Critical parameters for the partial coalescence of a droplet

T. Gilet, 1,

∗

K. Mulleners, 1 J.P. Le omte, 2 N. Vandewalle, 1 and S. Dorbolo 1,

†

1

GRASP, Physi s Department B5,

University of Liège, B-4000 Liège, Belgium

2

Dow Corning S.A.

Par Industriel-Zone C

B-7180 Senee - Belgium

(Dated: De ember19,2008)

Thepartial oales en eofadropletontoaplanarliquid/liquidinterfa eisinvestigated

experimen-tallybytuningthevis osities ofbothliquids. Theproblemmainlydependsonfourdimensionless

parameters: theBond number(gravity vs. surfa etension), theOhnesorge numbers(vis osityin

bothuidsvs. surfa etension),andthedensityrelativedieren e. Theratiobetweenthedaughter

droplet sizeand themother dropletsize is investigatedasa fun tionof thesedimensionless

num-bers. Globalquantitiessu hastheavailablesurfa eenergyofthedroplethasbeenmeasuredduring

the oales en e. The apillarywavespropagationanddamping arestudiedindetail. Therelation

betweenthese wavesand thepartial oales en e is dis ussed. Additionalvis ousme hanismsare

proposedinordertoexplaintheasymmetri roleplayedbybothvis osities.

PACSnumbers: 47.55.df,47.55.D-,47.55.db

Keywords: Dropletphysi s,PartialCoales en e,Surfa e-ten sion-dr ivenows

I. INTRODUCTION

Liquid droplets are more and more studied in the

framework ofmi rouidi appli ations. Indeed,they

al-lowtomanipulateandtransportverysmallquantitiesof

liquids. Droplets oales en e is probablythe most

on-venient way to mix liquidsin mi rodevi es withoutany

power supply [1, 2℄. Both ontrol and reprodu ibility

of the droplet sizes are therefore very important. For

instan e, one of the main di ulties en ountered is to

obtain a single small droplet, with a typi al size lower

than100

µm

. Su ha hallengeisalsopresentwhen deal-ingwiththepetrolinje tion in ar motors. Thesmaller

thedropletsare,themoree ientthe ombustionis[3℄.

Thepartial oales en eisasuitablewaytoa hievesu h

agoalsin eitprogressivelyemptiesadroplet[4℄.

Inordertoinvestigatepartial oales en e,Charlesand

Mason [5℄ have studied a system omposed by two

im-mis ible uids 1 and 2 of dierent densities (uid 1 is

heavier thanuid 2). This experimentis s hemati ally

representedin Fig.1. A thi k layerof thelighter uid2

is pla edoverthe bulkofuid 1. A millimetri droplet

of uid 1 is then dropped over the system. It rosses

theuid2layerandstaysontheinterfa ebetweenuid

1 and uid 2 for a while depending on the vis osities.

Suddenly,thedroplet oales es. Ingiven onditionsthat

willbedis ussed,thedropletexperien esapartial

oales- en e. Asmallerdropletremainsabovethe1,2interfa e.

The pro ess may o urseveral times, like a as ade: it

ispossibleto getasmu has sixpartial oales en es

be-fore the nal total one [4℄. Note that this experiment

∗

Ele troni address: Tristan. Giletulg.a .be

†

FIG.1: Experimentalsetup: A ontainer is lled with two

immis ibleuids. A dropletofuid1isfallingthroughuid

2,before oales ing withthe uid 1 bulkphase. Thesame

experiment anbe ondu tedwhenuid1islighterthanuid

2(thedropletisgoingup).

istotallyequivalentto adropletofuid 1(lighter than

uid2) omingupthroughuid2,whenapparentgravity

(gravity+buoyan yfor es)is onsidered.

Charles and Mason attempted to explain the

o ur-ren e of partial oales en e by onsidering the ratio of

dynami al vis osities in both uids. But this only

pa-rameteris not su ient to fully understand partial

o-ales en e me hanisms. This experiment was studied in

1993 by Leblan [6℄. He was interested in the

stabil-ity of emulsions. Indeed, partial oales en e

onsider-ably slows down the gravity-driven phase separationof

two immis ible liquids, for instan e when dealing with

impor-his work remained unpublished. In 2000, Thoroddsen

andTakehara[7℄studiedthisphenomenoninmoredetails

foran air/waterinterfa e. Same on lusions as Leblan

emerged on erningtheimpa tof vis osityontheow.

In the sametime, partial oales en e with a surfa tant

addition was des ribed by Pikhitsaand Tsargorodskaya

[8℄. More re ently, in 2006, Blan hette and Bigioni [9℄

haveexplainedtheme hanismofpin h-oinpartial

o-ales en e by using a subtle ombination of both

exper-imental and numeri al te hniques. The onvergen e of

apillarywavesonthetopofthedropletseemedtohave

a ru ialimportan eonthe oales en eout ome. Several

other studieshave been madeabout partial oales en e

in 2006: HoneyandKavehpour[10℄have onsideredthe

boun ing height of the daughter droplet. Aryafar and

Kavehpour [11℄ have fo used on the time s ales of the

partial oales en e. Chen, Mandre, and Feng [12℄

at-temptedto model these times ales,as wellas theratio

between the daughter and the mother droplets. They

re entlyextended theirstudytopolymeri liquids[13℄.

