Sedimentation of oblate ellipsoids [Elektronische Ressource] / vorgelegt von Frank Rodolfo Fonseca Fonseca

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Sedimentation of oblate ellipsoids [Elektronische Ressource] / vorgelegt von Frank Rodolfo Fonseca Fonseca

universitat_stuttgart

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Sedimentation of Oblate Ellipsoids¨ ¨Von der Fakultat Mathematik und Physik der Universitat Stuttgartzur Erlangung der Wurde¨ eines Doktors derNaturwissenschaften (Dr. rer. nat.) genehmigte Abhandlungvorgelegt vonFrank Rodolfo Fonseca Fonsecaaus Bogota´ ColombiaHauptberichter: Prof. Dr. H. J. HerrmannMitberichter: Prof. Dr. Ing. Rainer HelmigTag der mundlichen¨ Prufung:¨ 14. Mai 2004Institut fur¨ Computeranwendungen 1 der Universitat¨ Stuttgart2004To Laura Soﬁa,a little part of heavenon earth ...Contents1 Deutsche Zusammenfassung 11.1 Simulationsmethode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Die Phasen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3¨1.3 Der stationare¨ Zustand und periodische Phasen: Ahnlichkeitgesetz. . . . . 3¨1.4 Zustandsdiagramm und Ubergange.¨ . . . . . . . . . . . . . . . . . . . . 41.5 Sedimentationsgeschwindigkeit fur¨ oblate Ellipsoide . . . . . . . . . . . 51.6 Orientierungsverhalten . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.8 Raumliche¨ Korrelationen . . . . . . . . . . . . . . . . . . . . . . . . . . 8¨1.9 Anderungen im Volumenanteil und Kollaps . . . . . . . . . . . . . . . . 9¨ ¨ ¨1.10 Anderung der Behaltergroße . . . . . . . . . . . . . . . . . . . . . . . . 102 Introduction 132.1 The falling objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.

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Published by	universitat_stuttgart
Published	01 January 2004
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EllipsoidsOblateofSedimentation

VonderFakult¨atMathematikundPhysikderUniversit¨atStuttgart
zurErlangungderW¨urdeeinesDoktorsder
Naturwissenschaften(Dr.rer.nat.)genehmigteAbhandlung

vorgelegtvon

FrankRodolfoFonsecaFonseca

Hauptberichter:Mitberichter:

ausColombiaa´Bogot

Prof.Prof.DrDr.H..-Ing.J.RainerHerrmannHelmig

Tagderm¨undlichenPr¨ufung:14.Mai2004

Institutf¨urComputeranwendungen1derUniversit¨atStuttgart

2004

aLaur

little

Soﬁa,

part

earth

...

heaven

Contents

ZusammenfassungDeutsche1.1Simulationsmethode............................
1.2DiePhasen.................................
¨1.3Derstation¨areZustandundperiodischePhasen:Ahnlichkeitgesetz.....
¨1.4ZustandsdiagrammundUberg¨ange.....................
1.5Sedimentationsgeschwindigkeitf¨uroblateEllipsoide...........
1.6Orientierungsverhalten...........................
1.7Diffusion..................................
1.8R¨aumlicheKorrelationen..........................
¨1.9AnderungenimVolumenanteilundKollaps................
¨1.10AnderungderBeh¨altergr¨oße........................

oductionIntr2.1Thefallingobjects.............................
2.2Manyparticlesedimentation........................
2.3Drivensuspensionandhydrodynamicdispersion.............
2.4Steadysedimentationandtheﬂuidizedbedgeometry...........
2.5LowReynoldsnumberﬂow.........................
2.6Velocityﬂuctuationsinhard-spheresedimentation.............
2.6.1CaﬂischandLuke’swork......................
2.6.2Resumeofexperimentsandsimulations..............
2.6.3Sometheoreticalapproaches....................
2.7Non-Sphericalparticles...........................
2.8Overview..................................

Model3.1Navier-Stokesequations..........................
3.1.1Thegeneralequationforthedynamicsoftheﬂuid........
3.1.2ThedimensionlessformoftheNavier-Stokesequation......
3.2Boundaryconditions............................
3.3Themodel..................................
3.4Contactfunction...............................

Phenomenology4.1Trajectoriesofafallingoblateellipsoid..................
4.2Steady-fallingoblateellipsoid.......................
4.3Oscillatoryoblateellipsoid.........................

113345678910

131315161719202021212223

27272829303033

35353739

Contents

4.4Chaoticoblateellipsoid...........................
4.5ComparisonwithMahadevan´smodel...................
4.6Vortex....................................
4.7ConclusionsandOutlook..........................
Phases5.1Steady-FallingPhase.............................
5.1.1Changeintheinitialfallingheight..................
5.1.2Dependenceonthekinematicviscosity...............
5.1.3Changeintheellipsoidaspect-ratio.................
5.2PeriodicPhase................................
5.2.1Changeintheinitialorientation..................
5.2.2ComparisonwithBelmonte’sresults................
5.3Chaoticregime................................
5.3.1Sensitivitytothechangeintheinitialorientation.........
5.3.2Powerspectra,autocorrelation,Poincaresection..........
5.3.3Lyapunovnumber..........................
5.4ConclusionsandOutlook..........................
transitionsphaseandlawSimilarity6.1Steady-FallingOblateEllipsoid:Similaritylaw...............
6.2Periodicbehavior:Similaritylaw......................
6.3PhaseDiagram................................
6.4TransitionfromSteady-fallingtoOscillatoryphase............
6.5TransitionfromSteady-fallingtochaoticphase...............
6.6ConclusionsandOutlook..........................
articlesPMany7.1Results....................................
7.1.1Sedimentationvelocityforoblateellipsoids............
7.1.2Orientationalbehavior.......................
7.1.3Orientationalchanges........................
7.1.4ModerateReynoldsnumber....................
7.2OutlookandConclusions..........................
Diffusion8.1Introduction.................................
8.2Results....................................
8.2.1Changeindensity,viscosityandaspect-ratio...........
8.2.2Orientationaldiffusion.......................
8.2.3Non-diffusivedynamicalbehavior.................
8.2.4Similarity..............................
8.3OutlookandConclusions..........................
FluctuationselocityV9.1Spatialcorrelations.............................
9.1.1Changeinthevolumefraction...................

404041424343434447484850525253535457575959616263656565697273747777808184859192939394

Contents

9.1.2Collapsingofthespatialcorrelations.
9.2Changeofthecontainersize.........
9.3OutlookandConclusions...........

Conclusion1010.1OneOblateellipsoid.....
sedimentationellipsoidsyMan10.210.3Outlook............

yBibliograph

...

iii

9598101

103103104106

109

Contents

1Kapitel

ZusammenfassungDeutsche

Newtonzeigte,dassK¨orpermiteinerkonstantenBeschleunigungaufdieErdefallen,aber
trotzderunleugbarenAnziehungderSchwerkraftbewegensichnichtallefallendenGe-
genst¨andeaufgeraderFlugbahnabw¨arts.DieBetrachtungderFl¨ussigkeitbeinhaltetsehr
schwierigeundnichtlineareInteraktionen.TrotzderbahnbrechendenBem¨uhungendurch
Maxwellallgemeine(1853),ProblemderalsohneerstesL¨osung.dieK¨orper-AndererseitsFl¨ussigkistdieeit-InteraktionSedimentbildungbetrachteteineshat,bliebSystemsdas
vonProblemPartikinelnderineinerHydrodynamikFl¨ussigkeitundinunterderdemStatistischenEinﬂussderPhysik.SchwerkraftDieseseinProblemsehrhatwichtigesviele
ten,derAnwendungenBiophinysik,dendergrundleKlimaforschunggendenWsowieissenschaftenaufdemwieinGebietdenderLuftftechnischenahrtwissenschaf-chemi-
schenReaktoren,z.B.derAusbreitungvonVerschmutzung,inTintenstrahldruckern,den
druckaufgeladenenWirbelschichtsystemen,etc..DiesesProblemweistschwierigeMul-
tik¨orper-InteraktionenwegenderweitreichendenHydrodynamikauf,dief¨urKugeln¨uber
weinenamyBereich(2001).vTonrozΨ/edieseabf¨allt,breitenwobeieAnwendungmderAbstand¨oglichkzwischeneitenfehltdenPderartikPartikelnist,elsediment-Ramas-
bleibtbildungeinungelweiterhin¨osteseineProblem.statistisch-mechanischeundhydrodynamischeBeschreibungund

Simulationsmethode1.1

DasModellwurdedurchH¨oﬂerandSchwarzer(2000)entwickelt,erweitertdurchKuu-
selaHerrmannet.al.((1)(2001)2004)undwirdangewinandt.einigenDieBeStudienwevgungonKvonuuselaFl¨et.ussigkal.eiten(2003)wirdundgelF¨ost,onsecaindemand
mandieinkompressiblenNavier-StokesGleichungenaufeinemdiskretenGitterl¨ost:

∂∂iv+Σv∙)v=−p+,E()v+Ω(1.1)
Φ=v∙wterneobeivKraftdiedarstellt,GeschwindigkdieineitderunseremFl¨ussigkProblemeitist,diesowieSchwerkraftpdenist.DruckDieundReΩeineynoldszahlex-

240(a1)180Position in Y12060

(b1)

(I)

(c1)

Simulationsmethode1.1.1

−258118178238298
Position in X

Abbildung1.1:TypischeFalltrajektorieninunserenSimulationen.Wirzeigendasun-
ver¨zillationanderlich-fmitν=allendeΦ.ΦΩ)Re,Δegime,=ΦΣ.NΨ)Ψ´´,.mitΣPΨ)Δ,eund=Φ.dieΩ),ν=chaotischeΦ.Φ´´;BeΣOweΨ)gungdiemitΔperiodischee=Φ.Os-Ω),
ν=Φ.Φ´´.

ES=iΩEMρM/Σν),wobeiidievertikaleoblateEllipsoidgeschwindigkeitist,ΩEMder
gr¨oßteDurchmesserdesoblatenEllipsoids,ρMdieDichteundν=µ/ρMdiekinemati-
istschedabeiViskdieosit¨at(µSchwerkraft.istdieDieScherviskosit¨Grenzbedingungenat).DieFroudezwischenZahlistder>Fle¨=ussigkΣi))eit/ΣUundΣΩEdenM)),ob-g
Fllaten¨ussigkeitaufEllipsoidpartikderPelnartiksindeloberﬂerf¨¨ullt,achewennabh¨angigmanvinonderBetrachtrutschfreienzieht,dassdieBeGrenzbedingungwegungist,der
vΣx)=vj+rΣx)CM×ω,wobeividietranslationaleGeschwindigkeitdesEllipsoidsist,
rWΣxinkCM)derVelgeschwindigkektorveitondesseinerEllipsoids.MittezumPunktxanderEllipsoidoberﬂ¨ache,undωdie

DieInteraktionzwischenderellipsoidalenOberﬂ¨acheundderFl¨ussigkeit,dieandasEl-
lipsoidangrenzt,wirderhalten,indemmaneinewiederherstellendeKrafteinf¨ugt,die
eineDiese“VVerteilungskrafterteilungskraft”inahmtdiedieVGegenwolumenkraftartdesderNaEllipsoidsvier-StokinesdemSinneGleichungnach,vdasserursacht.die
Fl¨ussigkeitinnerhalbdesEllipsoidssichwieeinsteiferK¨orperbewegt.Einenichtreiben-
deKraftwirdausge¨ubt,wenndiePartikelschabloneunddersteifeK¨orpernichtinder
gleichenPositionsind(H¨oﬂer(2000)).

DierepulsiveKraftzwischendenEllipsoidenwirdproportionalzuihrer¨Uberlappung
gekurzew¨ahlt.AbstW¨ennandedievermeidenoblatendieEllipsoidehnichtydrodynamischen¨uberlappendKr¨afte,sind,dieistdiedasVKraftnull,orhandenseinund¨uberder
Fldiese¨ussigkKrafteitwirdbeschreiben,eineKdenKontaktfunktionontaktgewzwischen¨ahlt,dendiePartikeingehendereln(KinuuselaKet.uuselaal.et.al.(2001)).(2001)F¨ur
undinPerramandRasmussen(1996)erkl¨artwird.DieGeometriedesoblatesEllipso-
idsminimalenwirddurchRadiusseinEbSeitenzumverh¨maximalenaltnisΔegekRadiusEMennzeichnet,,somitΔedeﬁniert=Ebals/EdasM.Verh¨altnisdes

ZusammenfassungDeutsche

System L"System L

66System LSystem L"System LSystem L5544Vertical velocity vyVertical velocity vy33221100101418225791113
Time t(ts)Time t(ts)

Abbildung1.2:AusgangsbedingungenimSystem:θd=Ω+.+Φ,Vd=ΩΩΦ,Δe=Φ.Ψ)und
kinematischeViskosit¨atν=Φ.Φ1´.InderAbbildungstellenwirdievertikaleGeschwin-
digkeitgegendieZeitinbeidenSystemen(durchgezogeneLinie),A,(c=Ω,gestrichelte
LiniemitQuadraten)dar.DieSuperpositionwirddurchAnwendungderUmkehrtranfor-
mationdurchgef¨uhrt,dieinTabelleΨ.Ψbeschriebenwird.
SystemA→SystemASystemA→SystemA
ASdg→c∗ASdgG∗→c∗G∗
EM→c∗EMiSdg→vchor
Eb→c∗EbivEgi→vvcert
U→1∗Rc
Tabelle1.1:TransformationsregelnderSedimentation.cisteinereelleZahl.

PhasenDie1.2

WirfandendreiunterschiedlicheArtenvonBewegung,genanntPhasen,inunserenSimu-
lationen:Derstation¨areZustand,dieseitlicheoderperiodischeBewegung(Belmonteet.
al.(1998))undeinechaotischeBewegung,dieinAbb.1.1gezeigtwerden.

1.3Derstation¨areZustandundperiodischePhasen:
¨eitgesetz.Ahnlichk

DieGrenzgeschwindigkeitiwirdbestimmtdurchdas√Gleichgewichtzwischenden
Tr¨agheits-undViskosit¨atskr¨aften>T=>C,daheristi∼UEb.Wirnehmenan,dassdie
charakteristischePeriodederBewegungvonderseitlichenAbmessungASdgdesBeh¨alters
abh¨angtundvondervertikalenAbmessungunabh¨angigist.Folglichnehmenwiran,dass
diePeriodesichwieG∗∼ASdg/i¨andernsollte.DieTransformationinTabelleΨ.Ψl¨aßtdie
FroudeundReynoldszahlkonstantunddieDynamikindenzweiSystemenist¨aquivalent,

1.1.4Zustandsdiagrammund¨Uberg¨ange.

110-1Ref. [14]Ref. [13]This workTumbling10
PSfragreplacementsefghPSfragChaoticreplacementsChaotic
10-2bcd∗I
I∗∗ISteadyPeriodicaIE∗I*100
FallingΨΨ2ΦΦ10-3ΨΨ2ΦΦSteady−Falling Periodic
Ψ3ΦΨ3Φ
ΨΨΨΦ102E103104ΨΦ101Re102103
Abbildung1.3:DielinkeAbbildungzeigtdasZustandsdiagrammderfallendenScheiben
wiesenrauminf¨urBelmontedasfet.allendeal.Ψ221oblateberichtet.EllipsoidIndardes,dasrechtenwirinAbbildungunserenstellenSimulationenwirdererhaltenPha-
haben.

wennsichdiedynamischenVariablenwieinTabelleΨ.Ψ(Abbildung1.2)¨andern.Auch
dieDynamikf¨urdenstation¨arenZustandunddieperiodischenPhasenwerdenaufdieser
Skalaunabh¨angigvoneinander.

1.4Zustandsdiagrammund¨Uberg¨ange.

ImPhasenraumdeﬁnierenwireindimensionslosesTr¨agheitsmomentI=15ρρfellipluidΔe,das
dasVerh¨altnisdesTr¨agheitsmomentsdesoblatenEllipsoidszuderseineskugelf¨ormigen
¨AquivalentsbeidergleichenReynoldszahlES=>)νgM)ist.Esistwichtigzuerw¨ahnen,
dassdasExperimentf¨ureinefallendeScheibemitkleinenSeitenverh¨altnissendurch-
gef¨uhrtwurde,undwirerwarten,dassdieDynamikdesSystems¨ahnlichdereinesoblaten
wird.seinEllipsoidsWennwirunserDiagramminAbb.1.3rechtsmitdenexperimentellenResultatenlinksin
Abb.1.3(Fieldet.al.(1997))vergleichen,sehenwir,dassinbeidenAbbildungendieVer-
teilungderPhasen¨ahnlichist.DieUnterschiedebez¨uglichunseresDiagrammsmitdenen
vonFieldet.al.(1997)sind,dasssieScheibenbenutzenundnichtoblateEllipsoideund
dasTaumel-RegimenichtinunserenResultatenvorkommt.DieKoexistenzderdynami-
schenPhasenistvonderanf¨anglichenOrientierungdesoblatenEllipsoidsunabh¨angig.
LinksinAbbildung1.4zeigenwirdasVerhaltendercharakteristischenZeitG∗/G.Mit
Erh¨ohungdieReynoldszahlESgehtdiecharakteristischeZeitnachNullbeiESB≈´))
(Abb.1.3).NachdiesemPunktﬁndenwirdieperiodischePhase,diesichwiediePhase
desunver¨∗andlichenFallensmiteinerunendlichencharakteristischenZeitverh¨alt.Folglich
k¨onnenwirG/GalsOrdnungparameterbetrachten,undderKontrollparameterf¨urdie-
sen¨UbergangistdieReynoldszahl.Dieser¨UbergangistwieeinPhasen¨ubergangzweiter
Ordnung.DieinnereAbbildungstelltdasPotenzgesetzmiteinemkritischenExponenten

ZusammenfassungDeutsche

0.039

*/Ty=0.53*x−6.1*30T/TT0.0340.024/T)*ln(T150.009268.5818.58)ln(Re−Rec0.00911021031000.180.20.220.24
ReRecΔ r Δ rc

Abbildung1.4:DielinkeAbbildungstelltdenG∗/GgegendenOrdnungparameterES,
mitEExponentenSB≈´))nahedarbei.ΦDie.).DieinnererechteAbbildungAbbildungzeigtzeigtdasdenG/GPotenzgesetz-V∗gegendenerhaltenmitOrdnungspa-einem
rameterΔe,mitΔeB≈Φ.ΩΩ.
≈Φ.)dar.ImFalldesoberenTeilsdes¨Ubergangs,Abb.1.3,deutetdieVer¨anderungdes
oblatenEllipsoid-Seitenverh¨altnissesdie¨Anderungvoneinenstation¨arenZustandFallen
zueinemperiodischesRegimean.
Derrechts¨Uberdarganggestellt.vomWirstationv¨arenerwendenZustandG/G∗F,allend.h.diezumInversechaotischendeszuvReorgimevwirderwendeteninAbb.Ord-1.4
nungsparameters,umden¨UbergangzubeschreibenundalsKontrollparameterdasSei-
tet.tenvDieerh¨altnisΔecharakteristische.BeiΔeBZeitwirdGein/G∗vendlichererschwindetSprungwegendesdernicht-reOrdnungsparametersgelm¨assigenbebeobach-we-
sind.gungen,Dieserdie¨geUbergengangkleinescheintSchwdaherankungeneinderPhasenanf¨¨uberganglichenangersterOrientierungOrdnungzusehrsein.empﬁndlich

1.5Sedimentationsgeschwindigkeitf¨uroblateEllipsoide

InderlinkenAbbildung1.5zeigenwirdiemittlerevertikaleSedimentationsgeschwindig-
keitiΣg)alsFunktiondesVolumenanteilsΦVf¨urEllipsoideundKugelnundvergleichen
dannmitdemph¨anomenologischenRichardson-ZakiGesetzvΦ)=ΣΨ−ΦV)c(Richard-
sonandZaki(1954))mitc=).).DerGrenzwertΦV→Φv0entsprichtdemeinzelnen
fallendenEllipsoid,daswirinFonsecaandHerrmann((1)2004)studierten.Esistinter-
essantzuunterstreichen,dassdieSedimentationsgeschwindigkeitdesEllipsoids,diedem
ph¨anomenologischenRichardson-ZakiGesetzfolgt,verglichenmitderder¨aquivalenten
Kugel,kleinist.F¨urEllipsoidegehtdievertikaleMittelgeschwindigkeitdurcheinlokales
MaximumbeiΦV≈Φ.Φ).DiesesMaximumistziemlichinteressant,daesnichtf¨urKu-
gelnbeobachtetwird.¨Ahnlichenicht-monotoneSedimentation¨ubernicht-kugelf¨ormige
K¨orper(z.B.Fasern)istexperimentelldurchHerzhaftandGuazzelli(1999)undf¨urpro-

0.9 (vs)||0.6<v>0.3

Richardson−ZakiSim. Oblates3Sim. Spheres

2.4)s(v||<v>

1.8

erhaltenOrientierungsv1.1.6

SphEllip

00.050.1ΦV0.150.200.040.08ΦV0.120.16
dasAbbildungoblate1.5:EllipsoidLinkerPlot(gestricheltezeigtdieLiniemitmittlereQuadraten)unddieSedimentationsgeschwindigkKugel(gestrichelteeitiΣg)Linief¨ur
mitPunkten),alsFunktiondesVolumenanteilΦVbeiES=(∗ΨΦ−).DierechteAbbil-
dungzeigtdiemittlereSedimentationsgeschwindigkeitbeiES≈0.DasoblateEllipsoid-
Seitenverh¨altnisistΔe=Φ.Ω),die¨aquivalentKugelhatEEfjT=Φ.20.

lateEllipsoideinSimulationenvonKuuselaet.al.(2001)berichtetworden.
die¨Abbildungaquivalenten1.5Krechtsugelnstellt(◦)diealsmittlereFunktionvdesertikaleVolumenanteilGeschwindigkbeieitf¨urniedrigerEllipsoideRe(ynoldszahl)und
dar(ES≈7).IneinervorhergehendenArbeitistdieseSimulationsmethodemitErfolg
biszuES≈ΨΦ(H¨oﬂerandSchwarzer(2000)undKuuselaet.al.(2003))verwendet
worden.DasZwischenmaximumf¨urdieEllipsoidewirdnichtinAbb.1.5rechtsbeob-
achtet,wieinAbb.1.5,linksbeiderniedrigenReynoldszahlgezeigt.EinVergleichmit
demph¨anomenologischenRichardson-ZakiGesetz(durchgezogeneLinieimAbb.1.5
rechts)zeigteinenExponentenvonc<eS=´.Ωf¨urKugelnundcEaaTe=(.Φf¨urEllip-
soide.InbeidenF¨allenfolgendieDatendemRichardson-ZakiGesetzrechtgut.Diese
Exponenten(c=´.Ωundc=(.Φ)liegenzwischendemGrenzwertniedriger
Partikel-T<eSeilchen-Zahlen(c≈(E.)a)aTeundeinemturbulentenPartikelsystem(c≈>Ω.)),
(1954).ZakiandRichardson

erhaltenOrientierungsv1.6

DiemittlerevertikaleOrientierung(MVO)θwirdalsFunktiondesVolumenanteilsin
Abb.1.6linksgezeigt.F¨urkleinereVolumenanteilezeigtdasMVOeinest¨arkereAus-
mitrichtungdermitSchwerkraftderSchwerkraft,beobachtet,undimwelchesGrenzfdemallΦV→OrientierungsvΦwirdeineerhaltenfgenauere¨ureinoblatesAusrichtungEl-
lipsoidentspricht,dasinFonsecaandHerrmann((1)2004)beobachtetwurde.Wirsehen
auchf¨urdasMVOeinZwischenmaximumbeiΦV≈Φ.Φ),welchesdaslokaleMaxi-
mumdervertikalenGeschwindigkeitbeigleichemVolumenanteilerkl¨arenk¨onnte,dasin

ZusammenfassungDeutsche

900.50.280)−1>−0.1θ70Θ=<2cos(−0.460Ψ−0.7500.4/2.4−10.4/1.60.4/0.800.05ΦV0.10.1500.05Φ0.1V0.150.2
Abbildung1.6:DieVerteilungsfunktionPΣPdfΣθ))f¨urdiemittlerevertikaleOrientierung
θdief¨urReynoldszahlunterschiedlicheES=V(∗ΨΦ−olumenanteile.).DasOrdnungparameterEllipsoid-SeitenΨalsverhFunktion¨altnisdesistVΔe=Φolumenanteils,.Ω)und
ΦVf¨ur,dreiunterschiedlicheSeitenverh¨altnisseΔe=Φ.(/Ω.(;Φ.(/Ψ.+;Φ.(/Φ.1.

Abb.1.5dargestelltist.DiesesZwischenmaximumexistiertnichtf¨urKugeln.F¨urgr¨oßere
WertedesVolumenanteils(ΦV>Φ.Φ1)zeigtdieAbbildungeinemonotoneAbnahme.
GrUm¨oßedieΨ=σOrientierungΩPdfΣθ)der−Ψ>oblatenein,dieEllipsoideauchinKquantitatiuuselavet.zual.bestimmen(2003)alsf¨uhrenOrdnungspa-wirdie
rameterderOrientierungmitdenWerten−Ψ,Φoder+Ψverwendetwurde,wennalle
oblatenEllipsoidezurSchwerkraftsenkrecht,zuf¨alligorientiertoderentlangderSchwer-
kraftausgerichtetsind.Abbildung1.6rechtszeigtdasVerhaltenvonΨgegenΦV,f¨ur
kleinereVolumenanteilenimmtderOrdnungsparameterΦV≈Φ.ΦΦΨ¨−Φ.Φ1negative
Wertean,wasdieAusrichtungentlangderSchwerkraftbeweistundinUbereinstimmung
mitdemGrenzwert,ΦV→Φ(einoblatesEllipsoid)FonsecaandHerrmann((1)2004),
dieist.Ungeforientierung¨ahrbeiΦdeutetV≈beiΦ.Φ1positiistvderemOrdnungsparameterOrdnungparameterNull.senkrechtF¨urgr¨zuroßeresΦSchwerkraft.V≥Φ.Φ1Im,
woBereichwirveinonΦlokalesV≈Φ.MaximumΦΦΨ−Φin.Φ1,Abbhat.1.6ΨeinrechtslokalesundinAbbMinimum.1.5nahelinksfbeiΦanden.V≈DieΦ.Φ)Si-,
Δg=mulationenΦ.(/Φ.1;Φwurden.(/Ω.(mit,undzweiwiranderenbeobachteten¨unterschiedlichenahnlichesVL¨angenerhalten.verh¨altnissenwiederholt,

Diffusion1.7

Abbildung1.7(a)zeigtdieAbleitungdermittlerenquadratischenTeilchenverschiebung
RE/Rgf¨ursedimentierendeEllipsoide,wobeiEΣg)=σ[kΣg)−ΣkΣΦ)+σi>g)4)>
ist.DieKlammernbedeuten,dasseinMittelwert¨uberdasEnsemblejenerPartikel,die
sichimunterenBereichdesBeh¨altersbeﬁnden,gebildetwird.σi>istdiemittelere
GeschwindigkeitallerPartikelmiti=Φ.DieGraphikenstellendiegr¨oßenAnisotropie
denparallelen()undsenkrechten(⊥)Anteiledar.ImAllgemeinenstellenbeideAnteile

replacementsPSfrag

210⊥ll110log(dR/dt)010

110))log(t(ts

1.1.8R¨aumlicheKorrelationen

210

Abbildung1.7:Abbildungzeigtdieparallelen()undsenkrechten(⊥)AnteilederM.S.D.
f¨ursedimentierendeEllipsoideineinerlog-logSkala.DiedickengepunktetenLinienda-
zwischenstellendasWachstumindenballistischenundnicht-diffusivenRegimendar.
DerKugelradiusistEEfjT=Ψ.ΦΨ.DieReynoldszahlistΩ×ΨΦ−)und0dasSeitenverh¨altnis
Δe=Φ.(/Ψ.+.DieZahlderEllipsoideistvonderGr¨oßenordnungΨΦ.
ZeitAdimensionalZeit→igi→sσk)>→1θk,22eqρ
viv→sTabelle1.2:Transformationsregelnf¨urdieSedimentation

einanf¨anglichessogenanntesballistischesRegimedar,wieinAbbildung1.7(a),durch
eineparalleledickegepunkteteLiniezwischendenKurvenbildlichdargestelltist.Dieses
ballistischeRegimeistzug)proportional.Wirﬁndeneinnicht-diffusivesVerhaltendeﬁ-
αniertEllipsoiddichte,durchEΣg)der≡Vgiskundosit¨atundExponentendemSeitenzwischenverh¨Ψ.altnis(≤abhα¨≤angen.Ω.),derenWertevonder
InTabelle1.2stellenwirdieTranformationsregelndar,dieverwendetwerden,umdie
TGraphikranformationsreeningelnAbbildungunver1.8¨andert¨ubereinandergelassen.zuDieseslegen.ErDiegebnisRegiltf¨urynoldszahlkleinewirdESdurchΨΦ−).die

1.8R¨aumlicheKorrelationen

WirbeginnenunsereAnalyse,indemwirdier¨aumlichenKorrelationenindenGeschwin-
digkonsfunktioneitsﬂuktuationendesparallelenstudieren(||)(imAnteilsFderolgendenSCVF).GeschwindigkDienormalisierteeitsﬂuktuationenAutokwerdenorrelati-wie
folgtdeﬁniert(Segr´eet.al.(1997)):

ZusammenfassungDeutsche

450400400350350300(e)300(f)2502>*2002>*250
<x<x2001501501001.00.4/1.6100500.8/1.61.2/1.6501.41.2
1.61.6/1.60050010001500200000500100015002000
**time ttime t

Abbildung1.8:AlleKurvenliegenrelativgut¨ubereinander,wasdieTransformationsre-
gelninTabelleΩrechtfertigt.

