camera,operatingat50FPS.Theimagescapturedbythe camerawerepost-processedusingFijiImageJ(Ruedenetal.,2017;

Schindelinetal.,2012)toobtaintheslopeangleandtheavalanche

amplitude,asdepictedinFig.1.In theﬁrstprocessingstep, theirrelevantpartsoftheimagesequence(i.e.thebackgroundand thedrum walls)wereremovedandthelightandcontrastwere adjusted.Subsequently,theTrackmate(Tinevezetal.,2017) pack-agewasusedtodetecttheparticlesbasedonthedifferenceof Gaussiandistributions(Lowe,2004).Thedetectedparticleswere storedintabularformandimportedintothedatavisualization pro-gramParaview(Ayachit,2015).Finally,anin-housePythonscript wasemployedtodeterminethe slopeangle andthe avalanche amplitude.

Numericalmodel

WeemployedthesoftwarepackageLIGGGHTS(Kloss,Goniva,

Hager,Amberger,&Pirker,2012)tosimulatewetparticlesin a

rotarydrum, usingthediscreteelementmethod.Particle trajec-toriesarecalculatedbysolvingNewton’sequationsofmotion.The movementofanarbitraryparticlei,thatisincontactwithNa

adja-centparticles,isdescribedas mi¨xi= Na

j=1

Fc_ij+Fl_ij

+mig, (2) Ii¨i= Na

j=1

hij×

Fc_ij+Fl_ij

+q_ij

, (3)

withmassmi,positionvectorxi,contactforcevectorFcij,cohesive

forcevectorFl_ij,gravityvectorg,massmomentofinertiaIi,angular

positionvectori,distance-to-contactpointvectorhij and

addi-tionaltorquevectorqij(toaccountforrollingresistance).Notethat

variablesprintedinboldrepresentvectors. Contactmodel

Aspring-dashpotmodel(DiRenzo&DiMaio,2005;Klossetal.,

2012;Silbertetal.,2001)wasusedtocomputetheforceFc_ijbetween

twocollidingparticlesiandj.F_ijcisnonzeroonlyifthereisapositive normaloverlapınbetweentheparticles,whichisdeﬁnedas

ın=

ri+rj

−

xi−xj

·nij, (4)

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whereristheparticleradiusandnijtheunitnormalvector.When ın>0,Fcijiscomputedas Fc_ij=

knın−n˙ın

nij+

kı−˙ı

ij, (5)

withtangentialoverlapı,unittangentialvectorij,elasticspring

constantkandvisco-elasticdampingconstant.Thesubscriptsn anddenotenormalandtangentialcomponentsrespectively.

The Hertz–Mindlin model (Kloss et al., 2012; Mindlin &

Deresiewicz,1953),basedonHertztheoryinnormaldirectionand

theMindlinno-slipimprovedmodelintangentialdirection,was employedtocalculatetheelasticanddampingconstants.knisgiven

by kn= 4 3

1−2 i Ei + 1−2 j Ej

−1

r_∗ın, (6)

withmodulusofelasticityE,Poisson’sratioandeffectiveradius r∗=rirj/

ri+rj

.Furthermore,niscomputedas n=− 2√15ln (e) 3

ln2(e)+2

1 ir3i + 1 jr3j

−1 kn, (7)

withcoefﬁcientofrestitution(CoR)eandparticledensity.Asfor thetangentialcomponents,kisdeﬁnedas

k=8

2₋_i Gi + 2−j Gj

−1

r_∗ın, (8)

whereGistheshearmodulusandisdeﬁnedas

=− 2 √ 10ln (e) 3

ln2(e)+2

1 ir_i3 + 1 jr_j3

−1 k. (9)

A constantdirectionaltorquemodel(Ai,Chen,Rotter,&Ooi,

2011;Klossetal.,2012)wasimplementedtoaccountforrolling

resistance,andcontributeswithanadditionaltorquevectorqijto

Eq.(3),whichisgivenby qij=−RFknınr∗

˙rel

˙_rel

, (10)

withcoefﬁcientofrollingfriction(CoRF)RF.Thevector ˙relisthe

relativerotationalvelocitybetweenparticlesiandj.Inthecaseof particle-wallcontact,Eqs.(6)–(10)aretakeninthelimitofradius rjgoingtoinﬁnity.

