Physics of particulate flows: From sand avalanche to active suspensions in plants

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active suspensions in plants

Yoel Forterre, Olivier Pouliquen

To cite this version:

Yoel Forterre, Olivier Pouliquen. Physics of particulate flows: From sand avalanche to active suspen- sions in plants. Comptes Rendus. Physique, Académie des sciences (Paris), 2018, 19 (5), pp.271-284.

�10.1016/j.crhy.2018.10.003�. �hal-02080014�

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PrixMergier–Bourdeix2017del’Académiedessciences/PrixErnest-Déchelle2017del’Académiedessciences

Physics of particulate flows: From sand avalanche to active suspensions in plants

La physique des écoulements de particules : des avalanches aux suspensions actives dans les plantes

Yoël Forterre ^∗ , Olivier Pouliquen ^∗

AixMarseilleUniversité,CNRS,IUSTI,Marseille,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Availableonlinexxxx

Keywords:

Granularflows Suspensions Colloids Rheology Activematter Complexfluids

Mots-clés :

Écoulementsgranulaires Suspensions

Colloïdes Rhéologie Matièreactive Fluidescomplexes

Flows of granular media inair or ina liquidhave been a researchfield for physicists forseveraldecades.Sometimessolid,sometimesliquid,theseparticulatematerialsexhibit peculiarbehaviors,whichhavemotivatedmanystudiesatthefrontiersbetweennonlinear physics,softmatterphysicsand fluidmechanics.Thispaper presentsasummaryofthe recentadvancesinthefield,withafocusonthedevelopmentofcontinuousapproaches, which make it possible to treat granular media as a complex fluid and to develop a granularhydrodynamics.Wealsodiscusshowthebetterunderstandingofgranularflows wehavetodaymayhelptoaddressmorecomplexmaterials,suchascolloidalsuspensions orsomebiologicalsystems.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s u m é

Les écoulements de milieuxformés degrains dans l’air ou dans unliquide intéressent les physiciensdepuis plusieurs décennies. Tantôtsolides, tantôt liquides,ces matériaux divisés ontdescomportementssinguliers quisont aucœurde nombreuses études àla frontièreentrelaphysiquenonlinéaire,laphysiquedelamatièremolleetlamécanique des fluides. Cet article se proposede faire un point sur lesavancées récentes dans le domaine,en seconcentrant surledéveloppementd’approchescontinues quipermettent de traiter le milieu commeun fluide complexeet de développerune hydrodynamique granulaire.Nousdiscutonségalementenquoilacompréhensionplusfinedesécoulements granulairesquenousavonsaujourd’huipermetdemieuxappréhenderlesmatériauxplus complexescommelessuspensionscolloïdales,voirecertainsmilieuxbiologiques.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

*

Correspondingauthors.

E-mailaddresses:yoel.forterre@univ-amu.fr(Y. Forterre),olivier.pouliquen@univ-amu.fr(O. Pouliquen).

https://doi.org/10.1016/j.crhy.2018.10.003

1631-0705/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Fig. 1.Examples of granular media and illustration of the different interactions playing a role as a function of the sizedof the particles.

1. Introduction:grainsatdifferentscales

What isincommonbetweenthemortar preparedby aworker inamixer andaplant, whichrecoversa verticalpos- ture afterbeingtilted by thewind?Althoughvery different, both situationsinvolveto some extentthe flowofa pileof grains. Coarsesandand gravelsare avalanching inthe rotatingmixer while microscopic starch grainsare avalanching in the statocytes, the specific cells in plantsdedicated to the detectionof the gravity direction. Understanding the dynam- icsofgrain flowsisthusimportantinmanydifferentsituations,fromindustrialapplications(forexampleinconstruction industry, pharmaceuticalindustry,foodindustry) tonaturalprocesses (forexample,avalanches,debris flows,soil erosion, plantgravitropism), andatverydifferentscales.The descriptionofdensegranular media stillrepresentsa realchallenge, mainlybecausethismulti-bodysysteminvolvescomplexparticleinteractions[1–3].Thequestforacontinuum description for particulateflowsis notfinished, althoughadvances havebeen madeduring thelast 20years,which arediscussed in thispaper.

ThepropertiesofdenseparticulateflowsdependontheparticlesizeasillustratedinFig.1,themainreasonbeingthat interaction forces betweensolid grainsstronglydepend on their size.For particles typicallylarger than d

>

100 μm, the dominant forcesaredueto directmechanicalcontactsinvolvingnormalcompression andtangential frictionalforces,and when the grainsare immersedin aliquid, tohydrodynamics interactions induced by themotionof theinterstitial fluid.

For smaller particles,typically d

<

100 μm, other colloidal forcescome into play, such asvan der Waals orelectrostatic interactions, or entropic forces induced, for example, by polymer brushes at the surface of the grains. At even smaller sizes,whend

<

1 μm,particlesstarttobesensitivetothethermalagitationofthesolvent,andBrownianmotionbecomes important.Inthisreview,westartbyconsideringlargeparticlesforwhichonlycontactsandhydrodynamicinteractionsplay arole,beforediscussingthecaseofsmallparticles,wherecomplexityarisesfromtheexistenceofotherinteractionforces.

Inthecourseofthispaper,wewilldiscussresultsfromexperimentsusingmodelparticles(oftensphericalbeads),andalso resultsfromdiscretenumericalsimulations,whichconsistsinsolvingthemotionofeachindividualgrainfromNewtonlaws usingmodelforceinteractionstodescribethecontacts,andinsolvingtheflowoftheliquidbetweenthegrains[4].

2. Largeparticles:granularmediaandsuspensions

Drygranular mediaorgranularsuspensionsrefertomediamadeofgrainslargerthan100 μminairorfullyimmersed inaliquid.Inthiscase,thedynamicsiscontrolledbythecollisionsandthefrictionalcontactsbetweenthegrainsand,for suspensions,also bythehydrodynamic interactions. Thecaseofpartially immersedgrains,which correspondsto systems wherecapillaryforcescomeintoplay,isnotdiscussedhere,butbelongstotheclassofcohesivematerials,whichmayshare resemblancewiththepowdersdescribedinsection3.1.Althoughtheylooksimpleatfirstsight,granular mediastillresist acompletedescriptionabletopredicttheirbehaviorinthewholerangeofobservedregimes,fromverydiluteandagitated media to verydense andjammedsystems.In thefollowing,we discusshowa firstapproachto therheology ofgranular flowscanbeinferredfromsimpledimensionalarguments.