Littleisknownaboutmi roowso urringduringthe

oales en epro ess. First,traditionalexperimental

te h-niques aredi ultto beapplied at droplets ale.

How-ever,PIVexperimentsondroplet oales en eweremade

byMohamed-KassimandLongmire[15℄. Unfortunately,

the droplet was too large to allow partial oales en e.

Thesimplestwaytogetexperimentalinformationabout

mi roowsistoobservetheinterfa eposition,as in[16℄

forinstan e. On anumeri al pointof view,freesurfa e

owswith hangingtopology(su has oales en e,

break-down,andpin h-o)arereallydi ultandexpensiveto

ompute. Despite these di ulties,alot of information

havealreadybeen olle tedaboutmi roowsina

oales- en epro ess.

The present paper onsists in an experimental study

made over many oales en e events, with various

ini-tial droplet sizes, vis osities, and densities of both

u-ids. Two main aspe ts are investigated: the ratio

be-tweenthedaughterandthemotherdropletsandtherole

of apillary wavesonpartial oales en e riteria.

More-over,usingimagepro essing,globalvariablessu hasthe

availablepotentialsurfa eenergyarere ordedduringthe

oales en epro ess. Theirevolutionwithtimeraisenew

questionsandnew hallenges. Therelationbetween

ap-illarywavesand partial oales en eis dis ussed. In

ap-pendix, the numberof su essive partial oales en es is

theoreti ally omputedfordierentpairsofuids.

II. EXPERIMENTALSETUP

Ourexperimentalsetup onsistsinaglassvesselwhi h

is opened at the top. This ontainer is half lled with

theuid 1. A entimetri layerof uid 2isthen gently

poured over the uid 1 (see s hemati view in Fig.1).

A horizontalguide grooveisdug in theverti alborders

of the ontainer in order to redu e themenis us at the

oil/water interfa e (and in order to get a quasi-planar

Liquid

ρ(kg/m

3

₎

_ν(cSt)

W

1000

0.893

87.5%W+12.5%G

1030

1.16

75%W+25%G

1063

1.64

62.5%W+37.5%G

1093

2.65

50%W+50%G

1127

4.74

25%W+75%G

1195

30.1

90%W+10%E

983

1.35

80%W+20%E

969

1.82

70%W+30%E

954

2.23

60%W+40%E

934

2.47

DC-0.65 St

760

0.65

DC-1.5 St

830

1.5

DC-5 St

915

5

DC-10 St

934

10

DC-50 St

960

50

DC-100 St

965

100

80%DC-10 St+20%DC-100 St

940

19.5

60%DC-10 St+40%DC-100 St

946

30.4

40%DC-10 St+60%DC-100 St

953

46.4

20%DC-10 St+80%DC-100 St

959

69.7

TABLE I: Experimental setup: Fluid properties

(G=Gly erol, E=Ethanol, W=water, DC=Dow Corning

200oil).

interfa e),asinterfa e urvatureisknowntomodifythe

drainage time. Moreover, a ording to Blan hette [9℄,

it an alsoinuen e theresultof partial oales en e. A

syringeisusedto reateasmalldropletofuid1, olored

bymethyleneblueforvisualizationpurpose. By hanging

theneedlediameter,theinitialdropletradius

R

i

an be hanged. Betweenea hexperiment,thesetupiswashed

witha etonetoavoidinterfa e ontamination.

The as adeof partial oales en esis studied using a

setof dierentsili on oilsfortheuid 2(Trimethylsilyl

terminated polydimethylsiloxane, under the tradename

Dow Corning 200 uid). Their kinemati vis osity

ν

2

an be easily tuned from

ν

2

=0.65to 100 St. Mixtures aremadeinordertoobtainintermediatevis osities. The

uid1ismadeofamixtureofwaterandal ohol(gly erol

orethanol). Thekinemati vis osity

ν

1

maybemodied by the al ool/water ratio. Densities are also hanged

(seeTableI). Theinterfa ialtensionismeasuredbythe

pendant drop method. It is approximately 45 mN/m

for any oil/water interfa es (a 5mN/m error is

onsid-ered for ea h error bar). It is roughly 40mN/m when

dealingwith agly erol/water mixture. Theaddition of

ethanol greatly de reases the interfa ial tension. It is

approximately 25mN/m for 10%-ethanol, 19mN/m for

20%-ethanol,13mN/mfor30%-ethanol,and9mN/mfor

40%-ethanol. Wehaveexplored thedierentregimes of

partialandtotal oales en e.

In order to ensure that no signi ant mi roows are

presentat thebeginning ofthe oales en epro ess,the

experimentsforwhi hthelmdrainagetimeislessthan

onese ondarereje ted. Thevesselisbigenoughtoavoid

parasiteree tionsof apillarywavesonthewallsduring

the oales en e.