,Σr)≡δiΣΦ)δiΣr)(1.2)
δiΣΦ))
wobeidieKlammern...einenEnsembledurchschnitt¨ubermehrereindividellunter-
schiedlicheKonﬁgurationenimRaumundindenOrientierungen(Ellipsoide)darstellen.
DabeistelltδvT=vT−vhECdieFluktuationeninderGeschwindigkeitundvhEC=iT
dieMittlereGeschwindigkeitderKonﬁgurationdar.FallsderAbstandrinderRichtung
genommenwird,diezurSchwerkraftparallelist,dannnennenwirdieparalleleKompo-
nente,Σk)=,oderfallssenkrechtdazu,dannnennenwirdiesenkrechteKomponente
,Σz)=,⊥.

1.9¨AnderungenimVolumenanteilundKollaps

Abbildung1.9zeigt,dassalleKurvendesSCVFf¨urKugelnundEllipsoideim(||)Anteil
zurSchwerkraft¨ubereinstimmen.HierbeiwirdderAbstandEEfΦV−(/0benutzt,wievon
¨esSegralse´et.bemerkal.(1997)enswert,vordassgeschlagen.sief¨ur¨BeiderAnderungendesUbereinstimmungVolumenanteilsderKurvbisenzumbetrachten60-fachenwir
g¨ultigist.DievonunsgefundenenKorrelationsl¨angenk¨onnenwiefolgtangegebenwer-
den:ξ⊥,heS=Ω2eEfφ−(/0;ξ,heS=Ψ´eEfφ−(/0,wassichnichterheblichvondenResul-
tatenﬁndetvonmanSegrauche´et.eineal.sehr(1997)gute¨unterscheidet.UbereinstimmungImderEllipsoidfKurvallen.(sieheDieWerteAbbildungf¨urdie1.9,Korre-(b)),
lationsl¨angesindξ⊥,Eaa=Ω)EEfΦ−(/0;ξ,Eaa=ΨΦEEfΦ−(/0.DieKorrelationsl¨angef¨ur
EllipsoideistinbeidenAnteilenkleineralsf¨urKugeln.
DietuationenAmplitudeistinderAbbildungparallelen1.10()darundgestellt.senkrechtenDie(⊥Graphik)enAnteilewerdenderineinerGeschwindigklog-logeitsﬂuk-Skala
dargestellt.F¨urΦ.ΦΦ)≤ΦV≤Φ.Φ0sindFluktuationengefundenworden,diesowohlf¨ur

1.20.8Sph,||0.4C

N()

1.20.0020.0040.0200.1200.8Ell,||0.4C

1.1.10¨AnderungderBeh¨altergr¨oße

0.0020.0040.0200.120)O(

replacementsPSfragPSfragreplacements−0.402040(N)−0.402040
−1/3−1/3r/(ReqΦV)r/(ReqΦV)
Abbildung1.9:Kollapsderr¨aumlichenKorrelationsfunktionf¨urden(||)Anteilf¨urKugeln
(b).Ellipsoideund(a)

. δδ||⊥ Ellip Ellip

010. δδ⊥|| Sph Sph . δδ||⊥ Ellip Ellip
−110−1102>Vδ⊥2Vδ>⊥
2>,<Vδ<||2>,<Vδ<||10−2
−210−310−31010−2ΦV10−110−2ΦV10−1
Abbildung1.10:DieoberenAbbildungenstellendier¨aumlichenKorrelationsfunktionen
f¨urKugelnundEllipsoideindenparallelen()undsenkrechten(⊥)AnteilenalsFunktion
)−EdesEfjTV=Ψ.ΦΨolumenanteilsderKinugelneinerunddaslog-logSkalaEllipsoid-Seitendar.DievReerh¨altnisynoldszahlistΔeist=Ω×Φ.(/ΨΦΨ.+,.derRadius

Kugeln,alsauchf¨urEllipsoidewie≈ΦV(/0(diegeradeLinie¨uberdenDaten)wachsen,
(Segre´et.al.(1997)).F¨urgr¨oßereVolumenanteileΦV>Φ.Φ0)werdendieFluktuationen
inbeidenAnteilen,f¨urKugelnundf¨urEllipsoide,verringert.

1.10¨AnderungderBeh¨altergr¨oße
WirstudierendieGeschwindigkeitsﬂuktuationenwieindervorhergehendenArbeitvon
Segre´et.al.(1997)undver¨anderndieKorrelationsl¨ange,umGr¨oßeneffektezuunter-
suchen.DieResultatesindinderAbbildung1.11f¨urKugeln(gestrichelt-punktierte

ZusammenfassungDeutsche

1.2

0.8>s/v||Vδ<0.4

0.4

>s/v⊥Vδ<0.2

0SphEllip0SphEllip
2060L/(ReqΦ100−1/31401804080L/(ReqΦ−1/3)120160
Abbildung1.11:DielinkeAbbildungzeigtdenparallelenAnteilderGeschwindigkeitsf-
luktuationenbeiver¨anderlicherBeh¨altergr¨oße.DierechteAbbildungzeigtdenvertikalen
Anteil.DieReynoldszahlistΩ×ΨΦ−),derRadiusderKugelnistEEfjT=Ψ.ΦΨ,unddas
Ellipsoid-Seitenverh¨altnisistΔe=Φ.(/Ψ.+.

Linie)undEllipsoidegezeigt(durchgezogeneLinie).DieBeh¨altergr¨oßenwerdenmit
A/ΣEEfjΦ−(/0)normalisiert.WieimFallvonSegre´et.al.(1997)undvonH¨oﬂer(2000)
zeigendieGeschwindigkeitsﬂuktuationeneineanf¨angliche¨Ubergangsregion,dieeine
starkeAbh¨angigkeitvonderBeh¨altergr¨oße,zwischenΩΦ≤A/ΣEEfΦ−(/0)≤ΨΦΦhaben.
DanachsinddieSimulationsdatenunabh¨angigvonderBeh¨altergr¨oße.ImAllgemeinen
zeigenEllipsoideundKugelndasgleicheVerhalten,abermiteinemkleinerenGesamt-
wertf¨urEllipsoide.WieerwartetwirdistdasVerh¨altnisdesParallelanteilsderGe-
schwindigkeitsﬂuktuationzumVertikalanteil⊥≈Ω.)f¨urKugelnundEllipsoide,Segre´
et.al.(1997).DergleicheWertf¨urbeidePartikelnformenbeweist,dassdasanisotro-
peVerhaltenderGeschwindigkeitsﬂuktuationvonderPartikelformunabh¨angigist.Die
durchdieSchwerkraftverursachteSymmetriebrechungwirktgleichermaßenaufKugeln
Ellipsoide.undCaﬂishandLuke(1985)fanden,dassdieGeschwindigkeitsﬂuktuationenbeiZunahme
derBeh¨altergr¨oßedivergierenk¨onnen.AndererseitswurdewederinExperimenten,Si-
mulationen,nochinderTheorieeinBeweisf¨ursolcheDivergenzengefunden.InH¨oﬂer
(2000)wirdargumentiert,dassSysteme,diedurchW¨andebegrenztsind,nichteinever-
gleichbareSkalierungderGeschwindigkeitsﬂuktuationenaufweisen.Stattdessenzeigen
sieeineS¨attigung,sobalddiekleinsteAusdehnungdesBeh¨alterseinekritischeGr¨oße
¨ubersteigt.EsbestehtdieSchwierigkeit,eineinzigesSkalierungesetzzuﬁnden.Inunse-
renSimulationenwurdedieBeh¨altergr¨oßevariiert,indemmandiegesamtequadratische
Unterseite¨anderte.DieResultateinAbb.9.7(a)und(b)zeigenkeineDivergenzderGe-
schwindigkeitsﬂuktuationenf¨urKugelnoderEllipsoideundbewegensichsehrnahean
denResultaten,diedurchSegre´et.al.(1997)gegebenwerden.

1.1.10

¨Anderung

der

Beh¨altergr¨oße

2Chapter

oductionIntr

objectsfallingThe2.1

toThefollotreewleathevesshortestﬂuttertopath.theThegroundwayininwhichautumn,eobjectsxhibitingfallatothecomplexgroundmotionhasandbeenrefusingstudied
sinceGreeks.antiquityDuring.ObjectsRenaissance,wereGalileothoughttoGalileireturntodropped“theirtwonatural”metalballsplacesfrombythetheleaningancient
TNeowerwtonofshoPisawed,andthatshothewedthatbodiesthefyallfallonatearththedrisamevenratebyadespiteconstanthavingdifaccelerationferentandmasses.he
alsoobservedthecomplexmotionofobjectsfallinginbothairandwater(VietsandLee
(1971)).

Butdespitegravity’sundeniableattraction,notallfallingobjectstraveldownwardsin
straighttrajectories.Theconsiderationofﬂuidsurroundingtheobjects(ﬁgure2.1),in-
troducesaverycomplicatedandnonlinearinteractionbetweentheobjectandtheﬂuid.
TheﬁrstpioneeringeffortwasmadebyMaxwell(1853),whowastheﬁrsttoconsiderthe
ﬂuid-objectinteractionandproposedamodelforafallingpaperstrip.

Inthebeginning,theoreticiansmadefewassumptions
a)constrainedmotionin2-dwastakenintoaccount
c)b)vorticesconsideredinaﬂuidﬂuidwerewithzeroignoredviscosity
Basedontheseassumptions,GustavKirchhofshowedthattheproblemreducestoasim-
pliﬁedsetofequationsthatcanbesolvedforsimpleparticlegeometries.Thismethod
alsoappearsintheHoracelamb’sclassictreatiseonhydrodynamics,(Lamb(1932)).

Adeeperunderstandingofthemotionoffallingobjectsinaﬂuidisofgreattechnical
importance,andhasbeeninvestigatedinavarietyofcontexts,includingmeteorology
(Kajikawa(1982)),aircraftstability(Mises(1945)),powergeneration(Lugt(1983)),
chemicalengineering(Marchildonet.al.(1964)),andalsointhestudyofstabilityof
submarinesandthecentrifugationofcellsbybiologicaltechniques.

Inhoffthe’sninetiesequationsArefthatandtheJonestrajectory(1993),ofanfoundobjectbymomeansvingofthroughnumericalansolutionsincompressible,ofinKirch-vis-

objectsfallingThe2.2.1

Figure2.1:Imagesofthevorticessheddingby(a)arisingairbubble(KellyandWu1997)
theandﬂuid(b)amotionmetalduestriptoasaitfallsdispersionthroughofwsmallater(S.aluminumFieldetalparticles1998).intheItwisater.possibleThevtoorteseex
shedsineachcasearesynchronizedwiththezigzagmotionoftheobject.

cidandirrotationalﬂuidischaotic.TanabeandKaneko(1994,95)andMahadevanet.al.
(1995)usedaphenomenologicalmodelforthefallingofapieceofpaperin1-d.Theyin-
cludedtheliftandkinematicviscosity,butneglectedtheinertiaoftheﬂuid,anddescribed
ﬁvefallingregimesofwhichtwowerechaotic.Further,Mahadevan(1996)implemented
anolderworkpresentedinHoraceLamb’sclassictreatiseonhydrodynamicstotheprob-
lemoffallingcardsthattakesintoaccounttheeffectoftheﬂuidasashape-dependent
renormalizationofthemassandthemomentofinertiatensor.Tumblinganddriftmotion
wereobservedwhichareaconsequenceoftheanisotropyinmassandviscosityinthe
model.Furthermore,Mahadevanet.al.(1999)alsomadeanexperimentbydroppingho-
rizontalcardsofthicknessRandwidthjshowingthatthetumblingfrequency,Ωscales
asΩ≈R(/)j−(,consistentwiththedimensionalargumentthatbalancesthedragagainst
.vitygra

inaﬂuidAdditionallyand,Fieldidentiﬁedet.al.different(1997)invdynamicalestigatedreegimesasxperimentallyafunctiontheofbehatheviorofmomentfallingofinertiadisks
andtheReynoldsnumber.Theyobtainedexperimentalevidenceforchaoticintermittency
f(Bauerallinginet.aal.ﬂuid,(1992)).observedtwBelmonteoet.motions:al.side(1998),tosideinaneoscillationxperiment(ﬂutter)withandthinﬂatend-ovstripser-
endinertialrotationdragand(tumbling)liftwhichﬁgureΨ.reproducesΩ.TheythisproposedmotionaandyieldsphenomenologicaltheFroudemodelsimilarityincludingwhich
describesthetransitionfromﬂutteringtotumblingregime.

Inspiteofthislargeeffort,thegeneralproblemofthefallingofabodyininteractionwith
thesurroundingﬂuid(ﬁgure2.1),remainswithoutsolution.

oductionIntr

Figure2.2:Theimagesshowacollageofconsecutivevideoﬁelds(Belmonteet.al.1998)
oftwofundamentalmotions:side-to-sideoscillation(ﬂutter)andend-over-endrotation
(tumble).

sedimentationparticleMany2.2

Thesedimentationofasystemofparticlesinaﬂuidundertheactionofgravityisavery
importantprobleminﬂuiddynamicsandstatisticalphysics.Thisproblemhasmanyap-
plicationsinbasicsciencessuchasaerospacesciences,biophysics,environment,etc,and
intheﬁeldofengineeringe.g.chemicalreactors,contaminationspreading,ink-jetprint-
ing,ﬂuidizedbeds,etc.Manynaturalprocessesinvolveparticlesimmersedinﬂuidsfor
example,bloodﬂux,particlesintheatmosphere,diffusionstudiesoforganellestransport
inlivingcells,papermaking,sedimentationinriversandlakes.Tounderstandsuchsys-
tems,abetterknowledgeinsedimentationandabilitytosimulatethesedimentbehavior
isverymuchrequired.Withthiswiderangeofapplications,theparticlesedimentation
stilllacksastatisticalmechanicalandﬂuiddynamicaldescriptionandremainsanopen
problem.

Sedimentationandstatisticalphysicshavealonganddistinguishedcommonhistory,be-
ginningwiththeclassicaltheoretical(Einstein(1906))andexperimental(Perrin(1916))
studiesofBrownianmotion.Theearlyworkdealtprimarilywithsuspensionsatornear
thermalequilibrium.Thismeantthatthesourceofﬂuctuationsinthesystemisathermal
bathcharacterizedbythetemperature.Thecorrelationandresponsefunctionsofphysical
observablesaretightlylinkedbyﬂuctuationdissipationrelations.Theinterestofphys-
icistsinthestatisticalmechanicsandhydrodynamicsofsuspensions(Pusey(1997))has
continuedtothepresentdaywiththefocusshiftingprogressivelytotheproblemsofsys-
temsfarfromequilibrium.Theeffectofshearﬂowonthestructureandcrystallizationof
suspensionshasreceivedagreatdealofattention(AckersonandClark(1984));thecon-

162.2.3Drivensuspensionandhydrodynamicdispersion
whereceptuallythereissimpleronavstateerageofnorelatisedimentationvemotion(Russeloftheet.al.particles.(1989),BlancandGuyon(1991))
2.3Drivensuspensionandhydrodynamicdispersion
Systemspresentingsteadilysedimentingsuspensionsareinanon-equilibriumstateand
thereforehavepropertiesqualitativelydifferentfromthesysteminthermalequilibriumin
twoimportantways.Firstly,thesuspendedparticlesaredenserthantheﬂuidandalsoin
ordertobalancebetweengravityandviscousdissipationonaveragethereisadownward
relatidisplayvespeed.randomparticleSecondly,motionirrespectievenveofwhenthetheparticlethermalbeingBrowniansedimentedmotionorisnesheared,gligible.they
Eachparticleinﬂuencestheotherinsuchawaythatthedynamicsishighlysensitiveto
initialconditions.TheresultingchaosasobservedinBradyandBossis(1985)andJ´anosi
et.al.(1997),impliesthatthetime-evolutionofthecoarse-grainedquantitiesmustbe
describeddynamicsinusingthedifabsencefusionofcoefthermalﬁcientsBroandwniannoisemotionsources,iseentirelyventhoughdeterministic.theThismicroscopicphe-
nomenonofdiffusivebehaviorinducedbytheﬂowduetotheobjectsdriventhroughthe
ﬂuid,intheabsenceofathermalnoiseiscalledhydrodynamicdiffusionorhydrodynamic
dispersion.Thisplacesmanyquestionsaboutthelarge-scalestructureandlong-timedy-
namicsofsedimentingorshearedsuspensionsmainlyinthedomainofnon-equilibrium
statisticalphysicsratherthantraditionalﬂuiddynamics.
Ingeneral,thesuspendedparticlesareacteduponbyBrownianandotherforces.If<iS
isthethermalBrowniandiffusivityofasoluteparticlewithradiusNinaﬂowwithtypical
velocitygradientγ˙aroundtheparticle,thenthePS´cletnumberPS,
PS=γ˙N)(2.1)
<iSisdifafusionindimensionlessthesuspension.measureofFortheaparticleimportancemoofvingparticle(foreﬂowxample,comparedsedimenting)tothewiththermala
speedithroughanunboundedﬂuid,
i(2.2)γ˙≈NFoforsolvaBroentwniandisplaced)sphereLof=bradius,UNsettlingandbuothroughyantaweightviscous(i.e.ﬂuidweightatminustemperaturetheY.Gweightin
energyunits,thesettlingspeed,ihis:
ih=b,U(2.3)
ΓHere,UisaccelerationduetogravityandΓisthecoefﬁcientofviscousdragoftheﬂuid
(Γ=+πNνforasphere).TheEinsteinrelationtellsusthatthediffusivity,<iS=Y.G/Γ.
Thus,PS=b,UN/Y.Gisindependentofthekineticcoefﬁcientsofthesystem.A

oductionIntr

replacementsPSfragFhpSecNcgNcg
>eSS−fSggaWcU
FSRWbScg
U

Batch2.3:Figuresedimentation.

suspensioninwhichPSisextremelylarge,(ΨΦ)ormore)isanon-Browniansuspension.
Insuchsuspensionsthephysicsisdominatedbythedrivingforce(gravityinthecaseof
sedimentation)andhydrodynamicswhereasthermalﬂuctuationsplayanegligiblerole.

InelectricalcontrastorDethermalGrootandcurrents,Mazurﬂuctuations(1984),andhnon-equilibriumydrodynamicsteadydiffusistatesvitiesinsuchdriasvenﬂownon-of
Browniansuspensionsaredeterminedbythedrivingforce.Theyhavenothingtodowith
thethermodynamictemperatureofthesystemandarehencenotconstrainedbyﬂuctu-
ationsdissipationrelations.EveninsuspensionswithPS=Ψ,therewillbesubstantial
non-equilibriumcontributionstodiffusion,ﬂuctuationsandlinearresponseofthehydro-
interactions.dynamic

2.4Steadysedimentationandtheﬂuidizedbedgeometry

Oneofthemostfamiliaraspectsofsedimentationistheseparation(ﬁgure2.3),ofasus-
pensionintosedimentandsupernatantwithafree-settlinglayerinbetween(Russelet.al.
(1989))understeadysedimentation.Thisstatecaninprincipleberealizedbystudying

2.2.4Steadysedimentationandtheﬂuidizedbedgeometry

bed.Fluidized2.4:Figure

thefree-settlingregionalone,whilefeedinginparticlesfromthetoptocompensatethose
eouswhichformperpetuallythesediment.settlingstateAniselegtoantmowveaytheofachiereferencevingthisframeideawithinathespatiallysettlinghomogen-particles
inthe’ﬂuidizedgeometry’(referenceXueet.al.(1995))asfollows.Subjectthesus-
pensiontoanupwardﬂowofspeediΦfrombelow(ﬁgure2.4).Thenumberdensity,cΦ
isspatiallyuniformandiscompatiblewiththeﬂowrate.Forsampleswhoselineardi-
mensionsarelargeinalldirectionsthebehavior,apartfromachangeofreferenceframe,
shouldsettlingbewithidenticalspeediinΦinthethebulktolaboratorythatofaframeincollectionanofunboundedparticlesﬂuid.withAlthoughnumbernotdensityallctheΦ
experimentsthathavebeenperformedintheﬂuidized-bedgeometryprobethestatistical
propertiesofsteady-statesedimentation,itisanidealsettingforsuchstudies.Thema-
jorityoftheexperiments,eventhoseperformedinconventionalbatchsedimentationare
arycarried.outThroughoutunderthethemanimplicity-ellipsoidassumptionpartofthatthisthethesisunderlyingwearestateconcernedisstatisticallywiththestation-nature
ofsuchasteadilysedimentingstateandﬂuctuationsaboutthisstate.

oductionIntr

Figure2.5:(a)Apairofparticlessettlingsidebysidesettlesfasterthananisolated
particle;(b)apairwithobliqueseparationvectorsettleswithasmallhorizontalcompon-
enttoitsvelocity.

2.5LowReynoldsnumberﬂow

TheReynoldsnumberES=IΣΩ∗EM)/νisameasureoftheratioofinertialtoviscous
forcesintheﬂowofaﬂuidwithν=µ/ρthekinematicviscosityandµtheshearviscosity;
ρthemassdensity;withIandAbeingtheverticalvelocityandthelargestradiusofthe
ellipsoid,respectively.Forthesuspensions,ESrangesfromΨΦ−0upto(dependingupon
thevaluesofρandν.

IfweworkintheStokesianlimit,ES=Φ,severalimportantfeaturesofStokesianﬂowin
thepresenceofparticles(Russelet.al.(1989))canbesummarized.

(i)Theequationsoftheﬂuidﬂowinthislimitarelinear.

(ii)AnisolatedsingleparticleofbuoyantweightWsettlingundergravityinanunbounded
containergivesrisetoavelocityﬁelddecayingasL/ewithdistance,efromtheparticle
toanypointintheﬂuid.

(iii)tlingAparticleslocalizedlikedensitywiseproducesﬂuctuationaΨ/eaboutvaelocitybackgroundﬂuctuation.ofuniformconcentrationofset-

(izero:v)Thetheyrelatineithervevelocityapproach,ofanrecedeisolatedfrompairnorofrotatesettlingaboutparticleseachinotheran.Iftheunboundedystartatﬂuidtheis
sameheight,theyfalltogether(ﬁgure2.5(a))ataspeedgreaterthanthatofanisolated
singleparticlesubjectedtothesameforce.Iftheyareinitiallyseparatedbothvertically
andhorizontally,theircenterofmassfallsnotverticallybutobliquelywiththevelocity
pointinginadirectionbetweentheverticalandthevectorjoiningtheupperparticleto
thatatthelowerposition(ﬁgure2.5(b)).

(v)Thedynamicsofthreeormoreparticlesiscomplexandchaotic(J´anosiet.al.(1997)).

2.2.6Velocityﬂuctuationsinhard-spheresedimentation

2.6Velocityﬂuctuationsinhard-spheresedimentation

AnapproximatetheoryinsedimentationbeginswiththepioneeringworkbyBatchelor
(Batchelor(1970)).Hecomputedtheparticlesedimentationvelocitybyassumingaho-
mogeneousspatialdistributioninthedilutelimittogetthemeanparticlesedimentation
velocity.Thetheorypredictsthatatsmallvolumefraction,φthesedimentationvelocity
be:will

(2.4)

IΣφ)/IΦ=Ψ−+.)φ+DΣφ))(2.4)
withIΦbeingtheequilibriumvelocityofasphere.
Ingeneral,thisproblemrevealsaverycomplicatedmulti-bodyinteractionduetothe
long-rangehydrodynamicinteractionthatdecaysforasphereasΨ/e(Batchelor(1970))
wherenumbere.isThetheparticledistancevfromelocitiesthecanparticleﬂuctuateforaaroundspatiallytheinﬁnitemean,systembothatalongsmalltheRegraynoldsvity
(theaboutvtheertical)natureandofthethesevperpendicularelocityﬂuctuationsdirectionRamasw(horizontalamyplane).(2001).Verylittleisunderstood
Apartfromrecentadvancesintheﬁeldthisoldsubjectofsuspensiondynamicsposesex-
citingproblemsinthefrontiersofnon-equilibriumstatisticalphysics.Muchprogresshas
eralbeenpuzzlesmadeonremainthisasproblemdivergingbotheviewsdoxperimentallyontheandbasicissues.theoreticallyWe.Atdescribethesameeachtime,problemsev-
ﬁrstandoutlinetheprogressmade.

2.6.1CaﬂischandLuke’swork

AsevereprobleminthestatisticalphysicsofsedimentationwaspointedoutbyCaﬂisch
andLuke(hereafterCLCaﬂishandLuke(1985)).Averybriefresumeisgivenhere.
Considerasteadilysedimentingﬂuid-likesuspension(seesection1.3)ofhardspheres.
Aconcentrationﬂuctuationneartheorigininthissuspensionisapoint-likeforcedensity
andinthreedimension,givesrisetoavelocityﬂuctuationdecayingasΨ/ewithebeing
ultingfrommanyspatiallydistributedconcentrationﬂuctuationsissimplythesum,iT
distancefromtheorigin.ThelinearityofStokesﬂowimpliesthatthevelocityﬁeldres-
oftheindividualcontributions.Iftheseﬂuctuationstakeplaceinarandomandspatially)
uncorrelatedmannerthroughoutthesuspension,theresultingvarianceinvelocity,σvat
anypointinthesuspensionwouldbeclearlythesumofthesquaresoftheindividualcon-
tributions.ThisiT)hasC≈A0termsifthereareCsoluteparticlesinacontainerof
lineardimension,Ainalldirections.TheΨ/eformmentionedaboveforthevelocityﬂuc-
tuationproducedbyalocalizedconcentrationﬂuctuationmeansthatiT)≈A−)sothat
σv)≈A.SuchadivergingvarianceintheinﬁniteAlimit,posesseriousproblemsinthe
calculations(Batchelor(1970)),ofthemeansettlingspeedinanunboundedsuspension.
MostexperimentsﬁndsizeindependenceunlikethepredictionbyCaﬂischandLukebut
thereareseriousquestions(Brenner(1999)),thatcanbeaskedabouttheinterpretationof
themeasurements.Itisfairtosaythatexperimentshaveneitherconﬁrmedthe,Apredic-
tionsnordeﬁnitelyruledthemout.Itshouldofcoursebenotedthat,Af’predictions’

oductionIntr

saythatiftheconcentrationﬂuctuationsarestatisticallyindependentfromonepointto
anotherinspacethenthevelocityvariancediverges.Sufﬁcientlystronganticorrelations
intheparticleconcentrationﬁeldatlargelengthscaleswillsuppressthe,Adivergence.
Clearlywhatisnecessaryisatheorythattacklesconcentrationandvelocityﬂuctuations
onthesamefootinginsteadofpostulatingtheoneandinferringtheother.