2.1. Rheologyofgranularmediaandsuspensions

Tostudytherheologyofcomplexfluids(i.e.theflowbehaviorinresponsetoanappliedforce),thetraditionalapproach consistsinconsidering theplane shearconfiguration.The fluidofinterest isconfinedbetweentwoplates separatedbya

distance h,the top platemoving ata velocity u so that the fluid issheared ata constant shear rate

γ ˙ ₌

u

/

h. Classical rheology consistsinmeasuringhowtheshearstress

τ

exerted bythefluidonthetop platevarieswiththeshearrate

γ ˙

. In the caseof granular materials and suspensions,another crucial parameterhas to be considered, namely the granular pressure P^p.When sheared, the packingof grains pusheson the wall and exerts a normal stress P^p on the wall. As a consequence,thereexisttwowaystostudytherheologyofgranularsystems.Intheclassicalway,thematerialissheared ataconstantvolume,keepingthedistancebetweenthetwoplatesconstant.Inthiscase,onehastomeasurehowboththe shearstress

τ

andthenormalgranularstress P^pvarywiththeshearrateforagivenvolumefractionofgrains

φ

,where

φ

is theratioofthevolumeoccupiedbythegrainstothetotalvolumeofthesample.However,asecondmethodhasprovedto berelevantforgranularmediaandconsistsinshearingthematerialbyimposingthegranularpressure(moreprecisely,by imposingonthetopplateoftheshearcelltheverticalnormalstressonthegrains).Inthelattercase,thevolumeoccupied bythegrainsisfreetoadjust.Inthefollowing,westartbydescribingthepressure-imposedrheologybeforepresentingthe volume-imposedrheologyanddiscussingtheconnectionbetweenthetwo.

2.1.1. Pressure-imposedrheology

Theconceptofpressure-imposedrheologyhasbeenintroducedinitiallyfordrygranularmedia[5] andgeneralizedlater tosuspensions[6].Themotivationwasthat,inmanysituationsofinterestinapplications,thevolumefractionofthegrains isnotcontrolled.Atypicalexampleisthecaseofgranularavalancheswhengrainsflowdownaplane(inairorinaliquid) [7]. In thiscase, gravity imposes the stress distribution, butthe volume occupiedby the grains adapts,and theflowing granularlayermaydilateorcontractdependingontheconditions(thegrainsbeingmoreorlessagitated).Pressure-imposed rheologyhasbeenfirstdevelopedtodescribethepropertiesoffreesurfaceflowsandconsistsinprescribingashearrate

γ ˙

keepingthegranular pressure P^p constant. Thisisachievedina shearcell,wherethetopplateisfree tomovevertically andsubmitted toaconstant verticalforce (Fig.2a).Thetopplateimposed thegranular pressure P^p onthegrains[5].The twomeasuredquantitiescharacterizing therheologicalbehaviorofthematerialinthisconditionaretheshearstress

τ

and thevolumefraction

φ

,whichbothdependontheshearrate

γ ˙

andontheimposedpressure P^p.Interestingly,inthelimit ofinfinitelyrigidparticles ofdensity

ρ

p anddiameterdinteractingonly throughfrictionalcontacts,dimensionalanalysis imposes that the system iscontrolled by a single dimensionlessparameter calledthe inertial number: I

= ˙ γ

d

/

P^p

/ ρ

p. Theinertialnumberisequaltotheshearratemadedimensionlessusinganinertialtimescalebasedongranularpressure.

Consequently,fromdimensionalanalysisagain,theshearstress

τ

hastobeproportionaltothepressureP^p (theonlystress scaleintheproblem),withacoefficientofproportionality(amacroscopiccoefficientoffriction)beingafunctionof I.For thesamereason,thevolumefraction

φ

hasalsotobeafunctionoftheinertialnumberI only,suchthat:

τ =

^Pp

μ (

I

)

and

φ = φ (

I

),

(1)

whereI

=

_P^γ˙p^d/ρp

Theevolutionofthetwodimensionlessfunctions

μ (

I

)

and

φ (

I

)

asafunction oftheinertialnumber I aresketchedin Fig.2a [5]. The macroscopicfriction coefficient

μ (

I

)

starts ataconstant value

μ

c when I

→

^{0, and}experimentssuggest thatittendstoasecond constant

μ

2 whenI

→ ∞

^{.Thevolumefraction}

φ

isadecreasingfunctionoftheinertialnumber, withamaximumvalue

φ

c whenI

→

^{0 [8].}

The caseofasuspension when theparticles are immersedina liquidofthe samedensityandofviscosity

η

f can be addressedfollowingthesameformalism(Fig.2b).Thesuspensionisshearedbyaporoustopplate,freetomovevertically.

The mesh of the plate is smaller than the particle size and the liquid can flow through it. In the limit of the viscous regime,wheninertiaplays norole,therheologyiscontrolledbyasingledimensionlessnumbercalledtheviscousnumber J

= η

f

γ ˙ /

P^p,beingequaltotheshearratemadedimensionlessusingaviscous timescalebasedonthegranularpressure.

Thefrictioncoefficientandthevolumefractionarethengivenbythefollowingrelations:

τ ₌

P^p

μ (

J

)

and

φ = φ (

J

),

(2)

where J

=

^η_P^fγp^˙

Thefrictioncoefficient

μ (

J

)

andthevolume fraction

φ (

J

)

are sketchedinFig.2b.Thefrictioncoefficient isan increasing functionoftheviscousnumberstartingatthesamevalue

μ

c asinthedrycaseandincreaseslinearlywhen J

→ ∞

^.The volumefractionisequaltothesamemaximumvalue

φ

c asinthedrycaseinthequasi-staticlimit J

→

^{0,anddecreases} when J increases.Direct experimental measurements of

μ (

J

)

and

φ (

J

)

have been possible in a rheometer specifically designedtoimposeaconstantnormalstress[6,9].

Therheologyofdrygranular mediaandsuspensionsunderpressureimposedconditionsthussharesimilarities andare both described interm of a frictional law. In the quasi-static limit, when I

→

^{0 or J}

→

^{0, no} distinction can be made betweenadrygranularmedium andasuspension:themaximumflowingvolume fraction

φ

c andthequasi-static friction coefficient

μ

c arethesame,suggestingthatthislimitiscontrolledbycontactpropertiesonly.Experimentsandnumerical simulationsshowsthat

φ

c

≈

⁰

.

59 forspheresinteractingwithfrictionalcontacts(lessthan

φ

rcp

=

⁰

.

64,thevolumefraction ofrandomclosepacking)andthat

μ

c

≈

⁰

.