A fast videore order(Redlake MotionPro) is pla ed

nearthesurfa e. Aslighttilt(lessthan10

◦

)isneededto

bythe weight of thedroplet. Moviesof the oales en e

havebeenre ordedupto2000framesperse ond. Sin e

the partial oales en e, s aled on the apillary time, is

about10ms, awhole oales en epro essisusually

ap-tured by about 20 images. The pixel size is about 30

µm

. Innormal onditions,amillimetri dropletis repre-sentedbyabout30pixels. About150 oales en eevents

havebeenre orded. Thedroplet/ ameradistan eis

on-stant. Distan esonthepi turearemeasuredwitha

rel-ativeerrorlessthan5%. Sin ethis errorondistan es is

onstant during a single oales en e experiment, ratios

of distan es are hara terized by an error signi antly

smallerthanfore asted.

Initialandnalhorizontalradiiaremeasuredforevery

experiment. Inorder to getinformation about the

par-tial oales en e pro ess, the whole interfa e is tra ked

on ea h snapshot for ea h experiments. Of ourse, we

suppose that theinterfa e (and even thewhole ow) is

axisymmetri . Itisthenpossibletoestimateglobal

mea-surementssu hasthevolumeabovetheinterfa eorthe

surfa epotentialenergy. An original,highlyrobust,and

threshold-lessinterfa edete tionmethodhasbeen

devel-oped. Notethat theree tedlight an bedete tedas a

part of the interfa e, generating high dis ontinuities in

interfa e shape. These dis ontinuities are dete ted and

deletedin thepost-pro essing.

III. DIMENSIONALANALYSIS

On ethethin lmof uid2belowthedropletis

bro-ken, the dynami s of partial oales en e is

ma ros op-i ally governed by only three kinds of for es:

interfa- ial tension, gravity, and vis osity for es in both uids.

There are seven ma ros opi parameters: surfa e

ten-sion

σ

, densities

ρ

1

and

ρ

2

, kinemati vis osities

ν

1

and

ν

2

, gravitya eleration

g

, andthe initial dropletradius

R

i

(i = 1, 2)

.

As Dooley et al. [17℄ or Thoroddsen and Takehara

[7℄haveshown,thepartial oales en epro essisusually

s aled by the apillary time. This time is the result of

thebalan ebetweeninterfa ialtensionandinertia. It is

givenby

τ

σ

=

r

ρ

m

R

3

i

σ

,

(1)

where

ρ

m

isdened as themeandensity

ρ

1

+ρ

2

2

. There-fore,theinterfa ial tensionhastobethemain for e for

partial oales en etoo ur. Inotherwords,aself-similar

pro essisonlypossiblewhenone for eisdominant(the

surfa e tension). That means that there is no natural

lengths alerelatedto thebalan eoftwofor es.

A ordingtothe

π−

theorem(Vas hy-Bu kingham),it ispossibleto buildonly fourindependentdimensionless

numbers. Threeofthemarederivedfromtheratioofthe

hara teristi times alesofthedierentfor es withthe

apillarytime. Thegravitytimeisgivenby

τ

g

=

s

R

i

g

′

,

(2) (

g

′

is theapparent gravityexperien ed by the droplet),

andthevis ositytimesby

τ

ν1,2

=

R

2

i

ν

1,2

.

(3)

The Bond numberis the square of the ratiobetween

the apillarytimeandthegravitytime

Bo =

(ρ

1

− ρ

2

)gR

2

i

σ

.

(4)

TheOhnesorgenumbersaretheratiobetweenthe

ap-illarytimeandthevis oustimesin bothuids

Oh

1,2

=

ν

1,2

√

ρ

m

√

σR

i

.

(5)

Thelastdimensionlessnumber anbetherelative

dif-feren eofdensity

∆ρ =

ρ

1

− ρ

2

ρ

1

+ ρ

2

.

(6)

Ideally,anydimensionlessquantity anbeexpressedas

afun tion ofthese four parameters,espe ially theratio

Ψ

between thedaughter dropletradius and the mother dropletradius

R

f

R

i

= Ψ



Bo, Oh

1

, Oh

2

, ∆ρ



.

(7)

For large droplets, gravity is known to be as

impor-tant as surfa etension. Therefore, gravity signi antly

a elerates the emptying of the mother droplet [6, 9℄.

Moreover,it attens the initial droplet. The ratio

Ψ

is monotoni allyde reasingwith

Bo

,anditsuggeststhata riti alBondnumber

Bo

c

(Oh

1

, Oh

2

, ∆ρ)

shouldexistfor whi h

Ψ = 0

for

Bo > Bo

c

: oales en ebe omestotal.

As shown by Blan hette and Bigioni [9℄, the partial

oales en epro essis mainly due to the onvergen e of

apillary wavesat the topof thedroplet. These waves

aregeneratedatthebeginningofthe oales en e,bythe

re edinginterfa ebelowthedroplet. Thevis osityfor es

inbothuids,mainlypresentforsmallestdroplets,damp

these apillarywavesandinhibit thepartial oales en e

[6, 7, 9℄. Theratio

Ψ

also monotoni allyde reaseswith bothin reasingOhnesorgenumbers. Criti alOhnesorges

numbers

Oh

1c

and

Oh

2c

maybedened; beyondwhi h oales en ebe omestotal.

