Eu ophys. Le .,54 (1), pp. 45–50 (2001)
EUROPHYSICS LETTERS 1 Ap il 2001
Veloci y fluc ua ions and hyd odynamic diffusion
in sedimen a ion
M.-C. Miguel and R. Pas o -Sa o as
The Abdus Salam In e na ional Cen e o Theo e ical Physics (ICTP)
P.O. Box 586, 34100 T ies e, I aly
( ecei ed 13 No embe 2000; accep ed 23 Janua y 2001)
PACS. 45.70.Qj – Pa e n o ma ion.
PACS. 05.40.-a – Fluc ua ion phenomena, andom p ocesses, noise, and B ownian mo ion.
PACS. 82.70.Kj – Emulsions and suspensions.
Abs ac . – We s udy non-equilib ium eloci y fluc ua ions in a model o he sedimen a ion
o non-B ownian pa icles expe iencing long- ange hyd odynamic in e ac ions. The complex
beha io o hese fluc ua ions, he ou come o he collec i e dynamics o he pa icles, exhibi s
many o he ea u es obse ed in sedimen a ion expe imen s. In addi ion, ou model p edic s
a final elaxa ion o an aniso opic (hyd odynamic) diffusi e s a e ha could be obse ed in
expe imen s pe o med o e longe ime anges.
Despi e he ac ha he s udy o sedimen ing suspensions has a long and well-dese ed
his o y o hei ubiqui ous na u e and applica ions [1], a en ion o he non-equilib ium
densi y and eloci y fluc ua ions in hese sys ems has only been paid la ely. In pa icula , he
na u e o non-equilib ium fluc ua ions in he sedimen a ion p ocess has been a subjec o a long
con o e sy. While heo e ical a gumen s [2] and ex ensi e compu e simula ions [3] sugges ed
ha eloci y fluc ua ions should di e ge wi h he sys em size, he a ailable expe imen al
esul s [4, 5], and he heo e ical analysis in e . [6], ound no e idence o such di e gences.
These appa en ly con adic o y obse a ions may ha e ound a easonable in e p e a ion a e
he expe imen al e idence in e . [7], and he heo e ical s udy by Le ine e al. [8].
Ano he s iking piece in he puzzle o sedimen a ion was ecen ly added by he expe imen-
al wo k o Rouye e al. [9]. In hei expe imen , he au ho s analyzed he ajec o ies and
eloci ies o non-B ownian [10] pa icles sedimen ing in a quasi– wo-dimensional (2d) fluidized
bed, and showed he in insic non-Gaussian na u e o eloci y fluc ua ions. The main conclu-
sions o his wo k a e he non-Gaussian o m o he p obabili y densi y unc ions (PDFs) o
he eloci y fluc ua ions; he aniso opic cha ac e o he pa icle ajec o ies (diffusi e along
he ho izon al di ec ion and supe diffusi e along he e ical one); and he p esence o e y
long- ange co ela ions in he eloci y fluc ua ions along he g a i y di ec ion. New e idence
along some o hese lines is also p o ided in a ecen pape by Cowan e al. [11].
The esul s o e s. [7, 9, 11] pose new ques ions ega ding he p ocess o sedimen a ion,
which ha e no been add essed by p e ious heo e ical app oaches. Ou pu pose in his le e
is o ackle hese ques ions om he poin o iew o he pa icle’s dynamics o asce ain
he chie physical mechanisms unde lying such fluc ua ion phenomena. In o de o do so,
we p opose a model o sedimen a ion in which pa icles expe ience long- ange hyd odynamic
in e ac ions. We s a om he solu ion o he linea Na ie -S okes equa ion o he suspension
in an unbounded incomp essible fluid [1]. We conside a sys em o Npa icles obeying a sys em
c
EDP Sciences
46 EUROPHYSICS LETTERS
o coupled diffe en ial equa ions which we sol e nume ically. In he solu ion o he equa ions,
we keep ack o bo h posi ions and eloci ies o he pa icles, and compu e se e al ele an
s a is ical p ope ies. In ou model, we obse e mos o he expe imen al ea u es epo ed
in e s. [7,9,11], namely, slow and as pa icles, and swi ls and channels in he eloci y field,
which, o e all, yield non-Gaussian eloci y dis ibu ions and a slow ime elaxa ion o he
eloci y au oco ela ions. In addi ion, ou model p edic s a final elaxa ion o an aniso opic
(hyd odynamic) diffusi e s a e, no obse ed in e . [9].