4.Itisimportanttonotethat

φ

c increaseswhentheinterparticlefrictioncoeffi- cient

μ

p decreasestozero,andforthecaseoffrictionlessparticles(

μ

p

=

^0),numericalsimulationssuggestthat

φ

ccoincides

Fig. 2. a)Pressure-imposedrheologyfordrygranularmedia,andsketchesofthefrictioncoefficientμandthevolumefractionφ asafunctionofthe inertialnumberI;b) pressure-imposedrheologyforsuspensions,andsketchesofthefrictioncoefficientμandvolumefractionφ asafunctionofthe viscousnumber J;c)volume-imposedrheologyforgranularmedia,andsketchesofthetwoBagnoldfunctions fsand fn asafunctionofthevolume fractionφ;d)volume-imposedrheologyforsuspensions,andsketchesoftheshearandnormalviscositiesηsandηnasafunctionofthevolumefractionφ.

with

φ

rcp [10].The quasi-staticfrictioncoefficient alsodecreases whendecreasingtheinterparticlefrictioncoefficient,but remainsfiniteandequalto

μ

c

≈

⁰

.

1 forfrictionlessparticles[10,11].

2.1.2. Volume-imposedrheology

The moreconventional rheological approach consists in shearing the material keepingthe distance betweenthe two platesconstant,i.e.keepingthevolumefraction

φ

oftheparticlesconstant.Inthisconfiguration,themeasured quantities aretheshearstress

τ

andthegranularpressure P^p,whicharefunctionsoftheshearrate

γ ˙

andofthevolumefraction

φ

. The rigidity of the particles implies that there exists no intrinsic stress scale nor intrinsic time scale. For dry granular material madeofparticlesofdiameterdanddensity

ρ

p,dimensionalanalysisthusimpliesthat theshearandthenormal stress

τ

andP^p aregivenbythefollowingexpressions:

τ = ρ

pd²

γ ˙

² f_s

(φ)

and P^p

= ρ

pd²

γ ˙

²f_n

(φ)

(3) where f_s

(φ)

and f_n

(φ)

are two dimensionlessincreasing functionsof

φ

, whichappearto diverge closeto themaximum volumefraction

φ

c(Fig.2c).Thevariationofthestresseswiththesquare oftheshearrateisoftencalledtheBagnoldlaw, followingthe pioneeringworkofBagnold[12],whofirst experimentallyevidencedthesquare variationofthestresswith theshearrate.

Viscoussuspensions,madeofthesamerigidparticlesbutnowimmersedinaliquidofviscosity

η

fcanalsobeanalyzed within thesameframework. Thesuspensionpreparedatavolumefraction

φ

isshearedata shearrate

γ ˙

(Fig.2b).Inthe viscous regime,whentheinertiaoftheparticlesandofthefluidisnegligible,dimensionalanalysisimpliesthat theshear andnormalstressesvarylinearlywiththeshearrateas:

Fig. 3.Examplesofsimulationsofgranularflowsusingacontinuumdescription;a)realisticsimulationsofflowofsandmanipulatedbyhandsfrom[16,17];

b)predictionoftheflowinasilofrom[18];c)fingeringinstabilityobservedwhenagranularfrontmadeoftwosizesofparticlesflowsdownaninclined planefrom[19];d)flowofasuspensioninapipe,leadingtoamigrationoftheparticlestowardthecenter(from[20]).

τ = η

f

γ η ˙

s

(φ)

and P^p

= η

f

γ η ˙

n

(φ)

(4)

where

η

s

(φ)

and

η

n

(φ)

arecalledtheshearandnormalrelativeviscosityrespectively,andaretwodimensionlessincreasing functionsof

φ

,whichalsodivergeclosethemaximumvolumefraction

φ

c [13,6,14].

Theexistenceoftwodifferentdescriptions(volume-imposedandpressure-imposed)canbeperturbingfortheneophyte interestedingranularrheology.Thematerialisdescribed asafrictionalmaterialinthepressure-imposedapproach,witha yield stressanda shearstress proportionalto thepressure,whereas it isdescribed asaviscous orBagnoldliquidinthe volume-imposed approach, withno yield stress. This illustrates the fact that the behavior ofgranular materials strongly dependsonthe waytheyare manipulated.The two approachesareofcourse fullyequivalent, andonecan easily switch from one description to another, the two functions

μ (

I

)

and

φ (

I

)

(respectively

μ (

J

)

and

φ (

J

)

for suspensions) being relatedtothetwo functions fs

(φ)

and fn

(φ)

(respectively

η

s

(φ)

and

η

n

(φ)

).Assumingthatthefunction

φ (

I

)

(resp.

φ (

J

)

) is monotonic, it can be inverted to obtain I

(φ)

(resp. J

(φ)

). When injected in the definition of the inertial number I (resp. viscous number J), it comes that fn

(φ) =

¹

/

I

(φ)

² (resp.

η

n

(φ) =

¹

/

J

(φ)

) and fs

(φ) = μ (φ)/

I

(φ)

² (reps.

η

s

(φ) = μ (φ)/

J

(φ)

).Theequivalencebetweenpressure- andvolume-imposedrheologiesgivesalsoinformationaboutthedivergence ofthefunctionsfs

(φ)

, fn

(φ)

,

η

s

(φ)

,and

η

n

(φ)

closeto

φ

c.Theobservationthatthefrictioncoefficienttendstoafinitevalue inthequasi-staticlimit I

→

^{0 or J}

→

^{0 impliesthatthetwofunctions f}s

(φ)

and fn

(φ)

divergeinthesamemannerwhen

φ → φ

c (thesamewiththetwofunctions

η

s

(φ)

and

η

n

(φ)

),toinsure thattheratioremainsfinite.Finally,thedivergence closeto

φ

ccanbewrittenasapowerlaw,i.e. fs

(φ) ∝

^fn

(φ) ∝ (φ

c

− φ)

^−αgran,

η

s

(φ) ∝ η

n

(φ) ∝ (φ

c

− φ)

^−αsus.Ourexperiments showthat

α

sus iscloseto2forfrictionalparticles[13,6,9],butmaybe larger,about2.8, forfrictionlessparticles[15].For thecaseofdrygranularmaterial,thedivergenceobservedinnumericalsimulationsisalsocloseto

α

gran

∝

^2.

2.2. Towardshydrodynamicsofgranularmedia

Theknowledge oftheresponse ofagranularmedium submittedto aplaneshearhasbeenusedasastarting pointto develop afulltensorial rheologicalmodel,able todescribecomplexflowconfigurations withshearindifferentdirections.

For drygranular media, the generalization ofthe frictional approach interms of a friction coefficient depending on the inertialnumberseemsa promisingroute.Forsuspensions,thepresenceoftheliquidphase hastobetakenintoaccount, andtwo-phasesflowapproacheshavebeendeveloped.