Ontheotherhand,whentheBond(resp. Ohnesorge)

number is mu h smaller than the riti al Bond (resp.

Ohnesorge)number,gravity(resp. vis osity) anbe

on-sidered as negligible. The

Ψ

fun tion doesnot depend on

Bo

(resp.

Oh

)anymore. For dropletswithnegligible

BondandOhnesorgenumbers,

Ψ

be omesafun tion

Ψ

0

that only depends on

∆ρ

, the relative dieren e of in-ertialee tsgeneratedbytheinterfa ialtensioninboth

uids. Inthese onditions,thepro essisself-similarsin e

thedensitiesdonot hange betweentwosu essive

par-tial oales en es.

Thepreviousdimensionalanalysisis orre twhenthe

dropletisatrestatthebeginningofthe oales en e. The

radiusistheonlykeyparametertodes ribethedroplet.

Su h an hypothesis is true when mi roows due to the

previousdroplet fall are damped out. The times ale of

theseows anberoughlyestimatedasthetimeneeded

for the thin lm of uid 2 to be drained out. When

thistimeismu hlargerthanthe apillarytime,the

ini-tialmi roows anbe onsideredasnegligible ompared

to the mi roowsgenerated by the oales en e pro ess.

Asalreadymentionned,this onditionhasbeen he ked

for ea h experiment of oales en e. Obviously, the lm

drainagetimeisshorterforadropletsurroundedbyagaz

thanforadropletsurroundedbyaliquid. Therefore,itis

easiertoworkwithaliquid/liquidinterfa easeviden ed

byMohammed-KassimandLongmire[15℄.

IV. RESULTS

A. Qualit ative des ription

Thedierentstagesofa oales en eareshownonthe

snapshots of Fig. 2. Four experiments are presented.

Theyhavebeen hosentoeviden etheinuen eof

Ohne-sorgenumbersonthe oales en e.

Therstexperiment(Fig. 2(a)) orrespondstoaquasi

self-similarpartial oales en e,where theBondnumber

hasaslightinuen eontheresult,whiletheOhnesorge

numbersarenegligible.

Thelm rupture usuallyo urs where the lmis the

thinnest. This latter is o- enter, as it was shown

ex-perimentallybyHartland[18℄andtheoreti allybyJones

and Wilson [19℄. This o- entering ee t is mainly due

to the presen e of a surrounding uid, whi h generates

anoverpressureatthe enterwhileitisdrainedoutward.

Whenthelmisbroken,it qui klyopensdue tohigh

pressure gradients reated by surfa e tension near the

hole. Thoroddsen et al. [16℄ have mentioned that for

water droplets surrounded by air, the hole opening is

dominated by inertia instead of vis osity. In this ase,

Eggersetal. [20℄haveproposedthefollowings alinglaw

fortheholeradius

r

h

∼

 σR

i

ρ

1



1/4

t

1/2

.

(8)

Thislawisbasedonabalan ebetweeninertia and

sur-fa etension;thebiggest urvatureisgivenby

R

i

/r

2

h

. Itis possibletorewritethisequationwiththe apillarytime

 r

h

R

i



2

∼

_τ

t

σ

.

(9)

Measurements of Men ha a et al. [21℄, Wu et al. [1℄,

and Thoroddsen et al. [16℄ onrm this s aling. When

the surrounding uid is vis ous oil instead of air, one

an expe t that vis osityee ts are as mu h important

asinertiaee ts, andthementioneds aling an be ome

obsolete.

Duringtheretra tion,apartofthebulkphaseofuid

1 omes up into the droplet, as shown by Brown and

Hanson [22℄. They observed a slightly olored

daugh-ter droplet when using a olorless mother droplet on a

oloredbath. Thisupwardmovementisdepi tedin[17℄.

Next,theemptyingdroplettakesa olumnshape,and

thenexperien esapin h-othat leadstothe formation

of the daughter droplet. Charles and Mason [5℄ have

suggestedthis pin h-o was due to a Rayleigh-Plateau

instability. Re ently, Blan hette and Bigioni [9℄ have

shownthatthishypothesiswaswrong. Theyhavesolved

Navier-Stokes equations for a oales ing water droplet

surroundedbyair. Whenthetopofthedropletrea hes

its maximum height, they stop the numeri al

simula-tion, set velo ities and pressure perturbations to zero,

andrestartthe omputation. The oales en e,normally

partial,wasobservedtobetotalwiththisowreset. The

pin h-o annotbeduetoaRayleigh-Plateauinstability.

Leblan [6℄, Thoroddsen et al. [7℄, and

Mohamed-Kassim et al. [15℄ have already noted the existen e of

apillary wavesgenerated at the bottom of the droplet

afterthelmbreak-down. Apartofthesewaves

propa-gates faraway, onthe planarinterfa e. The other part

limbs over the droplet and onverges at the top.

A - ordingtoBlan hette[9℄, su ha onvergen egreatly

de-forms the droplet and delays its oales en e, the

hori-zontal ollapse(thepin h-o) ano urbeforethe

emp-tying. Gravity balan es this delay by a elerating the

emptyingofthedroplet. Inotherwords,toomu h

grav-ityee tsleadtoatotal oales en e.Vis ositiesofboth

uids damp the apillary waves, the onvergen e ee t

isredu ed, and oales en e an also be ome total (Fig.