2.6.2Resumeofexperimentsandsimulations

Experimentsonthevelocity-ﬂuctuationsproblemuseawiderangeoftechniquesinclud-
ofingthetrackingparticlestheinvanelocityﬁeldilluminatedbyreparticlegion(Toryimaginget.(Seal.gre´et.(1992)al.and(1997)),Leiet.directal.counting(2001)),
’diftagged’fusing-waparticlesveinaspectroscopsuspensiony(Xueofet.otherwiseal.(1995)),indexmatchedtrackingthespheresmotion(Toryofet.al.individual(1992,
Nicolai(2000)).et.Ithasal.been(1995))claimedandbysingleSegrase´wellet.al.as(1997)multipleandNicolaisound-scatteringand(CoGuazelliwanet.(1995),al.
awithonesize-independentsigniﬁcantvealuebxceptionutthisofToryinterpretationet.al.has(1992),beenthatthecriticizedbyﬂuctuationsBrennersaturate(1999).to
NumericalsimulationsbyLadd(1996)showclearevidenceofsize-dependenceoverthe
theserangeofstudiesAewerexplored,probingalthoughscalesitisarsmallerguedbythanSeagrlare´geet.al.screening(1997)thatlength.thisH¨woﬂerasbecause(2000)
citypointedoutﬂuctuations.thattheIfthesmallestsmallestdimensiondimensionoftheiscontainerincreased,thecontrolvtheelocitymagnitudeﬂuctuationsofthevincreaseelo-
uptoalimitandareindependentofthecontainersize.
ThereisalsoaclassofexperimentsbyRouyeret.al.(1999),whichseparatestheprob-
lemofhydrodynamicdiffusionandnon-equilibriumstatisticalbehaviorinﬂuidizedbeds
fromthequestionofwhetherthevelocityvariancediverges.Thisisaccomplishedby
workingwithasuspensioninaneffectivelytwo-dimensionalgeometry,i.e.withlength,
Aandwidth,Lmuchlargerthanthethickness,δ,andδslightlylargerthantheparticle
size.Thisyieldsasystemwhoselocalhydrodynamicsisthreedimensional,sothathy-
drodynamicdispersiontakeplacewithlong-rangeeffectsincludinganypossibleCaﬂisch
ityLukedistribdiverutiongenceofthescreenedvelocityoutonﬂuctuationslateralscalesofh>>yperdifδ.fusiTheveparticlemeasurementsintheseofethexperimentsprobabil-
thestilllackaparticle-imagingtheory.ThevworkelocimetryofXueofSeet.gral.´eet.(1995)al.(1997)usingdifalsofusing-wbroadlyavfeallinspectroscopthiscateygoryand.
Butwiththethesequestioninterestingoftheanddiverimportantgentveelocityxperimentsvariance.doThenotconﬁnedspeciﬁcallyexperimentalconcernthemselvgeometryes
hoandwethever,isnotionof(Serelevgre´anceetal.since(2001))thedeofvanelopmenteffectivofeourtemperaturesimulationsisarecentralmadetothebasedonstochasticthat
P</approachofLevineet.al.(1998).

2.6.3Sometheoreticalapproaches

al.Apart(2000)fromseemthetoideashavinevruledolvingoutparticleprettyorconclusiﬂuidvelyinertia,therewhichhaveebeenxperimentspreciselybyCofourwanthe-et.

particlesNon-Spherical2.2.7

(∂Foretical)(KochattemptsandtogoShaqfehbeyond(1991))whatwereCaﬂischtheﬁrstandtoLukare,gue,Athathavaedone.mechanismKochandanalogousShaqfehto
screeningoftheCoulombinteractioninelectrolytesmightworkinsedimentingsuspen-
sions.dynamicallyTheystartedinteractingfromaparticlesmicroscopicandshowedstatisticalthatdescriptionthree-particleoftheencounterssedimentingcouldhleadydro-to
aentscreeningmeasurementsofthe,Acoulddiverpredictgence.whetherTheyahagivveennot,hosuspensionwever,isscreenedmentionedorthatnot.independ-Brenner
ar(1999)guingthatassumedthethe,AinterpretationmechanismoftheebutxperimentsquestionedbySethegree´et.videnceal.in(1997)favorareofgreatlyscreening,com-
plicatedbytheproximatewallsofthecontainer.Thecoarse-grainedapproachofLevine
et.locityal.ﬁelds(1998)ofaconsistssedimentingofthestochasticsuspension.equationsThisofretainsmotiononlyforthosethetermsconcentrationwhichanddominateve-
atlargelengthscalesassumingnorelationsamongstthephenomenologicalparameters
otherunderlyingthanthosetheoriesforcedofbydynamicalthesymmetriescriticalofthephenomenaproblem.(HalperinTheandspiritisHohenberidenticalgto(1977)),those
thehydrodynamicsoforderedphases(Martinet.al.(1972))orindeedtheﬂuctuating
Navier-Stokesequations(LandauandLifshitz(1969)).
Thestationaryimportantdifconﬁgurationferenceisthatprobabilitiesaresedimentationnotgiisvenabyanon-equilibriumBoltzmann-Gibbssteadystate,distribwhereution
withrespecttoanenergyfunctionbutcanbeobtainedbysolvingtheequationsofmotion.
Thisapproachyieldsaphasediagramforsteadysedimentationcontainingan’un-
screened’phaseinwhichthevelocityvarianceΩidivergesasaasobservedin,A,and
a’nally,Tscreened’ongandphaseAckinersonwhich(1998)Ωimadesaturatesanfor’intriguingA’greaterobservthanationthethatthescreeningmodellengthξequations.Fi-
forthermalconsedimentationvectionatatlarlargegePSPrandtlandsmallandReRayleighynoldsnumbers.numberESTheareidenticalsedimentationtothoseproblemfor
dealswiththeconcentrationﬁeldwhereastheconvectionproblemdealswiththetem-
peratureRayleigh-Bernardﬁeld.TheyturbthenulencetotranscribedargueforresultsscreeningfromandKraichnan’hence,sforamixing-lengthﬁnitevelocitytheoryvari-for
anceinsteadylowReynolds-numbersedimentation.Animportantdifferencebetween
TanonguniformandAckersonconcentration,(1998)whileandthewhereaseconxperimentsvectionisisthatdrithevenlatterbyanwereimposedundertakingtemperaturewith
gradient.

2.7particlesNon-Spherical

Asmentionedabove,manyoftheinvestigationshavebeenmadeonspheresandina
reducedmanneronslenderbodies,(e.g.ﬁbers)byRamaswamy(2001)andBatch-
elor(1970).Ellipsoidalparticles(ﬁgureΨ.+),havefoundanapplicationinthemodeling
ofthebloodﬂowOlla(1999).Fibers,ontheotherhand,havenumeroustechnological
applications,thepapermanufacturing,transportandreﬁningpetroleum,pharmaceutical
smallerprocessingvolumeandenfractionsvironmentalwhichwisastecharacterizedtreatment..byThetheyshowsedimentationorientationalvelocitytransitionthathasfora

oductionIntr

Figure2.6:Snapshotoftheoblateellipsoidsfallinginaﬂuid.Thepictureshowsthe
distance“cluster”isgivformationeninunitsalongofthetheflaralling.gerTheradius,Eellipsoidandtheaspect-ratioReynoldsisΔe=numberΦ.(/isΨ.E),Sthe=
M(∗ΨΦ−).

globalmaximumaroundthetransitioninstationarystate,Kuuselaet.al.(2003).

Withthisbackground,theinvestigationofthesedimentationofnon-sphericalparticles
isfundamental.Anunderstandingofthesettlementmovement,orientationandspatial
distributionofparticlesindrivenﬂowsiscrucialforthistask.

viewerOv2.8

InthisthesisIpresentanumericalstudyofthedynamicsofoneandmanyfallingob-
lateellipsoidsparticleinaviscousﬂuid,inthreedimensions,usingaconstrained-force

viewerOv2.2.8

Figure2.7:Asectionofatridimensionalfallingoblateellipsoid.Thesystemsizeis
Re)Φ×ynoldsΨ1Φ×)ΦnumbertheisoblateΩ∗ΨΦ−),aspect-ratioandtheisΦratio.Ω/ofΨ.Φtheinﬂuidunitsofdensitythelarovgerertheellipsoidellipsoidradius.densityThe
.).´is

technique(H¨oﬂerandSchwarzer(2000);H¨oﬂer(2000);Kuuselaet.al.(2001)),under
gravitytablishing.Wethestudytypestheoffallingdynamicalmotionsbehaviorandforﬁndingaatypicaldosimilaritywnwlaardwformotion,regularﬁg.Ψ.0motions.,es-
Wemotion.proposeInamanmechanismy-particlesfor(ellipsoids)understandthesedimentationtransitions(ﬁgurebetweenΨ.+),Ithestudydiftheferentssettlingtypesvofe-
locityandtheaverageorientationoftheellipsoidsasafunctionofthevolumefraction.
Wealsoinvestigatethediffusivebehaviorofasedimentingellipsoidatlowandmoderate
Reellipsoidsynoldsandnumberspheres.Finallyonthe,wesizeshowcontainerthe.dependenceofthesedimentationdynamicsof

oductionIntr

Inchapter2webrieﬂypresenttheﬂuiddynamicalbasisofthemodelandweexplain
thebasicingredientsofthemodel.Inchapter3weshowthebasicphenomenologyof
thefallingofoneoblateellipsoidineachregime(steady-falling,periodic-oscillationand
chaoticmotion)ofthesystemandwecompareourresultswithpreviousworks.Inchapter
4weinvestigatethedynamicsbehindeachregime.Fortheperiodicandsteady-falling
regimeweﬁndasimilaritylawderivedfromtheinvarianceoftheReynoldsandFroude
number.Inthechaoticregimethetrajectoryoftheoblateellipsoidischaracterizedbya
highsensitivitytotinyvariationsintheinitialorientation.
Inchapter5aphasediagramispresentedandcomparedtotheresultsofFieldet.al.
(1997).Thetransitionfromoscillatorytosteady-fallingoccursatESB=´)),where
thetransienttimeofoscillationinthesteady-fallingregimetendstoinﬁnity,beyondthis
valuethesystemisoscillatory.ThetransienttimehasapowerlawdivergenceatESBwith
anexponentof0.5.Thetransitionfromsteady-fallingtochaoticregimebecomesabrupt,
foracertainvalueoftheellipsoidradii.
Inchapter6westudythesettlingvelocityandtheaverageorientationoftheellipsoidsas
afunctionofvolumefraction.Weseethatthesettlingvelocityshowsalocalmaximum
attheintermediatedensitiesunlikethespheres.Theaverageorientationoftheellipsoids
alsoshowsasimilarlocalmaximumandweobservethatthislocalmaximumdisappears
astheReynoldsnumberisincreased.Also,atsmallvolumefractions,weobservethat
theoblateellipsoidsexhibitanorientationalclusteringeffectinalignmentwithgravity
accompaniedbystrongdensityﬂuctuations.Theverticalandhorizontalﬂuctuationsof
theoblateellipsoidsaresmallcomparedtothatofthespheres.
Inchapter7weinvestigatethediffusivebehaviorofsedimentingellipsoidsatlowand
moderateReynoldsnumber.Webeginintroducingthecommontheoreticaltoolsused
tostudythediffusivebehaviorofsedimentingparticles.Wealsodiscusstheresultsfor
ellipsoidsincomparisontotheequivalentspheres.Moreover,westudythebehaviorofthe
sedimentingspheresandellipsoidsundervariationsofthekinematicviscosity,ellipsoid
densityandaspect-ratio.Furthermore,wepresenttheorientationaldiffusionbehavior.
Additionally,weshowtheanomalousdiffusionforellipsoids.
Finallyinchapter8wepresentthedynamicalbehaviorofsedimentingellipsoidsand
spheresundervariationsofthecontainersize.Westudytheinﬂuenceonthespatialcor-
relationsastheparticulatevolumefractionischanged,comparingtheresultsforellipsoids
andspheres.Wealsopresentthestudyofthevelocityﬂuctuationsasafunctionofthe
volumefraction.Afterthat,weinvestigatethedivergenceofthevelocityﬂuctuationsas
changed.issizecontainerthe

2.2.8

viewerOv

3Chapter

Model

Thischapterlaysoutthegeneraltechniqueusedformodelingthephysicsofoneand
manysedimentingellipsoids.Weintroducetheimportantquantitiesandterminologies
thatareemployedintherestofthethesis.Intheﬁrstsectionwepresentbasicresultsof
ﬂuiddynamics.Next,adescriptionofﬂuidandparticlecouplingispresented.Finally,
theellipsoidcontactmethodfortheellipsoid-ellipsoidinteractionissketched.

3.1Navier-Stokesequations

Fluidisdeﬁnedasthestateofmatterthatcannotsustainanyshearstress.Foranelastic
solid,thestrainincreaseswithtimeandattainsasteadyvaluewhereas,foraﬂuiditin-
creasesindeﬁnitelywithtime.Inthemathematicaldescriptionofaﬂuidﬂow,theﬂow
quantitiessuchasvelocity,pressureareassumedtovarycontinuously.Inviewofthe
particlenatureofmatter,thevalidityofthisassumptioncanbequestioned.Ifthemean
freepathofthemoleculesiscomparablewiththecharacteristicdimensionofthemac-
roscopicﬂow,thecontinuousapproximationbreaksdown.Standingbythiscontinuous
hypothesis,onecanderivetheconservationofmassintheﬂuidﬂowbywritingtheglobal
equationforthemassoftheﬂuidinsideaﬁxedvolume.ConsideranarbitraryvolumeK,
ﬁxedinareferenceframeusedfordescribingtheﬂowofﬂuidandboundedbyaclosed
surfaceF.Ateveryinstantoftime,ﬂuidentersandexitsfromthisvolume.Therate
ofchangeofthemass,bcontainedwithinthevolumeisequalandoppositetotheﬂux
leavingtheboundarysurface.Thenwehave
RbR
Rg=RgVρMRK=−<ρMv∙nRF(3.1)
wherenistheoutwardunitvectornormaltothesurface,FoftheboundaryandKisthe
volume.SinceKisﬁxedwecaninterchangetheorderofintegrationanddifferentiation
withrespecttotime.Further,byapplyingtheGauss’sdivergencetheorem,weobtain:
∂∂ρgM+∙ΣρMv)=Φ(3.2)
27

3.3.1Navier-Stokesequations

Makinganexplicitdifferentiationoftheterm∙ΣρMv)andgroupingweget,
ρ∂Σ∂gM+v∙ΣρM))+ρM∙Σv)=Φ(3.3)
Theﬁrsttermoftheaboveequationisknownastheconvectivederivativecorresponding
totheLagrangiandescriptionandthereforewecanrewrite3.3as
RρM+ρM∙Σv)=Φ(3.4)
Rgwhichshowsconservationofmass.

3.1.1Thegeneralequationforthedynamicsoftheﬂuid
WesolveNewton’sequationofmotion

RgRρMvRτ=ρMΩRτ+[σ4∙nRΣ(3.5)
<VVwhereRτisthedifferentialvolumeofﬂuid,RΣisthesurfaceelementoftheclosedsur-
face,FthatboundsK,and[σ4isthetensorofallthe(pressureandviscosity)forcesacting
onRΣ.ThevolumeforceΩperunitmassofﬂuidcouldbethegravitationalforce.
Thewiththederivatiﬂuid,vethenR/RgtheisaproductLagrangianρRτisderivaticonstant,ve,easvitaluatedmovesinthealongthereferencelocalvframeelocitymoﬁeldving
Moftheﬂow.ApplyingGaussdivergencetheoremtothesecondtermontherighthandside
3.5wecanwriteequation3.5as
Mρ
VRgvRτ=VρMΩRτ+V∙[σ4Rτ(3.6)
Takingthelimitasvolumetendstozeroanddividingbythevalueofthevolumeelement,
weobtainthelocalequationofmotionforaparticleofﬂuid:
ρMv=ρMΩ+∙[σ4(3.7)
Rg[σ4canbewrittenas[σ4=[σ4−pδTU.
evhaweThen

pδ∂TU∙[σ4)T=∙[σ4T−∂kU

(3.8)

Model

becomes3.7equationThen

ρMRv=ρMΩ−p+∙[σ4(3.9)
RgThisequationisapplicabletoanyﬂuid.IfwestudythemotionofaNewtonianﬂuidthen
wecanexpressσas

)∙[σ4=µ∂iT+Σξ+µ)∂∂ia(3.10)
∂kT∂kU´∂kT∂ka
Onsubstitutingineq.3.6,weobtaintheequationofmotionforacompressibleorincom-
ﬂuid:wtonianNepressible

ρM∂v+ρMΣv∙)v=ρMΩ−p+µ)v+Σξ+µ)Σ∙v)(3.11)
´g∂Ifthecompressibilityeffectsarenegligibleintheﬂuidﬂowthen∙v=Φ.Theresulting
equationistheNavier-Stokesequations:

ρM∂∂gv+ρMΣv∙)v=ρMΩ−p+µ)v(3.12)

3.1.2ThedimensionlessformoftheNavier-Stokesequation
WecanalsowritetheNavier-Stokesequation,eq.3.12,intermsofdimensionlesspara-
meters(thatarelabeledwith’primes’).LetAandIbetherespectivescalingfactors
forthespatialcharacteristiclength(particlediameter)andvelocityofthesedimenting
particle.Thenwehave:

and

r=r,v=v
IA

(3.13)

Fp−p
g=A/I,p=Ψ/ΩρMI)(3.14)
Indeﬁningp,thevalueofpΦhasbeensubtracted,whichisthehydrostaticpressure.After
dividingeachsidebyρMI)/A,theNavier-Stokesequationbecomes:

∂∂gv+Σv∙)v=−p+EΨS)v+>e)Σ−g)(3.15)

conditionsBoundary3.3.2

Intheaboveequation,theinverseoftheRe)ynoldsnumberES=AIρM/µassociatedwith
theﬂowappearsasafactorofthetermv.Itisfoundthatthisnumberrepresentsthe
ratioofthenon-linearconvectivetermΣv∙)vtotheviscoustermµ)v.Wedeﬁne
>e)=UA/I)astheFroudenumber.Fromtheaboveequation,thevelocityandpressure
ﬁelds(vandp)thatsatisfytheappropiateboundaryconditionsforagivenproblem,are
form:theof

v=FΣk,l,z,g,ES,>e)
p=GΣk,l,z,g,ES,>e)

(3.16)(3.17)

where>andGarefunctionsthataredependentontheﬂow.Wemakeuseofthisin
.)Chapter

Boundary3.2conditions

teThegrationcompleteofthesolutionequationforofthemotionmotionofoftheaﬂuidﬂuidvparticleselocityandﬁeld,thevΣr,g)speciﬁcationincludesofboth,boundarythein-
conditions;i.e.,thevalueofthevariablesatalltheboundariesoftheﬂuid.
Boundaryconditionsvarydependingonwhethertheboundaryissolidoraﬂuid.Inthe
casesolidofarequiressolidthatwallthebeingcomponentaboundaryofthe,vthefelocityactthatnormalthetoﬂuidthecannotboundarypenetratesurfaceintoshouldthe
beequalfortheﬂuidandthesolid:

modelThe3.3

viodad∙n=vﬂkad∙n

(3.18)

Thegeneralideaofourapproach,proposedbyFogelsonandPeskin(Fogelson(1988)),
istoworkwithasimplegridbyresolvingtheﬂuidmotionatalltimesandrepresentthe
particlesnotasboundaryconditionstotheﬂuid,butbyavolumeforcetermorLagrange
multipliersintheNavier-Stokesequations.Thistechniquewasdevelopedintheworkby
Schwarzeret.al.(H¨oﬂerandSchwarzer(2000)),H¨oﬂer(2000),Kuuselaet.al.(Kuusela
et.al.(2001)),Wachmann,etal.(WachmannandSchwarzer(1998)).Thisemploys
anumericalsolverforthedynamicalsimulationofthree-dimensionalrigidparticlesina
Newtonianﬂuid,boundedbyarectangularcontainer.Theequation3.12isdiscretizedona
regular,marker-and-cellmeshtosecondorderprecisioninspace.Forthetimeevolution,
weemployanoperator-splitting-techniquewhichisexplicitandaccuratetoﬁrstorder.
Thesuspendingﬂuidissubjectedtono-slipboundaryconditionsatthesurfaceofthe
particles.MoredetailsarepresentedinKuuselaet.al.(2001),H¨oﬂerandSchwarzer
(2000).oﬂer¨H(2000),

Model

Figure3.1:Thepictureshowsthedistributionforcebetweenparticleandtheﬂuid

Anoblateellipsoidisrepresentedbyarigidtemplateconnectedtotheﬂuidtracer
straintparticles.ontheThisﬂuidisindoneorderbytothebodydescribeforcetheoblateterm,intheellipsoid.NavierThe-StokforceesdensityequationΩB,isasachosencon-
tobespringlike.WedeﬁnethisforcedensityΩBas:

ΩB=TBΣxab+Σxab))=−YΣxab)(3.19)
wherexabisthedisplacementﬁeldoftheseparationbetweenthemarkersWandtheir
correspondingreferencepointX.ThestiffnessconstantY,mustbechosenlargeenough
sothat|Σxab)|V,Vbeingthesizeofthegrid,holdsforalliterations.
Ingeneral,thedisplacementﬁeldΣxab)isdeﬁnedas:

aΣxabb)=xabb−xabg(3.20)
Thevector,xabbisthepositionofaﬂuidtracerwhosemotionisdeterminedbythelocal
i.e.,,elocityvﬂuid

x˙abb=IΣxbTU)(3.21)
Thexabgarethereferencepointsassociatedwiththetemplatehavingtheshapeofthe
particle:ysicalph

xabg=xT+DTΣg).rTU

(3.22)

HerexTisthecenterofmassofthetemplate,DTΣg)istherotationmatrixthatdescribes
theinstantaneousorientationoftheoblateellipsoidandrabdenotetheinitialpositions

The3.3.3model

ofthereferencepointswithrespecttothecenterofmass.Rotationisdescribedbythe
quaternionformulationasinAllenandTilsdesley(1987).Theequationsofmotionofthe
are:templateparticle

and

F˙=UB

˙I=Ω

(3.23)

(3.24)

whereBisthemassofthetemplateparticle;UandΩarethelinearandangularvelocities
ofthetemplateparticle,respectively;Iisthemomentofinertiaoftheoblateellipsoid
withonlythreenon-zeroelements,I((,I)),I00whereI((>I))=I00(Goldsteinet.al.
(2002))andisthetorque,(Kuuselaet.al.(2001),H¨oﬂerandSchwarzer(2000)).The
boundaryconditionsnearthecontainerwallare:thenormalvelocitycomponentofthe
ﬂuidiszero,thewallsareassumedtobeimpenetrable,becauseofano-slipcondition
forthetangentialcomponent(H¨oﬂerandSchwarzer(2000),WachmannandSchwarzer
(1998)).Theinteractionbetweentheoblateellipsoidandthewallsisdeﬁnedthrougha
contactforce(PerramandRasmussen(1996)andKuuselaet.al.(2001)),wherethewalls
aretreatedasaparticlewithinﬁnitemassandinﬁniteradius.Avelocity-Verletintegrator
(Presset.al.(1992))servestointegratetheequationsofmotionforthetranslationanda
Gear-predictorintegrator(AllenandTilsdesley(1987))fortherotationofthetemplate:

F=−BUˆX+ρMKUˆX+ΩTB+ΩTe(3.25)
TwhereˆXistheunitvectoralongtheverticalw.r.tthetemplate’scenterofmassrBb.

(3.26)

=Σxa−xTm)×ΩTB(3.26)
Twithrespecttothetemplate’scenterofmassrBb.
Thedeﬁnedasgeometrytheofratiotheoftheoblatesmallestellipsoidradius,isEbtocharacterizedthelargestbyΔe,radius,itsEM:aspect-ratiowhichis
Δe=EEb(3.27)
MWedeﬁneanequivalentspherefortheoblateellipsoid,asthespherethathasthesame
volume,withanequivalentradius:
EEfjT=3EbE)M(3.28)


Model

TherespectiveStokesvelocity:

ΩUE)EfjTΣρρfellip−Ψ)
ih=2ν(3.29)
withρEaa,Tetheellipsoiddensity,ρMbeingtheﬂuiddensity,νtheﬂuidkinematicviscosity
andUthegravity.
WedeﬁnetheunitStokestimeghasthetimeneededforanisolatedellipsoidtogoesover
adistanceofone-ellipsoidlargerradius,EMwithavelocityih,as:
EMgh=ih(3.30)

functionContact3.4

Thereisaconsiderableamountofliteratureinwhichmodelsforsoftpotentialsbetween
nonsphericalparticles(e.g.,oblateellipsoids)areemployedinordertosimulatethebeha-
itviorisofnecessarysimpletomoleculesdetermineandtheliquidforce,>ecrystals.duetoThetheinterestinellipsoid-ellipsoidthepotentialcontacts.arisesbecause
TThelubricationcontacttheoryatlowReynoldsnumberestablishesthatthepresenceof
ﬂuidsmoothlyavoidsthetouchingofellipsoid-ellipsoidsurfaces.Inthealgorithmthese
forcesarecapturedcorrectlyonlyonscaleslargerthanthegridresolution.Sincewe
arewellipsoidalorkingsurfwithacesdilutearerare.systemsWithatvtheeryaimlowofvolumerestrictingfraction,signiﬁcantcloseparticleencountersoverlappingbetween
arepulsiveforceisintroducedbetweentheellipsoidsandischosentobeproportional
totheirintermediateoverlap.distancesIfthethehoblateydrodynamicellipsoidsareforcesnon-odescribingverlapping,theethexistenceforceofisthezeroﬂuidaandvoidat
thecontactbetweentheparticles(Kuuselaet.al.2001).
WeconsidertwoellipsoidslabeledAandBwithsemi-axesN,O,PandN,O,P,respect-
ively.Therotationalstateoftheellipsoidisexpressedbythe()sets,0h(,h),(h0)and0i(,i),i0
ofareatrorthonormalandrunitrespectivvectorsely,thealongtheintercenterprincipalvectoraxisisofdeﬁnedthetwas:oellipsoids.Ifthecenters
ﬂ

andthematricesψand.are:

R=r−rﬂ

ψ=NV−)IcIcT
V.=OV−)vcvcT
V

(3.31)

(3.32)(3.33)

3.3.4functionContact

Inthecasewherenoneofthesemi-axesvanishes,thesematriceshavetheinverses

ψ−(=NV)IcIcT
V

(3.34)

.−(=OV)vcvcT(3.35)
VPerramandWertheimhavederivedtheobjectfunction,FΣλ)(PerramandRasmussen
1996)and(Kuuselaet.al.2001):

FΣλ)=λΣΨ−λ)E/[ΣΨ−λ)ψ−(+Σλ).−(4−(E(3.36)

FΣλ)=λΣΨ−λ)E/[ΣΨ−λ)ψ−(+Σλ).−(4−(E(3.37)
whereλisaparameter.FΣλ)isnon-negativeforλ∈[Φ,Ψ4.Thenthecontactfunction
>Σψ,.)forthetwoellipsoidsis

>Σψ,.)=[bNkFΣλ)|λ[ΦΨ44(3.38)
If>Σψ,.)σΨ,thetwoellipsoidsoverlap,if>Σψ,.)>Ψ,theydonotandif>Σψ,.)=
Ψ,thetwoellipsoidsareexternallytangent.