2.2.1. Continuummodelingofdrygranularmedia

Fordrygranularmedia,thepressure-imposedapproachappearstobethemostrelevant,asinmostconfigurations,the volumefractionofgrainsisnotprescribedandthesystemisfreetodilateorcontract.Thishasmotivatedthedevelopment ofa simpledescription,based ona three-dimensionalgeneralization ofthe

μ (

I

)

friction law,assuming that thematerial is incompressibleandthat theshear stress tensor iscollinear withtherate-of-deformation tensor [21]. Thisapproach is equivalenttoavisco-plasticdescription,inwhichboththeyieldstressandtheviscosityarepressure-dependentquantities.

Suchadescriptionhasbeenimplementedinfluidmechanicscodes,andquantitativepredictionshavebeenmadeforflows oninclinedplane,flowsinsilo[22,18],granularcollapsesofcolumns[23,24].Stabilitypropertiesforflowoninclinedplanes have been predictedwithin this framework [25]. A modified version ofthe

μ (

I

)

rheology preventing negative pressure hasbeen recentlydeveloped,and givesrealistic resultsin situationsascomplex ashands manipulatingdrysand [16,17]

(Fig.3).Ingeophysics,manyapplicationsinvolvegranularlayersonslopes,withatypicalthicknessmuchsmallerthanthe flowlength.Intheseconfigurations,adepth-averagedapproachisrelevant,whichconsistsinwritingmassandmomentum equationsintegratedovertheflowthickness.Awholecorpusofworkshasbeendevotedtothisapproach,implementingthe

frictionalrheology ofgranularmediaindepth-averaged equations[26–28].Recentstudies areabletodescribesegregation phenomenaandtheappearanceofcomplexfingeringpatternatthefrontofanavalancheflowingdownaslope[19] (Fig.3c).

Although severallimitsexist tothisapproach that willbe discussed later inthispaper, an increasing numberofstudies show that thissimplefrictionalrheology

μ (

I

)

capturesimportantfeatures ofgranular flows incomplexsituations.Alast importantremarkconcernstheexistenceofnonphysicalinstabilitiesincertainrangeofparametersofthegeneralized

μ (

I

)

rheology[29].Ithasbeenshownthatashort-waveinstabilitymayappearclosetothequasi-staticregime,orathighinertial number,whichisreminiscentofanill-posednessofthesystemofequations.

2.2.2. Continuummodelingofsuspensions

The caseof suspensionsis morecomplex than thecase ofdrygranular media dueto thepresence ofthe interstitial fluid.Inmanysituations,arelativemotiontakesplacebetweentheliquidandthegrains,andthedynamicsofbothphases needs tobeproperlydescribed.Two-phaseflowmodelshavebeendeveloped,thatconsistinconsideringthefluidandthe grains astwo intricate continuum phases, andinwriting themass andmomentum equationsforeach of them[30–32].

The difficultyliesinthechoiceofthestressesforeach phaseandinthechoice oftheinterphaseforce[33].Thegranular rheologydiscussedintheprevioussectionprovidesexpressionsforthegranularstresses,whichcanbeinjectedintwo-phase flowmodelstogivepredictionsindifferentconfigurations.

Fortheexpressionofthegranularstresses,wehavethechoicebetweenthedescriptionintermsofthevolume-imposed rheology (the viscous description) orintermsofthepressure-imposed rheology (the frictionaldescription). Althoughthe two descriptions areequivalent, depending onthe configuration,itmaybe moreconvenientto useone ortheother.For example, theflow of asuspension ina pipe whenthe particles andthe fluid havethe samedensityis typically tackled usingthevolume-imposedrheologyandthedescriptionintermsofaneffectiveviscousfluid,becausethevolumefraction iswellcontrolled,whereastheparticlestressadjusts.Bycontrast,theflowofdensegrainsimmersedinalighterfluidand avalanching down an inclined plane is aconfiguration wherethe volumefraction adjusts,andstresses areprescribed by gravity,meaningthatthefrictionaldescriptionofthepressure-imposedrheologyismoreconvenient.

Anillustrationofthesuccessofthetwo-phaseflowapproachistheso-calledmigrationphenomenonsketchedinFig.3d [20].Aninitiallyhomogeneoussuspensionisinjectedinapipe.Themediumbeingequivalenttoaneffectiveviscousliquid, the velocityprofile isa parabola, reminiscentof aPoiseuilleflow. However, aftera certaindistance,the particlesmigrate tothecenter,andtheconcentrationishigherinthemiddleofthepipethanonthewall,andthevelocityprofilepresents a pseudoplug regioninthecenterpart.Qualitatively,themigrationcan beexplainedbyconsideringtheeffectofparticle pressuregradients.Initially,whenthesuspensionishomogeneous,theshearrateismaximumclosetothewallandvanishes in thecenter.Fromthe expressionofthegranular pressure–Eq. (4) –thisimpliesthat P^p ismaximumon thewalland vanishes atthe center.This gradientofgranular pressure createsa netbody force on theparticlephase fromthe region ofhighpressuretotheregion oflow pressure,i.e.fromthewall tothecenter,explainingthemigration.Thisinwardflux of particlesis compensatedbya outward flux ofliquid.The migrationstopswhen gradients ofvolumefractionare such thatthegranularpressureishomogeneousacrossthesection.Thisphenomenoncanbecapturedusingthetwo-phaseflow approach[34,33,35].

2.3. Microscopicoriginoftherheology

We have discussed inthe precedingsections theprogress madein the continuum descriptionof granular media and suspensions.However,theconstitutivelawsusedintheseapproachesremainempirical.Relatingthemacroscopicbehavior ofthemediumtothedynamicsandtothepropertiesoftheindividualgrainsremainsachallenge.Forexample,predicting the valueofthemacroscopicfrictioncoefficientfromthepropertiesofthegrains(shape, interparticlefrictioncoefficient), isstilloutofreach.However,numericalsimulationsusingdiscreteelementmethodshasbeenapowerfultooltoinvestigate thedynamicsofthegrainsandtheircollectivemotion,givingkeysfordevelopingtheoreticalanalysis.

The most advanced theory hasbeen developed forthe dilute andagitatedflow regime of granular media, calledthe granulargases[38].Inthisregimetheparticlesinteractthroughcollisions.Themaindifferencewithamoleculargasliesin thedissipativenatureofthecollisions.Inthe1980s,akinetictheoryforgranulargaseshasbeendeveloped[39],whichhas beenmuchimprovedsince,leadingtoquantitativepredictionsforshearflowingaseousstates[40].However,thestandard kinetic theoryfailstoproperlydescribedense granularflowswhenapproachingthemaximumvolumefraction. Whenthe volumefractionincreases,thesystemleavesthepurecollisionalregimeandparticlesexperienceenduringcontacts,leading tocooperativemotions.Afirstapproachtocapturethedenseregimeconsistsinmodifyingthekinetictheory.TheExtended Kinetic Theory developed in [41] introduces a correlation length L, andconsiders that the dissipation is controlled by collisionsbetweenclustersofsizeL.Thistheorycorrectlypredictsflowsdowninclinedplanes.