The eloci y o a pa icle nin a dilu e suspension is gi en by he exp ession Un=
mHnm ·Fm, whe e he sum is ca ied ou o e all he pa icles min he suspension, Hnm is
he mobili y enso , and Fm, he ex e nal o ce ac ing on each pa icle, is g a i y g, o ien ed
along he posi i e z-axis [1]. The simples o m o he enso Hco esponds o dilu e suspen-
sions o poin -like pa icles. In his case, he solu ion o he s a iona y Na ie -S okes equa ion
in an unbounded medium yields he so-called Oseen enso H0( )=(I+ ⊗ / 2)/8πη [12],
whe e Iis he iden i y ma ix, he ope a o ⊗s ands o he enso ial p oduc , and ηis he
fluid iscosi y.
We s udy a suspension o monodispe se non-B ownian pa icles ( o which ine ial e -
ec s a e i ele an in a iscous fluid) a e y low concen a ions, whe e he poin -pa icle
assump ion is indeed a good app oxima ion. Ini ially, pa icles a e placed on he same e -
ical xz plane. The o m o he mobili y ma ix in he Oseen app oxima ion ensu es ha
pa icles in such a configu a ion will ne e lea e ha plane. Simula ions a e pe o med
on a sys em o Npa icles in a squa e cell o size L(co esponding o a concen a ion
c=N/L2). Pe iodic bounda y condi ions (PBCs) a e imposed in all di ec ions (including
he y-di ec ion pe pendicula o he ini ial plane) in o de o gua an ee he uni o mi y o he
suspension [13]. To a oid he discon inui ies a ising om unca ing long- ange hyd odynamic
in e ac ions, imposing PBCs amoun s o conside ing Oseen in e ac ions wi h an infini e se
o images o he o iginal sys em [14]. In his way, he eloci y o each pa icle is w i en as
Un=mdH0( nm +d)·g, whe e he index m uns h ough all he pa icles inside a cell
o olume V, nm indica es he ela i e posi ion o a pai o pa icles wi hin ha cell, and d
uns h ough he posi ions o he images o min an infini e numbe o cell eplicas along he
x,y,andzaxes.
Imposing PBCs along he y-axis is ma hema ically equi alen o imposing slip bounda y
condi ions o he fluid eloci y field on effec i e walls pa allel o he sedimen a ion plane,
and loca ed a dis ances ±Ly/2, whe e Ly<L. Sedimen a ion expe imen s a e usually ca -
ied ou wi hin a hin fluid slab confined by pa allel glass pla es. A ealis ic modeliza ion
o his p ocess should hus include wall effec s by imposing no-slip bounda y condi ions on
he walls. By doing so, hyd odynamic in e ac ions become exponen ially sc eened o leng h
scales la ge han he slab hickness Ly. One hen expec s ha exponen ially damped in e -
ac ions in oduce an ex e nal cha ac e is ic leng h in o he p oblem, Ly, which will go e n
he dynamics o he sys em, a ac ha has no been poin ed ou in he expe imen s. On he
o he hand, sho - ange in e ac ions se e ely es ic he ex en o he co ela ions, and ende
he dynamics essen ially diffusi e on all leng h scales, p e en ing he sys em om showing he
collec i e beha io epo ed in he expe imen s. We ha e checked his las poin by pe o ming
simula ions o a sys em wi h eal walls a dis ances compa able o he a e age in e pa icle
sepa a ion. In pa icula , wi h ou ini ial condi ions one ob ains a sum o modified Bessel
unc ions which decay exponen ially as o leng h scales g ea e han Ly. As expec ed, a e
a sho ballis ic ansien , we obse e an essen ially diffusi e beha io , qui e diffe en indeed
om he da a epo ed in e s. [7,9,11]. We hus conclude ha long ange in e ac ions mus
be p ese ed in o de o accoun o he scale compe i ion obse ed in he sys em. Ou model
is based on his simple conside a ion.