4Chapter

Phenomenology

Inthisthechapterprecedingwewillchapterapplywethishavemodelandpresentedseethethemodelcommonofphtheysicalsedimentingphenomenaellipsoids.associatedIn
tothefallingoblateellipsoids.Wewillpresentthethreephasesthatwefound,andthe
terminologiesbidimensionalthatonewillbeelaboratedusedbyintheMahadenextvan.chapters.ThevWeelocityalsoﬁeldcompareforthethisﬂuidmodelaroundwiththethe
fallingellipsoidisshownandcomparedwiththeworkbyBelmonteet.al.(1998).We
presentinaqualitativewaythepresenceofvorticesintheﬂuid.

4.1Trajectoriesofafallingoblateellipsoid

Wefoundthreedifferentkindsofmotioninoursimulations:steady-falling,side-to-side
orwhichareshoperiodic-oscillation,wninﬁg.kno4.1.wnTheseasﬂutterkindsof(Belmontemotionareet.al.called(1998))patterns,andreagimes,chaoticormotion,phases
intheliteratureandweshallusethetermphaseinthisthesis.

eIngeneral,xperimentalthesetup.numberButofitisphasespossibledependstoonclassifythethespeciﬁcphasesmodelintoandtwothelargeconditionssetsofnamelythe,
a)regularandsmoothoscillationsandb)irregularandchaoticoscillations.Forexample,
Tsionalanabeandmodel:Kanekthreeoregular(1994,95),andtwoidentiﬁedchaotic.ﬁvInefaneallingxperimentalphaseswusingorkawithsimpliﬁeddroppingbidimen-disks,
asFieldshoet.wnal.inﬁg.(1997)4.2.reportedAndlastlyfour,phasesBelmonteofet.whichal.three(1998),werereobservgularedaandreonegularwasoscillatorychaotic
androtationalphaseintheirexperimentswiththinstrips.

Inthemajorityofcases,thetrajectoriesinoursimulationsdependstronglyontheinitial
cosityconditionsνandandthetheoblatepropertiesaspect-ratiooftheΔesystem).In(oblate’orderstoinitialreducetheorientationparameterΘd,spacekinematicinvis-our
system,weﬁxρMajTC=ΨBbR3andρd∂aiE=´.)BbR3inoursimulations.

240(a1)180Position in Y12060

4.4.1Trajectoriesofafallingoblateellipsoid

(b1)

(I)

(c1)

−258118178238298
Position in X

Figure4.1:Typicalfallingtrajectoriesobtainedinoursimulations.FigureIshowsthe
componentsYΣiSegWPNa)−MΣVdeWzdcgNa)inthesteady-fallingregime,ΣNΨ),withinitial
conditionsθd=Ω+.+Φ,Δe=Φ.Ω),ν=Φ.Φ´´andΣN)Vd=Ω(Φ;ΣOΨ)periodic-oscillation
withinitialconditions:ν=Φ.ΦΩ),Vd=Ω(Φ,Δe=Φ.Ψ´´andθΦ=+´.(d.ΣPΨ)chaotic
motion,withinitialconditionsVd=Ω(Φ,Δe=Φ.Ω),ν=Φ.Φ´´andΣN)θΦ=Ω+.+d.

TheFigureimages4.2:Twererajectoriesobtainedoffallingfromthedisks,sidebypresentedusinginathewvideoorkbycamara.BelmonteTheetal,trajectory(1998).in
(a)correspondstothesteady-fallingregime,(b),periodic-oscillatingmotion,(c)chaotic
motionand(d)thetumblingmotion.

Phenomenology

Comparison vertical velocity and trajectoryChange of Vertical Velocity4.8TrajectoryVyFirst peak−to−peak amplitudeL14210Fourth AmplitudeL23.2Terminal VelocityL3140Vertical Velocity VyPosition in Y2.4L4L51.670L60.8L7Decreasing peak−to−peak amplitude amplitude0048Position in X & Vertical velocity12160510Time t(t15)202530
s1

0.60.2Horizontal Velocity Vx−0.2−0.6

−107142128
)Time t(ts

theFiguresame4.3:height.(Left)(Right)ComparisonDecreasingbetweenamplitudethevoferticalthevvelocityerticalvandelocitythe.spatialInitialtrajectoryconditionsat
ofthesystemareθd=Ω+.+Φ,Δe=Φ.Ω),ν=Φ.Φ´´.FallinginitialheightVd=ΩΩ1in
gime.reallingsteady-fthe

oblateSteady-falling4.2ellipsoid

Whentheellipsoidbeginstofall,theverticalvelocityischaracterizedbyadampedwaver-
ing(transientoscillation)intimeresemblingthebehaviorofadampedoscillator(Gold-
steinet.al.(2002)).Ataverylongtime,theverticalcomponentofvelocitybecomes
ellipsoidconstant.Theapproachesamplitudetheofbottomthevoferticalthecontainercomponentﬁg.ofv4.3.elocityThevdecreaseserticalastrajectorytheoblateﬁg.
4.3(right),alsoshowsthedampedwavering.Thiswaveringinthetrajectorycurveis
composedofsuccessiveturningpoints.Eachoneofthem,inturn,correspondstopoints

ellipsoidoblateSteady-falling4.4.2

Figure4.4:Euleranglesφ,θandψusedforthedescriptionoftheoblate’sorientational
.viorbeha

wheretherateofchangeintheverticalvelocityiszero.Thisvanishingvalueintherate
ofhorizontalchangeoflinesthevL2,...erticalL5,vwithelocityviserticalevidentvelocityin.ﬁgureThere4.3isan(left),byinterestingtherelationintersectionofbetweenthe
Ftheorevnumbererytwofoturningsuccessivepointsturninginthepointstrajectoryintheandvtheerticalverticaltrajectoryvelocitythere(seeareﬁg.four4.3,turningleft).
pointsintheverticalvelocity.Ifacharacteristiclengthisassociatedwiththevertical
trajectory,thenthecharacteristiclengthcorrespondingtotheverticalvelocityisreduced.
Theandinhorizontalgeneraltheyhacomponentsvereofgularthevoscillationselocityfollo(seewﬁg.a4.3,similarbottom).behaviorastheverticalones

Theoblate’sorientationisdescribedthroughthethreerotationaldegreesoffreedom,
calledtheEulerangles,ﬁg.4.4.Wepresentthetimeevolutionoftheanglebetween
theoblate’snormalandtheverticaldirection.Thiswecalltheverticalorientation,ﬁg.
4.5(top).Θ=Φimpliesthattheoblate’sprincipalaxiswillbeparalleltothecontainer’s
bottomﬁg.4.5(bottom).Atthebeginningofthemovement,thereisalargeangular
changeoftheoblate’snormalwiththevertical,ΔΘﬁg.4.5(bottom),whichischarac-
terizedbyalargepeak-to-peakamplitudeΘe−e.inﬁg.4.5weillustratethedeﬁnitionof
thepeak-to-peakamplitudeasthedistancebetweensuccessiveturningpoints,whichde-
creasesastheoblatesinks.Inthesteady-fallingregime,boththepeak-to-peakamplitude,
Θe−eandthechangeΔΘ=ΘM−ΘT,decaysastheoblateapproachesthebottomofthe
container.Theoblatetendstoalignitsmajoraxisalongthevertical(Huanget.al.(1998))
havingaverylowresistancetoitsdescentintheﬂuidacquiringtheterminalvelocityﬁg.
(top).4.3

Phenomenology

Angular VariationChange of Vertical Orientation8590θiChange vertical orientation Θ(t)
80Terminal Orientationθ75Θ756070Δθ=θf−θiθft
Vertical Orientation 30Position in Y60Θ=0
45Θ65First peak−to−peak amplitude p−p

60Decreasing peak−to−peak amplitude15550051015Time t(t20)2530354050−5051015202530
sPosition in X

Figure4.5:(left)Decreasingpeak-to-peakamplitudeintheangularoscillation.(right)
AngularchangeΔΘalongasegmentoftheverticaltrajectory.Initialconditionsofthe
systemθd=Ω+.+Φ,Δe=Φ.Ω),ν=Φ.Φ´´andVd=ΩΩ1Pb.
Vertical Orientation & time100150II)27090110240807021070Vertical velocity & time306018026101418
4.1150503.3(I)2.512040III)1.790304812162024
24681012141618261014
Figure4.6:InitialconditionsVd=Ω´Φ,Ak×Az=Ω)×Ω),Δe=Φ.Ψ´´,ES=(´).
InitialorientationΘΦ=+´d.I)VerticaltrajectoryII)Theverticalvelocity.III)The
orientation.erticalvellipsoidoblateOscillatory4.3

Inthedynamicalvoscillatoryariables.phase,Wetheobserveellipsoidsuchisabehacharacterizedviorinourbyaresimulationsgularatoscillationsmallinkinematicallthe

°Θ=00540Θ0=(10−2)°
360Position in Y

180

12
Position in Z

Position in Z9

ellipsoidoblateChaotic4.4.4

°=0Θ0Θ0=(10−2)°

01.56.511.516.561113151719
Position in XPosition in X

Figure4.7:Theleftpicturesho−)wsthechangeintheverticaltrajectorywhentheinitial
orientationchangesΔΘΦ=ΨΦ.Therightpictureshowsthehorizontalcomponents.
TheinitialconditionsVd=+(Φ,Ak×Az=Ω)×Ω),Δe=Φ.(´,ES=ΩΩ).
viscosityandsmallaspect-ratios.Inﬁg.4.6weshowtheellipsoid’sverticaltrajectory,
velocityandorientation.ThisoscillatoryphasehasbeenstudiedexperimentallybyBel-
monteet.al.(1998).Thisphasehasadistinctivecharacterthattheperiodofoscillation
oftheverticalvelocityishalftheperiodofoscillationoftheverticalorientation.Oursim-
ulationagreesverywellwiththeexperimentwhichwillbediscussedinthefourthsection
4.chapterof

Chaotic4.4ellipsoidoblate

inWeﬁg.observ4.7.eThechaoticﬁgshobehawsviortheforvprominentariationschangeintheinthevellipsoid’erticalsandinitialhorizontalorientationastrajectoriesshown
astheinitialorientationchangesbytheorderofΨΦ−).Inoursimulationsthechaotic
isphaseimportanthasbeentopointfoundoutforthatlargethesensitiaspect-ratiosvitytoeq.small3.28vandariationsforinalltheReinitialynoldsconditionsnumbers.ofIt
fallingbodies(inourcasetheellipsoidalorientationΘd)hasnotbeenreportedneitherby
simulations,theorynorbyexperimentsuntilnow.

4.5ComparisonwithMahadevan´smodel

Ashadevanmentionedet.al.in(1995)chapterandΨ,thereMahadearevanseveral(1996))modelsthatha(TveanabegivenandgoodKanekinsightotothe(1994,95),Ma-prob-

Phenomenology

−5

−15Vertical position−25

Vertical trajectory

−5

−20Vertical position−35−50

−0.80.1Horizontal position11.9−1.2

Vertical trajectory

2.41.20Horizontal position

Figure4.8:Steady-fallinginitialconditionsΔed=β=Φ.Ω),α=Φ.Ω2,ν(=Φ.Ψ,ν)=
ΦΦ..+Ω),ν,0α==ΦΦ..0,Ω2ν,TνT==Φ.ΦΦ.,λΦ´´,=νTΦ.=ΦΦ+Φ.,Φ,Θλd==+∗Ω+.+ΨΦ.−7F,orΘdthe=2Φd.oscillatoryThesolidregime:andΔdottede=βlines=
showMahadevan’sandourresults,respectively.

lem.WeuseasimpliﬁedmodelformulatedbyMahadevan(1996)thatoffersagood
absencequalitativofequantitaticomparisonvetoouragreementsimulationshastobeintheunderstoodoscillatoryintermsandofsteady-fthedifallingferencephase.betweenThe
itethetwsizedocmodels.ylinder,whereMahedevtheancylinderdescribesaxistheismotionassumedofatofbeallingstripperpendicularofpapertotheinvanertical.inﬁn-
Theﬁeld.systemTheisviscosityassumedistaktoenbeintobidimensionalaccountinwithtermsanoftheincompressiblegeneralizedﬂuidRayleighinagradissipationvitational
appliedfunction.alsoThisbywLamborkisa(1932),continuationwhosetheofthecentralmethodapproachusedbyKconsistedelvininandignoringKirchhofthefvandor-
ticesandassumingthattheﬂuidhaszeroviscosity.Thenewequationsbecomecomplex
sincethereareΩΦnewly’addedmass’terms(Lamb(1932),Mahadevan(1996)).Inour
case,wehaveathreedimensionalmodelwithboundariesasexplainedinchapterΩ.
teThegratorequationsandweofusemotiontheinitialforthisconditionsmodelareforsolvwhichedbysteady-fusingallingRunge-Kanduttaoscillatoryfourthreordergimesin-
areobserved,ﬁg.4.1.Theresultsareshowninﬁg.4.8.

ortexV4.6

Thevortexproductionisanimportantpartoftheﬂuiddynamicsthatmustbetakeninto
accountintheformulationofatheoryforfallingbodies.Willmarthet.al.(1964)made
apioneeringworkwheredye-laddendisksweredroppedintowaterandvorticeswere

OutlookandConclusions4.4.7

Figure4.9:Therightpictureshowsthevortexstructureoftheverticalandhorizontal
componentsoftheﬂuidvelocityﬁeldΣh,i)generatedbythefallingoblatewithadiameter
3.2inacontainerofΩΦΦ×´Φ×´ΦandReynoldsnumber,ES=ΨΩ1,aspect-ratio,
Δe=Φ.).TheleftpictureshowssheddingvorticesreportedbyBelmonteet.al.,ref.[12],
strip.allingfafor

clearlyobserved.Inoursimulationsthefallingellipsoidgeneratessheddingvorticesin
theﬂuidalongitstrajectoryasshowninﬁgure4.9(right).Itshowstheﬂuidvelocityﬁeld
aroundtheoblateandthevortexislocalizedjustintheregionabovetheoblateandwhere
itexperiencesalargeangularchange,ΔΘasshowninﬁg.4.5(top).TheReynolds
numbercalculatedfromtheoblate’sdiameterandterminalvelocity,ES=ΨΩ1.Wepoint
outthatthevortexisobtainedalsointheworkbyBelmonteetal(Belmonteet.al.(1998))
whereasheddingvortexiscreatedbythezigzagmotionofthefallingstripasshownin
4.9.ﬁgureoflefttheAsystematicstudyoftheboundarylayeroftheellipsoidisnotpresentedinthethesisand
itwillremainasafuturework.

Conclusions4.7Outlookand

Thetainerhasmotionbeenofastudied.singleWoblateefoundellipsoidthreebasicsettlingreingimesaﬂuidforintheathreedynamicsofdimensionalthesystemcon-
(steady-falling,oscillatory,andchaotic).Wefoundthatourresultsareingoodqualitative
agreementwiththesimpliﬁedmodelproposedby(Mahadevan(1996)).Thesameinitial
conditionshavebeenusedinourworkinthestudyofoscillatoryandstead-fallingre-
gimes.Moreworkhastoberealizedinfutureinordertounderstandtheroleoftheﬂuid
pressureandvelocityﬁeldsbetter.

5Chapter

Phases

Inthischapterweinvestigatethedynamicsofthephasesintroducedintheprevious
chapter.Weexploreeachphasebychangingdifferentparametersinthesystem.We
introducesomeimportantconceptsanddeﬁnitionswhichwewilluseinthenextchapter.
Intheﬁrstsection,wediscussthesteadyfallingphasewhentheinitialheightofthefall,
aspectratiooftheellipsoidandthekinematicviscosityarechanged.Next,wedepictthe
oscillatoryphase.Furthermore,westudytheeffectonthedynamicsduetochangesinthe
initialorientationoftheellipsoid.AcomparisonwiththeexperimentsbyBelmonteet.
al.(1998)hasbeenmadeandoursimulationsagreequitewellwiththeexperiments.Fi-
nally,wepresentthechaoticregime,whichishighlysensitivetothechangeintheinitial
orientationoftheellipsoid,asshownintheLyapunovexponent.

Steady-F5.1Phase.alling

5.1.1Changeintheinitialfallingheight.

Figure5.1(I),showsthetrajectoriesfordifferentinitialfallingheightsbyﬁxingallother
parameters.Itisevidentfromtheﬁgurethatthereisnochangeinthepeak-to-peakamp-
litude.Foralltheheights,thetrajectoriesbehavelikethatofadampedoscillator.

Inﬁg.5.1(II),weseethattheellipsoidapproachesthesameterminalvelocityindependent
ofzerotheasftheallingoblateheight,touchesatthethesamebottomtime.oftheAlso,thecontainerverticalwhichviselocityindependentsharplyofconvtheerfgesallingto
height.

inFiguretheinitial5.1(III)fshoallingwstheheight.vFerticaloralltheorientationaltrajectoriesbehaviourshownofintheﬁg.ellipsoid5.1(I),withwetheseeinchangeﬁg.
5.1which(III)isthatseenthebytheorientationsemi-majoroftheaxisellipsoidgettingoffersalignedwithminimumgravity.resistanceagainsttheﬂuid,

)hgΣgreplacementsPSfrag04allingSteady-F5.5.1Phase.

0.75(c)0.5Vertical velocity0.25(II)(c)(b)(a)

(I)(b)

(a)

75Vertical position5025

100

Inthissection,westudythedynamicsofthesteady-fallingellipsoidasthekinematicvis-
θΦcosity=isΩ+.+d.changed.ItglidesThedoellipsoidwnwardsstartsanditstotheswingingsidemotionacquiringwithasomegivenamplitude,initialwhileorientation,the
kinematicviscosityνacts,thereby,reducingthesubsequentamplitudesofoscillationas
5.2(I).ﬁg.inwnshoFig.orientation5.2(I,ofII,theIII)shoellipsoidwthefordiftrajectoryferent,verticalkinematicandviscosities,horizontalvrespectielocitiesvely.andCurvtheevNerticalwith

5.1.2Dependenceonthekinematicviscosity.

ΦTFigurerajectory5.1:hasInitialdifferentconditionsinitialofheighttheΣN)system.Vd=θd2+,=ΣO)Ω+V.d+,=Δ0+e,=ΣP)ΦV.dΩ)=,ν)+.=I)Φ.TheΦ´´.spatialEach
trajectoryintheverticalplane.II)Verticalvelocityvs.time.III)Verticalorientationvs.
time.

PSfragreplacements005101520253035
gΣgh)Time t(ts)

0replacementsPSfrag68Horizontal position1012gΣgh)051015Time t(t20)253035
s(a)100θ(c)(b)75Vertical Orientation 50(III)25

(.Φ−(.Φ0.Φ5.Φ7.Φ9.Φ)hgΣgreplacementsPSfrag (a)←Vertical Orientation Vertical Orientation vs. Time120110100908070605040302035302520151050(.Φ−(.Φ0.Φ5.Φ7.Φ9.Φ)hgΣgreplacementsPSfragVelocities Vx,Vy(a)(b)Velocity & Time(II)(c)Vy2520151050(.Φ−(.Φ0.Φ5.Φ7.Φ9.Φ)hgΣgreplacementsPSfrag(I)(c)(b)(a)Vertical PositionHorizontal PositionOblate Vertical Trajectory90807060504030201001614121086445

ν=Φ.ΦΩ),curveOwithν=Φ.Φ´´andcurvePwithν=Φ.Ψ.Weobservefromallthe
threeplotsthatasviscosityincreases,theattenuationbecomesstrongertherebybehaving
likeadampedharmonicoscillator.
Theattenuationintheamplitudeoftheverticalvelocitywithtimeandthetimeperiod
betweentwoconsecutiveturningpointsarenotverydifferentfromeachotherasthe
viscositychanges.Also,weseethatboththehorizontalandverticalvelocitiesconverge
tothesamevalue(eg.ik≈Φandil≈Φ.+)forallthevaluesofviscosity.
Thisstrongattenuationforlargeviscosityishighinthehorizontalcomponentofthe

ΦFiguretrajectory5.2:hasaInitialdifferentconditionskinematicintheviscositysystem.ΣθN)dν==ΦΩ+..+ΦΩ),,VΣdO)ν==1ΦΦ,.ΔΦ´´e,=ΣP)νΦ.Ω)=.Φ.EachΨΦΦ.
(I)(bottom)Projectionvelocitiesofthevs.time.trajectory(III)inVtheerticalverticalorientationplane.vs.(II)timeV.ertical(top)andhorizontal

Phases

← (b) (c)←(III)

1.8)oln(A/A1.6

Characteristic time & viscosity

1.4sν/ν0.71.40120145170195
T/ts1494)Time(ts

allingSteady-F5.5.1Phase.

(1)(2)(3)(4)(5)

2419

Figure5.3:Initialconditionsofthesystem.θd=Ω+.+Φ,Vd=ΩΩΦ,Δe=Φ.Ψ).Inthe
ﬁgureeachlineshowsthedecreasinglogarithmicamplitudeofoscillationagainsttime
fordifferentvaluesofkinematicviscosityν(=Φ.Φ(,ν)=Φ.Φ0,ν0=Φ.Φ1,ν1=Φ.Ω
andν5=Φ.).Intheinsetweplotthecharacteristictimeτofeachlinevsthekinematic
.viscosity

velocityasseenintheﬁgure.Thiscanbeexplainedbythefactthattheinteractionbetween
thewallsandtheoblateislesswhentheﬂuidislessviscousBrenner(1961).Thisisalso
trueforlargeangularvariationsΔΘinﬁg.5.2(II)cforsmallvaluesofviscosity.

Inﬁg.5.2(III)weseetheorientationalbehaviourastheviscositychanges.Theﬁrstpeak-
to-peakamplitudesarenearlythesameforallviscosities.Forν=Φ.ΦΩ)thesubsequent
peak-to-peakamplitudesapproachaconstantvalueofΘe−e∼Ψ)dwhileforhighervis-
cosities,theoscillationsgetdampedandtheellipsoidreachesaconstantorientation.We
alsoobservethattheﬁnalorientationoftheellipsoidgetsalignedwithgravity.

Figure5.3explainsthattheamplitudeofoscillationdecaysexponentiallywithtimewhich
isrevealedbytheloglinearplot.Thedimensionlesscharacteristictime,τdecayslinearly
withthedimensionlesskinematicviscosity,νasshownintheinsetofisﬁgure5.3.The
solidlineintheinsetshowsalinearﬁtwithνsequation,iτs=−Φ.Φ´νν0+).).Theﬁgure
clearlyshowsthatviscosityplaystheroleasadampingfactorthatdeterminesthedecay
rateoftheverticalvelocityandposition.

Phases

100

8060Vertical Position402003

Increment in the Aspect−Ratio1

Vertical Velocity vs. Time

(a)0.75(b)(c)Vertical Velocity Vy0.5(c)II)I)(b)0.25(a)ΔΔΔ r=0.18 r=0.22 r=0.28
579005101520
)Time t(tHorizontal PositionsVertical Orientation vs. Time140 (c)←

100θAngle 60

(a) ↑ (b)↑

III)2008Time t(t16)24
s

Figure5.4:Initialconditionsofthesystem.Θd=Ω+.+Φ,Vd=2+,Ak×Al×Az=
ΣOΨΦ)Δ×e=ΨΦΦΦ×,ΩΩ,ΨΦ,ΣνP)Δ=eΦ=.Φ´´Φ,.Ψ1.EachI)Trajectorytrajectoryforhasthedifvferenterticalplane.aspect-ratioII)VΣN)Δerticalev=Φ,elocityΩ2,
vs.time.III)Verticalorientationvs.time.

5.1.3Changeintheellipsoidaspect-ratio.

Whentheellipsoidbeginstofall,itgainsalargeamplitudeofoscillation.Thetrajectory
ΣP),plottedintheﬁgure5.4I,isforanellipsoidwithaspectratio,Δe=0.18.Weseean
oscillatorytrajectorycharacterizedbyaconstantpeak-to-peakamplitudeof3.Asthe
behaaspectvior.ratioIfistheaspectincreasedratiotois0.22,furtheri.e.,increased,trajectorytheΣO)intrajectorytheﬁgure,showsshoirrewsgularasteadyoscillations.falling
Atthebeginningofthefall,theellipsoidwithlargeaspectratio(trajectoryN),showsa
rapiddampingamplitudeintheﬁrsthalfofthetrajectorybutinthesecondhalf,doesn’t

P5.5.2Phase.eriodic

showanysteadyfallingbehavior,insteadchangesabruptlyintoanon-stationarystate.
Inaspectﬁgureratio,5.4(I),Δeforallincreasestrajectories,from0.18thetoﬁrst0.29.amplitude,ψΦdecreasesfrom3.0to1.8asthe
As(minortheaxisofaspect-ratio,theΔoblateeisisﬁxedincreasedinourfrom0.18simulations)to0.29,therebytheprooblate’svidingarealessbecomesresistancesmallto
theﬂuid.Therefore,theﬁnalverticalvelocitydecreasesastheaspect-ratioisincreased,
ﬁg.5.4II.Alsoastheaspect-ratioisincreasedthepeak-to-peakamplitudeofthevertical
decreases.elocityvTheratiodecreasespeak-to-peakortheamplitudeoblate’sforareathevincreases.erticalFororientationΔe=ΘΦ.e−Ω2etheincreasespeak-to-peakwhentheamplitudeaspect-
ddisforΘeΔ−ee==ΦΨ).Ψ1,.ﬁg.Inall5.4cases(IIIa),theandoblatemuchatthesmallerendcomparedorientsvtoΘerticallye−e=ﬁg.0Φ,5.4ﬁg.(III).5.4As(IIIc),the
whichellipsoidwillbeaspectratio,discussedΔeindetailincreases,inthechaptersystem).transitsfromregulartoirregulardynamics

Phase.eriodicP5.2

Wesmallerﬁndperiodicaspect-ratiobeha(Δevior=forΦ.Ψ´´smaller).Thekinematicdynamicsofviscositythef(ν)alling=Φ.ΦΩ)ellipsoid→EisSgo=v(1Φerned)andby
inertialeffects.Inﬁgure5.5(I),weshowthetransitionfromaquasi-periodic,oralong
steady-fallingtrajectory(ν(=Φ.ΨΦΦ),toaperiodicbehaviorﬁg.5.5(I,ν)=Φ.ΦΩ)),when
thekinematicviscosityisvariedfromν(=Φ.Ψtoν)=Φ.ΦΩ).Thetrajectorypresented
inﬁg.5.5I,withkinematicviscosityν)=Φ.ΦΩ)hasawavelengthof20.
Theverticalvelocityshowninﬁg.5.5(II),hasthesametransitionfromalongsteady-
fhasallingareoscillationgimewithperiodaﬁnalofa´v.´.eragevelocityof´.Φtotheperiodicregimewherethevelocity
Theverticalorientationshowninﬁg.5.5(III)hasalsothesametransitionfromalong
oscillatesteady-fallingaroundreΘgime=to2Φdperiodicwithangularbehaviorwithpeak-to-peakaperiodofamplitude6.6s,Θand=the+Φd.angularvalues
e−eΦtheThesystemdecreasingfrominthesteady-fkinematicallingtoviscosityoscillatory,foraphase.muchWesmallerwillinvestigaspect-ratioateΔthiseσΦ.transitionΨ,takines
.)chapter

5.2.1Changeintheinitialorientation

Inthissection,weinvestigatetheperiodicphasewithdifferentinitialorientations.The
correspondingtrajectoriesareshowninﬁg.5.6(I)forΘΦ=Ω+dandforΘΦ=2Φd
andthepeak-to-peakamplitudesare2.3and0.4,respectivelyandshowanoscillatory

=0.100ν=0.025νVertical OrientationVertical Orientation & Time1301109070)hgΣgreplacementsPSfrag−132241680)hgΣgreplacementsPSfragHorizontal Position25141185249

behavior.Thepeak-to-peakamplitudeoftheoscillationinthetrajectoryverticalvelocity
andorientationﬁg.5.6(I,II,III)decreasesastheinitialangleoforientationincreases.
Theﬁnalverticalvelocityandorientationforallthethreeinitialorientationsisonan
averageΦ.+and1)d,respectively.Thismeansthattheaveragelimitingvaluesofthe
verticalvelocityandorientationarenotaffectedbythevariationoftheinitialorientation
.ΘdWeobservealargepeak-to-peakamplitudeofoscillationintheverticalvelocityandver-
ticalorientation,equaltoΨ.Φand0Φd,respectively,withaninitialorientation,ΘΦ=Ω+d
anditreducesastheoblate’sinitialorientation,ΘΦtendsto2Φd.