However,todescribeandunderstandpreciselythephysicsofgranularfloworsuspensionsclosetothemaximumvolume fraction, moredetailedanalysisofthefluctuatingmotionandtheassociatedcorrelationshavebeennecessary.Fig.4shows boththecontactforce networkbetweentheparticles(Fig. 4a)andthefluctuatingvelocityaroundthemeanappliedlinear shear profile(Fig.4b) fortwo differentvolumefractions.Forcechains areobserved,whichbecome morepronouncedand extendonalongerdomainwhenthevolumefractionapproachesthemaximumvolumefraction. Thevelocityfluctuations are also much moreimportant andform vortex-likepatternwhen approaching

φ

c. Theoretical approachesinspired from statisticalphysicsofjammed systemshavebeendevelopedtocharacterizethishighlycorrelatedfluctuatingmotionandto

Fig. 4.Evolutionoftheforcenetwork;(a)(datafrom[36])andofthefluctuatingvelocityfield;(b)(datafrom[37])fortwodifferentvolumefractions approachingthemaximumvolumefraction;(c)cartoonillustratingtheamplificationofthefluctuationswhenapproachingthemaximumvolumefraction (theso-calledlevereffect).

predictscalinglawscloseto

φ

c [42,43].Thekeyingredientisthatthedissipationisenhancedbecausefluctuatingmotions areamplifiedclosetojamming.Thisiscalledthelevereffect[44,15]:becausethesystemisclosetoarigidtransition,there existonly few degreesoffreedom to deform,meaningthat, to satisfy themacroscopicimposed deformation, thesystem exhibitsmoreandmoretortuousmotionswhenapproaching

φ

c.ThecartoonofFig.4cillustrates thelevereffectwiththe simplesystemcomposedoftwobeads,onebeingfixedandtheotherbeingpushedhorizontally.Therigiditytransitioncor- respondstothetwoparticles beingalignedhorizontally.Inthiscartoon,asystemfarfrom

φ

c isrepresentedbya particle farfromhorizontal(leftinFig.4c).Imposingahorizontaldisplacement(theredarrowinthefigure)leadstoalargerreal displacementdisplayed ingreen,duetothenon-penetrability condition.Thecaseofa systemcloseto

φ

c corresponds, in thiscartoon,toaspherealmostalignedwiththehorizontal(ontherightinFig.4c),wherefromgeometricalargumentsitis straightforwardtoshowthattherealdisplacementimposedbythesamehorizontaldisplacementisamplified.Theamplifi- cationofthefluctuationstheninducesmorecollisionsinthedrycase,andmoreviscousdissipationinthesuspensioncase.

Thelevereffecthasbeenusedasthestarting pointintheoreticalapproachespredictingthedivergenceoftheconstitutive lawswhenapproaching

φ

c [42,43].No doubtthattheprogressmadeintheunderstandingofthemicroscopicdynamicsat thegrainscalewillhelpdevelopingmoreaccurateconstitutivelawsinthefuture.

2.4. Beyondsimplerheology

The constitutive laws discussed in the previous sections are based on the generalization of the properties of steady uniformplane shear flows.Although realsuccesses havebeen obtainedinpredicting complex configurations,thissimple approachfailstoproperlycapturethedetailsofunsteadyandnon-uniformflows.

Afirst missingingredient concerns transient flows. It is well knownthat a granular material can be preparedat rest atdifferentvolumefractions,higherorlower thanthemaximumvolume fractionmeasuredunderasteadyshear

φ

c.This meansthatwhensheared,amediumpreparedatavolumefractiondenserthan

φ

c hastodilate(thewell-knownReynolds dilatancy),andthatapackinglooserthan

φ

chastocompact.Asaresult,apileinitiallypreparedinadensestatedoesnot behave thesame asa pilepreparedin a loosestate [45]. The influenceof thepreparationis evenmore pronounced for immersedgranularmedia,asillustratedinFig.5a.Acolumnofgrainsimmersedinwaterinitiallypreparedinaloosestate spreadsfarandfastwhenreleased[46].When preparedinadensestate, thecollapseofthe columnismuch slowerand stopsearly.Inthisexample,thecouplingwiththeliquidphaseplaysamajorrole:duringthedilatationphaseofthedense case, liquid is sucked in betweenthe grains, creating an additional compressive stress, which enhances the friction and decreasesthemobility.Fortheinitiallyloosecase,thephenomenonisreversed.Theinitialcompactionphaseisassociated

Fig. 5.a)Submarinecollapseofgranularcolumnshowingthattheinitialpreparationofthepackinginadenseorinaloosestatestronglyinfluencesthe dynamics[46].b)Evidenceofnon-localeffectsinaCouettegeometry:arodsubmittedtoaforceFandimmersedinthemediumfarfromtheshearband startsmovingassoonastheinnercylinderrotates,althoughnomacroscopicmotionisobservedaroundit[52].c)Starting(opencircles)andstopping (blackdots)anglesofalayerofgrainsonaninclinedplaneasafunctionofthethicknessh/dofthelayer[53].

withliquidbeingexpelledfromthemedium,whichdiminishesthecompressivestress,enhancingmobility.Tocapturethose effects,anadditionallawdescribingthedilatation/compactionprocesshasbeenproposed andcoupledwiththetwo-phase flow equations,leading topredictions fortheinitiationof submarinegranularavalanches [47],orforthe dynamicsofan objectimpactingadenseoraloosesediment[48].

Taking intoaccountthe variationsofthe volumefractionrepresentsonlya firststeptowardsa detaileddescriptionof transient flows.The microstructureofthemedium,i.e.theorientation ofthe contacts,alsoevolvesduring transientflows beforereachingthesteadystate.Thisis,forexample,evidencedinexperimentswherethedirectionoftheshearisreversed [49–51]. When a suspension initially sheared in a direction is suddenly sheared in the reverse direction, the viscosity suddenlydropsbeforeincreasingagaintowarditssteadyvalue.Thedropcorrespondstotheopeningofthecontacts,which were oriented in the directionof the initial shear andrequire a finite deformation to reorient along the newdirection whentheshearisreversed.Fewattemptsexists todevelop rheologicalmodelsthattakeintoaccounttheevolutionofthe microstructure[51].