Liquidcohesionmodel

Acompositionofmodels,assuggestedbyEasoandWassgren

(2013),wasusedtodeﬁnethecohesiveforcebetweenparticlesi

andj.Itisassumedthatasurfaceliquidﬁlmexistsoneach parti-cle,allowingforaliquidbridgetoformuponcontactwithanother particle. Thebridgeruptureswhen

xi−xj

exceeds therupture

distanced0,whichisgivenby

d0=

₁ 2+ ˛i+˛j 4

V_bond1/3 , (11)

withcontactangle˛andliquidbondvolumeVbond(Lian,Thornton,

&Adams,1993).ItisassumedthatVbondequals5%ofthecombined

liquidﬁlmvolume,whichdistributesevenlyoverthetwo parti-clesafterbridgerupture.ThecohesiveforcevectorFl_ijbetweenthe particlesisdeﬁnedas

Fl_ij=Fcap_ij +Fvis_ij , (12)

withcapillaryforcevectorFcap_ij andviscousforcevectorFvis_ij .

Thecapillaryforceactsinnormaldirectionandisdeﬁnedby

Soulie,Cherblanc,ElYoussouﬁ,andSaix(2006)as

Fcap_ij =−

rirj

c+exp

a−ın rmax+b

nij, (13)

withliquidsurfacetensionandrmax=max

ri,rj

.Parametersa, bandcaregivenby

a=−1.1

Vbond r3 max

−0.53 , (14) b =[−0.037ln(Vbond r3 max )−0.24](˛i+˛j)2 −0.0082ln(Vbond r3 max )+0.48, (15) c=0.0018ln

Vbond r3 max

+0.078. (16)

Theviscousforceconsistsofatangentialandnormalcomponent andisdescribedbyNase,Vargas,Abatan,andMcCarthy(2001)as

Fvis_ij =6r_∗˙ı

₈ 15ln

_r ∗ −ın

+0.9588

ij +6r∗2˙ın −ın nij, (17)

withliquidviscosity.Thevalueforthenormaloverlapisbounded by

−0.1(ri+rj)≤ın≤−0.01(ri+rj), (18)

topreventlimitlessdevelopmentofthecohesiveforces.Inthecase ofparticle-wallcontact,Fcap_ij andFvis_ij aresetequaltozero.

Tosummarize,thetotalforcevectorFijcanbedecomposedin

normalandtangentialcomponentsas Fij =Fcij+Fijl =Fnij+Fij =

knın−n˙ın−

rirj

c+exp

a−ın rmax+ b

+6r∗2˙ın −ın

nij+

kı−˙ı +6r∗˙ı

₈ 15ln

_r ∗ −ın

+0.9588

ij, (19)

withtotalnormalforcevectorFn_ij andtotaltangentialforce vec-torF_ij.Furthermore,thetangentialcomponentofthecontactforce Fc_ij·ijistruncatedtofulﬁll

Fc_ij_·_ij

_≤F

Fnij

, (20)

withcoefﬁcientoffriction(CoF)F,toensurethesatisfactionof

Coulomb’slaw.

DEMsimulationparameters

ThemodelparametersintheDEMsimulationsweresettomimic thephysicalexperiments.Thematerialpropertiesoftheparticles anddrumweresettoresembleglassandpoplarwoodrespectively

(Gray,1972;Green,Winandy,&Kretschmann,1999;Oberg,Jones,

Horton,&Ryffel,2000;Serway&Jewett,2004;SigmundLindner,

2018),andaresummarizedinTables2and3.