Figure5.5:Trajectoriesgeneratedforν(=Φ.ΨΦΦ,ν)=Φ.ΦΩ).(I)Trajectoryinvertical
andhorizontalposition.(II)VerticalorientationΘvstime.(III)Verticalvelocityilvs
time.TheinitialconditionsareVd=2+,Δe=Φ.Ψ´´,θΦ=+´.(d.

Phases

III)PSfragreplacements500816243240
Σg)gh

Vertical Velocity & Time4=0.025ν3=0.100ν=0.025ν2Vertical Velocity10II)

Oblate Trajectory10085=0.100ν70Vertical Position5540I)

o26o63o90

Inthissection,wecompareourresultswithBelmonte’sresults.Figure5.7(a)(simulation)
showsthetimedependenceoftheverticalorientationwithν)=Φ.ΦΩ)andthevalueof
thepeak-to-peakamplitudeisΘd=+Φd.
Theverticalvelocityasshowninﬁg.5.7(b)(simulation)reachesitsmaximumvalue´.+
asΘapproachesΘbkthereby,showingaminimumdrag.Theminimumverticalvelocity
il=Ω.)isachievedatΘbTc∼0Φd.
Thebutterﬂyshapedcurve(ﬁg.5.7(b))wasalsomeasuredintheexperimentalworkby
Belmonteet.al.(1998)inwhichtheverticalorientationΘoscillateswithtwicetheperiod

5.2.2ComparisonwithBelmonte’sresults.

Figure5.6:I)Trajectoriesforthreeinitialorientations.(a)ΘΦ=Ω+d(b)ΘΦ=+´d(c)
ΘΦ=2Φd.II)Thecorrespondingverticalvelocities.III)Theverticalorientations.Initial
conditionsVd=ΨΦΦ,Ak×Az=ΨΦ×ΨΦ,Δe=Φ.Ψ´´,ES=(´).

Phase.eriodicP5.5.2

Vertical Velocity & Time

0.890c)b)a)0.660Vertical VelocityVertical Position0.4II)300.2I)00replacementsPSfrag2Horizontal Position11gΣgh)05Time t(t10)
s Vertical Orientation & Time140a)120100c)Vertical Orientation8060b)40III)replacementsPSfrag20051015gΣ20gh)25303540

Oblate Trajectory1

)hgΣgreplacementsPSfrag

51Phases80130120601104010020900(Degrees)Θ80 [degrees]Θ−20
70)θ−40a)60−60a)50024681012141618−8000.20.40.60.8
T(s)t[sec]7043.8603.6503.43.240Vy[cm/s]Vy(cm/s)3302.82.6202.4b)b)102.2250607080Θ90(Degrees)1001101201300−80−60−40−20Θ0 [degrees]20406080
601.550140300.520100Vx[cm/s]Vx(cm/s)0−0.5−10−20−1−30−40−1.5c)c)−50−25060708090100110120130−60−80−60−40−20020406080
[degrees]Θ(Degrees)ΘFigure5.7:ComparisonwiththeresultsofBelmonteetal.ref[12]forafallingstripin
theperiodicregime,forthe(a)VerticalorientationΘvstime.(b)Verticalvelocityilvs
Θ.(c)HorizontalvelocityikvsΘ.TheinitialconditionsareVd=2+,Δe=Φ.Ψ´´,
ν(=Φ.ΦΩ),θΦ=+´.(d.

Oblate Trajectory

egime.rChaotic5.5.3

105907560Position in Y→45c) →30a) b)←15044.555.566.577.5
Position in XFigure5.8:InitialconditionsVd=2+,Δe=Φ.Ω),ν=Φ.Φ´´andtinyvariationsofthe
initialorientationΣN)θΦ=Ω+.+d,ΣO)θd=Ω+.+ΦΦΨd,ΣP)θd=Ω+.+ΦΦΦΦΦΨd.

.ioflThehorizontalvelocityoscillatesaroundzerowithconstantperiodofoscillationandits
maximumvalueik,bk=Ψ.)andtheminimumik,bTc=−Ψ.)atΘ∼2Φdasseenin
ﬁg.5.7(c).Whenthehorizontalvelocityiszerotheoblatetakesitsmaximum(ΨΩΦd)and
minimum(+Φd)valuesinΘwithanon-zeroverticalvelocityil=´.Ω.

egime.rChaotic5.3

5.3.1Sensitivitytothechangeintheinitialorientation

Wenowdiscusssthesensitivityofthedynamicstotinychangesintheinitialorientation.
Wehavesimulatedthreetrajectoriesshowninﬁg.(.1,whichhaveslightlydifferent
initialorientation.Atinyvariationintherelativeorientation(Δθd=ΨΦ−0)producesa
signiﬁcantvariationintheshapeofallthecurves.Thesechangesareappreciatedinthe
lowerpartofthetrajectories.
Inordertogetabetterpictureofsensitivity,wehaveincrementedthefallheightVdto166.
Theresultingfourtrajectoriesforfourslightlydifferentinitialorientationsinthevertical
planeareplottedinﬁg.(.2andweobservehighsensitivityto−0theinitialorientation.For
thefourtrajectoriestherelativeangularvariationisΔθd=ΨΦ.

Phases

150

Trajectory Sensitivity to the Initial Orientation

125100b)a)Position in Y7550

d)c)

02468101214161820
Position in XFigure5.9:InitialconditionsVd=Ψ++,Δe=Φ.Ω),ν=Φ.Φ´´andtinyvariationsofthe
initialorientationΣN)θΦ=().´1(d,ΣO)θd=().Φ´´d,ΣP)θd=((.21Ψd,ΣR)θd=((.20+d.

5.3.2Powerspectra,autocorrelation,Poincaresection.

Duetothesensitivitytosmallchangesintheinitialorientation,weproceedtoanalysethe
systembytheFourierpowerspectrumtimeseriesofthehorizontalcoordinatekΣg),kΣg+
δg),kΣg+Ω∗δg),...andinourcaseδg=Φ.Φ)´)++.Abroadspectrumoffrecuencies
appearsasshowninﬁg.5.10IIthereby,indicatingachaoticmotion.
Theautocorrelationfunction,forthesametimeseries(ﬁg.5.10I),doesnotfallquicklyto
zeroratherdecreaseslinearlywithtimeandthepointsarenotindependentofeachother.
Intheﬁgure5.10III,wepresentslicesorPoincare´sectionsΣpk,k),correspondingtothe
trajectoriesinﬁg.(.2awhicharequiteirregular.Theorbitsarequasi-periodicinthe
notsenseclosedthattheandyarepassnotrepeatedlyassociatedandwithirreagularlyparticularthroughtimetheperiod.wholeThedomain.sensitiThevitytoorbitsinitialare
conditionsisclearinthesefourﬁgures.Asmallchangeintheinitialorientationresultsin
largechangesinpositionandvelocity.

5.3.3Lyapunovnumber.

Wenowinvestigatequantitatively,thissensitivitybystudyingtheincrementintheEuc-
(.2lidean(a)anddistance,(c).Re1Figuree2=(.ΨΨ,Σk(sho−wsk)))that+theΣl(−ldistance)))betweenbetweenthenearbycurvespointspresentedhasaninovﬁg.er-
allexponentialtimedependenceRΣg)∼expΣλg)andtheﬁtgivesanestimateforthe

OutlookandConclusions5.5.4

Power Spectra Z10^2

10^1P−S

10^0

II)

Power Spectra ZAutocorrelation Function10^210.90.80.7II)10^10.6P−SAutocorr0.50.4I)0.310^00.20.10238028803380i38804380488010^110^2i10^310^4
Phase Space10.8III) θ0=45,033o
0.60.40.2Velocity component Vz0−0.2−0.4−0.61.522.533.544.555.56
Position in Z

Figure5.10:Detectionofchaos.I)AutocorrelationfunctionforthetimeseriesofkΣg)
forthetrajectoryofﬁg.14a.II)PowerSpectraofﬁg.14aIII)PoincaresectionΣpk,k)for
14a.ﬁg.oftrajectorythe

Lyapunovexponentλ=Φ.Φ)Ω±Φ.ΦΦ).ThepositiveLyapunovexponentgivesaclear
Chaos.ofindication

Conclusions5.4Outlookand

Wehaveoscillatory(periodic),observedthreeandphaseschaoticofrethegimes.Thedynamicssteady-foftheallingsystemandthenamely,periodicsteady-fregimealling,ex-
hibitBelmonteasimilaret.al.ph(1998).ysicalWbehaehaviorveasobservcharacterizededfortheﬂatteneddynamicsbodiesofthebyFieldsteady-fet.allingal.re(1997),gime
whenthekinematicviscosity,droppingheight,andoblate’saspect-ratioarechanged.
Someconclusionscanbedrawnfromthispartofthework.
(a)ThespatialtrajectoriesΣk,l)arecomposedofoscillationsthatcorrespondtoadamped

Phases

0.5

0ln(dist)−0.5−1

−1.5

Increment of the Distance in Time

PSfragreplacements−201020304050607080
)gΣghFigure5.11:Increasinglogarithmicbehaviorfortheseparationdistancebetweenthe
trajectoriesΣN)andΣP)inﬁg.14,thatslightlydifferintheinitialorientationangleby
ΔΘ=Φ.(Φ´d.

harmonicmotion.ThisregimeispresentforsmallvaluesofI≈Φ.)−Ψ,ES≈ΨΦΦ
andisshowninﬁg.5.1-5.4Thereisnovariationinthetrajectorieswhenweincreasethe
initialheight.Theviscosity,foraconstantsmallaspect-ratio,determinesthedecayofthe
positionandthevelocityoftheellipsoidasshowninﬁg.5.2.Whentheaspect-ratiois
changed,thetrajectoriesvarysigniﬁcantly(ﬁg.5.4).
(b)Theﬁnalverticalvelocity,ildoesnotdependontheinitialfallingheightandthe
.viscositykinematic(c)Theverticalorientation,Θoftheellipsoid,undergoesarotationalmotionuntillits
majoraxisgetsalignedwithgravity.Thistendencyofexhibitingminimumresistance
againsttheﬂuidexistsforallReynoldsnumbersintherangeES≈Ψ−)ΦΦ.
TheperiodicbehaviorinoursimulationsisfoundforES∼(ΦΦandsmallaspectratios
ΣΔe≤Φ.Ψ).Theverticalorientation,Θoscillateswithtwicetheperiodofoscillation
oftheverticalvelocityilandatthesameperiodofthehorizontalvelocityik.This
periodicmotionhasalsobeenobservedby(Belmonteet.al.1998)andthisshowsthat
oursimulationsareessentiallycorrect.Inthisregimetheinitialorientationdetermines
thevalueoftheamplitudeofoscillationinthespatialtrajectoryΣk,l),velocityiland
orientationΘ,andplaysthesameroleasthephaseangleintheoscillatorymotion.For
Θd=2Φd,theamplitudesoftheabovequantitiesapproachasmallvalue.
Thechaoticbehaviorispresentforlargeraspect-ratios(Δe≥Φ.´)andintheentirerange
ofReynoldsnumbersusedinthesimulation.Theseparationbetweenthespatialtraject-
oriesofthefallingoblateellipsoiddivergesforsmallvariationsintheinitialorientation

andwsgroΦ)Ω.Φ±

xponentiallye.ΦΦ).Φ

intime.Thealuev5.5.4

foundforConclusions

theyapunoLand

veOutlook

xponentis

6Chapter

lawSimilaritytransitionsphaseand

Inphasethetwandothevprecedingariationinchapterseachwephasehavewithseenrespectthetothecharacteristicphysicalparameters.phenomenologyInoftheeachﬁrst
sectionofthischapter,wediscussthesimilaritylawthatdescribesintrinsicallythedy-
namicsofthesteady-fallingandoscillatoryregime.Aphasediagramhasbeencompared
totheresultsofFieldet.al.(1997).Furthermore,wepresentanovelmechanismthatex-
plainsthetransitionfromtheoscillatorytothesteady-fallingphaseandthentothechaotic
phase.

6.1Steady-FallingOblateEllipsoid:Similaritylaw.

Itiswellknownforlargevelocitiesthattheinertialdragisgivenby,

>C=,ρMK)F

(6.1)

where,istheformfactoroftheinertialdrag,Fisthecross-sectionalareaoftheoblate
ellipsoid,ρMtheﬂuiddensityandKtheellipsoidvelocity.Theweightoftheoblate
toproportionalisellipsoid

>T≈ρEaEa)MEbU(6.2)
withρEaabeingtheellipsoidaldensity,EMandEbbeingtheminorandmajorradius,
respectively.Theterminaldownwardvelocityisdeterminedbytheequilibriumbetween
thesetwoforcesandsincebothdensitiesareﬁxed,theterminaldownwardvelocityis
,byenvgi

>T=>C=⇒K∼UEb
57

(6.3)

654Vertical velocity vy32

6.6.1Steady-FallingOblateEllipsoid:Similaritylaw.

System L"System L

018141022)Time t(ts

654Vertical velocity vy32

System L"System L

05791113
)Time t(ts

ΦFigurekinematic6.1:viscosityInitialν=conditionsΦ.Ψ.Ininthetheﬁguresystem.(left)θdwe=plotΩ+.the+,vVdertical=v1+,elocityΔe=agΦ.ainstΨ)timeand
inapplyingboththesystemsinvAerse(solidline)transformationandAof(dottedtableΨ.line),andthesuperposition(right)performed

Wesupposethatthecharacteristicperiodofoscillationdependsonthelateraldimension
whichinoursimulationsisASdgΣ=Ω))andindependentoftheverticaldimension(falling
height).Therefore,weassumethattheperiodofoscillationshouldchangeas,

G∗∼ASdg(6.4)
KThedynamicsofthesystem,ingeneral,dependsontheReynoldsandtheFroudenumbers
(Sec.Ω.Ψ.Ω,Belmonteet.al.(1998)).WecanrescaletheparametersofasystemofsizeA
toasystemofsizeAthrough

SystemA→SystemA
ASdg→c∗ASdg
EM→c∗EM
Eb→c∗Eb
U→1∗Rc

Table1.Transformationsrules.

transitionsphaseandlawSimilarity

ThetransformationspresentedintableΨkeeptheFroudeandReynoldsnumberscon-
stantandthedynamicsinthetwosystemsbecomeequivalentifthevelocitycomponents
changeasintableΩ.Asaconsequenceofthistransformation,thedynamicsinsystemA
undergoesachangeintheellipsoidperiod,theverticalandhorizontalcomponentsofthe
velocity,asshownintableΩ.

ASystemASystem→G∗→c∗G∗
VhorKSdgc→KvEgi→Vvert
c

Table2.PeriodandvelocityapplyingthetransformationrulesinTable1.

Inﬁg.6.1wepresentthesuperpositionoftheverticalvelocitybyapplyinganinverse
transforminAwithn=2.Thethreecurvescoincidequitewellinagreementwiththe
scaling.

6.2Periodicbehavior:Similaritylaw.

Figure6.2showstheverticalvelocityagainsttimeforthethreesystemsA(solidline)
(cand=AΩ),(n=2)andasandinAthe(n=4),caseoftherespectivsteadyely.fWallingeapplyregime,thethetransformationdynamicsforrulestheofsystemstableAΨ,
andA,intheoscillatoryregime,arerelatedbyasimilaritylaw.

Diagram.Phase6.3

∗Inthephasemomentspace,ofweinertiadeﬁneoftheaoblatedimensionlessellipsoidtomomentthatofofitsinertiasphericalIwhichequivisalenttheatratiosameof
Reynoldsnumber.WedoasimilaranalysisforourresultsasdonebyFieldet.al.(1997).
Itisimportanttoremarkthattheexperimentwasforafallingdisk,withsmallaspect-ratio
andweexpectthedynamicsofthesystemtobeclosetothatofanoblateellipsoid.
Thedeﬁnitionsofthedimensionlessvariablesforoursystemare:
I=Id∂aiE=)ebρEaaTe=)ρEaaTeΔe(6.5)
IheSEgE(eMρMajTC(ρMajTC
ES=IΣΩeM)(6.6)
ν∗ReFigureynolds6.3leftnumbers,sho(highwsthekinematicresultsintheviscosity),log-logthescale.motionAtisolovwervalues-dampedofandItheandoblatesmall

543Vertical velocity vy21

002.51.50.5Horizontal velocity vy−0.5−1.50

, r g, rMm g/8, 2*rm, 2*rM 5
4Vertical velocity vy321

Diagram.Phase6.6.3

System L|System L||System L

7.51522.5300012345678
Time t(ts)Time t (ts)
System L g, rm, rM System L|
g/8, 2*rm, 2*rM System L||
1.8Horizontal velocity vy−0.2

(d)−2.27.51522.5300246810
Time t(ts)Time t(ts)

ΦFigurekinematic6.2:viscosityInitialν=conditionsΦ.ΦΩ2.inIntheﬁguresystem.(left)θwed=plotΩ+.the+,vVdertical=v1+,elocityΔeag=Φ.ainstΨ)timeand
forapplyingsystemstheAinv(soliderseline)andtransformationAof(dottedtableline).Ψ.Therightﬁgureshowsthesuperposition

ellipsoiddropstothebottomofthecontainerwithoutanyoscillationandthisisthebe-
ginningofthesteadyfallingregime.Thisisseeninthelefthandbottomcornerofthe
diagram.WhentheReynoldsnumberisincreasedΣES≥ΨΦ))andI∗isﬁxed,thetraject-
oryiscomposedofsuccessiveoscillationsthatdecreaseinamplitudeandeventuallythe
oblatereachesthebottomofthecontainer.Thisisthesteadyfallingregime.
ForsmallvaluesofI∼ΔeσΨ,wehaveaﬂatellipsoid.ForReynoldsnumber,
ES≥(ΦΦ,thetrajectory,velocityandtheorientationarecharacterizedbyoscillations
thatareperiodicintimeandspace.Thisphaseiscalledastheoscillatoryregime.As
weincreaseI,theobjectbecomesaspherethatisslightlyﬂattenedatthepolesandits
dynamicsbecomessensitivetosmallvariationsintheinitialorientationtherebyexhibiting
.trajectorychaotica

transitionsphaseandlawSimilarity

This work10-1Ref. [14]Ref. [13]Tumbling

110

Chaotic

Steady−Falling

Periodic

efghChaoticChaotic
PSfragreplacements10-2bPSfragcdreplacements
∗Ia∗II*100
SteadyPeriodicEFallingI∗10-3I∗Steady−Falling Periodic
Ψ2ΦΨ2Φ
ΨΨΦ3ΦΨΨ3ΦΦ
Ψ102103104Ψ101102103
ΨΦEΨΦRe
et.Figureal.6.3:(1997).TheInlefttherightpictureplotshowewsthepresentphasetherediagramgimesofofftheallingphasedisksspacereportedfortheinfallingField
oblatedimensionlessellipsoidmomentobtainedofininertiaourI∗,andsimulations.theInhorizontalbothaxispictures,ESisthethevReerticalynoldsaxisnumberisthe.

IfwecompareourdiagramwiththeexperimentalresultsobtainedbyFieldet.al.(1997),
ﬁg.6.3leftweseethatinbothpictures,thedistributionofphasesisequivalent.The
differencesinourdiagramwiththatofField’sarethattheyusedisksandnotoblate
ellipsoidsandthetumblingregimeisnotpresentinourresults.Thecoexistenceofthe
dynamicalphasesasexplainedaboveisindependentoftheinitialorientationoftheoblate
ellipsoid.

6.4TransitionfromSteady-fallingtoOscillatoryphase

Inﬁg.6.4weshowthebehaviorofthecharacteristictimeG∗/G(Sec.(.Ψ.Ω),adimen-
sionalizedusingeq.).(,asweincreasetheReynoldsnumberES,thecharacteristictime
goestozeroatESB≈´)),inﬁg.6.3.Beyondthispointweﬁndtheoscillatoryregime,
thatbehaveslikeasteady-fallingregimewithaninﬁnitecharacteristictime.
Therefore,wecanconsiderG∗/Gastheorderparameterandthecontrolparameteristhe
Reynoldsnumberforthistransition.Thistransitionislikeasecondorderphasetransition.
Theinsetexhibitsthepower-lawbehaviorwithacriticalexponent≈Φ.).Inthecaseof
theupperpartofthetransition,inﬁg.6.3,thevariationintheoblateellipsoidaspect-ratio
impliesthechangefromsteady-fallingtooscillatoryregime,whichisalsosupportedin
ﬁg.5.4wherethetrajectoriesvaryfromsteady-fallingtooscillatoryastheaspect-ratiois
decreased.

0.039

6.6.5TransitionfromSteady-fallingtochaoticphase.

/Ty=0.53*x−6.1*T0.034/T)0.024*ln(T0.009268.5818.58)ln(Re−Rec0.009310210110ReRec

∗TheFigureinset6.4:shoThewstheorderpower-laparameterwbehaGvior/Gvs.withtheanecontrolxponentcloseparametertoΦE.)S,withESB≈´)).
60

*30T/T

00.180.2Δ r 0.22Δ r0.24
c

Figure6.5:TheorderparameterG/G∗vs.thecontrolparameterΔe,withΔeB≈Φ.ΩΩ.
6.5TransitionfromSteady-fallingtochaoticphase.
Thetransitionfromsteady-fallingtochaoticregimeispresentedinﬁg6.5.Weusethe
orderparameterG/G∗,i.e.,theinverseoftheoneusedbeforeinordertodescribethe
transition,andthecontrolparameteristheaspect-ratioΔe.AtΔeBaﬁnitejumpinthe
orderparameterisobserved.ThecharacteristictimeG/G∗disappearsduetothenon-
regularoscillationsthatareverysensitivetosmallvariationsintheinitialorientation.
Thistransitionseemstobethereforeofﬁrstorder.

transitionsphaseandlawSimilarity

Conclusions6.6Outlookand

Forthesteady-fallingandoscillatoryregimeweobtainasimilaritylawexpressedintables
ΨandΩ,whichisadirectconsequenceoftheinvarianceoftheReynoldsandFroude
numbers.

Theconstructionofthephasediagramshowsthreewell-deﬁneddynamicregionsas
shobehawnviorbyisFieldassociatedet.al.with(1997).theThetransitiondiftoferencechaosinthethroughaboveintermittencreferenceyiswhichthatisthenotchaoticseen
inoursimulations.Thephasediagramisindependentoftheinitialorientation.

Ouritionfromsimulationssteady-fshowallingthattothechaotictransitionregimefromcanbesteady-funderstoodallingtoasoscillatorysecondandandﬁrstthetrans-order
phasetransition,respectivelyandthecharacteristictransienttimebeingtheorderpara-
.meter

6.6.6

Conclusions

and

Outlook

7Chapter

articlesPMany

Intheprecedingchapterswepresentedthephysicsbehindthefallingofoneoblateel-
lipsoid.Now,inthischapterwearegoingtoexposethedynamicsofmanysedimenting
ellipsoidsinaﬂuidandundertheactionofgravityusingthemodelgiveninchapterΩ,at
smallandmoderateReynoldsnumbers.Westudythesettlingorsedimentingvelocityas
functionofthevolumefraction.Also,weexaminetheverticalandhorizontalﬂuctuations
oftheellipsoidscomparedtospheres.Additionally,weconsiderthebehaviorofpaircor-
relationfunctionsasafunctionofthevolumefraction.Nextwepresentanorientational
studyusingtheEulerangles,andtheangulardistributionsandtheorientationalbehavior
withvolumefraction.Thesamestudyforthesedimentingvelocityandorientationare
madeastheReynoldsnumberincreases(ES≈0).Furthermore,wepresentthebehavior
oftheorientationalorderparameterwiththevolumefraction.Finally,aconclusionis
en.vgi

Results7.1

7.1.1Sedimentationvelocityforoblateellipsoids

vInolumeﬁg.7.1fractionweshoΦVw,thewithinmeanavrangeerticalof0.01sedimentationto0.21,vforelocityoblateiΣg)asellipsoidsafunctionandofspheresthe
andthencompareittothephenomenologicalRichardson-ZakilawvΦ)=ΣΨ−Φ)c
RichardsonandZaki(1954)withc=).).ThelimitofΦV→Φcorrespondsv0totheVsingle
fallingellipsoidwhichwestudiedinthepreviouschapter.Itisinterestingtopointout
thatsphere,theeq.Ω.sedimentationΩ1,whichvfolloelocitywsofthetheellipsoid,phenomenologicalissmallcomparedRichardson-Zakitothatoflawthe.Thisequivisalentnot
thecaseforﬁbers(elongatedellipsoids),whereitisfoundthatthesedimentationvelocity
hasamaximumforsmallervolumefractionwhichcanexceedtheterminalvelocityofa
singleﬁberKuuselaet.al.(2003).
Foroblateellipsoidsthemeanverticalsedimentationvelocitypassesthroughalocalmax-
imumatΦV≈Φ.Φ).Thismaximumisquiteinterestingsinceitisnotobservedfor
65

0.9

(vs)||0.6<v>

0.3

Richardson−ZakiSim. OblatesSim. Spheres

Results7.7.1

00.050.1Φ0.150.2
VandFiguresphere7.1:Mean(dash-dotline),sedimentationasvfunctionelocityofiΣgthe)vforolumetheoblatefractionΦellipsoid.Theoblate(dash-squaredellipsoidline)
Vaspect-ratioisΔe=Φ.Ω),the−)radiusoftheequivalentsphereEEfjT=Φ.20andthe
ReynoldsnumberES=(∗ΨΦ.

spheres.Similarnon-monotonicsedimentationofnon-sphericalbodies(e.g.ﬁbers)has
beenreportedexperimentallybyHerzhaftandGuazzelli(1999)andforprolateellipsoids
insimulationsbyKuuselaet.al.(2001)duetoanorientationparalleltogravity.
Wechoosethedensityoftheﬂuid,theStokesvelocityandthelargerradiusoftheellipsoid
equaltounityinoursystem.Inallcasesthecontainerhasaheight,A=1)andabase
ofΨΦ×ΨΦ,andthelatticeconstant,V=Φ.´.WechangeΦVaddingellipsoidsandthe
maximumnumberofellipsoidsinoursimulationsisoftheorderofseveralthousands.
Theratiobetweenthedensityoftheoblateellipsoidsandtheﬂuidis(.
Inﬁg.7.2wepresenttheparallel()andperpendicular(⊥)componentsofthevelocity
ﬂuctuationswithrespecttogravityasafunctionofthevolumefractionΦVwhichare
as:deﬁnedδi)=σi)>−σi>)(7.1)
δi)⊥=σi)⊥>(7.2)
Theangularbracketsindicateaveragingovertheellipsoidsthathavenotreachedtheﬁnal
bottompositionatthecontainer.Theaveragesweremadeoveratleast50realizations
startingwithdifferentrandompositionsandorientations.Inﬁg.7.2thevertical(parallel
togravity)ﬂuctuationsforspheresandforellipsoidsaremuchlargerthantherespective
horizontalcomponents.Theﬂuctuationsforellipsoidsdecreasewiththevolumefraction.
Foranequivalentsystemofspheres,theﬂuctuationsshowamaximumatintermediate
volumefractions(ΦV≈Φ.Φ0)(Kalthoffet.al.(1996)andNicolaiet.al.(1995)).In
allcasestheﬂuctuationsforthespheresareconsiderablylargerthantheﬂuctuationsfor
ellipsoids.oblate

articlesPMany

0.080.06>2⊥Vδ>,<2||0.04Vδ<0.02

. δδ⊥ Ellip Ellip
|| Sphδ. ⊥ Sphδ ||

0.010.060.11ΦV0.160.21
theFigurevertical7.2:V(solidelocityline),andﬂuctuationshorizontalforellipsoids(dash-dotline)(circle-line)andcomponentsspherescorresponding(squared-line)toforﬁg.
Ψ.Theoblateellipsoidhasanaspect-ratioofΔe=Φ.Ω)−,)theequivalentradiusofasphere
isEEfjT=Φ.20,andtheReynoldsnumberES=(∗ΨΦ.