M.-C. Miguel e al.:Fluc ua ions in sedimen a ion 47
To compu e Un, we eso o he Ewald summa ion me hod [14], which yields he ollowing
exp ession:
M( )≡
d
H0( +d)=
=1
8πη
d
e c | +d|
2β
| +d|I+
e c | +d|
2β
| +d|+e−| +d|
2β2
√πβ
( +d)⊗( +d)
| +d|2
+
+1
ηV
G
eiG· e−β2G2
G2I−(1 + β2G2)G⊗G
G2.(1)
He e he unc ion e c(x) is he complemen a y e o unc ion, and βis a pa ame e which
con ols he con e gence o bo h he dand Gsums. The ecip ocal space ec o s Ga e such
ha G·d=2πk, whe e kis an in ege . The e ms p opo ional o Ia e he same as o he
Coulomb po en ial [14]; he o he e ms a e in insic o he Oseen enso . As in he Coulomb
case, he e m G= 0 in eq. (1) yields a di e gen con ibu ion. In he elec os a ic case, his
infini e con ibu ion cancels ou a e imposing an o e all cha ge neu ali y condi ion. In he
sedimen a ion p oblem, he G= 0 e m cancels ou a e sub ac ing he mean sedimen a ion
eloci y UM=(N/V )d3 H0( )·g, and hus wo king wi h eloci y fluc ua ions un=
Un−UM. By doing so, he pa icle posi ions a e on a e age fixed, as in he sedimen a ion
expe imen s in a fluidized bed.
To ollow he e olu ion o he ajec o ies and eloci ies o Npa icles, we in eg a e nu-
me ically he 2Ncoupled equa ions d n/d =mM( n− m)·g−UM, whe e Mis gi en by
eq. (1), using an adap i e s ep-size fi h-o de Runge-Ku a algo i hm [15]. We ha e chosen
a con e gence pa ame e β=L/12; o he alues o βwe e also es ed, yielding equi alen
esul s. Simula ions s a om a configu a ion o Npa icles andomly placed on a squa e cell
o size L. Since he Oseen app oxima ion is no alid a sho dis ances, o a oid singula i ies
in he eloci y field we ha e in oduced an ad hoc e y sho - ange epulsi e ha d-co e e m
o he o m exp[−( −2a)/ρ], whe e ais he adius o he pa icles and ρis a small pa ame e
ha we selec equal o 0.1.
The concen a ions desc ibed by ou model a e se e ely limi ed by bo h he ange o
alidi y o he Oseen app oxima ion and he a ailable CPU ime. In ou simula ions, he e o e,
we ha e conside ed concen a ions c≤1%, and cell sizes anging om L= 100a o L= 200a.
We shall see, howe e , ha ou esul s o dilu e concen a ions al eady exhibi mos o he
salien ea u es epo ed in he li e a u e. A e ages we e made o e a leas 100 ealiza ions
s a ing wi h diffe en andom ini ial condi ions.
In he e olu ion o ou model, pa icles build up a complex and highly fluc ua ing pa e n
o eloci y swi ls and channels, e y simila o hose expe imen ally obse ed [7,9]. In fig. 1a),
we show a snapsho o a sys em wi h concen a ion c= 1% and cell size L= 200a.The
numbe o swi ls and hei sense o flow (clock- o an iclock-wise) esul om he collec i e
in e ac ions, and ha e he cons ain o ze o global o ici y n∇×un= 0, as ollows om
he symme ies o he Oseen enso .
A la ge imes, he a e age oo -mean-squa e eloci y fluc ua ions (RMSVF) along he
ho izon al, ¯ux, and e ical, ¯uz, di ec ions g ow wi h he concen a ion c. As nai ely expec ed
om he symme y b eaking induced by g a i y, fluc ua ions a e aniso opic. We measu e
a a io ¯uz/¯ux≃2.5, which seems independen o co L. This obse a ion ag ees wi h he
esul s epo ed in e . [7].
Nex , we ha e measu ed he p obabili y densi y unc ion (PDF) o he eloci y fluc ua ions,
p(uz)andp(ux), no malized so as o ha e ze o mean and uni y s anda d de ia ion. In fig. 2 we
48 EUROPHYSICS LETTERS
a)
20 40 60 80 100
x
50
150
250
350
z
b)
Fig. 1 – a) Snapsho o he eloci y fluc ua ions, showing bo h swi ls and channels. b) T ajec o ies
o a as (le ) and a slow ( igh ) pa icles (see ex ). Uni s gi en in pa icle adii.
plo he in eg a ed dis ibu ion unc ions P+(u)=∞
up(u)du o he downwa d ( igh wa d),
u>0, eloci y, and P−(−u)=u
−∞ p(u)du o he upwa d (le wa d), u<0, eloci y, o
bo h e ical and ho izon al fluc ua ions. In pa icula , he plo s co espond o alues o
c=1%andL= 200a. We obse e ha he ho izon al fluc ua ions a e le - igh symme ic
and e y well app oxima ed by a Gaussian dis ibu ion (solid line in fig. 2a)). On he o he
hand, e ical fluc ua ions a e ai ly asymme ic and appa en ly non-Gaussian.