Inallsimulations,thedrumwasﬁlledto35%ofitsvolumewith monodisperseparticlesandrotatedataconstantrotationalspeedof =0.5rpm,provenlowenoughtoobserveintermittentavalanches inthedryand lowviscositycases.Theparticleshadaradiusof

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Table2

Propertiesoftheparticles.

Parameter Value

Radius,r(mm) 1.25,2.0,3.0 Density,(kg/m3₎ ₂₅₀₀

Young’smod.,E(MPa) 63 Poisson’sratio,(–) 0.21 CoR,epp_(–) _0.95 CoF,pp_F (–) 0.10(wet) 0.40(dry) CoRF,pp RF(–) 0.01 Table3

Propertiesofthedrum.

Parameter Value

Radius,R(m) 0.0605

Width,w(m) 0.022

Young’smod.,E(MPa) 12 Poisson’sratio,(–) 0.35 CoR,ewp_(–) _0.72 CoF,wp_F (–) 0.15(wet) 0.30(dry) CoRF,wp RF(–) 0.01

r=1.25,2.0or3.0mm.Weperformedsimulationsunderdry con-ditions,andsimulationswithsevenwater–glycerolmixtures, as presentedinTable1.WeusedareducedCoFforwetparticlesto mimiclubricationeffectsinthegranularsystem.

Ineachsimulation,wetrackedtheparticlepositionsand veloci-tiesfor50seconds,toexaminetheevolutionoftheslopeangleand velocityproﬁles.ThesoftwarepackageParaView(Ayachit,2015) andin-housepythoncodeswereusedtoevaluatetheslopeangle ,theavalancheamplitudeandthepackingfraction ,deﬁned asthevolumeofallparticlesdividedbythevolumeofthepacking. Furthermore,weevaluatedthegranulartemperatureTg(Campbell,

2006;Zhou&Sun,2013),deﬁnedas

Tg= 1 3Np Np

i=1

Ui

2 , (21)

whereNpisthetotalnumberofparticles.Theﬂuctuationvelocity

ofparticleiisgivenby

U_i₌u_i_{− ¯u}_i, (22)

whereuiisthevelocityofparticlei,and ¯uithemeanvelocityofthe

particlessurroundingi,withinasphericalregionofradius7ri/3.

Thus,Uiindicatesaparticle’svelocitywithrespecttoits

surround-ingparticles.

Thevelocityproﬁlesoftheparticlesinthebedwereobtained usingaGaussianKernelpointvolumeinterpolationscheme(Jarray,

Magnanimo,&Luding,2019;Price,2012;Schroeder,Lorensen,&

Martin, 2004), witha kernel radiusof8mmfor thecaseswith

r=1.25mmandr=2mm,andakernelradiusof11mmforthecases withr=3mm.

Resultsanddiscussion Modelvalidation

Tovalidateournumericalmodel,weﬁrstcomparethesimulated andmeasuredtime-averagedsurfaceangleandtime-averaged avalancheamplitude.Fig.2(a)showsasafunctionofthe interstitialliquid viscosity.Bothexperimentsandsimulations indicatethathasnosigniﬁcanteffectonintheinvestigated viscosityrange.However,wetparticlesyieldahigherin com-parisontodryparticles,duetheformationofcapillarybridges.Inall

Fig.2. Experimentalandnumericaltime-averagedavalanchepropertiesasa

func-tionofliquidviscosity,withaparticleradiusofr=1.25mm.(a)Time-averaged

surfaceangle.(b)Time-averagedavalancheamplitude.Symbolsinthegreen

zonerepresentsimulationsandexperimentsunderdryconditions.

wetcases,simulationsyieldaslightlyloweraveragedslopeangle thantheexperiments.In theDEMmodel,theliquid bridge vol-umewassettoalwaysdistributeevenlyoverbothparticlesafter bridgerupture.Thus,weproposetoattributetheobserved differ-encebetweensimulationsandexperimentstotheoccurrenceof in-homogeneousliquidmigrationbetweenparticlesinthe experi-ments.