>122⊥Vδ>/<2||8Vδ<4

Ratio SphRatio Ellip

0.010.06Φ0.11V0.160.21
Figure7.3:Ratiooftheverticaltothehorizontalvelocityﬂuctuationsforspheresand
oblateellipsoidsasfunctionofthevolumefraction,ΦV.Theoblateellipsoidaspect-ratio
isΔe=Φ.Ω),andtheReynoldsnumberES=(∗ΨΦ−).

Inﬁg.7.3wepresenttheratio,δi||)/δi)⊥forspheresandoblateellipsoids.Forspheres
theratioshowsamaximumaroundΦV≈Φ.Φ0(Kalthoffet.al.(1996)andNicolaiet.

Results7.7.1

18y=88x+0.7611>2⊥Vδ4>/<2||20Vδ<−32>V⊥15
y=33x+3δ10−102>/<Vδ||50
<0.010.060.11ΦV0.160.21
0.010.060.11Φ0.160.21
VFigure7.4:Ratiooftheverticalvelocityﬂuctuationsforspherestothatoftheoblate
ellipsoidsasfunctionofthevolumefraction,ΦV.Theinsetshowsthecorresponding
ratioforthehorizontalﬂuctuations.−)Theoblateellipsoidaspect-ratioisΔe=Φ.Ω),and
theReynoldsnumberES=(∗ΨΦ.

al.(1995)).Forellipsoidstheratioshowsaslightlylargervaluethanthatofthespheres
forverysmallvolumefractionsandhasanoverallmonotonicdecreasewiththevolume
fraction.Wedisplaytheratiooftheverticalvelocityﬂuctuationsforspherestothatoftheellipsoids
inﬁg.7.4.Thequotientexhibitsalinearbehaviorwiththevolumefractionfollowing
approximatelytherelationΣδiv)Egi,heS/δiv)Egi,Eaa)Te=11∗ΦV+Φ.0+.Theinsetshowsthe
horizontalcase,alsoalinearbehaviorΣδi)Sdg,heS/δi)Sdg,Eaa)Te=´´∗ΦV+´isfound.
Inﬁgure7.5wepresenttheparallel,τandperpendicular,τ⊥componentsoftheautocor-
relationtimeforbothellipsoidsandspheres.Weusethedeﬁnitionofthecorrelationtime
as:∞ΨτB=,ΣΦ)Φ,Σg)Rg(7.3)
Where,Σg)istheparticlevelocityautocorrelationfunctionwhichisdeﬁnedas,Σg)=σ
δiΣg)δiΣΦ)>.HereδiΣg)=iΣg)−σi>isthelocalvelocityﬂuctuation,whereσi>
wastakenasthemean(horizontalorvertical)velocity.
Theverticalcomponentshowsanoveralllargevaluefortheoblateellipsoidscompared
tothatofthespheres.Thehorizontalcomponentsoftheautocorrelationtimebetween
oblateellipsoidsandspheresareindistinguishable.
Inﬁgure7.5thefourcurvesdecayaspowerlawsgivenbyτ≈Φ−α,Weseethatthe
valuesofαfortheparallelandperpendicularcomponentsforellipsoidsareα≈Φ.ΨΨ
andα⊥≈Φ.Ψ+respectivelyandthatforthespheresareα≈−Φ.ΨΦandα⊥≈Φ.ΩΩ

articlesPMany

))s(tτd d →→
log(102c c →→
→b →b →a →a

Ellip ||⊥Ellip Esf ||⊥Esf

10−2log(Φ)10−1
VΦVFigurefor7.5:oblateellipsoidsAutocorrelationandthetimesτequiv(inalentunitsofspheresgh)assplitfunctionintoofcomponentsthevolumeparallelfraction(||)
andequivalentperpendicularsphere(⊥system)tohasgraavity.radiusTheEEfjToblate=Φ.20ellipsoidandtheEaspect-ratioS=(∗isΨΦΔ−e).=TheΦ.Ω),resultsthe
areplottedinalog-logscale.

respectively.Theseexponentsarevalidinthedilutelimit.
WecalculatethepaircorrelationfunctionfordifferentvolumefractionsΦVandtheresults
areshowninﬁg.7.6.Thepaircorrelationfunction,forsmallervolumefractionsΦ=
Φ.ΦΦΨ,clearlyshowslargeinhomogeneitiesinthesensethatthereisa“packingformation”V
asseeninﬁg.Ψ.+,ofoblateellipsoids.Theseinhomogeneitiesdisappearforlargevolume
thefractionpairΦVcorrelation≥Φ,Ω.functionFurthermore,thattheintheﬁrstpeakintermediateisclosecasetotheforΦVorigin,=Φ.locatedΦ)weatcane=seeΦ.+in,
whichisalsopresentatΦV=Φ.Ωbutsmaller.
ThisadditionallargerpeakatΦV=Φ.Φ)couldberelatedtothelocalmaximuminthe
sedimentationvelocityﬁg.7.1.Bylookingatthesnapshotsastheoneshowninﬁg.Ψ.+
oneseesthatentirebundlesofalignedparticlesseemtodetachandmovedownfaster
beenwhichobservmightedwellinthebethesedimentationoriginofofthisﬁbers,peak.ThisHerzhaftkindandof“bGuazzelliundlebeha(1999),vior”wherehasthesealso
bundlessettlefasterthantheindividualﬁbers.

viorbehaOrientational7.1.2ForthemeasurementoftheorientationweusetheEuleranglesdescribedinﬁg.4.4.The
meanverticalorientation(MVO),θ,asafunctionofthevolumefraction,isshownin
ﬁg.7.7.ForsmallervolumefractiontheMVOshowsmorealignmentwithgravityandin

g(r)

0015

002015

10g(r)

Pair correlation function=0.2ΦV

12r(R)345
MPair correlation function=0.05ΦV

12r(R)345
MPair correlation functionΦ=0.001V

Results7.7.1

0012r(R)345
MFigure7.6:Pairdistributionfunctionsforoblateellipsoidsfordifferentvolumefractions,
ΦV.TheReynoldsnumberES=(∗ΨΦ−).

articlesPMany

00.05ΦV0.10.15
Figure7.7:Meanverticalorientationθforoblateellipsoidsasafunctionofthevolume
fractionΦV.−)Theoblateellipsoidaspect-ratioisΔe=Φ.Ω),andtheReynoldsnumber
ES=(∗ΨΦ.

theorientationallimitΦV→behaΦaviorcloserforonealignmentoblatewithellipsoidgravityobservisedinobservedsection(which.Ψ.Ω;(.Ψ.corresponds´.tothe

WtheealsolocalvseeerticalforthevMVelocityOanmaximumintermediateatthesamemaximum,volumeatΦVfraction≈Φ.Φ)sho,wnwhichinﬁg.could7.1.explainThis
ΦV>intermediateΦ.Φ1theplotmaximumshoiswsanotpresentmonotonicfordecrease.spheres.Forlargervaluesofthevolumefraction,

Figure7.8showstheorientationaldistributionfunctionPΣPdfΣθ))fortheverticalangle,
θ,fordifferentvolumefractions,ΦV.Forsmallervolumefractions,ΦV=Φ.ΦΦ1the
orientationaldistributionshowsamaximumaroundPdfΣθ)≈Φ.Ψinagreementwithﬁg.
7.7.Thelimitingcase(ΦV→Φ),i.e.,onesedimentingoblateellipsoid,studiedbyusin
section(.Ψ.Ω;(.Ψ.´,presentsaverticalalignmentwithgravity(θ≈2Φd),andinﬁg7.7we
canseeavalueofθ≈1)d.
Asthevolumefractionincreases,thedistributionsﬂatten,andatΦV=Φ.Φ)thedistri-
butionshowsamoderatemaximumaroundPdfΣθ)≈Φ.(),correspondingtothesimilar
intermediatemaximuminﬁgures7.7and7.1.Weconcludethattheverticalvelocityis
inﬂuencedsigniﬁcativelybytheorientationalbehavioralonggravity,asitiswellknown
forotherspheroidsystemsKuuselaet.al.(2003).

Figure7.9showstheorientationaldistributionfunctionPΣPdfΣφ))fortheangleφ,for
differentvolumefractions,ΦV.Theorientationaroundtheverticalslightlyincreasesfor
smallervolumefractions,anddecreaseswithlargervolumefractions.Similarbehavioris
alsofoundforthethirdEulerangleψ.WeconcludethattheEuleranglesφandψarenot
muchinﬂuencedbythevolumefraction.

0.0080.050.173

Results7.7.1

0.008210.050.1731815))θ12P(cos(96300.20.4cos(θ)0.60.81.0
Figure7.8:ThedistributionfunctionPΣPdfΣθ))forthemeanverticalorientationθfor
difnumberferentEvS=olume(∗ΨΦ−fractions.).Theellipsoidaspect-ratioisΔe=Φ.Ω),andtheReynolds

))φP(cos(

0.0080.0500.173

00.20.4cos(φ)0.60.81.0
Figure7.9:ThedistributionfunctionPΣPdfΣφ))forthemeanverticalorientationφfor
oblateellipsoids.Theaspect-ratioisΔe=Φ.Ω),andtheReynoldsnumberES=(∗ΨΦ−).

changesOrientational7.1.3ToquantifytheorientationoftheoblateellipsoidsweintroducethequantityΨ=σ
ΩPdfΣθ)−Ψ>thatwasalsousedinKuuselaet.al.(2003),HerzhaftandGuazzelli(1999)

articlesPMany

0.5

0.2)−1>−0.1θ=<2cos(−0.4Ψ−0.70.4/2.4−10.4/1.60.4/0.800.05Φ0.10.150.2
VFigure7.10:OrderparameterΨasafunctionofthevolumefraction,ΦVforthreediffer-
entaspect-ratiosΔe=Φ.(/Ω.(;Φ.(/Ψ.+;Φ.(/Φ.1.

aswereorientationalperpendicularordertogravityparameter,.randomlyItwouldorientedgive−orΨ,Φalignedor+Ψwithifallgrathevityoblaterespectivelyellipsoids.
Figure7.10showsthebehaviorofΨagainstΦV,forsmallervolumefractions,ΦV≈
Φ.ΦΦΨ−Φ.Φ1theorderparametertakesnegativevaluesevidencingthealignmentalong
gravityandinagreementwiththelimit,ΦV→Φ(oneoblateellipsoid,section
(Φ.Ψ.Ω;≥(.Φ.Ψ.´Φ1).apositiveApproximatelyorderatparameterΦV≈Φ.impliesΦ1thetheorderorientationparameterisisperpendicularzero.Fortolargragerv-
V.ityIntherangeofΦV≈Φ.ΦΦΨ−Φ.Φ1,ΨhasalocalminimumclosetoΦV≈Φ.Φ)wherewe
foundalocalmaximuminﬁg.7.7andﬁg.7.1.Thesimulationswererepeatedwithtwo
otherdifferentaspectsratiosΔg=Φ.(/Φ.1;Φ.(/Ω.(andweobservedsimilarbehavior.In
thecaseofoneoblateellipsoid(ΦV→Φ)theorderparameterΨhasavalueverycloseto
−Ψastheellipsoidaspect-ratioisincreased.

numberReynoldsModerate7.1.4Figure7.11presentsthemeanverticalsedimentationvelocityforoblateellipsoids(
squaredline)andtheequivalentspheres(◦circlelined)asafunctionofthevolumefrac-
tionatmoderateReynoldsnumber(ES≈7).Inourpreviousworkthissimulationmethod
hasbeenusedwithsuccessuptoES≈ΨΦH¨oﬂerandSchwarzer(2000)andKuuselaet.
al.(2003).Theintermediatemaximumfortheellipsoidsisnotobservedinﬁg.7.11as
seeninﬁg.7.1atlowReynoldsnumber.
AcomparisonwiththephenomenologicalRichardson-Zakilaw(continouslineinﬁg
7.11)showsanexponentaroundc<eS=´.ΩforspheresandcEaaTe=(.Φforellips-
oids.Inbothbothcases,thedatafollowtheRichardson-Zakilawratherclosely.These

2.4)s(v||<v>

1.8

Outlook7.7.2Conclusionsand

SphEllip

00.040.08Φ0.120.16
Vline)Figureand7.11:asphereMean(dash-circlesedimentationline),vaselocityfunctioniΣg)offorthethevolumeoblatefraction,ellipsoidΦV.The(dash-squaredoblate
ellipsoidaspect-ratioisΔe=Φ.(/Ψ.),theequivalentspherehasEEfjT=Φ.20andthe
ReynoldsnumberES≈0.

elimitxponents(c≈((c.)<)eSand=´a.ΩturbandulentcEaaTeparticle=(.Φ)systemare(cbetween≈Ω.)),thelowRichardsonparticleandReZakiynolds(1954).number
Figure7.12(top)presentstheverticaldistributionfunction,PΣc6sΣθ))atmoderateReyn-
oldsnumber.Forallvolumefractions,PΣc6sΣθ))presentsalargerdistributionaround
PdfΣθ)≈Φ(θ≈2Φd),whichtendstobemuchﬂatter(PdfΣθ)≥Φ.Ψ))thaninﬁg.7.8.For
thec6sΣψ)other≈Φ,angularandforvlarariables,gervφolumeandψ,fractions,thedistribtheyutionsfollowshoawaconstantpeakbehaaroundvior.c6sΣφ)≈Φ,
Thebottomofﬁg.7.12showsthebehavioroftheorientationalparameterΨatmoderate
Reynoldsnumber.Foroneoblateellipsoid(ΦV→Φ),thevalueofΨiscloserto−Ψ
(verticalalignment),asinthecaseoflowReynoldsnumberﬁg.7.9.Theintermediate
maximum,forΔe=Φ.(/Ψ.),isnotobservedinﬁgure7.10,andthepointatwhichthe
(bottom).orientationalThisshiftparameterinΨΨisvalsoanishes,seenisinshiftedthecaseslightlyoftoﬁbers,thewhenright(theΦVRe≈Φ.ynoldsΨ)ﬁg.number7.12
increases,byafactorof5,Kuuselaet.al.(2003).

ConclusionsandOutlook7.2

articlesPMany

0.60.4)θcos(

0.0080.1100.173

1.00.8

0.110180.1731512))θP(cos(96300.00.20.4cos(θ)0.60.81.0
0.30.1)−1>−0.1θ−0.3=<2cos(−0.5Ψ−0.7−0.900.05Φ0.1V0.150.2
Figure7.12:Thetoppictureshowsthedistributionfunction,PΣPdfΣθ))ofthemeanver-
ticalorderorientationparameteratbehadifvesferentwithvtheolumevolumefractionfractionΦV.ΦV.TheThebottomoblatepictureellipsoidshowsaspect-ratiohowtheis
Δe=Φ.(/Ψ.),theequivalentspherehasEEfjT=Φ.20andtheReynoldsnumberES≈0.

Thislocalmaximuminthevelocitycanberelatedtothenonmonotonicbehaviorofthe
verticalorientationoftheoblateellipsoidsalonggravity,whichisshowninﬁgures7.7,
and7.8,andcanbeexplainedbythe“cluster”formationshowninﬁg.2.7,whichisalso
foundinﬁber-likesuspensionsHerzhaftandGuazzelli(1999).
AtlowReynoldsnumbertheorientationalorderparameterΨvanishesaroundΦV≈Φ.Φ1
ﬁg.7.7.AsΦVdecreasestheorientationalalignmentwithgravityincreasesasshownin
ﬁg.7.7and7.12(bottom),asforlowandmoderateReynoldsnumberandinthelimit
ΦV→Φasingleellipsoidalignswithgravity,whichisadistinctivefeatureofthesteady-
stateregimeforasingleoblateellipsoidasreportedinsections4.1.2,4.1.3andGaldiet.
(2001).al.

7.7.2ConclusionsandOutlook

WealsopresentdataatmoderateReynoldsnumber(ES≈0)forthesedimentationve-
locityofoblateellipsoidsasthevolumefractionΦVisincreased.Asinthecaseoflow
Reynoldsnumbertheellipsoidshaveasmallersedimentationvelocitythantheequivalent
spheres,ﬁg.7.1and7.11.ThedataforellipsoidsandspheresfollowtheRichardson-
ZakilawRichardsonandZaki(1954)withexponents(cEaaTe≈´.Ω,ES=ΨΦ−))and
(c<eS≈(.Φ,ES=0)respectively.ThePΣc6sΣθ))distributionpresentsalargeralignment
ofellipsoidswithgravitycomparedtothosewithsmallReynoldsnumber.Thevanishing
oftheorderparameterisslightlyshifted(ΦV≈Φ.Ψ)totherightastheReynoldsnumber
increases(seeﬁg7.12,bottom).Thealignmentwithgravityispresentforsmallandmod-
erateReynoldsnumberasΦV→Φ,asshowninﬁg.7.12(top)andﬁg.7.8,whichisin
agreementwiththeorientationalbehaviorofasingleellipsoid(section4.1.2;4.1.3).All
thesimulationsinthisworkarelocatedinthesteady-fallingregime,chapters´and(,for
ellipsoid.oblatesinglea

8Chapter

Diffusion

ThislipsoidschapteratlowisanddedicatedmoderatetotheReinvynoldsestigationnumbers.oftheWedifwillfusivbeeginbehawithvioranofintroductionsedimentingtheel-
commontheoreticaltoolsusedtostudythediffusivebehaviorofsedimentingparticles.
eWexplorediscussthethebehaviorresultsofforthesystemellipsoidsunderinvcomparisonariationsoftothetheequidynamicalvalentspheres.viscosity,Also,ellipsoidwe
densitytionally,andwepresentaspect-ratio.theanomalousFurthermore,difwefusiongivefortheoblateorientationalellipsoids.difLastlyfusion,webehavior.summarize.Addi-

oductionIntr8.1

AswasexplainedinchapterΨ,inasedimentingsuspensiontherearelargechangesin
thesiondominatesconcentration,overthatthecanthermaleitherBrobewniantemporaldiforfusion,spatial.intheThissystemhconsideredydrodynamic-likhere.edifThefu-
dimensionlessquantitythataccountsfortherelativeimportanceofthishydrodynamicdi-
fussionprocessoverthethermaldiffusionistheP´ecletnumberPS.WhenPSismuch
largerthan1,BrownianmotioncanbeneglectedRamaswamy(2001),KuuselaandAla-
Nissila(2001).ForsmallervaluesofPSσσΨ,inturn,thehydrodynamicdiffusionisnot
ant.vreleInaregimewhere(PS>>ΨandESσσΨ),sedimentingspheresundergolong-ranged
hydrodynamicﬂuctuations(seechapter1,sec.1.6).Theycauseinthelongtimelimit,
thattheﬂuctuatingparticlemotionbecomesdiffusive,Nicolaiet.al.(1995).There-
fore,wecandeterminethelongtimebehaviorofthisrandom-likeparticlemotions,by
ebyxaminingcomputinghowthetheparticleparticlevvelocityelocityﬂuctuationbecomesuncorrelated.autocorrelationThisefunction,xaminationΣg),(seeisKrealizeduusela
andAla-Nissila(2001)andchapter+,sec.+.Ψ).Thisquantitywasdeﬁnedinsec.+.Ψ,as
ws:follo

,Σg)=σΔiΣg)ΔiΣΦ)>

(8.1)

oductionIntr8.8.1

wheretheaverageistakenoveralltheparticlesinmotion.HereδiΣg)=iΣg)−σi>
isthelocalvelocityﬂuctuation,whereas,Σg)isusedtodeﬁnethehydrodynamicdiffu-
sioncoefﬁcient<,inanalogywiththeBrowniantracerdiffusioncoefﬁcientofBrownian
particlesimmersedinaﬂuid(KuuselaandAla-Nissila(2001)).<isdeﬁnedbytheGreen-
as:formulauboK

∞Ψ<=RΦ,Σg)Rg
whereRisthespatialdimension,andforthediscretecaseasfollows:

∞ΨΨ<=R[Ω,ΣΦ)+,Σc−τ)4
=(c

(8.2)

(8.3)

whereτisthetimestep.
Anothermethodtoinvestigatethelongtimebehaviorofasedimentingparticles,istocom-
putethemeansquareparticledisplacement(hereafterM.S.D.),Kalthoffet.al.(1996),
i.e.thesecondmomentoftheparticledisplacement,HerzhaftandGuazzelli(1999).Itis
ws:folloascalculated

σk)Σg)>3=σ[kΣg)−ΣkΣΦ)+σi>g)4)>(8.4)
herethebracketsindicateanaverageovertheensembleofthoseparticlesthatareabove
theﬁnalbottompositioninthecontainer,andσi>istheirmeanvelocitytakenofall
particleswithi=Φ.Asimplediffusivebehaviorischaracterizedbyalineargrowthwith
time.Theself-diffusivity<,(Nicolaiet.al.(1995)),canbedeterminedfromtheslopeof
line.thisAponentslargetograanosotropvity,yhasbetweenbeenthereportedM.S.D.byforothertheparallelauthors()(Nicolaiandet.perpendicularal.(1995);(⊥)Laddcom-
(1996);Kalthoffet.al.(1996)),whofound<>><⊥.However,thisanisotropybe-
comeslessimportantforlargeReynoldsnumbersandvolumefractions(Kalthoffet.al.
(1996),Nicolaiet.al.(1995),KuuselaandAla-Nissila(2001)).
Both,<and<⊥areobtainedbyexamininglongtimebehavioroftheparticledisplace-
mentbythefollowingrelations:

σkΣg)⊥>)≡Ω<⊥g;σkΣg)>)≡Ω<g.(8.5)
Despitetheclearevidencefornormaldiffusivebehaviorfoundinthecurrentresearch,
someexperiments,(Rouyeret.al.(1999))andsimulations(MiguelandPastor-Satorras
(2001)),hadshownthatthevelocitiesandtrajectoriesofnon-Brownianparticlesina
quasicomponentbidimensionalandasuperdifﬂuidizedfusivebed,behaexhibitvioraparalleldiffusitovegrabehavity.viorThisalonglatterthesuperdifperpendicularfusive

Diffusion

200

7260

48vertical36

201612horizontal

Figure8.1:(a)Abidimensionalvelocitymapshowingfasterandslowerellipsoids.(b)
TypicalBrownian-liketrajectoriesinellipsoidssedimentationinthelaboratoryreference
frame.TheReynoldsnumberisΩ×ΨΦ−)andtheellipsoidaspect-ratioisΔe=Φ.(/Ψ.+.
behaviorisalsoknownas,“anomalousdiffusion”,anddeﬁnedasthegrowthofthesecond
by:momentumorder

σk)Σg)>≡gα

(8.6)

withα=Ψ.
Furthermore,wecouldextendequation8.5totheangularcase,anddeﬁnetheM.S.D.as:

σθ)Σg)>≡Ω<θg(8.7)

Withregardtosedimentingparticleswithnon-sphericalshape,apart,fromafewexcep-
tions(e.g.Kuuselaet.al.(2001)andchapters(,)),verylittleisknownaboutthediffusive
behavior,beitfromsimulation,experimentortheory.Non-sphericalparticles,withro-
tationalsymmetryaroundanaxis(oblateorprolateellipsoids),givesrisetorotational
degreesoffreedom,andgenerallyplaysanimportantroleinthesedimentation,(Kuusela
et.al.(2003)andchapter+).

600⊥ll450(a)>*2300<x150

3060)time t(ts32⊥Sph⊥Ellip24(c)>*216<x

210⊥ll110log(dR/dt)(b)(b)010

19010))log(t(ts600llSphllEllip450(d)>*2300<x

150

Results8.8.2

210

204060306090120
time t(ts)time t(ts)

Figure8.2:Figure(a)showstheparallel()andperpendicular(⊥)componentsofM.S.D.
forsedimentingellipsoids.Picture(b)showstheslopebehaviorintime,ofthecurvesin
pictureballistic(a),andinanon-diflog-logfusivescale.regimes.ThethickThelodottedwerlinespictures,in(c)between,and(d)presentshowthethegrowthcomparisoninthe
betweenellipsoidsandsphereswithregardtotheperpendicular⊥(c)andparallel(d)
componentsoftheM.S.D..ThesphereradiusisEEfjT=Ψ.ΦΨ.TheReynoldsnumberis
Ω×ΨΦ−)andtheaspect-ratioΔe=Φ.(/Ψ.+.Thenumberofellipsoidsisoftheorderof
0.ΨΦResults8.2Inﬁgure8.1(a),wepresentbidimensionalsnapshotofthevelocitymapforsedimenting
ellipsoids.Themapischaracterizedbycomplexandhighlyﬂuctuatingvelocityvectors

Diffusion

pointinginawiderangeofdirections,withverydifferentmagnitudes.Thesnapshot
reﬂectsthecomplexityoftheswirlsandchannelsthatareverysimilartothosereported
veolumexperimentallyfractioninofSethegre´etparticlesal.in(2001)theandsysteminisΦsimulations=Φ.Ω.byRouyeret.al.(1999).The
VInﬁgure8.1(b)wepresentatypicalellipsoidtrajectoriesinacontainerscaledinunits
ofthecharacteristicsmayorradiiobservEMed.byTheNicolaitrajectorieset.al.are(1995)ratherandPecomplicatedyssonandandeGuazzellixhibitman(1999),yofsuchthe
asthepresenceofloopsandastronganisotropybetweentheparallelandperpendicular
ﬂuctuationsintheellipsoidtrajectory.
exhibitPicturesthe(a)larandge(b)inanisotropﬁgurey8.2betweenshowthetheparallelM.S.D.(for)andsedimentingperpendicularellipsoids.(⊥)Thecomponents.graphics
Inﬁguregeneral,8.2(b),bothinalog-logcomponentsscale,presentbyaanparallelinitialthicksocalleddotted-lineballisticinrebetweengime,asofthedepictedcurves.in
Thisballisticregimeisproportionalto≈g).Then,wefoundanon-diffusivebehavior
whichisquiteinterestingandwillbediscussedinsection)ofthischapter.
Anotheraspectofthelong-timebehavioristhat,asaconsequenceoftheﬁnitesize
container,theellipsoidsreachasaturationregime.Thisischaracterizedbyﬂuctuations
aroundtheﬁnalpositionofeachellipsoidinthecontainerbottom)(Kalthof)fet.)al.(1996)).
Thetimeingraphicsunitsofaregh,(seenormalizedeq.Ω.in´Φ).bothcomponentsaccordingtok∗≡k/EEfjandthe
ThecomparisonbetweentheM.S.D.forellipsoidsandthatforspheresisgiveninﬁgure
8.2,numberin(theES⊥≈(c)ΨΦand−))and(d)forﬂatcomponents.ellipsoids,Inourthatthesimulations,componentswefoundforatloellipsoidswerReareynoldsmuch
largerthanthoseforspheres.Thisphenomenoncanbeexplainedbythepresenceofa
muchlargernetdisplacementforellipsoidsthanspheres,ascanbeseeninﬁgure8.1.
Itisinterestingtopointout,thattheperpendicularcomponentoftheM.S.D.,ﬁg.8.2
(c),foraveryﬂatellipsoid,experiencesapeakwhichthendecreasesabruptly,beforethe
saturationregimeisreached.Thisbehavioriscompletelynewandcanonlybeassociated
totheparticleshape,sincethespheresonlyechibitamonotonicincrement,(seeﬁg.8.2
with(c)).aThisﬁxedlargepeakaspect-ratio,inthebutvellipsoidariableviscosityperpendicularorparticlecomponent,densityis..presentinallsimulations

8.2.1Changeindensity,viscosityandaspect-ratio

Theslopeforeachoneofthecurvesinﬁg.8.3(a)and(b)becomeslargeastheellipsoid
densityisincreased,keepingconstanttherestofparametersinthesystem,showingthe
ponentsincreaseofincreasestheinertiamuchinfasterellipsoidsthanforandspheres.spheres.ForFortheparallelellipsoids,thecomponentslopetoingrabothvity,com-the
denserellipsoidsdiffusemuchfaster,quicklyreachingthesaturationregime.Thiscan
be8.3(c)seenandby(d)wecomparingpresentthethecurvesM.S.D.forρ(beha=).viorΦandasρthe1=ΨΨ.kinematicalΦinﬁgureviscosity8.3(a).isInchanged.ﬁgure
Forallofthecases,theincreamentintheviscositydecreasestheslopeofeachonthe

Results8.8.282400600(a)(b)4002>*2>*200
<x<x200ρ1=5.0ρ1=5.0
ρ2=7.0ρ2=7.0
ρ3=9.0ρ3=9.0
ρ4=11.0ρ4=11.0
020406080100120020406080100120
time t(ts)time t(ts)
µ1,Ell=7.7µ1,Sph=7.7
µ2,Ell=6.7300µ2,Sph=6.7
400µ3,Ell=5.9(c)µ3,Sph=5.9
µ4,Ell=5.3µ4,Sph=5.3
2>*2>*200(d)
<x<x20010001530450306090120
time t(ts)time t(ts)
6006001.00.4/1.61.20.8/1.64801.6/1.61.2/1.64801.61.4
3603602>*(e)2>*(f)
<x<x2402401201200408012016020024028004080120160200240280
time t(ts)time t(ts)
Figure8.3:Figuresa)showtheparallelcomponentoftheM.S.D.forellipsoidsand
equivalentspheresb),astheparticledensitychanges.Figuresc)andd)presentthevari-
ationforthedynamicalviscosityandﬁgurese)andf)showthechangeintheoblate
aspect-ratioandtheequivalentradius,respectively.TheReynoldsnumberisES≈ΨΦ−),
thespheresradiusisEEfjT=Ψ.ΦΨandtheellipsoidaspect-ratioΔe=Φ.(/Ψ.+.