Asecondlimitoftherheologybasedonthegeneralizationofsteadyuniformshearconcernednon-localeffects.Astriking illustration isgiven bythe experimentsketched inFig. 5b [52].A granular material confinedin betweentwo concentric cylinders is sheared when the inner cylinder is put in rotation at an angular velocity

. In the quasi-static regime, a shear bandisobservedclosetotherotatingcylinder,whichextendstypically over10particlediameters.Averticalrodis immersed inthe apparent staticregion farfromthe shearband andsubmitted toa force F.When

=

^{0,everything is} static,andaminimalforce F_c isnecessarytomovetherod,whichotherwiseremainsstatic.However,assoonastheinner cylinderisputintomotionandthatashear banddevelops,therodslowlymovesintothemedium,althoughtheforce F isbelowthecriticalforce F

<

Fc.Thisobservationmeans thattheflowclosetotheinner cylinderinducesaflow further away closetothe rod.Thisis aclearevidenceofnon-localeffects,in thesensethat theshear atone positioninfluences the rheology further awayin regions whereno macroscopicflow isapparent. The appearance ofnon-local effectsinthe dense-flow regime is not a surprise, when considering the existence of highly correlated motion and long force chains

describedinsection 2.3.However, takingthemintoaccountinrheologicalmodelsisstilla challenge.Thedevelopmentof non-localmodelsisaveryactivedomainofresearch[54,55].

Alastobservationnotdescribedbysimpleconstitutivelawsistheexistenceofahysteresisinthetransitionfromsolid toliquidregime.Astaticlayerofgrainsrestingonaroughplanestartstoflowwhentheinclinationreachesacriticalangle

θ

start,whereasaflowinglayerwillstopwhentheslopegoesbelowacriticalangle

θ

stop

< θ

start(Fig.5c)[53].Onaninclined plane,thecriticalanglesdependonthethicknessofthelayer,aneffectreminiscentofnon-localeffects.Thesamehysteresis isobservedonapileorina rotatingdrum.Totriggeran avalanche,thepilehastoreacha criticalangle,whichishigher than theangleobserved aftertheavalanche. Althoughthe hysteresisofthe avalanche anglesis avery well-known effect studiedformorethan40years,noconsensusexistsonitsphysicalorigin.

3. Smallparticles:powdersandcolloidalsuspensions

Inthefirstpartofthisreview,we haveseenthat forparticlesinteractingonlythroughhardcontactsorhydrodynamic interactions,dimensionalanalysisstronglyconstrainsthepossiblesteadyrheologicalresponsesofgranularmatter.Inpartic- ular,whenshearedatfixedvolume,theshearstress mustbeproportionalto

γ ˙

² intheinertialregime (Bagnoldbehavior) andto

γ ˙

intheviscousregime(Newtonianbehavior),withapre-factorthatonlydependsonthevolumefraction

φ

butnot onthe shearrate. However,realgranular media andsuspensionsoftendeviatefromthisidealrheologicalresponse. They can displaya minimumstress to flow (yieldstress)atconstant volume,a shear-ratedependent viscosity(shear thinning orshearthickeningbehavior)oratime-dependentrheology (thixotropy).Thisisespeciallytrueformaterialscomposedof

‘small’particles,typicallyofdiametersbelow100 μm,likepowdersorcolloidalsuspensions(Fig.1).Thereasonforthisde- partureliesintheexistenceofotherforce scalesattheparticlelevel,suchascohesiveforces,short-rangerepulsiveforces, orthermal fluctuations.Unifying all thesephenomena ina unique rheologicaldescription isstill a challenge [56]. Inthe following,we give a briefoverviewof themain rheological behavior ofthesemore complexgranular media, withsome recentprogressinthefield.

3.1. Attractiveinteraction

Cohesion in granular media and suspensions can have different physical origins. Forinstance, it is well known that addingasmallamount ofliquidtoadrysandis sufficientforbuildingsandcastles,asignature ofcohesionarising from the capillary bridges between grains. Another source of cohesion comes from the van der Waals interactions between moleculesthataffectallparticlesatsmallscales.Theseattractiveforcesareresponsiblefortheaggregationofsmallcolloids insuspensionswhen nostabilizingrepulsiveforces exist.Thefirst mainconsequenceofcohesionin powdersorcolloidal suspensions is the appearance of a yield-stress in the absence of any external confining stress. In addition, the critical flowingpackingfraction

φ

cofcohesivemediaisusuallysmalleratlowstressthanatlargestress,meaningthattheresponse at fixed volume is now shear-thinning. Finally, in colloidalsuspensions, the aggregation process itself is often slow and limitedbythermaldiffusionsuchthat,intheabsenceofexternalforcing,thestructureofthemediumevolvesintime.The rheologicalresponse thereforedependson thewaiting time beforethe suspensionis sheared.At largeconcentration, the network ofaggregativeparticles canpercolate throughoutthewhole system,providing thesystema yield-stress (gel-like response).Undershear,thenetworkofparticlescanbebroken.Thiscompetitionbetweentheagingofthestructureinduced byaggregationanditsrejuvenationinducedbytheshearrategivesrisetoarichrheologicalresponse,combiningthixotropy andnon-linearshearratedependence.

3.2. Repulsiveinteraction

Topreventaggregation,colloidalsuspensionsareoftenstabilizedbyashort-rangerepulsiveforcebetweenparticles.This repulsiveforcecaneitherstemfromelectrostaticsurfacechargesontheparticlesurfacesorfromaspecificpolymercoating.

Thisisthecaseofmodernconcrete,wheretheadditionofpolymers(superplastizers)improvestheworkabilityofthefresh concrete andits final strength. Itwas recentlyproposed that such a short-range repulsive force Frep induces africtional transitioninthesuspension,whichhasdramaticconsequencesontherheology [57,58].Atsmallshearrate(small stress), therepulsive force preventsthe particlefrommaking contact.The suspension thus flowsasifparticles were frictionless, withaviscositythatdivergesatacriticalpackingfraction

φ

^μc^p=0

= φ

rcp

≈

⁰

.

64 corresponding tofrictionlessparticles(see

§2.1.1) (Fig. 6a). Conversely, atlarge shear rate(large stress),the hydrodynamic forces overcome the repulsive force and particlesarepressedintofrictionalcontact.Therheologyofthesuspensionthusswitchestothatofafrictionalsuspension whose viscosity divergesat a critical packingfraction

φ

^μc^p=0

≈

⁰

.

59

< φ

^μc^p=0. A suspension of frictional particles witha short-rangerepulsive force Frep thereforehastwo possiblerheologicalbranches:a low-viscosityfrictionlessbranchatlow stressandahigh-viscosityfrictionalbranchatlargestress(Fig.6b).Thetransitionbetweenthetwobranchesgivesrisetoa shear-thickeningbehavior,whichhasbeenstudiedbothnumerically[57,59] andtheoretically[58].Dependingontheinitial packingfraction

φ

,theshear-thickening transitioncanbe continuous, discontinuousor evenleads tothejamming ofthe suspension,asillustratedinFig.6b.Thediscontinuoustransitionoccursatacriticalshearstress

τ

c andacriticalshearrate

˙

γ

c givenby

Fig. 6.Frictionaltransitionandshear-thickeningindensesuspensions.a)Frictionaltransitionmodelforfrictionalparticleswitharepulsiveforce.b)(left) Viscositybranchescorrespondingtothefrictionlessandfrictionalstates;(right)sketchoftheshearstressasafunctionoftheshearrateforthethree differentconcentrationsindicatedbyarrowsontheleftgraph.