Fig.2(b)showstheaveragedavalancheamplitudeasafunction of.Weobserveadecreasingtrendofwithliquidviscosity inboththenumericalandexperimentalcases.

Comparable resultsare obtainedfrom the experiments and numericalsimulations,whichconﬁrmthevalidityoftheadopted model.Sinceexperimentalresearchdidnotallowinvestigationof theavalanchebehaviourforhighviscosityliquids(>370mPas), asdiscussedinSection“Experimentalmethod”,andtogetmore insightsintothemicro-mechanicsofgranularavalanches,wewill henceforthfocusontheresultsobtainedfromtheDEMsimulations. Simulationresults

Timeevolutionofthegranularﬂow

Fig.3(a)showsatypicalevolutionofthesurfaceangleduring severalsuccessivedryavalanches.Maximumstabilityandrepose anglesmandrrespectivelycorrespondtothelocalmaximaand

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Fig.3.Timeevolutionofgranularavalancheswithaparticleradiusofr=2mm.(a)

Surfaceangleand(b)granulartemperatureTgforthedrycasen=0.(c)and(d)

Tgforthelowviscositywetcasen=1andhighviscositywetcasen=7.

areverypronounced;rigidbodymovementoftheparticulatebed isalternatedwithparticlesﬂowingdowninthesurfacelayer.Thisis furtherillustratedbytheevolutionofthegranulartemperatureTg,

showninFig.3(b).WeobservepeaksinTgduringtheavalanches,

whichindicatelargerelativemovementsbetweenparticles,that coincidewithdropsin.Inbetweenavalanches,Tgdropstonearly

zero,whiletheparticulatebedisliftedagainasarigidbody. Qualitatively similar behaviour is observed for wet gran-ular avalanches with an interstitial liquid of low viscosity (=1.01mPas, n=1),shownin Figs.1(c)-(d)bythecontinuous blueline.However,forhighviscosity(=1410mPas, n=7),the evolution of thesurfaceangle doesnotshowa steady periodic trend. Smaller ﬂuctuations occur more frequently, while dis-tinctavalanchesdisappear,implyingamorecontinuousﬂow.We observeasimilartrendinthedevelopmentofTg.Incomparisonto

thedrycasen=0andlowviscositycasen=1,peaksforthecase n=7aresigniﬁcantlysmaller.Furthermore,Tgdoesnotdropas

sig-niﬁcantly withtheincreaseof,indicatingthat thereisalways relativemovementbetweenparticles,andtheparticulatebednever behavesasarigidbody.

AninterestingfeatureoftheavalanchesshowninFig.3(a)and (c)istheratherlargevariationintheamplitudesofsuccessive avalanches.Aslowerrotationalspeedofthedrumislikelyto reducethesevariationsandprovideanevenmoreperiodictrend

(Balmforth&McElwaine,2018;Fischeretal.,2008).

Appendix“Additionaldataforotherparticlesizes”depictsthe time evolutionof andTg forsimulationswithr=1.25mmand

Fig.4.Velocityproﬁlesduringgranularavalancheswithaparticleradiusofr=2mm.

(a)Drycasen=0.(b),(c)and(d)Wetcasesn=1,3,7respectively,withviscositiesof

=1.01mPas,=219mPasand=1410mPas.(e)Particlevelocityvasafunction

ofpiledepthdforn=0,a,3,7,wheredisthedistancealongtheinward-pointing

normaltothefreeﬂowingsurface,startingfromitsmiddle.

r=3mm.Inthesecasesweobservequalitativelysimilarbehaviour tothosewithr=2mmshowninFig.3.

Velocityproﬁles

Fig.4(a)–(d)showstypicalvelocityprofilesforavalancheswith lowtohighliquidviscosity.Inallcases,ahighvelocityflowlayer nearthesurfaceisobserved,whilebelowthislayerparticlesareat restwithrespecttothedrum,i.e.onlymoveduetoitsrotational speed.Thedrycasen=0yieldsalargerflowlayervelocity

v

than thewetcases,forwhich

v

decreaseswithliquidviscosity.Thisresult hasalsobeenobservedbyBrewster,Grest,andLevine(2009),who foundthatcohesiveforceslimitthedevelopmentofthevelocityin theﬂowlayer.