2(b) and 2(d)). The pre ise role played by vis osities

in the apillary waves damping will be detailed in this

paper.

The timeat whi h thepin h-oo urs vary withthe

BondandtheOhnesorgenumbers. Itisusuallybetween

1.4 and 2

τ

σ

. After the pin h-o, the uid below the dropletiseje ted downward,due to thehigh remaining

pressuregradients. This reates apowerfulvortexring

that go down through the uid 1(Fig. 2(a)and 2( )).

Su havortexiswelldes ribedin[23℄,[24℄and[25℄. Sin e

thisvortexisformedwhenthepartial oales en eis

n-ished, it annot have any inuen e on the oales en e

out ome.

The shapeevolution oftheinterfa eis relatively

on-stant from one experiment to another. However, when

Oh

2

isimportant(Fig. 2( )and2(d)),theinterfa etakes a usp-likeshapebeforediving away.

(a)Partial oales en e:

Bo = 0.15

,

Oh

1

= 0.002

,

Oh

2

= 0.013

(b)Total oales en e:

Bo = 0.24

,

Oh

1

= 0.017

,

Oh

2

= 0.005

( ) Partial oales en e:

Bo = 0.03

,

Oh

1

= 0.003

,

Oh

2

= 0.16

(d)Total oales en e:

Bo = 0.03

,

Oh

1

= 0.003

,

Oh

2

= 0.34

FIG.2: Partialandtotal oales e n esofdropletsatliquid/liquidinterfa es. Thewholepro essiss aledbythe apillarytime

τ

σ

. Therst snapshot(Fig. 2(a)) orrespondsto a partial oales e n e when bothOhnesorges are small. The se ond(Fig. 2(b))isatotal oales en emainlyduetothehighvalueof

Oh

1

. Thebothlastsnapshots,Fig. 2( )and2(d),depi tapartial oales en eforanintermediate

Oh

2

,andatotal oales en ewhenthe

Oh

2

ishighenoughrespe tively.

B. The

Ψ

law

Inthisse tion,theratiobetweenthedaughterdroplet

radiusandthemotherdropletradius(the

Ψ

fun tion)is investigated.

1. Asymptoti regime

Thanks to the dimensional analysis, we have shown

thatfornegligible

Bo

,

Oh

1

,and

Oh

2

(surfa etensionis theonlydominantfor e),

Ψ

onlydependsonthedensity relative dieren e. Experimental results show that one

has

In our experiments, the relativedieren e in density is

alwayslessthan20%,andnorelevant orrelationbetween

∆ρ

and

Ψ

has been underlined. Maybethe inuen e of the density relative dieren eis higherfor a liquid/gaz

interfa e.

2. Inuen eof theBondnumber

FIG.3:Bondinuen eontheradiusratio

Ψ

. TheOhnesorge numbersaresmallerthan

7.5 × 10

−3

. Dashedlinesareguides

fortheeyes.

Figure 3 shows the observed inuen e of the Bond

number on

Ψ

, when the Ohnesorge numbers are neg-ligible (i.e.

< 7.5 × 10

−3

). For small Bond numbers,

the asymptoti regime is obtained. As notedin the

di-mensional analysis,anin rease ofthe Bondnumber

re-sults in a de rease of

Ψ

. A ross over is observed at

Bo ≃ 6 × 10

−2

. A ordingtoourdata,we annotassess

abouttheexisten eofany riti alBondnumber

Bo

c

for whi h

Ψ

be omeszero. Mohamed-KassimandLongmire [15℄ have reported total oales en es for Bondnumbers

equal to 10. However, both Ohnesorge numbers were

greaterthan0.01,andvis osityplayedasigni antpart

ontheout omeofthese oales en es.

3. Inuen eoftheOhnesorgenumbers

InFig.4,thedependen eof

Ψ

onOhnesorgenumbersis plotted: Triangles orrespondto

Oh

1

,andbla ksquares to

Oh

2

. Forthe

Oh

1

(resp.

Oh

2

) urve,thesele teddata points orrespond to

Bo < 0.1

and

Oh

2

< 7.5 × 10

−3

(resp.

Oh

1

< 7.5 × 10

−3

). Vis ous dissipation in uid

2leadsto asmoothde reasingof

Ψ

with

Oh

2

, starting fromtheasymptoti regime

Ψ ≃ 0.45

. The riti al Ohne-sorge for partial oales en e

Oh

2c

is about

0.3 ± 0.05

. Leblan [6℄ haveobserved

Oh

2c

≃ 0.32

, whi h is in a - ordan e with our results. The behavior of

Ψ

with in- reasing

Oh

1

istotallydierent: thede reaseoftheratio

FIG.4: Ohnesorge1(

△

)andOhnesorge2(



)inuen es on the radius ratio

Ψ

. TheBond numberis smaller than 0.1, andtheotherOhnesorgenumberissmallerthan

7.5 × 10

−3

.

Dashedlinesareguidesfor theeyes.