τ

c

= β

^Frep

d² and

γ ˙

c

= β

^Frep

η

^μ_s^p=0

(φ)

d²

(5)

with

β ≈

⁰

.

04 [59].

The above mechanismprovidesa coherentframework forexplainingthe dramaticshearthickening behavior observed insomenon-Browniansuspensionsofparticles,suchascornstarchinwater.Experimentalsupportforthismechanismwas recentlyobtainedfromdirectmeasurementsofthefrictionalpropertiesofshearthickeningsuspensions.Atthemacroscopic level,Clavaudetal.[11] usedrotatingdrumexperimentsandamodelsuspensionwheretherepulsiveforcecanbetunedto deducethefrictioncoefficient μ ofthesuspension fromtheavalanche angle.Theyshowedthatthepresenceofarepulsive forcebetweenparticlesleadstoafrictionlessstateatlowstressandashear-thickeningrheology,whicharebothsuppressed when the repulsive force is missing [11]. At the microscopic level, Comtet et al. [60] measured the friction coefficient

μ

p betweenpairs ofparticles ofashear thickeningsuspension usinga tuning-fork-based AtomicForce Microscope. They evidenced africtionaltransitionabovea criticalnormalload Frep,ingoodagreementwiththeshear-thickeningtransition measuredforthemacroscopicrheology[60].

3.3. Browniansuspensions

Forsuspensionswithparticlediametertypicallybelow1 μm,thethermalagitationoftheparticles (Brownian motion) isnolongernegligible.Thisintroducesanewdimensionlessparameterin therheology,thePécletnumber,whichdescribes thecompetitionbetweentheadvectiontimescale,

γ ˙

⁻¹ andthediffusiontimescale,d²

/D

,where

D =

^kBT

/(

3

π η

fd

)

isthe Stokes–Einsteindiffusioncoefficient.ThePécletnumberisgivenby:

P e

= η

f

γ ˙

d³

k_BT (6)

For P e

^{1,the diffusiontimeis muchsmallerthan theadvectiontime.The rheology then}corresponds tothelinear re- sponse closetothethermalequilibrium, withashear-rate-independentviscosityandparticlepressure,whichare bothof Brownian origin. Adistinctive property ofBrownian(frictionless)hard particles isthat the thermalviscositydramatically increases, bytensoforderofmagnitude,asthepackingfractionapproachesa value

φ

g

≈

⁰

.

58

−

⁰

.

60 [61,62].Remarkably, thisvalue issmaller thanthemaximal flowingpackingfraction

φ

c^μp=0

≈

⁰

.

64 discussed previously fornon-Brownian fric- tionlessspheres.Thislargeincreaseoftheviscosityat

φ

gisanalogoustothecaseoftheglasstransitionobservedinsome molecularliquidswhentheyarecooledbelowacriticaltemperature.Itcomesfromthedramaticslowingdownofthepar- ticlediffusiondynamicscloseto

φ

g(the diffusiontimefortheparticletoescapethe‘cage’ofitsneighbors).Above

φ

g,the viscosityissolargethatthesuspensiondevelopsayieldstressonexperimentaltimescales.

Interestingly, thermalfluctuationsinBrownian hard particles seta force scale, F_ther

=

^kBT

/δ

where

δ

is theinterpar- ticle gap, which acts asan analogue of a repulsive force between particles [63,64]. As the Pécletnumber increases, the hydrodynamic forcesenables toexplore deeperthethermal repulsiveshellaround theparticles, yielding anintermediate shear-thinningbehavior. However,atlargePécletnumbers,whenthehydrodynamicforcesovercomethethermalrepulsive

forces,theparticlesmakecontact.Thesuspensionthusswitchesfromfrictionlesstofrictionalandthesameshear-thickening mechanismasdescribedabovecanapply[65,66].WhenbothBrownianmotionandrepulsiveforcearetakenintoaccount, thecriticalshearstressfortheshear-thickeningtransitionisgivenby

τ

c

= β

^Frep d²

+ α

^kBT

d³ with

α ∼

^d

δ

^{∗ (7)}

Inthisexpression,

δ

^∗ isthetypicalinterparticlegapatwhichcontactoccurs, whichmaydependonparticleroughnessor elasticity.TherespectiveimportanceofthermalmotionorrepulsiveforceindenseBrowniansuspensionsthereforedepends onthedetailedphysicsatthecontactscale.

4. Activegranularmedia:anexampleinthevegetalworld 4.1. Introduction

So far, we have discussed the flowingbehavior of passive granular matter, that is a medium made of inertparticles that aredriven through interactionswithother particles andhydrodynamic stresses.However, thelivingworld alsooffer fascinatingexamplesof‘granular’assembliesthataremadeofnon-passiveoractive‘particles’,suchasschoolsoffish,bac- teriacolonies ororganellesinthecytoskeletonofthecells[67]. Inthesesystems,themedium iscomposedofself-driven unitsthat consumeorextract energyfromthesurroundingfluid, such thateach particleis animated by itsown motion, whichcanbe directionalorrandom.Similarexamplesinthenon-livingworldincludemotilecolloids,collectionofrobots or shakengranular media made ofanisotropic particles. Strikingly, this input of energyat the local scale can havedra- maticconsequencesonthecollectivebehaviorandflowresponseofthisactivegranular matter,includinggiantfluctuation of density,out-of-equilibriumphase transition, and emergentpatterns. The topicofactive matter initially emerged from theoreticalandnumericalstudiesonthecollectivemotionofflockofbirds,byanalogywithphasetransitionsincondensed- matterphysics[68].Morerecently,thefieldgainsstrongmomentuminthesoftmatterandbiophysicscommunity,withthe studyofmotilecolloidsandactivegels.Inthefollowing,wedescribeapeculiarexampleofactivegranularmatterthatwe recentlyuncoveredinthevegetalworld:thestatolithsthatgiveplantsthesenseofgravity.