Fig.4(e)showstheparticlevelocityalongthepiledepthd,which isalongtheinward-pointingnormaltothefreeflowingsurface, startingfromitsmiddle.Inallcases,thevelocityoftheflowing layerdecreasesslowlywithdinitially,nearthesurface;thenmore rapidly,inthemiddleoftheflowinglayer;butthendeclines,until thevelocityoftheparticlesispurelyduetotherotationalspeedof thedrum.Theslopeofthevelocitycurvedecreaseswithviscosity, andissteepestforthedrycase.However,thethicknessoftheflow layerappearstobeindependentofliquidviscosity.

The velocity proﬁles for particle radii of r=1.25mm and r=3mm,presentedinAppendix“Additionaldataforotherparticle sizes”,showqualitativelysimilarbehaviourtothosewithr=2mm

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Fig.5.Time-averagedavalanchepropertiesplottedasafunctionofliquidviscosity

,forseveralparticlesizes.(a)Avalancheamplitude,wherethesymbolsinthe

greenzonerepresentsimulationswithdryconditions.(b)Dimensionlessavalanche

amplitudewet/dry.(c)Dimensionlessaveragedeviationhwet/hdry.

Theinsetpictureschematicallyshowsthemethodusedtoquantifytheﬂuctuations

inthesurfaceangle.

Fig.6.Avalanchepropertiesplottedasafunctionofviscosity,forseveralparticle

sizes.(a)Time-averagedgranulartemperatureTg.(b)Dimensionlessavalanche

timeparameterta/ts.Symbolsinthegreenzonerepresentsimulationswithdry

conditions.

showninFig.4.Theﬂowlayervelocitywasobservedtoincreasein magnitudewithparticlesize,yettheoveralltrendsremainsimilar. Avalancheamplitude

Fig.5(a)showsthetime-averagedavalancheamplitudeas afunctionofliquidviscosity,forseveralparticlesizes.Notethat asincreases,becomeslessameasureofavalanche ampli-tude,andmoreameasureofsurfaceangleﬂuctuationintime,as theﬂowtransitionsfromtheavalanchingregimetothecontinuous regime.Theavalancheamplitudeisshowntoincreasewithparticle radiusr.Thistrendisconsistentwithexperimentalresearchondry

(Balmforth&McElwaine,2018)andwet(CourrechduPontetal.,

2003)granularavalanches,inwhichlargeravalanchesareobserved astheratioofparticleoverdrumradiusr/Rincreases.Forall parti-clesizes,isobservedtodecreasewithviscosityandsaturate athighvaluesof,inagreementwithstatementsbyCourrechdu

Pontetal.(2003).

We can exclude the effect of the particle size on the wet avalanche amplitude by considering its dimensionless form wet/dry.Asaresult,thenormalizedavalancheamplitude

datacollapseonasingleviscosity-dependantmaster-curve,plotted inadashedlineinFig.5(b).Theﬁttingcurveisdescribedby wet/dry=a+bexp

−/c

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Fig.7. Avalanchepropertiesplottedasafunctionof

1/Gag,withgranularGalileo

numberGag,forseveralparticlesizes.(a)Dimensionlessavalanchetimeparameter

ta/ts.(b)Time-averagedavalancheamplitude.

where a and b are ﬁtting constants equal to 0.4 and 0.47, respectively.Furthermore,cisacharacteristicviscosityequalto

295mPas.Weinferthatfor<cthedecayofwet/dryis

large,whilefor>c,thedimensionlessamplitudesaturates.