Ψ

is verysharp. It immediately vanishes to zero when

Oh

1

> Oh

1c

≃ 0.02 ± 0.005

. This riti al valuewas al-readyobtainedbyBlan hette[9℄(a

√

2

fa torisneeded to have the samedenition of the Ohnesorge number),

andthirteenyearsagobyLeblan [6℄(whofound0.024).

Inthe apillarywavess enario[9℄, therolesofboth

vis- ositiesshouldbethesame. Thedieren ebetweentheir

inuen esisthenamazingandwillbedis ussedlater.

4. Conjugatedinuen es

In the previous se tions, the

Ψ

law has been investi-gatedby varying asingle parameter, both others being

negligible. Here, two parameters over three are varied,

andthethird oneistakennegligible.

The onjugatedinuen eofvis ositiesisshowninFig.

5. Whiletheboundarybetweenpartialandtotal

oales- en esisunambiguousforlargeandintermediate

Oh

2

,it isquite fuzzy for large

Oh

1

. It seemsthat, onsidering forexample

Oh

1

= 0.03

,the oales en eistotalforsmall

Oh

2

(as seen in the previous se tion). Then, it an be partialfor

Oh

2

around0.02,and itbe omestotalagain for greater values of

Oh

2

. This unexpe ted outgrowth ofthepartial oales en eregimeroughly orrespondsto

thestraightline

Oh

1

= Oh

2

.

Inordertotakebothvis ositiesintoa ount,Leblan

[6℄ suggestedthat the riti alparameterforpartial

oa-les en eo urren eisalinear ombinationofboth

Ohne-sorgenumbers. Partial oales en etakespla ewhen

Oh

1

+ 0.057Oh

2

< 0.02.

(11) Thislawis omparedto experimental resultsin Fig. 5.

Thelaw apturesthegeneraltrendofthepartial-to-total

transition, but fails in des ribing the large

Oh

1

partof thediagram.

FIG.5:Phaseplane

Oh

1

-

Oh

2

(

Bo <

0.1

). Experimental re-sults: Total oales en e (



)vs. partial oales en e(

♦

). The dashedline orrespondsto thelinearrelationship(11). The

solidline orrespondstotheequalityofbothOhnesorge

num-bers. Cir ledletters indi atethe positionof the four

snap-shots(Fig. 2).

Figure 6(a) and 6(b) indi ate that the riti al

Ohne-sorgenumbersareslowlyde reasingwhentheBond

num-ber be omes signi ant. This result agrees with data

fromBlan hette[9℄.

C. Theglobal variables evolution

Inordertogetmoreinformationaboutthepartial

o-ales en epro ess,wehavemeasuredglobalquantities

as-so iated to the dete tedinterfa e on ea h experimental

image: thevolume

V

ofuid1abovethemeaninterfa e levelandtheavailablesurfa epotentialenergy

SP E

(as in[16℄).

The available surfa eenergy (resp. the volume

V

of the dropletabovethe mean interfa elevel), normalized

by its initial value, is plotted as a fun tion of time in

Fig.7(a)(resp. inFig.7(b)). Resultsofthreeexperiments

with dierent surrounding vis osities are presented: a

partial oales en e (

Oh

2

≪ Oh

2c

), an attenuated par-tial oales en e (

Oh

2

.

Oh

2c

), and a total oales en e (

Oh

2

> Oh

2c

). TheBondandtheOhnesorge1are neg-ligibleforthethreeexperiments.

Figure 7(a) and 7(b) show that the volume and the

surfa eenergyde reasesare roughlyindependentof the

oales en eissue.

Duringthe main partof the oales en e pro ess, the

surfa ede reaseisremarkablylinear. Therefore,we an

assessthatsurfa epotentialenergyis onvertedinto

ki-neti energy at a onstant rate [32℄ This rate does not

seem to depend on

Oh

2

, and orresponds to a 40%-de reaseinsurfa eenergyduringthe apillarytime. Su h

alinearde reasein surfa eenergywasalreadyobserved

when dealingwiththe initial holeopening, as seen

pre-viously.

The dropletstarts to signi antly empty at

approxi-mately

t = 0.5τ

σ

. As seenin Fig. 7(b), it ispossibleto orre tlyt databythefollowingempiri allaw

V (t)

V (t = 0)

= 1 − C

V

 t

τ

σ



3

.

(12)

A ording to the tting, the oe ient

C

V

is approxi-mately 0.2, and experien es a slight in rease with

Bo

. Indeed, in Fig.7(b), the Bond number is higher for the

partial oales en e experiment than for the total one.

Therefore, gravityenhan es the dropletemptying. The

owrate

Q

throughthemeaninterfa elevel anbe de-du edbydierentiateEq.(12)

Qτ

σ

V (t = 0)

= 3C

V

 t

τ

σ



2

.

(13) D. Capillary waves

AsshownbyBlan hette[9℄, apillarywavesthattravel

along thedroplet are thought to be responsible for the

delay in the verti al ollapse. Sin e this delay allows

the pin h-o pro ess to o uron time (i.e. before the

ompleteemptying), apillary waveshavean important

inuen e on the

Ψ

fun tion. This se tion attempts to answerwhi hmodesmakethe onvergen epossible,and

howtheyaredampedbyvis osity.