4.2. Statoliths:anagitatedgranularmediumattheoriginofplants’sensitivitytogravity

Fromtinyshootstolargetrees,allplantsareabletosensegravityandreorienttheirgrowthtowardtheverticaldirection ofthegravitationalfield[69,70].Thisabilityisnotonlyimportantatearlystagesofdevelopmentforrootstoanchorinthe soil andshootstofindlight. Itisalsokey allalong theplant’slife,fortheplanttomaintainits uprightpositionandnot fallagainst itsownweight.Thedetectionofgravityinallplantsoriginatesinspecializedcells,calledstatocytes,inwhich starch-richparticlescalledstatolithsarepresent(Fig.7a).Thesemicro-sizegrainsaredenserthanthesurroundingcytoplasm andsedimentatthebottomofthecells,thusgivingthedirectionofgravity.Whentheplantistilted,thestatolithstrigger aseriesofbiochemicalreactions thatinduce an asymmetricgrowthbetweenthetwofaces oftheorgan. Thisdifferential growtheventuallyleadstothebendingoftheplanttowardtheverticaldirection(Fig.7a).

Howcellsdetectthestatolithsandhowthissensingisconvertedintoabendinggrowthresponseattheorgan’slevelare stilltheobjectofmanystudiesinplantbiology.Werecentlyshowedthatplantsareactuallynot sensitivetotheintensity ofthegravity field,butonlytotheinclination againstthedirectionofgravity [72,73]. Thegravisensorinplantsis thusa positionsensor,notaforcesensor.Thisfindingissurprising,becauseitsuggeststhatthepileofstatolithsatthebottomof thecellmoveandrespondtoeventhetiniesttilts.Atfirstsight,suchabehaviorcontradictsourknowledgeofthephysics ofgranularmedia,whichstipulatesthatanassemblyofgrainscannotmovebelowacriticalavalancheanglesetbyfriction andstericconstrainsbetweenparticles(see§2.1).

Toaddressthisissue,wedirectlyvisualized themotionofstatolithsingravisensingcells (wheatcoleoptile)inresponse to various cell inclinations (Fig. 7b) [71]. When a cell is tilted, statoliths first behave like a classic (immersed) granular avalanche:the statoliths flow in the bulk andthe pileangle rapidlyrelaxes towarda critical angle

θ

c ina few minutes (Fig.7c).However,thelong-timebehaviorofstatolithsstronglycontrastswiththatofaclassicalgranularmedium.Instead ofbeingstuckatthecriticalangle,thestatolithpilekeepsevolving andslowlycreeps. Eventually,the freesurface ofthe pilerecoversthehorizontalafterfewtensofminutes,asaliquidwoulddo.Investigationofstatolithmotionattheparticle levelsuggeststhat thislong-time liquid-likebehavior comesfromtheagitationofthestatoliths.Unlikeapassivegranular material,statolithsexhibitrandom andlargefluctuatingmotion,whichlikelyhelpsthegrainstounjamandflow evenfor verysmallinclinations.

Tounderstandtheoriginofthisagitation,wehavecomparedtheavalanchedynamicsofthestatolithswiththebehavior of inert particles of similar size in biomimetic cells. In this case, the only source of fluctuation is thermal fluctuation (Brownianmotion),whoseintensityischaracterizedbytheinversegravitationalPécletnumber

P e⁻_g¹

=

^kBT

m g d (8)

Fig. 7.An exampleofactivegranular matter:thegravisensors ofplants. a)Gravitropismresponseofaninclinedwheatcoleoptile andpictureofthe gravisensingcellswiththestatolithpileatthebottom.b)Time-lapseimagesofastatolithpileavalancheinresponsetocelltilting.c)Statolithpileangle asfunctionoftimeshowingafirstrapidgranularavalancheregimefollowedbyaslowcreepingregimetowardhorizontal.d)Statolithverticalfluctuation insideandoutsideofthecell,comparedwithBrownianfluctuationsofsilicamicroparticlesinwaterfordifferentgravitationalPécletnumbers.Adapted from[71].

wherekBT isthethermalenergy,m istheparticle’s masscorrectedby thebuoyancy, g is theintensityofgravity,andd the particle’sdiameter.In thisbiomimeticsystem, liquid-likeavalanchessimilarto thebiological onescan be observedif thethermalagitationishighenoughcomparedtothegravitationalenergy(large P e⁻_g¹).However,quantitativecomparison betweenbothsystemsshowsthatstatolithsflowmuchmorerapidlythanpurelyBrownianparticleshavingthesamevalue oftheinversegravitationalPécletnumber.Everythinghappensasifstatolithsintheplantcellswereagitatedbyaneffective temperaturetentimeshigherthanthephysicalthermaltemperature(Fig.7d).

The active nature ofstatolithagitation istherefore key toexplain the remarkablesensitivityof plantsto gravity[71].

The statoliths’agitation islarge enough toerasethe traditionalflow thresholdofgranular media, whilesmall enoughto maintainthegrainstogetheratthebottomofthecellandgivethedirectionofgravity.Ourresultsshowthatthisagitation isnotthermal,butcomesfromthebiologicalactivityofthecytoplasmthatsurroundsthegrains.Thisexampleshowshow activitycanstronglymodifytheflowandthetransportofdenseparticulatemediaatsmallscales.Understandingthephysics andtherheologyofsuchactivegranularmatterisanexcitingtopicforfutureresearch[74,75].

5. Moreopenproblems!

Inthisreviewwehavediscussedthephysicsofgranularflows,illustratingthroughafewexamplestherecentadvances made inthe field. However, ourpresentation isfar frombeingexhaustive,and some importantproblems havenot been discussed,whichrepresentnovelavenuesforfutureresearches.Amongthemwecanmention:

•

^{thetransitionbetweentheviscousandtheinertialregimein}suspensions,

•

^theflowofpolydispersedparticles,whichcouldleadtosegregationphenomena,

•

^{theflowofdeformableparticles,}

•

^{theinfluenceoftheparticlesattheinterfacesofliquid,withthedynamicsofwetting,ofdrops,offilms,}

•

^theelongationalrheologyofgranularmedia.

Acknowledgements

Thisarticlehasbeenwrittenundertheauspicesofthe‘Laboratoired’excellencemécaniqueetcomplexité’(ANR-11-LABX- 0092),theExcellenceInitiativeofAix-MarseilleUniversity –A^∗MIDEX (ANR-11-IDEX-0001-02)fundedby theFrenchGov- ernment“Investissementsd’avenir”programme,oftheEuropeanResearchCouncilundertheEuropeanUnionHorizon2020 ResearchandInnovationprogramme(grantagreementNo. 647384),oftheANR BlancGrap2(ANR-13-BSV5-0005-01). We wouldliketosincerelythanktheFrenchAcademyofSciencesforitsrecognitionofourresearchwork.Thisworkwouldnot havebeenpossiblewithoutthecollectiveinsightandworkofourcolleagues,post-docandstudents,atIUSTIandabroad.

Wearedelightedtohavetheopportunityheretowarmlythankallofthem.

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