Slopecurvature

Thecurvatureofthesurfacecanbeusedtodifferentiatebetween avalanchingandcontinuousflowbehaviour.Forgranularflowsin theavalanchingregime,thesurfaceangleisgenerallyobservedto beconstantalong thesurface(Fischeretal.,2008;Jaeger etal., 1989).Forcontinuousflowshowever,thesurfaceisobservedtobe “S-shaped”ifthecohesionislowand>5rpm(Jarray,Magnanimo, etal.,2019),orevenslightly convexifcohesiveforcesarelarge

(Brewsteretal.,2009).Signiﬁcantﬂuctuationsinthesurfaceangle

(i.e.thedeviationbetweenthelocalsurfaceangleandthe aver-ageangleofthewholesurface)areindicativeofcontinuousﬂow behaviour.

Wecanquantifythesefluctuationsusingthefollowingmethod. First,weapproximatewhichparticlesarepartofthesurfacelayer bydividingthedrumintoNbinsandconsideringthetopparticle ineachbin.Polyfittingalinearfunctionthroughthecoordinates ofallsurfaceparticlesyieldsaflataveragedsurface(seeinsetof

Fig.5(c)).Fluctuationsinsurfaceangleareevaluatedby

measur-Fig.8.Time-averagedpropertiesasafunctionofliquidviscosity,forseveral

par-ticlesizes.(a)Surfaceangle.(b)Packingfraction .Symbolsinthegreenzone

representsimulationswithdryconditions.(Forinterpretationofthereferencesto

colourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

ingtheaveragedeviationh,ofthedistancebetweenthesurface particlesandthepolyﬁttedsurface.Wethusdeﬁnehas h= 1 N N

i=1

xi−xﬁt_i

2 +

yi−yﬁt_i

2 , (24)

inwhich(xi,yi)denotethecoordinatesofparticlei,and(xﬁt_i ,yﬁt_i )

denotepointsonthepolyﬁttedsurface.Fig.5(c)showsthatindeed thetime-averagednormalizedsurfacedeviationhwet/hdry,

slightlyincreaseswithliquidviscosity,andconsistentlydecreases withincreasingparticlesize.

Granulartemperature

The time-averaged granular temperature Tg is plotted in

Fig.6(a),andshowsasimilartrendasthetime-averagedamplitude .Tgdecreaseswithviscosity,butincreaseswithparticlesize.

Asr/Rincreases,Tgisevaluatedoveralargerlocalvolume,thus

largerrelativemovementsareindeedexpected.Thegranular tem-peratureofthedrycasesisofanorderofmagnitudegreaterthan forthewetcases.Theliquid-inducedcohesivebondsbetween par-ticleslimitrelativeparticlemotion.Particlesclumptogether,and thenumber ofinter-particlescollisionsreduces. Thiseffectgets magniﬁedasincreasesandthecohesivebondsbecomestronger. Interestingly,the curves for r=1.25mm and r=2mmare quite

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Fig.9.Timeevolutionofgranularavalancheswithaparticleradiusofr=1.25mm.

(a)Surfaceangleand(b)granulartemperatureTgforthedrycasen=0.(c)and

(d)Tgforthelowviscositywetcasen=1andhighviscositywetcasen=7.

smooth,whileforr=3mmsomeoutliersappear.Thisislikelydue tointerlockingofparticles,causedbythehighparticleoverdrum sizeratio(r/R≈0.05).

Liu,Specht,andMellmann(2005)pointedoutthatthe

avalanch-ingregimeismarkedbyintermittentrigidbodymovementofthe particulatebed,duringwhichTgdropstonear-zero(see Fig.3).

Astheliquidviscosityincreases,thetimeintervalsofrigidbody movement become shorter, and the flow transitions from the avalanchingregimetothecontinuousregime.Incompleteabsence oftheseintervals,theflowcanbecharacterizedasfully continu-ous.Inordertoinvestigatethetransitionfromavalanchingflow tocontinuousflow,weevaluateavalanchingtimeratiota/ts.Here

taisthetotalavalanche duration(i.e.thedurationforwhichTg

≥2·10−9_m2_/s2_,_{corresponding}_to_non-rigid_body_movement_of_the

bed),andtsisthesamplingduration.Fig.6(b)depictsta/tsasa

func-tionofviscosity,wheretswassettotheﬁnal30softhesimulation.