We now u n ou a en ion o he wo- ime s a is ical p ope ies o he eloci y fluc ua ions.
Fi s , we conside he eloci y au oco ela ion unc ion gα( )=uα(0)uα( )/uα(0)2, o
α=x, z, whe e he b acke s deno e an a e age o e pa icles and ealiza ions, a a fixed ime .
In fig. 3 we depic gα( ) o wo diffe en concen a ions, c= 1% (sys em I), ep esen ed wi h
(◦), and c=0.25% (sys em II), ep esen ed wi h (×), in a box o size L= 200, as well as c=1%
in a smalle box o size L= 100 (sys em III) which we plo wi h (). The main plo ep esen s
ou da a as a unc ion o he escaled ime c ; aw da a a e shown in he inse . Fo bo h gx
and gz, we obse e an ini ial exponen ial decay o he co ela ions wi h a cha ac e is ic ime
p opo ional o c−1. This scaling o gαa sho imes can be unde s ood by means o a simple
mean-field–like a gumen : Gi en he exp ession o he Oseen enso , he eloci y co ela ions
can be w i en as u( )u(0)∼u(0)/ ( ), whe e is he sepa a ion be ween any pai o
0123456
10-6
10-4
10-2
100
012345
10-6
10-4
10-2
100
a) b)
Fig. 2 – In eg a ed dis ibu ions P+and P− o he (a) uxand (b) uz eloci y fluc ua ions in linea -log
scale. The solid line co esponds o an in eg a ed Gaussian dis ibu ion.
M.-C. Miguel e al.:Fluc ua ions in sedimen a ion 49
0 500 1000
0.0
0.5
1.0
0246810
0.0
0.5
1.0
Fig. 3
10-2 10-1 100101
10-1
100
101
Fig. 4
Fig. 3 – Veloci y au oco ela ions as a unc ion o ime. The cu es shown in he inse co espond o
he aw da a, whe eas in he main plo ime has been escaled by he cha ac e is ic ime τ∼c−1.
◦Sys em I, ×sys em II, sys em III (see ex ).
Fig. 4 – Time de i a i e o he mean-squa e displacemen in a double loga i hmic scale. The solid
line wi h slope 1 ep esen s he ballis ic egime.
pa icles. Taking a ime de i a i e, ∂ u( )u(0)∼∂ u(0)/ ( )∼−u( )u(0)/ 2, whe e in
he second s ep we ha e commu ed de i a i e and a e age. A u he simplifica ion conside s
∼c−1/2,i.e., he a e age sepa a ion be ween pa icles. Then, we ha e ∂ u( )u(0)∼
−u( )u(0)/c−1, yielding an exponen ial elaxa ion wi h cha ac e is ic ime τ∼c−1.
A e his ini ial decay, he xco ela ions o he mo e concen a ed sys em I show a
clea nega i e egion. Cu iously, his beha io esembles ha o a dense liquid. Nega i e
au oco ela ions in a dense liquid a e due o backsca e ing effec s a e collisions among
molecules. In ou sys em, howe e , nega i e co ela ions a e due o he pe manence o he
pa icles in a eloci y swi l. As a gued in [9], du ing he cou se o a simula ion some o he
pa icles become pa o eloci y swi ls and spend in hem a conside able amoun o ime.
They can be called slow pa icles and desc ibe coil-like ajec o ies. O he s ( as pa icles)
spend mo e ime inside he channels sepa a ing swi ls, and hei ajec o ies a e much mo e
elonga ed. Bo h channels and swi ls can be obse ed in fig. 1a). In fig. 1b), we plo ypical
ajec o ies o a as and a slow pa icle.
A la e imes he co ela ions o he xcomponen s oscilla e a ound ze o, whe eas he z
au oco ela ions go h ough a second egime o much slowe elaxa ion, and e en ually become
ze o owa ds he end o he simula ion ime. This enhancemen o he zau oco ela ions is
due o he e y exis ence o channels be ween swi ls, inside o which pa icles ollow ballis ic
ajec o ies wi h small fluc ua ions. Channels a e in e up ed by swi ls, bu since hese mus
be c ea ed in pai s o opposi e o ici y (due o o ici y conse a ion), hei c ea ion is cos ly
and only a ew a e p esen in a box o small size. A long ime is hus equi ed o he pa icles
ini ially in a channel o become pa o a ew eloci y swi ls and unco ela e om hei ini ial
condi ions.
To u he explo e he beha io o he sys em a long imes, we ha e also s udied he
mean-squa e displacemen o he pa icles, Rα( )=[ α( )− α(0)]2, which is an efficien
indica o o a possible effec i e diffusi e beha io o he sys em (hyd odynamic diffusion) [3].