1. Wavepropagation

Considering axisymmetri perturbations of an ideal,

irrotationnal,andin ompressibledropletofradius

R

i

,it ispossibletondthedispersionrelationshipofpotential

apillarywavesonaspheri alinterfa e[33℄

(ω

l

τ

σ

)

2

=

l(l

2

_{− 1)(l + 2)}

2l + 1 + ∆ρ

,

(14) where

τ

σ

is the apillary time

pρ

m

R

3

i

/σ

and

ω

l

is the pulsationofmode

l = k × R

i

.

Theinterfa eisgivenby

R(t, θ)

R

i

= 1 +

∞

X

l=0

A

l

P

l

(cos θ) cos ω

l

t,

(15)

where

A

l

istheperturbationamplitude omponent asso- iatedto

P

l

,theLegendrepolynomialofdegree

l

.

Assumingrelativelysmallwavelengths[34℄,we an

as-sess that

l = kR

i

≫ 1

and the dispersion relationship be omes

(ω

l

τ

σ

)

2

=

l

3

2

.

(16)

Sin ethisrelation isindependenton

∆ρ

, adieren e in densitybetweenbothuidsdoesnotgenerateany

(a)

Bo

-

Oh

1

(

Oh

2

< 7.5 × 10

−3

) (b)

Bo

-

Oh

2

(

Oh

1

< 7.5 × 10

−3

)

FIG.6: Phaseplane

Bo

-

Oh

i

. Experimentalresults: Total oales en e(



)vs. partial oales en e(

♦

).

(a) Availablesurfa eenergy (b)Volumeabovethemeaninterfa elevel

FIG. 7: Evolution of the global quantities for a self-similar partial oales e n e (

•

-

Oh

2

≪ Oh

2c

), an attenuated partial oales en e(



-

Oh

2

. Oh

2c

),andatotal oales en e(

N

-

Oh

2

> Oh

2c

).

Experimentally, apillary waves an be dete ted by

subtra ting two su essive images, as shown in Fig. 8.

Aprogressingfrontappearsinblue,whileare edingone

appearsin red.

A ording to these pi tures, the dominant mode an

beestimatedto

l ≃ 11±1

. Itis reatedduringtheinitial holeopening,roughly7.5

◦

belowtheequator. Thelinear

theorypredi tsaphasevelo ity

V

ϕ

τ

σ

R

i

=

ω

l

τ

σ

l

.

(17)

Pi tures give a onstant angular phase velo ity about

2.52radians per apillary time unit, whi h orresponds

to

l ≃ 11

.

The propagation time

t

W

of the waves from bottom totopisroughlyproportionalto

l

−1/2

. Thismeansthat

highermodes(

8 < l

)arriveontopinthesametime,while lowermodes (

l < 8

)arrivelater, separately. Therefore,

the onvergen e annot bedue to these modes. Indeed,

they are still on the way when the onvergen e is

ob-served.

The relative elevation

h

of the top of the droplet is plotted as a fun tion of time in Fig. 9. Su h a

pa-rameter was already studied by Mohamed-Kassim and

Longmire[15, 21℄. Atthebeginning of oales en e,the

relativeelevationisnearly zerosin eonlythebottomof

thedropletmoves. Whenthe apillarywavesarenottoo

damped (partial oales en e),they onvergeat thetop,

andtheelevation in reasesuntil

H

max

,sometimes more than30%oftheinitialradius. Thepropagationtime

t

W

anbeestimatedfromthesedataasthetimebetweenthe

lm rupture and the point of maximal elevation. Both

quantities

H

max

and

t

W

areillustratedinFig. 9. InFig.10,it anbeseenthatthistimeisroughly

0.9τ

σ

, anditisindependentof

Oh

1

and

Oh

2

.

FIG.8: (Coloronline) Capillary wave propagation obtainedbyimage subtra tion. Theblue interfa e isprogressing toward

FIG. 9: Evolution of the height of the droplet

h

for a self-similarpartial oales en e (

•

-

Oh

2

≪ Oh

2c

),anattenuated partial oales e n e(



-

Oh

2

. Oh

2c

),andatotal oales en e (

N

-

Oh

2

> Oh

2c

).

FIG.10: Propagatio ntimevs.

Oh

1

(△)

,

Oh

2

()

,and

Bo

(◦)

. Dashedlinesareguidesfortheeyes.

with in reasingBondnumber. For example,

Mohamed-Kassimand Longmire[15℄obtained apropagation time

approximatelyequalto0.6 apillarytimeunits. The

dis-persionrelationshipofsurfa e-gravitywavesonaplanar

interfa ebetweentwosemi-innitemediaisgivenby

(ωτ

σ

)

2

=

l

2



l

2

_{+ Bo}



,

(18)

where

l

istheangularmomentum

kR

i

,

k

thewave num-ber,and

R

i

the initial dropletradius. For

Bo < 1

, the inuen eof

Bo

onthisdispersionrelationshipis negligi-ble ompared to

l

2

. The phase velo ity is a priori

un-ae ted by the Bond number, and this annot explain

theobservedde reaseinthepropagationtime. Another

explanation ouldresidein theattening of thedroplet

on theinterfa ebefore the oales en e when

Bo . Bo

c

(espe iallysin ethehorizontalradiusisusedforthe

om-putationofthe apillarytimeunit).