Weﬁndthatta/tsincreaseslogarithmicallywithliquidviscosity.For

particlesizesr=1.25mmandr=2mm,continuousﬂowdevelops beyond=520mPasand =940mPas,respectively.The corre-spondingaverageavalancheamplitudesarebelow2.5◦.However, forr=3mm,continuousﬂowdoesnotseemtodevelopwithinthe rangeofviscositiesexaminedinourresearch.

Dimensionalanalysisandﬂowtransitionregion

In order toestimatetherelative importanceof gravitational forceswithrespecttotheviscousforcesactingonwetparticles, weintroducethegranularGalileonumber:

Fig.10.Timeevolutionofgranularavalancheswithaparticleradiusofr=3mm.(a)

Surfaceangleand(b)granulartemperatureTgforthedrycasen=0.(c)and(d)

Tgforthelowviscositywetcasen=1andhighviscositywetcasen=7.

Gag=

2_gr3

2 . (25)

TheGalileo number is anespecially relevant parameter, asthe displacementofparticlesduringavalanchesisdominatedby grav-itational forces. In Fig. 7(a), we plot ta/ts as a function of the

square-rootoftheinverseofGag.FromFig.6(b),wefoundthat

continuous ﬂow develops only for cases with r=1.25mm and r=2mm, beyond =520mPas and =940mPas, respectively, whereta/ts≈1.Weareabletodeﬁnearegioninwhichthe

transi-tionfromavalanchingﬂowtocontinuousﬂowtakesplace,marked bythebluezoneinFig.7(a).Thisregionroughlycorrespondstoa granularGalileonumberGag≈1,i.e.theviscousforcesstartto

sur-passthegravitationalforcesatthispoint.Wealsoobservethatthe datasetcollapsesontoasinglehill-typecurve,thattendstota/ts=1

athighvaluesof

1/Gag.

Havingestablishedtheavalanching-continuousﬂowtransition zone interms of the Galileonumber, we subsequently explore theeffectofthisnumberontheavalancheamplitudeinFig.7(b). For

1/Gag<10−2,theﬂowbehaviourisalmostunaffectedby

theviscousforces,butrathergovernedbycapillaryand gravita-tionalforces,resultinginarelativelyhighavalancheamplitude. As

1/Gag increases, theﬂow becomes more dilated and the

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Fig.11. Velocityproﬁlesduringgranularavalanches withaparticleradiusof

r=1.25mm.(a)Drycasen=0.(b),(c)and(d)Wetcasesn=1,3,7respectively,with

viscositiesof=1.01mPas,=219mPasand=1410mPas.(e)Particlevelocityv

asafunctionofpiledepthdforn=0,1,3,7.

Averagesurfaceangleandvolumefraction

Fig.8(a)showsthetime-averagedsurfaceangle,with stan-darddeviations,asafunctionof,forseveralparticleradii.Forthe drycase,increaseswithparticlesize,inagreementwith previ-ousobservations(Balmforth&McElwaine,2018).Forthewetcases, showsaslightlydecreasingtrendwithr,whichisinagreement withobservationsmadebyNowak,Samadani,andKudrolli(2005). Again,someoutlyingpointsappearforsimulationswithr=3mm, possiblytheresultofinterlocking.Weobservenocleardependency oftheaverageangleontheliquidviscosity.

Finally, thetime-averagedvolume fractionof theparticulate bed isplottedasafunctionoftheinterstitialliquidviscosity inFig.8(b).Here, isdeﬁnedasthesummedvolumeofthe parti-clesoverthevolumeoccupiedbythepacking.Weobservethatwet particlesyieldahighervolumefractionthandryparticles, indica-tiveofamorecompressedsystem(Shietal.,2020).Overall,itseems thatviscosityonlyhasaweakeffecton .However, decreases withincreasingparticlesize.Thisislikelyduetotheincreasing par-ticleoverdrumradiusratior/R,whichnegativelyimpactspacking efﬁciency.