Diffusion

>*2<x

1512

0.4/1.60.8/1.61.2/1.61.6/1.6

0306090120150180210240
)time t(tsFigure8.4:BehavioroftheperpendicularcomponentofM.S.D.forellipsoidsasthe
aspect-ratiochanges(Δe=Φ.(/Ψ.+,Δe=Φ.1/Ψ.+Δe=Ψ.Ω/Ψ.+andΔe=Ψ.+/Ψ.+).
TheReynoldsnumberisES≈ΨΦ−).

curvesInﬁgure8.3(c)and(d).TheincreaseoftheviscosityreducestheM.S.D.inthe
suspension.Also,asforparticledensityvariations,thechangesintheviscositylead,for
lowerReynoldsnumbers,toadirectlyproportionalincrementintheM.S.D..
Thepeaksintheellipsoidperpendicularcomponent(seeﬁg.8.4),ispresentinallofthe
simulationswherethekinematicalviscosityorparticledensityarechanged,withaﬁxed
aspect-ratioandlowerReynoldsnumber(ES≈ΨΦ−)).
Next,wepresenttheM.S.D.fordifferentellipsoidaspect-ratiosΔeandequivalentsphere
radiiEEf.Thevariationintheaspect-ratiogoesfromaﬂatellipsoiduptoasphere,while
thevolumefractionΦ≈Φ.Ωiskeptconstant.Theresponseundervariationsoftheoblate
aspect-ratio,presentsalinearincreaseintheslopeofeachcurve,inbothellipsoidsand
spheres,asisshowninﬁgures0.´(e)and(f),respectively.
TheslopegrowingintheM.S.D.astheaspect-ratioincreases,becomingone,showsthat
thesphereshaveamuchlargerM.S.D.thantheellipsoids.Theellipsoids,ontheother
hand,havemoreresistancetosediment.Thischaracteristicispresentinbothcomponents
(seeﬁgure8.3(e)and8.4).Asfortheparallelcomponentﬁg.8.3(e)and(f),spheresand
ellipsoidspresentthesamemonotonicbehaviorbeforetheyreachthesaturationregime.
Inﬁgure8.4,wecanseehowtheperpendicularcomponentoftheM.S.D.approachesthe
saturationregime,astheaspect-ratiogrows.Thepeakisshiftedtotheleftandbecomes

120

902>Θ<603000

452>φ<30(a)ρρ1=7.0=5.015
2=9.0ρ3=11.0ρ430609012000
)t(ts60

Results8.8.2

(b)=5.0ρ1=7.0ρ2ρ=9.03=11.0ρ410t(ts)2030

452>ψ<(c)3015ρρ12=7.0=5.0
=9.0ρ3=11.0ρ400612t(ts)182430
Figure8.5:AngularM.S.D.fortheEuleranglesθ-ﬁg.(a),φ-ﬁg.(b)andψ-ﬁg.(c)under
variationsofellipsoiddensity,ρ(=),0,2,ΨΨ.TheReynoldsnumberisΩ×ΨΦ−)andthe
ellipsoidaspect-ratioisΔe=Φ.(/Ψ.+.

muchlargerandsharperastheaspect-ratiobecomesΨ(thesphere).

diffusionOrientational8.2.2Figure8.5showstheangularM.S.D.,σΔΘ>)forthethreeEulerangles(seeﬁgure4.4
inchapter´),undervariationsoftheellipsoiddensity,whichcorrespondtothetransla-
tionalbehaviorshowninﬁgure8.3(a).
Itisinterestingtonote,thattheangularM.S.D.followsabehaviorwhichissimilarto
thecorrespondingtranslationaldegreesoffreedom,inthesensethattheangularM.S.D.
exhibitsatthebeginningofthesedimentationafastergrowthandthenapproachesto

Diffusion

thecomparedsaturationtotheregime.translationalThisapproachcomponentsto(seesaturationﬁg.is8.3quite(a)),fandasteris(seepresentﬁg.for8.5),theasitthreeis
angles.EulerAstheoblatedensityisincreased,theangularM.S.D.foreachoneoftheEulerangles
shiftingslightlyintheincreases.ﬁnalvThealuereleofvtheantvsaturationariationinrethegime(seeangularﬁg.M.S.D.8.5).Wisecanpresentobservinetheupw(approx-ard
imately),thatequalvariationsinellipsoiddensitycauseequalupwardincrementsinthe
alue.vsaturationIttheissamealsokindinterestingofupwtoardobservshiftinge,thatasthedescribeddecreaseaboofve,thebutwithkinematicalasmallerviscosityshiftinggivesvalueriseofto
theﬁnalsaturation(seeﬁg.8.6).
reducingIncreasingthethekinematicalimportanceofviscosityinertial,increasesforcesbytheusingsaturationahigherstateinmasstheofEulertheangles.ellipsoidor

8.2.3Non-diffusivedynamicalbehavior

EWΣeg)no=wσkstudy)Σg)the>,behaeq.vior(8.4).oftheThistimetypederiofvativeofcharacterizationtheM.S.D.,ofthedeﬁneddiffusiasveREΣgbeha)/Rgviorwithin
sedimentationisalsousedintheworkofRouyeret.al.(1999)andMiguelandPastor-
whoseSatorrasconstant(2001).vInalueaiscaseequivwherealentREto/Rgone≡halfofPdcfgNcgthe,difwefusionhaveaconstant,simpledif(seefusieq.ve8.5).behaIfviorwe,
ﬁndnon-diffusivebehavior,thenRE/Rg=PgSandthetemporalbehavioroftheM.S.D.is
givenbyeq.(8.6).

densityparticletheinChange

Figure8.7(a)presentstheperpendicularcomponent(⊥),ifthespheredensityischanged.
Forsimpleρ(=dif)fusi.Φvetheandevolutionsaturation.oftheAncurveinterestingintime,aspectexhibitsheretheconsistsabovinerethefgimes,actthatballistic,when
themuchparticleshorterindensitytime.ρisBasicallyincreased,the(ρ=)increment.Φ→ofρ=theΨΨ.Φinertial)theforces,simpledifreducesfusivetheretimegimeofis
thesimplediffusiveregime.Thisgeneralbehaviorforspheresisinagreementwiththe
resultspresentedby(Rouyeret.al.1999)and(MiguelandPastor-Satorras2001).
ForAfterthethat,aparallelnon-diffusicomponentvebeha()viorﬁgure8.7characterized(b),allbythegα,curvwithesaneexhibitxponentaαballistic=Ψ.(recangime.be
observed(Rouyeret.al.(1999)).Liketheperpendicularcomponent,thelengthintime
oftheparallelcomponentisreducedforlargespheredensities,asshowninﬁgure8.7(b).
Inﬁgure8.8(a)and(b)wecanobservethebehaviorofthetimederivativeoftheM.S.D.
aforballisticellipsoids.regimeForandthethenaperpendicularnon-difandfusivereparallelgime,whichcomponentsupto(seenowﬁgureisnot8.8)knowewnﬁrstinﬁndthe

120

90>2Θ<603001

50>2φ<25=5.0νEll,1=7.0νEll,2=9.0νEll,3=11.0νEll,4021110t(t)s60

Results8.8.2

=5.0νEll,1=7.0νEll,2=9.0νEll,3=11.0νEll,4302010)t(ts

45>2ψ<3015ννEll,2Ell,1=7.0=5.0
ννEll,3Ell,4=9.0=11.0
0010t(t)2030
sFigure8.6:AngularM.S.D.fortheEuleranglesundervariationsinthekinematicalvis-
cosity.TheReynoldsnumberisΩ×ΨΦ−)andtheellipsoidaspect-ratioisΔe=Φ.(/Ψ.+.

literature.TheexponentsinbothcomponentsareequalαegaaEa=Ω.),αeEge=Ω.)and
largerthantheslopeoftheballisticregime.
Incontrasttothespheres,thedensityvariationdoesnotmodifysigniﬁcativelythenon-
diffusiveregime’scharacteristictime.Thiskindofbehaviorforthediffusionofnonspher-
icalparticleswasneverbeforereporteduptonow,andtheexperimentalveriﬁcationwill
benecessary.Fromtheplot,wecanalsoextractthat<vEgi<Sdg,thereforethediffusive
anisotropic.highlyisgimereForbothellipsoidsandspheres,theM.S.D.inbothcomponentsaredisplacedupwardas
theparticledensitygrows.AsexpectedatlowReynoldsnumber,anincrementinthe
inertialforces,producesalargegrowthoftheM.S.D..

.elyvrespecti,viorbehaevfusinon-diftheandevfusidiftherepresent,linesdashedandsolidThe.+.Ψ/(.Φ=eΔisaspect-ratioellipsoidthe,)−ΨΦ×ΩisnumberynoldsReThechanged.isdensitytheasellipsoidsforM.S.D.theofevativderitimetheofcomponents(c)paralleland(a)⊥perpendicularthewshoﬁguresThe8.8:Figure)OΣ)NΣreplacementsPSfrag=11.04ρ=9.03ρ=7.02ρ=5.01ρ/dt)log(dR))slog(t(t110010−110−210−310210110010)OΣ)NΣreplacementsPSfrag=11.04ρ=9.03ρ=7.02ρ=5.01ρlog(dR/dt)))slog(t(t010−110−210−310210110010.elyvrespecti,viorbehaevfusinon-difande87

vDiffusion

fusi)NΣ

difOΣ)

therepresentlinesdashedandsolidThe.ΦΨ.Ψ=jTEfEradiusspheresthe,)−ΨΦ×ΩisnumberynoldsReThechanged.isdensitytheasspheresforM.S.D.theofevativderitimetheofcomponents(b)paralleland(a)⊥perpendicularthewshoﬁguresThe8.7:Figure)NΣreplacementsPSfrag=11.04ρ=9.03ρ=7.02ρ=5.01ρ/dt)log(dR))slog(t(t110010−110−210−310210110010replacementsPSfrag=11.04ρ=9.03ρ=7.02ρ=5.01ρ/dt)log(dR))slog(t(t010−110−210−310110010

010

110

Results8.8.2

ΣΣOO))

10−1ΣΣNN))100ΣΣOO))
log(dR/dt)log(dR/dt)−11010−2ν1,Sph=7.7ν1,Sph=7.7
=6.7ν=6.7νν2,Sph3,Sph=5.910−2ν3,Sph2,Sph=5.9
PSfragreplacements−3ν4,Sph=5.3PSfragreplacementsν4,Sph=5.3
10101102ΣN)101102
ΣO)log(t(ts))log(t(ts))
Figure8.9:Theﬁguresshowtheparallel(b)andperpendicular⊥(d)componentsof
RetheynoldstimederivnumberativeisofΩ×theΨΦ−)M.S.D.,theforspheresspheresradiusastheEEfjT=kinematicalΨ.ΦΨ.viscosityischanged.The

viscositykinematicaltheinChange

Nextwestudythebehaviorofthesystemunderchangesinthekinematicalviscosity.Fig-
ure8.9(a)and(b)showstheM.S.D.forspheresintheperpendicular(⊥)andparallel(||)
components.Figure8.9(a)showswelldeﬁnedballisticandsimplediffusiveregimes.The
plotsshowaslightincrementintimeforthesimplediffusiveregimeasthekinematical
viscositydecreases.Also,thereisaclearupwardshiftingasthekinematicalviscosityde-
creases.ThespheresincreasetheirM.S.D.asthekinematicalviscositybecomessmaller.
Inthecaseoftheparallel(||)component8.9(b),wefoundaballisticandnon-diffusive
regimes.Theexponentαforthenon-diffusivebehavioristhesameinallthecurvesand
α=Ψ.((Rouyeret.al.(1999)).Suchasfortheperpendicular(⊥)component,the
upwardshiftingisalsopresentfortheparallel()component.
Theellipsoidssystemisstudiedunderthesamekinematicalviscosityvariations.Figure
8.10showsthebehavioroftheperpendicular(⊥)(a)andparallel()(b)componentsof
M.S.D..theTheperpendicularcomponent(seeﬁgure8.10(a))exhibitsballisticandnon-diffusive
behavior.Thecurves,asinthecaseofspheres,showanupwardshiftingasthekinemat-
icalviscositydecreases.Astheviscositydecreasestheexponentαforthenon-diffusive
regimebecomesgraduallymuchsmaller:ν=0.0→α=Ω.(toν=).Φ→α=Ω.Ψ.
Fortheparallelcomponent(),thedifferencebetweentheballisticandthenon-diffusive
behaviorisnotpronounced,andforthelowerkinematicalviscosityν=).Φ→α=Ω.Φ
thenon-diffusivebehaviorinthesystempracticallydisappears.

.)−ΨΦ×ΩisnumberynoldsReThechanges.radiusspheretheasspheresforM.S.D.theofevativderitimetheofcomponents(b)paralleland(a)⊥perpendicularthewshoﬁguresThe8.11:Figure)OΣ)OΣ)NΣreplacementsPSfrag))slog(t(t−110210110)OΣ)NΣ)NΣreplacementsPSfrag))slog(t(t−210210110.+.Ψ/(.Φ=eΔisaspect-ratioellipsoidthe,)−ΨΦ×ΩisnumberynoldsReThechanged.isviscositykinematicaltheasellipsoidsforM.S.D.theofe89

vΣΣOO))

atiChangeintheellipsoidaspect-ratioandsphereradius
Inthecaseofvariationsoftheequivalentradii,eq.(3.28),ﬁgure8.11showsthebehavior
oftheparallel()andperpendicular(⊥)components.Astheequivalentradiidecreases,

v210

deriDiffusion

time110

the=1.0Req,1=1.2Req,1=1.4100RReq,1=1.6
eq,1log(dR/dt)−110

of2=1.0R10eq,1=1.2Req,1=1.4Req,1=1.6Req,1110log(dR/dt)010

components(c)⊥perpendicularand(a)parallelthewshoﬁguresThe8.10:Figure)NΣreplacementsPSfrag=5.34,Ellν=5.93,Ellν=6.72,Ellν=7.71,Ellνlog(dR/dt)))slog(t(t110010−110−210210110)OΣ)NΣ)NΣreplacementsPSfrag=5.34,Ellν=5.93,Ellν=6.72,Ellν=7.71,Ellνlog(dR/dt)))slog(t(t010−110−210−310210110

Results8.8.2

Furthermore,intheparallelcomponent(seeﬁgure8.12(b)),wecanseethechangeinthe
difslopefusivthatebehaviorcharacterizesforthespheresv(ariationΔe→fromΨ).Thenon-difefusixponentsvebehagoesviorfromfor(Δe=ellipsoidsΦ.Ω),toαthe=Ω.non-´)
oblateellipsoidsto(Δe=Ψ,α=Ψ.(,Rouyeret.al.(1999))spheres.

Inﬁgure8.12(a)and(b)wepresenttheM.S.D.behaviorforellipsoidsastheaspect-
ratioischanged.Herewewanttodrawattentiontoaninterestingchangefromanon-
diffusiveregimetosimplediffusivebehaviorthatcanbeobservedintheperpendicular
(⊥)component(seeﬁgure8.12(a))astheellipsoidaspect-ratiotendstoone(Δe→Ψ,
sphere).Theslopeofthecurveinthenon-diffusivepartchangesfrom(Δe=Φ.Ω),α=
Ω.Ω)to(Δe=Ψ,α≈Φ).

Fortheperpendicular(⊥)component,allthecurveslookquitesimilartoeachother,with
aregimeslightisdifmuchferencefasterbeingthanthatintheforcasespheresofEEfwith=EΨ.EfΦ.=TheΨ.e+thexponentαapproachforthistothenon-difsaturationfusive
regimeisα=Ψ.(.

wecanseeinﬁgure8.11(a)thesamekindofupwardshiftinginthecurves,asviscosity
ordensity,ischangedinthesystem.Asthesphereradiusisincreasedthepresenceofthe
simplediffusiveregimeisincreasedduringaverylongtime.

102Δ r1=0.4/1.6
=0.8/1.6 rΔ2=1.2/1.6 rΔ3=1.6/1.6 rΔ4110

log(dR/dt)010

=0.4/1.6 rΔ1=0.8/1.6 rΔ2100Δ r3=1.2/1.6
rΔ=1.6/1.64−110log(dR/dt)

.)−ΨΦ×ΩisnumberynoldsReThechanges.aspect-ratioellipsoidtheasellipsoidsforM.S.D.theofevativderitimetheofcomponentsparalleland(a)⊥perpendicularthewshoﬁguresThe8.12:Figure)OΣ)OΣ)NΣreplacementsPSfrag))slog(t(t−110210110)OΣ)NΣ)NΣreplacementsPSfrag))slog(t(t−210210110

Diffusion

600400(a)3004002>*2>*200
<x<x200ρρ1=7.0=5.0100
2=9.0ρ3=11.0ρ40501001502000
)time t(ts=7.7ν1,Ell=6.7ν2,Ell400νν3,Ell=5.3=5.9300
4,Ell2>*2>*200
<x<x200(c)100

01057035time t(t)s480

360(e)>*2240<x

0480

360>*2240<x

1201200.4/1.60.8/1.61.2/1.61.6/1.600900600300)time t(ts

(b)

=5.0ρ1=7.0ρ2=9.0ρ3=11.0ρ450100150time t(t)200250
s=7.7ν1,Sph=6.7ν2,Sph=5.9ν3,Sph=5.3ν4,Sph

(d)

50100time t(t150)200250
s

(f)

1.01.21.41.6900600300time t(t)s

Figure8.13:Weshowthecollapseofthecurvesfromﬁg0.´.Allthecurvescollapse
quitewelljustifyingthetransformationrule,table0.Ψ.
Similarity8.2.4

Inpresentedtable8.1,inweﬁgurepresent8.3.ThetheReynoldstransformationnumberrulesReisthatleftareinvusedarianttobythecollapsethetransformations.pictures,

ConclusionsandOutlook8.8.3

Time→AdimensionalTime
igi→sσk)>→1θk,22eqρ
viv→sTable8.1:Transformationrulesinsedimentation.

)−noTheseanalyticalresults,ehowexpressionver,toonlyholdaccountforthesmallterminalReynoldssettlingvnumberselocityE,SatΨΦmoderate,sinceRethereynoldsis
numbers.Therefore,itisimpossibletocomputetheStokestime,eq.(3.30)atmoderate
Re.

ConclusionsandOutlook8.3

Inthischapter,thediffusionofoblateellipsoidsinsedimentationwasstudied.Wehave
foundanon-diffusivebehaviorinbothcomponents(parallelandperpendicular),which
iscompletelynewandnotreportedintheliterature.Ourcomparisonwiththeequivalent
spheresystemtoellipsoidsshowsthatthereisasimplediffusiveprocessfortheperpen-
dicularandanon-diffusiveprocessfortheparallelcomponents.Thisresultregardingthe
diffusivebehaviorinspheresagreeswiththeworkof(Rouyeret.al.1999)and(Miguel
2001).-SatorrasastorPandInaddition,thebehaviorforellipsoidsandsphereswasinvestigatedastheparticledensity,
kinematicviscosityandellipsoidaspect-ratioorsphereradiiwerechanged.Itwasfound
thattheincrementoftheinertialforces,bymeansofthegrowthinparticledensityorthe
decreasinginthekinematicalviscosity,reducesthepresenceofthediffusivebehavior,
andthesystemgoesfasterfromtheballistictothesaturationregime.Whenthesphere
radiiarechanged,wecouldobserveasimilarbehavior:Asthesphereradiusisincreased
theinertiagrowsandthusalsotheparticleM.S.D..
ItisimportanttopointoutthebehaviourofthetimederivativeoftheM.S.D.inboth
componentsastheaspect-ratiochanges(seeﬁgure8.12).Fortheverticalcomponent,
withthegrowthoftheaspect-ratiotheM.S.D.goesfromthenon-diffusivebehaviorto
thesimplediffusiveregime.Intheparallelcomponent,theexponentαfornon-diffusive
regimeschangesfor(α=Ω.Ω,Δe=Φ(/Ψ.+)ﬂattenedellipsoidsto(α=Ψ.(,Δe=Ψ)
spheres.Last,wefoundasimilaritylawforthesedimentationprocesswhichisvalidatsmall
Reynoldsnumbers.Itispresentedintable8.1.

9Chapter

FluctuationselocityV

Inthislastchapterweexaminethedynamicalbehaviorofsedimentingellipsoidsand
spheresunderchangesofthecontainersize.Intheﬁrstsectionwestudytheinﬂuence
onthespatialcorrelationsastheparticulatevolumefractionischanged,comparingthe
resultsforellipsoidsandspheres.Wealsopresentthestudyofthevelocityﬂuctuationsas
afunctionofthevolumefraction.Afterthat,weinvestigatethedivergenceofthevelocity
ﬂuctuationsasthecontainersizeischanged.Finally,wesummarize.

Wechoosethedensityoftheﬂuid,theStokesvelocityandthelargerradiusoftheellipsoid
equalsquaretobaseunityofinsideourA=system.ΩΩeInxtendedallofupthetoAcases,=theΨ0+,andcontaineralatticeheightisconstantA=ofVΨ)Φ=andΦ.0a.
Theratiobetweenthedensityoftheoblateellipsoidsandtheﬂuidis(.

elationscorrSpatial9.1

Westartouranalysisbystudyingthespatialcorrelationsinthevelocityﬂuctuations(here-
afterSCVF).Thenormalizedautocorrelationfunctionoftheparallel(||)componentofthe
velocityﬂuctuationsaredeﬁnedas,(Segre´et.al.(1997)):

,Σr)≡δiΣΦ)δiΣr)(9.1)
δiΣΦ))
1cmwherethebrackets...representanensembleaverageoverseveralindividualdif-
ferentconﬁgurationsinspaceandorientations(ellipsoids).WhereδvT=vT−vhEC,
representstheﬂuctuationsinthevelocityandvhEC=iTisthemeanvelocityoverthe
conﬁguration.Ifthedistanceristakeninthedirectionparalleltogravity,k,thenwecall
theparallel,orperpendicular,l,,⊥component.
93

0.6||C

0.2

−0.201

0.6EllC0.2

||,Sph||,Ell1

(a)0.6⊥C

0.2

−0.26040200r,Ell⊥||,Ell1

elationscorrSpatial9.9.1

,Sph⊥,Ell⊥

(b)

604020r⊥,Sph||,Sph

(d)

−0.20204060−0.20204060
rrFigure9.1:Theﬁguresshowtheparallel(a)andperpendicular⊥(b)componentsof
theSCVFforellipsoidsandspheres.Figures(c)and(d)comparetheparallelandper-
pendicularcomponentsforellipsoidsandspheres,respectively.TheReynoldsnumberis
Ω×ΨΦ−),thespheresradiusEEfjT=Ψ.ΦΨandtheellipsoidaspect-ratioisΔe=Φ.(/Ψ.+.

9.1.1Changeinthevolumefraction
Inﬁgure9.1,wepresentthespatialcorrelationsofvelocitiesforellipsoidsandspheres.
Figure9.1showsthecomparisonbetweentheparallel(a)andperpendicular(b)compon-
entsforspheresandellipsoids.Inﬁgures(a)and(b)thecomponentsforspheresshowa
muchfasterrelaxationthanellipsoids.Figures(c)and(d)exhibittheanisotropy,charac-
teristictothesedimentationprocess,withaslowerdecayoftheparallelcomponentofthe
velocityautocorrelationfuntion.Thisanisotropyispresentinbothspheres,(Nicolaiand
Guazelli(1995),Segre´et.al.(1997),H¨oﬂer(2000))andellipsoids.

FluctuationselocityV

WepresenttheSCVFfortheparallel()andperpendicular(⊥)componentsﬁg.9.2for
ΦVspheres=Φ.(a,b)ΦΩ,Φ.andΦ(,Φ.Φ1,ellipsoidsΦ.Ψ+.(c,d)Theanisotropcorrespondingytobetweenfourtherepresentaticomponentsvesvisalsoolumepresentfractionsas
thevolumefractionΦVincreasesasisshowninﬁg.(a)and(b).

eThexponentialcomponentdecay,ofas,the||≈SVCFSkpΣfor−z/ξspheres,heS),(see(Nicolaiﬁgureand9.2(a),Guazelliapproximately(1995),Segrfolloe´et.wsal.an
(1997),H¨oﬂer(2000)).Ontheotherhand,the⊥component(seeﬁgure9.2(b))shows
avaluerapidofinitialthedecaycorrelationinarelength,gionξ⊥of,heSdifinferentthenegsystem.ativevSegralues.e´et.Thisal.(1997)minimummeasureddeﬁnesthethe
0/(−the(dependence)oneξof,heSthe=ΨΨNcorrelationΦ−(/0,lengthwhereinNisthethe(⊥)spherecomponentradiusas(inξ⊥our,heS=notationΩ0NΦN=,EEfand)andfor
fraction.olumevtheΦ

InordertomakeoursimulationscomparablewiththeexperimentalresultsofSegre´et.al.
(1997)andalsowithprevioussimulations,H¨oﬂer(2000),weuseasystemsizeofsquared
basewithasideΨ((andheightΨ)ΦandavolumefractionofΦV=Φ.Φ´.Segre´et.
al.(1997)showthatthecorrelationlengthoftheperpendicularcomponent(⊥)becomes
ξ⊥,heS≈0´NforΦV=Φ.Φ).Inoursimulations,weﬁndacorrelationlengthforthe
equivalentspheresoftheorderofξ⊥,heS≈01N.