Fo he la e , we expec Rα( )=2Dα ,i.e.,dRα( )/d =2Dα≡cons , whe e Dαis an
effec i e diffusion coefficien . In fig. 4 we ep esen he ime de i a i e o Rα( ) o he
displacemen s along he xand zdi ec ions. The pla eau a long imes clea ly indica es ha
50 EUROPHYSICS LETTERS
he displacemen along he x-di ec ion becomes pu ely diffusi e igh a e an ini ial ballis ic
egime (Rx( )∼ 2). The zdisplacemen also becomes e en ually diffusi e, bu a longe imes
scales. This final diffusi e beha io is compa ible wi h he as decay o he ails o he PDFs
shown in fig. 2. We obse e ha DzDx, hence he diffusi e egime is highly aniso opic.
A in e media e imes, we obse e ha Rz( ) can be fi ed o a powe law Rz( )∼ αwi h an
exponen wi hin he ange 1-2. Such beha io was epo ed in [9], whe e expe imen s could
no be un o long enough imes as in ou simula ions. We expec ha expe imen s ca ied
ou o e longe ime scales would also show he e en ual diffusi e beha io along he e ical
di ec ion.
To sum up, we p esen a model o he sedimen a ion o non-B ownian pa icles in an
unbounded fluid ha inco po a es long- ange hyd odynamic in e ac ions and PBCs in he
simples Oseen app oxima ion. This model exhibi s mos o he salien ea u es o he ex-
pe imen s epo ed in e s. [7, 9, 11]. Ou findings can be unde s ood wi hin he pic u e o
slow and as pa icles: Slow pa icles spend mos o he ime wi hin eloci y swi ls and con-
ibu e o he as elaxa ion o he eloci y co ela ions. Fas pa icles mo ing along eloci y
channels ha e s ongly co ela ed (quasi-ballis ic) ajec o ies and a e esponsible o he slow
elaxa ion componen . Fo sufficien ly long imes, all pa icles become pa o enough eloci y
swi ls, and ou model p edic s ha he sys em e en ually elaxes o a hyd odynamic diffusi e
egime, ha could be confi med by expe imen s pe o med o e longe ime spans.
∗∗∗
We hank S. F anz, M. Ka da , I. Pagonaba aga, M. Rub´
ı, and A. Vespigna-
ni, o help ul discussions. The wo k o RPS has been suppo ed by he TMR Ne wo k
ERBFMRXCT980183.
REFERENCES
[1] Russel W., Sa ille D. and Schowal e W.,Colloidal Dispe sions (Camb idge Uni e si y
P ess, Camb idge) 1995.
[2] Ca lisch R. and Luke J.,Phys. Fluids,28 (1985) 259.
[3] Ladd A.,Phys. Re . Le .,76 (1996) 1392.
[4] Nicolai H. and Guazzelli E.,Phys. Fluids,7(1995) 3.
[5] Xue J.-Z., He bolzeime E., Ru ge M. A., Russel W. B. and Chaikin P.M.,Phys. Re .
Le .,69 (1992) 1715.
[6] Koch D. and Shaq eh E.,J. Fluid Mech.,224 (1991) 275.
[7] Seg `
e P., He bolzheime E. and Chaikin P.,Phys. Re . Le .,79 (1997) 2574.
[8] Le ine A., Ramaswamy S., F ey E. and B uinsma R.,Phys. Re . Le .,81 (1998) 5944.
[9] Rouye F., Ma in J. and Salin D.,Phys. Re . Le .,83 (1999) 1058.
[10] Pa icles o which he diffusi e mo ion caused by he su ounding sol en is negligible.
[11] Cowan M. L., Page J. H. and Wei z D. A.,Phys. Re . Le .,85 (2000) 453.
[12] No e ha he sel -in e ac ion e m ∼I/6πηa has been omi ed since we a e in e es ed in eloci y
fluc ua ions.
[13] Spu ious esul s eme ge om he bounda y discon inui y o he densi y when imposing open
bounda y condi ions.
[14] Rapapo D.,The A o Molecula Dynamics Simula ion (Camb idge Uni e si y P ess, Cam-
b idge) 1995.
[15] P ess W. H., Flanne y B. P., Teukolsky S. A. and Ve e ling W. T.,Nume ical Recipes
in C: The A o Scien ific Compu ing, 2nd edi ion (Camb idge Uni e si y P ess, Camb idge)
1992.