Conclusions

Usingdiscreteelementmethodsimulations,wehave investi-gatedhowthedynamicsofwetgranularavalanchesina rotary drum are affectedby theviscosityof theinterstitial liquidand

Fig.12.Velocityproﬁlesduringgranularavalanches withaparticleradiusof

r=3mm.(a)Drycasen=0.(b),(c)and(d)Wetcasesn=1,3,7respectively,with

viscositiesof=1.01mPas,=219mPasand=1410mPas.(e)Particlevelocityv

asafunctionofpiledepthdforn=0,1,3,7.

particlesize.Simulationswerevalidatedagainstalimitedsetof experimentalmeasurementsandgoodagreementwasobtained.

Discreteavalanches wereobserved forthe drycase and the wetcaseswithrelativelylowviscosity.Asviscosityincreases,the flowbehaviourtransitions fromtheavalanchingtothe continu-ousregime.Thistransitionisindicatedbyadecreaseinavalanche amplitude,flow layervelocityand granulartemperatureandan increaseincurvynessoftheflowsurface.Theavalanchingregime ismarkedbyintermittentrigidbodymovementoftheparticulate bed,andnear-zerodropsinthegranulartemperature.For contin-uousflow,rigidbody movementofthebeddoesnotoccurand thegranulartemperatureassumesfinitevalueswellabovezero. Basedontheevolutionofthegranulartemperatureandavalanche amplitude,wewere abletoidentifyanavalanching-continuous flowtransitionregion,asafunctionofthedimensionlessgranular GalileonumberGag.

Astheparticlesize increases,theavalancheamplitude, ﬂow layervelocityandgranulartemperatureincreaseaswell,indicating anincreaseinrelativeimportanceofinertialeffectswithrespectto frictionalforcesandviscousforces.

Our ﬁndings suggest that theeffects of liquid viscosity and particlesizeshouldalwaysbeconsideredwhenstudying granu-laravalanches,particularlyfor

1/Gag>0.5,whetherlargescale

investigation,pilotscaleexperimentsornumericalsimulationsare ofinterest.

Theanalysispresentedinthispapercouldbeextendedto cor-relatetheﬂowbehaviour(i.e.theangleofreposeortheavalanche

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amplitude)ofthegranularmixturetoitsapparentshearviscosity, whichcanbeobtainedbyextractingtheratiooftheshearstressto theshearrate(Schwarze,Gladkyy,Uhlig,&Luding,2013)alongthe depthoftheﬂowinglayer.Additionally,itwouldbeinterestingto explorehowliquidviscosityaffectsthedynamicsofgranular mat-teratahigherrotationalspeedofthedrum,wheretheﬂowregime iscontinuousandintermittentavalanchesareabsent.

Conﬂictofinterest

Theauthorsdeclarenoconﬂictofinterest. Acknowledgements

WethankStefanLudingandBertJ.Scheperfortheirhelpand suggestions.

Additionaldataforotherparticlesizes

Figs.9and10 showthetimeevolutionofthesurfaceangle

and thegranulartemperatureTg forcases withr=1.25mmand

r=3mm, respectively. Similartocases withr=2mm, shownin

Fig.3,discreteavalanchesareobservedfordrycasesandlow viscos-itycases(=1.01mPas).Highlyviscousliquids(=1410mPas) yieldsignificantreductionsinsurfaceanglefluctuationsand fluctu-ationsinTg,andavalanchesarelesspronounced.Thiseffectappears

lessdominantforlargerparticleshowever,asther=3mmcasewith highviscositystillshowsdistinctavalanches.

Figs.11and12 showthevelocityproﬁlesforavalancheswith

lowtohighliquidviscosityforcaseswithr=1.25mmandr=3mm, respectively.Thevelocityoftheﬂowinglayerincreaseswith parti-clesize,whileitsthicknessremainsrelativelyconstant.

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