Inandthecaseperpendicularfor(d)ellipsoidsthecomponents,correlationweseealengthlarger(ﬁg.value9.2),ofwiththeregardcorrelationtothelengthparallelinthe(c)
parallelcomponentξ⊥,heS≈10Nwhereastheperpendicularcomponentdoesnotshowan
ference.difappreciable

9.1.2Collapsingofthespatialcorrelations

Figure9.3(a)and(b)showthe−(/0collapsingoftheSCVFforspheres,inbothcomponents,
wscalingorksquitethewell,distanceandwithweEEfconsiderΦVitasnotewasworthproposedythatitbyisSevgralide´et.foral.changes(1997).ofThethevcollapseolume
fractionbyupto60times.Thecorrelationlengthsthatwefoundare:

ξ⊥,heS=Ω2EEfΦV−(/0;ξ,heS=Ψ´EEfΦV−(/0

whichdoesnotreallydifferfromtheresultsofSegre´et.al.(1997).

(9.2)

10.75

0.5Sph,||C0.25

0010.750.5Ell,||C0.25

(a)

0.0020.0040.80.0200.120

0.4⊥Sph,C0

80120160200−0.40
r)1.20.0020.0040.0200.1200.8(c)⊥Ell,C0.4

corrSpatial9.9.1elations

(b)

4016012080r(d)

0.0020.0040.0200.120

0408012016020004080120160
rrFigure9.2:Spatialcorrelationsfunctionsintheparallel(a,c)andperpendicular(b,d)
componentsoftheSCVFforellipsoidsandspheres,withchangesofthevolumefraction.
TheReynoldsnumberisΩ×ΨΦ−),thespheresradiusEEfjT=Ψ.ΦΨandtheellipsoid
aspect-ratioΔe=Φ.(/Ψ.+.

Intheellipsoidcase(seeﬁgure9.3),wepresenttheSCVFfortheparallel(c)andperpen-
dicular(d)component,respectively.Thecollapsealsoworkswellinthiscase.Thevalues
forthecorrelationlengtharethefollowing:

ξ⊥,Eaa=Ω)EEfΦV−(/0;ξ,Eaa=ΨΦEEfΦV−(/0(9.3)

Thecorrelationlengthforellipsoids,inbothcomponents,issmallerthanforspheres.

elocityVFluctuations

(a)

1.20.0020.0040.0200.1200.8⊥Sph,0.4C0

(b)

0.0020.0040.0200.120

1.20.0020.0020.0040.0040.0200.02010.1200.1200.8(b)(a)0.75⊥Sph,||Sph,0.40.5CC0.250002040−0.40204060
−1/3−1/3r/(ReqΦV)r/(ReqΦV)
1.20.0020.0020.0040.0040.0200.02010.1200.1200.8(d)(c)0.75⊥CEll,||0.5CEll,0.4
0.2500010203040−0.402040
r/(ReqΦV−1/3)r/(ReqΦV−1/3)
Figure9.3:Collapsingofthespatialcorrelationsfunctionfordatainﬁg.1.Ω.

−0.404020r/(ReqΦV−1/3)
1.20.0020.0040.0200.1200.8(c)⊥Ell,0.4C

80604020r/(ReqΦV−1/3)
0.0020.0040.0200.120(d)

Theamplitudeintheparallel()andperpendicular(⊥)componentsofthevelocityﬂuc-
tuations(seechapter+eqs.+.Ψand+.Ω),ispresentedinﬁgure9.4.Theﬁguresarepresen-
ted(in/0alog-logscale.ForΦ.ΦΦ)≤ΦV≤Φ.Φ0,thevelocityﬂuctuationsgrowlinearlyas
≈ΦV(straightlinesuperimposedondata),asforspheresaswellasforellipsoids(Segre´
(1997)).al.et.Forlargervolumefractions(ΦV>Φ.Φ0),theﬂuctuationsarereducedinbothcomponents,
forspheresandforellipsoids.Thedecreaseinthevelocityﬂuctuationsasthevolume
fractioncontainerhasincreasesaﬁxedvcouldolumebeeandxplainedthebyparticlethegroencounterswthofaretheaparticledissipativeencounters,processinsincenature.the
Thereforethevelocityﬂuctuationstendtoreduce(Kalthoffet.al.(1996)).
Itwasmentionedinchapter+above,thatthevelocityﬂuctuationscomponentsforspheres
aremuchlargerthanforellipsoids.Theappearanceofrotationsaroundtheellipsoid

010−110>2⊥Vδ>,<2||Vδ<−210−310

−1−21010ΦV010−110>2⊥Vδ2V>,<||10−2
δ<

−310

Sphδ. ⊥ Sphδ ||−110>2⊥Vδ>,<−2102||Vδ<−310

9.9.2Changeofthecontainersize
. δδ⊥ Ellip Ellip
||

−1−21010ΦV Sphδ. ⊥. δδ|| Ellip Sph
⊥ Ellipδ ||

10−2ΦV10−1
Figure9.4:Thetoppicturespresentthevelocityﬂuctuationsforspheresandellipsoidsin
theparallel()andperpendicular(⊥)componentsasafunctionofthevolumefraction,
inalog-logscale.Thebottompicturecomparestheresultsforbothkindsofparticlesand
components.TheReynoldsnumberisΩ×ΨΦ−),thespheresradiusEEfjT=Ψ.ΦΨandthe
ellipsoidaspect-ratioΔe=Φ.(/Ψ.+.

centerofmass,impliesaworkagainsttheﬂuid,thenthedissipationofenergyispresent
nowfortranslationalandrotationaldegreesoffreedom.Therefore,thedecreaseofthe
velocityismoredramaticforellipsoidsthanspheres.

9.2Changeofthecontainersize
InﬁxedtheprecontainervioussidesectionofΨ0+weewherexaminedthethecharacteristicsystemasthecorrelationvolumelengthfractionwasfound,increaseseqs.with9.2a
9.3.andNowweturntotheanalysisoftheeffectsofvariationsinthecontainersizeonthevelocity

FluctuationselocityV

1(a)

0.6Ell,||C0.2

−0.220100r1(c)

0.6Sph,||C0.2

44881132176

0.6⊥Ell,C0.2

−0.24030044881132176

0.6⊥Sph,C0.2

(b)

10(d)

20r

4488132176

40304488132176

−0.20204060−0.2010203040
rrFigurecomponents9.5:ofSpatialthevelocitycorrelationﬂuctuationsfunctionsforfortheellipsoidsparallel(a,b)and(a,c)andspheres(c,d)perpendicularasa⊥function(b,d)
ofthecontainerside.TheReynoldsnumberisΩ×ΨΦ−),thespheresradiusEEfjT=Ψ.ΦΨ
andtheellipsoidaspect-ratioΔe=Φ.(/Ψ.+.

ﬂuctuations,asexaminedintheworkby(Segre´et.al.(1997)andH¨oﬂer(2000)).Weuse
asmallcontainersize(closetothecharacteristiccorrelationlength),inordertoinvestigate
thesizeeffectsontheﬂuctuations.
Inﬁgure9.5,weshowtheSCVFforspheres(a,b)andellipsoids(c,d)asthecontainer
sizeisincreased.Thecorrelationlengthξ⊥,heSforspheresandellipsoidsdecreasesin
bothcomponentsasthecontainerincreases.However,thedecayforspheresislargerthan
thedecayforellipsoidsintheparallelcomponent.Theperpendicularcomponent,onthe
otherhand,doesnotdisplayanydifference.
W9.6).ealsoFormeasurespherestheandpairellipsoidscorrelationweﬁndfunctionintheaspairthecontainercorrelationsideforisallchangedcontainer(seesizes,ﬁgurea
characteristiclengthoftheorder≈Ω,followedbyamonotonicdecaythatincreaseswith
thecontainerside(seeﬁgure9.6(a)and(b)).Itisinterestingtopointoutthattheratioof

100

(a)

9.9.2Changeofthecontainersize

1761328844(b)

2017617613213268888164444(b)5(a)124EllSphg(r)g(r)38241002.557.51012.5002.557.51012.5
rr617613258844(c)4Sph3g(r)210012345
rFigure9.6:Paircorrelationfunctionforspheres(a),andellipsoids(b)andtheratioofthe
(c).ellipsoidstospheres

thecontainerpairsidecorrelationΨ0+.forΨ´Ω(seespheres9.6to(c)).thatofThisthepeakellipsoidsrevealsemuchxhibitslarhighergerdensitypeakfortheﬂuctuationslarger
forspheresthanforellipsoids.Thebehaviorofthepaircorrelationfunctionsresemble
thepaircorrelationfunctionforliquids,wherethepositionsofneighboringmoleculesare
stronglycorrelated,leadingtoamodulationofthepaircorrelationfunctions(Barratand
Hansen(2003)),whichisverysimilartothatpresentedinﬁgure9.6(a)and(b).
Infromorderthetoinvcorrelationestigatelength,sizeefasinfectstheinwtheorkvofSeelocitygre´et.al.ﬂuctuations,(1997).weTheundertakresultsearevariationspresent
inﬁgure9.7forspheres(dashdottedline)andellipsoids(dashcontinuousline).The
containersidearenormalizedbyA/ΣEEfΦ−(/0).ThevelocityﬂuctuationsσδK⊥,/ih>
(Sepresentgre´et.anal.initial(1997)transitionandHre¨oﬂergion,(2000)),whichhavebetweenastrongΩΦ≤A/dependenceΣEEfΦ−(on/0)the≤ΨΦΦcontainer,aftersideit,
thespheressimulationshowadatasimilararebehaviorindependent,butofwiththeasmallercontainerovside.erallvInaluegeneral,fortheellipsoids.ellipsoidsTheratioand
oftheparallelvelocityﬂuctuationtotheperpendicular⊥component,forspheresand

elocityVFluctuations

1.2

0.8>s/v||Vδ<0.4

0.4

>s/v⊥Vδ<0.2

101

EllipEllipSphSph02060L/(RΦ100−1/314018004080L/(RΦ−1/3)120160
eqeqFigure9.7:Theleftgraphicshowtheparallelcomponentofvelocityﬂuctuationsasthe
numbercontainerisΩside×isΨΦ−),changed.theThespheresrightradiuspictureEEfshojTws=Ψthe.ΦΨvanderticalthecomponent.ellipsoidTheaspect-ratioReynoldsis
Δe=Φ.(/Ψ.+.

ellipsoidsis≈Ω.),inagreementwithSegre´et.al.(1997).
Thisequalvalueforbothkindofparticles,revealsthattheanisotropicbehavioronthe
velocityﬂuctuationcomponentsareindependentontheparticleshape.Thesymmetry
breakinginducedbygravityactsequallyonspheresandellipsoids.
Asmightisdiverdiscussedgewithinchapterincreasing1andbycontainerCaﬂishsize.andOnLuktheeother(1985),hand,theevelocityxperiments,ﬂuctuationssimula-
Htions¨oﬂerand(2000),theoryitsec.isarΨ.+guedhavethatfoundsystems,noevidenceboundedforbysuchwdialls,verdogence.notshoInawapreviouscomparablework,
scalingofvelocityﬂuctuationsbutasaturation,ifthesmallestextensionofthecontainer
exceedsacriticalsize,andthenthedifﬁcultytoﬁndauniquescalinglaw.Inoursimula-
tions,thevariationsofthecontainersize,weremadebychangingtheentiresquaredbase.
Theﬂuctuationsresults,ﬁg.neither9.7for(a)andspheres(b),nordon’tellipsoids,presenteandvidencebehaveforreallythedivcloseergencetotheoftheresultsvgielocityven
bySegre´et.al.(1997).

9.3ConclusionsandOutlook

WsmalleRestudiedynoldsthevnumberariationΩof×ΨΦthe−).Itcontainerwasfoundsizeforthatthespheresspatialandcorrelationsellipsoidsofthesedimentingverticalat
velocityforellipsoidsshowamuchslowerdecayastheperpendicularcomponent.The
equivalentspheresystemreproducesthesameanisotropy,thismatchingrathercloselyto
theresultsofSegre´et.al.(1997).
ThecollapsingemployedinthespatialcorrelationsforΦV≤Φ.Ψworksquitewellfor

102

ConclusionsandOutlook9.9.3

bothspheresandellipsoids,ﬁg.9.3and9.1.Thecorrelationlengthforellipsoidshavea

smallervaluethanforspheres,eq.9.2and9.3.

Thevelocityﬂuctuations,forbothkindsofparticles,alsorevealtheanisotropypresent

inbyusedtransformationcollapseThecorrelations.spatialthe

inthespatialcorrelations.Thecollapsetransformationusedby
ΦV≤ﬂuctuationsΦ.Ψ,agreesdecrease.quiteItiswellimportantwithourtodata,remark,ﬁg.that9.4.thevAfterelocityΦV

are

much

gerlar

for

Itisimportant

ellipsoids.

remark,

that

the

velocity

Se>grΦ.e´Ψ,et.allal.thev(1997),elocityat
spheresforﬂuctuations

ﬂuctuations

for

spheres

10Chapter

Conclusion

Theaimofthisthesisisthestudyofsedimentationofoblateellipsoidsusinganumerical
oblatesimulationellipsoidtechniqueatlowH¨andoﬂerhighandReSchwynoldsarzernumber(2000),andwhichmanyisoblateapplyedtoellipsoidstheatcaselowofandone
moderateReynoldsnumber,inthreedimensions.

OblateOne10.1ellipsoid

Themotionofapieceofpaperoraleaf,asitfallstotheground,isanoldandunsolved
probleminPhysics.Maxwell,HelmholtzandKelvinarejustsomewhohavestudiedthis
problem.Recentexperiments(Fieldet.al.(1997)andBelmonteet.al.(1998))and
simpliﬁedmodelsMahadevan(1996),conﬁrmthatthemotionoffallingobjectsisstillfar
understood.beingfrom

Inourworkthefallingobjectsareconsideredaveryﬂatoblateellipsoid,suchasleavesor
asheetofpaper,settlinginaﬂuid,inathreedimensionalcontainer.Wefoundthreebasic
regimesforthedynamicsofthesystem(steady-falling,oscillatory-periodic,andchaotic).

Thesteady-fallingexhibitsasimilarphysicalbehaviourasobservedexperimentallyby
Fieldet.al.(1997)andBelmonteet.al.(1998).Wehavecharacterizedthedynamics
ofthesteady-fallingregimewhenthekinematicviscosity,droppingheight,andoblate’s
aspect-ratioarechanged.Someconclusionscanbedrawnfromthispartofthework.This
regimeispresentforsmallvaluesofI≈Φ.)−Ψ,ES≈ΨΦΦandisshowninﬁg.5.1-5.4.
TheperiodicbehaviourinoursimulationsisfoundforES∼(ΦΦandsmallaspect-ratios
ΣΔe≤Φ.Ψ).Theverticalorientation,Θoscillateswithdoubletheperiodofoscillationof
theverticalvelocityilandatthesameperiodofthehorizontalvelocityik.Thisperiodic
motionhasalsobeenobservedexperimentallybyBelmonteet.al.(1998),showingthat
correct.essentiallyaresimulationsour

Wequalitatiﬁndvethatouragreementresultswithinthethecaseofsimpliﬁedthesteady-fmodelallingproposedandbyoscillatory(Mahadevanphases,(1996)).areingood

103

104

sedimentationellipsoidsMany10.10.2

Thechaoticbehaviourispresentforlargeraspect-ratios(Δe≥Φ.´)andintheentirerange
ofReynoldsnumbersusedinthiswork.Theseparationbetweenthespatialtrajectories
ofthefallingoblateellipsoiddivergesforsmallvariationsintheinitialorientationΘd,
andgrowsexponentiallyintime.ThevaluefoundfortheLyapunovexponentisλ=
Φ.Φ)Ω±Φ.ΦΦ).Itisworthwhiletopointoutthatwegiveaquantitativemeasureofthe
sensitivitytosmallchangesintheinitialstateofthesystem.
Forthesteady-fallingandoscillatoryregimeweobtainasimilaritylaw,whichisadirect
consequenceoftheinvarianceoftheReynoldsandFroudenumbers.Also,thesimilarity
expressestheindependenceofthephysicalresultsofthegridsize,whichisagoodtest
forthedynamicsofthesetworegimes.
Weconstructaphasediagramthatshowsthreewell-deﬁneddynamicregionsasisalso
shownbyFieldet.al.(1997).Thedifferencewiththeabovereferenceisthatthechaotic
behaviourisassociatedwiththetransitiontochaosthroughintermittencywhichisnot
seeninoursimulations.Thephasediagramisindependentoftheinitialorientation.
Thetransitionforsteady-fallingtooscillatoryandthetransitionfromsteady-fallingto
chaoticregimecanbeunderstoodassecondandﬁrstorderphase,respectivelyandthe
characteristictransienttimeanditsinversebeingtheorderparameter,respectively.

sedimentationellipsoidsMany10.2

Wehavesimulatedthesedimentationofoblateellipsoidsatsmallvolumefraction(ΦV≤
Φ.Ω)andsmallReynoldsnumber(ES≈ΨΦ−)).Wehavefoundthatatintermediatevolume
fractionthesettlingvelocityexhibitsalocalmaximumwhichtoourknowledgehasnever
literature.theinreportedbeenWealsopresentdataatmoderateReynoldsnumber(ES≈0)forthesedimentationve-
locityofoblateellipsoidswichfollowsamonotonicbehaviourasthevolumefraction
ΦVisincreased.AsinthecaseoflowReynoldsnumbertheellipsoidshaveasmaller
sedimentationvelocitythantheequivalentspheres.Thedataforellipsoidsandspheres
followtheRichardson-Zakilaw(1954)withexponents(cEaaTe≈´.Ω,ES=ΨΦ−))and
(c<eS≈(.Φ,ES=0)respectively.
Inaddition,thelocalmaximum,atlowReynoldsnumber,inthevelocitycanberelated
tothenon-monotonicbehaviouroftheverticalorientationoftheellipsoidsalonggravity,
andcanbeexplainedbythe“cluster”formation.Thisnon-monotonicbehaviourisalso
foundinﬁber-likesuspensionsHerzhaftandGuazzelli(1999).AtmoderateESthereis
alsoalargeralignmentofellipsoidswithgravity,comparedtothosewithsmallReyn-
oldsnumber.ThealignmentwithgravityispresentalsoatmoderateReynoldsnumber,
whichisinagreementwiththeorientationalbehaviourofasingleellipsoidFonsecaand
2004).((1)HerrmannAtlowReynoldsnumbertheorientationalorderparameterΨvanishesaroundΦV≈Φ.Φ1
ﬁg.7.7.AsΦVdecreasestheorientationalalignmentwithgravityincreases.Forlow
ReynoldsnumberinthelimitΦV→Φasingleellipsoidalignswithgravity.The

Conclusion

105

alignmentwithgravityisadistinctivefeatureofthesteady-stateregimeforasingleob-
lateellipsoidasreportedinreferencesFonsecaandHerrmann((1)2004)andGaldiet.
al.(2001).Thevanishingoftheorderparameter,atmoderateES,isslightlyshifted
(ΦV≈Φ.Ψ)totherightastheReynoldsnumberincreases(seeﬁg7.12,bottom).All
thesimulationsinthisworkarelocatedinthesteady-fallingphaseforasingleoblate
ellipsoid.

difThefusidifvefusionbehaofviouroblateinbothellipsoidscomponentsin(parallelsedimentationandwasperpendicular),studied.Wehawhichveisfoundacompletelynon-
newandnotreportedintheliterature.Ourcomparisonwiththeequivalentspheresystem
tonon-difellipsoidsfusiveshoprocesswsthatfortheretheisaparallelsimpledifcomponents.fusiveThisprocessresultfortheregardingperpendicularthediffusiandvae
behaviourinspheresagreeswiththeworkof(Rouyeret.al.1999)and(Migueland
2001).-SatorrasastorP

Furthermore,thebehaviourforellipsoidsandsphereswasinvestigatedastheparticle
density,kinematicsviscosityandellipsoidaspect-ratioorsphereradiiwerechanged.It
wasfoundthattheincrementoftheinertialforces,bymeansofthegrowthinparticle
densityorthedecreasinginthekinematicsviscosity,reducesthepresenceofthediffusive
behaviour.Whenthesphereradiiarechanged,wecouldobserveasimilarbehaviour:as
thesphereradiusisincreasedtheinertiagrowsandthusalsotheparticleM.S.D..

TheverticalcomponentoftheM.S.D.passesfromthenon-diffusivebehaviourtothe
esimplexponentdifαfusiforverenon-difgimefusiasvetheregimesaspect-ratiochangesforincreases.(α=FΩ.orΩ)theﬂattenedparallelellipsoidscomponent,to(αthe=
spheres.)(.Ψ

Last,wefoundasimilaritylawthatcollapsequitewellthediffusionprocessanditisvalid
numbers.ynoldsResmallat

Westudiedthevariationofthecontainersizeforspheresandellipsoidssedimentingat
smallReynoldsnumberΩ×ΨΦ−).Itwasfoundthatthespatialcorrelationsoftheparallel
velocityforellipsoidsshowasmallerdecayastheperpendicularcomponent.Theequi-
valentspheressystemreproducesthesameanisotropy,thismatchingrathercloselytothe
resultsofSegre´et.al.(1997).

ThecollapsingemployedinthespatialcorrelationsforΦV≤Φ.Ψworksquitewellfor
spheresandellipsoids.Thecorrelationlengthforellipsoidshasasmallervaluethanfor
spheres.

Thevelocityﬂuctuationsforellipsoidsrevealtheanisotropypresentinthespatialcorrela-
welltions.withTheourcollapsedata.AfterΦtransformationV>Φ.Ψ,usedallthebyvSegrelocitye´et.al.ﬂuctuations(1997),atΦVdecreases.≤Φ.ΨIt,isagreesimportantquite
toremark,thatthevelocityﬂuctuationsforspheresaremuchlargerasforellipsoids.

106

10.3Outlook

Outlook10.10.3

manDespiteyopenthefactsproblems.thatfallingUsingthisbodiesmodelhaveitbeenisnowstudiedpossiblefortosuchastudylongthetime,theresedimentationarestillof
differenttypesofparticlesandthestructureoftheﬂuidsurroundingthem.Futurework
couldbeaddressedinseveraldirections:

•

Thexperimentseillmarth(Wet.al.(1964),Belmonteet.al.(1998)andellyKandWu(1997))haveconﬁrmed,thatthepresenceofvorticesisfundamental,asthe
objectsfallorriseinaﬂuid.Vortexgenerationissuchanimportantpartofﬂuid
dynamicsthatacompletetheorymustbetakenintoaccountinordertounderstand
theroleoftheﬂuidinthemotionoffallingobjects.Then,anaturalfollowingwork
istoundertakeasystematicresearchaboutthevelocityandpressureﬁeldsinthe
dynamicsoffallingobjects.Thisisataskthatouralgorithmisabletogive.

•Theautorotationortheangularmotionthattheﬂatobjectexecutesasitfall,is
animportantresearchinalargenumberoflaboratorystudies.Theeffortscome

animportantresearchinalargenumberoflaboratorystudies.Theeffortscome
mainlybypracticalconsiderationsinmeteorologyastheformationofhailstones;
thedynamicsofaircraftafteritstall,etc.Theautorotationhasbeenfoundinsome
experiments(Fieldet.al.(1997)orMahadevanet.al.(1999))forveryﬂatobjects.
Ofparticularnoteistheﬂatobjectcanoscillateseveraltimesasitfell,increasingits
amplitudeineachoscillationuntilitcompletelyturnedover.Thesimulationofthis
typeofmotionrequiresthatthethicknessobject(smallerellipsoiddiameter)hasto
be,atleastoftheorderofthegridsize.Therefore,thelargerobjectdiameter(small
aspect-ratio)andthecontainersizewillimplyahugenumberofgridpointsinthe
simulationfortheinvestigationofthismotion.Inordertocarryoutthesimulation
ofautorotation,theaboveconditionsmustdemandahighcomputationaleffort.
Thesimulationtechniqueusedinthisthesisinconnectionwiththeparallelized
algorithmversioncanovercomethesedifﬁculties.

•Inthedynamicsofonefallingﬂatobjectwefoundasimilaritylawforthesteady-
fallingandoscillatoryphases(Sections).Ψand).Ω).Theexperimentaldatasup-
portingtheserelationswouldbeveryimportant.

•Wefoundanon-monotonicbehaviourfortheoblateellipsoidsettlingvelocityas
thevolumefractionincreases.Alsotheincreasingellipsoidalignswithgravityas

thevolumefractionincreases.Alsotheincreasingellipsoidalignswithgravityas
thevolumefractiondecreases.Finallyinthedynamicsofmanyellipsoidssedi-
mentation,wefoundananomalousdiffusionbehaviourfortheparallelcomponent
togravity.Uptonow,wedon’tknowanexperimental,simulationortheoretical
resultrelatedwiththissettlinglaworanomalousdiffusion,andanexperimental
workratifyingthesebehavioursisnecessary.

•Suspensionsinnatureandindustrygenerallyinvolvethemixturesofparticlesof
differenttypes,shapesandsizes(e.g.,oblateellipsoids,prolateellipsoidsand

differenttypes,shapesandsizes(e.g.,oblateellipsoids,prolateellipsoidsand
spheres).Thebidispersesuspensionsarenormallymadeoftwodifferentsize
sphericalparticles,butnotalikeshaped.Inthestatisticalphysicsofbidisperse

Conclusion

107

hard-spheresmixturesisknown(AsakuraandOosawa(1954))the“depletioninter-

action”,whichitistheeffectofthelargeparticlestogetherincreasetheavailable

volume,thereforetheentropy,

thetodueinteraction”pletion

important

task

for

accomplish.

Then,particles.smallthefor

shape

and

not

only

olumev

studythe“de-thisof

ferencedif

another

108

10.10.3

Outlook

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II,wψouldΨandlikefortohisthankinvProf.aluableDr.supportHansJ.andHerrmannorientation,forthewithoutgreathim,opportunitythisworktowworkouldathavthee
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IalonghavethetoedistancexpressmyandingratitudeallthetoaspectsDianaofAlonso,lifethatmyaregreatmoreadvawentureayofpartnerPh,ysics.weAlso,were
alwaysside-by-sideinourlifeinGermany.

IamgratefultothemembersoftheICA1forthesupportandthecomradeship,specially
withReinmarM¨uckandStefanSchwarzer,whoguidedmeatthebeginningintheartof
simulations.IwouldliketothankT.Ihleforthephysicalandscientiﬁccriticsconcerning
tomywork.MyspecialthankstoV.R.IyengarandF.Raischelforwrittingcorrections
tothiswork,andalsotoMarliesParsons,whosupportedmeattheICA1concerningall
administrativeaffairs.

Iofwasvfrienshiperyluckinymytolife,meetalotspeciallyofR.friendsCruz,thatA.madeMora,myA.stayinEhl,J.GermanGaribayya,greatL.Le´opez,xpierienceJ.D.
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specialtotheColombiancommunity,forthewonderfultimethatwehadtogetheralong
years.these

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andconstantsupportduringthelasttimeofmythesis.
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Finally,IwanttoexpressmydeepestrespectandadmirationfortheGermanpeoplethat
makemepossiblelearnabitofthisgreatnation.

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