Asa on lusion, onvergen eismainlyduetotheee t

ofmodes

l ≥ 10

,thedominantone being

l = 11

. Waves onvergetothetopatapproximately

0.9τ

σ

.

2. Wavedamping

Sin etheamplitudeissmallduringthewave

propaga-tion,itisdi ulttoassessaboutwavedampingbyonly

observing Fig. 8. A better indi ator is the

dimension-lessmaximal elevation

H

max

of the top of the droplet. InFig.11,

H

max

isplotted with respe tto

Oh

1

,

Oh

2

or

Bo

(both other parametersbeing negligible). It an be seenthatbothvis ositiesplaythesamepartindamping

thewaves[35℄: vis osityinuid1isonly1.5timesmore

e ientthaninuid2.

FIG.11: Maximum droplet elevation vs. Ohnesorge 1

(△)

, Ohnesorge2

()

,andBond

(◦)

. Dashedlinesare guidesfor the eyes. The solid line orresponds to the fore ast of the

linearwavestheory(Eq. (26)).

Thevis ousdissipation anbeestimateda ordingto

thepotentialwavesolution[26℄ [36℄.

Thevelo itypotentialisgivenby

Φ ≃

 −A

l

ω

l

l

r

l

_P

l

(cos θ) sin(ωt)

in uid1

A

l

l+1

ω

l

r

−(l+1)

P

l

(cos θ) sin(ωt)

in uid2

.

(19)

Thelo alenergydissipationrateisanalyti ally omputed

as

D =

∂v

i

∂x

j

 ∂v

i

∂x

j

+

∂v

j

∂x

i



.

(20)

The dimensionless total dissipated power is then given

by

P = Oh

1

(1 + ∆ρ)

Z

V

1

D

1

dV + Oh

2

(1 − ∆ρ)

Z

V

2

D

2

dV.

(21)

Thetotalamountofme hani alenergyisgivenby

E

m

= (1 + ∆ρ)

Z

V

1

|v

1

|

2

dV + (1 − ∆ρ)

Z

V

2

|v

2

|

2

dV.

(22)

A ordingto Landau and Lif hitz[26℄, thewave ampli-tudeisde reasingas

A = A

0

e

−γ

l

t

,

(23) where

γ

l

=

|P |

2E

m

.

(24)

On aspheri alinterfa e, thedampingfa tor

γ

l

is given by

γ

l

=

2l + 1

2l + 1 + ∆ρ



(l

2

−1)Oh

1

(1+∆ρ)+l(l+2)Oh

2

(1−∆ρ)



.

(25)

Therefore, a ording to the lineartheory, one should

expe t

H

max

= H

max

0

e

−0.6γ

l

,

(26)

where

H

0

max

orresponds tothemaximalheightrea hed withoutvis ous dissipation(

Oh

i

≪ Oh

ic

). Asit anbe seenonthesnapshotinFig. 8,the0.6fa toristhetime

needed by the waves to travel from bottom to top (in

apillarytimeunits).

AsshowninFig. 11,thelineartheoryseemsto givea

representativepi tureofthewavedamping. However,it

isnotabletoexplainwhyuid2is1.5timeslesse ient

indampingwavesthanuid1. Additionalee tssu has

non-linearitiesanddissipationduringthewaveformation

haveto betakenintoa ount.

3. Capillarywavesandpartial oales en e

Althoughthedelayinverti al ollapseduetothe

on-vergen eof apillarywavesisimportantto getapartial

oales en e, it is not the only determinant fa tor.

In-deed,inFig. 12, oales en ewithanimportantvis osity

in uid 1, the oales en e is total, despite the presen e

of apillary waves. On the other side, in Fig. 13,

oa-les en ewith animportantvis osityin uid 2,the

oa-les en eispartial,althoughthe apillarywavesarefully

damped.

V. DISCUSSION

Theout omeofa oales en eisinuen edbythe

vis- osity of both uids. Criti al Ohnesorge numbers has

beenidentied for whi h apartial oales en e be omes

total.

Until now, the partial oales en e was thought to be

dire tlyrelatedto the onvergen eof apillarywaveson

thetopofthedroplet. The riti alOhnesorgetogetfully

damped apillary wavesis similarin uid 1and in uid

2, and is about

Oh

i

≃ 0.08

. Surprisingly, the riti al Ohnesorgeforthepartial-to-total oales en etransition

is about 4times higher in uid 2and 4times lower in

uid 1: it is possible to observe waves without partial

oales en eandvi eversa .

Therefore, the onvergen e of apillary waves annot

be the only me hanism responsible for partial

oales- en e. When

Oh

1

is high, a me hanism has to be a -tivatedtoenhan etheemptyingofthedroplet,resulting

in a premature total oales en e. Inversely, when

Oh

2

ishigh, anotherme hanismhastogiveadvantagetothe

horizontal ollapse. However,when

Oh

2

istoohigh,this advantagehavetobe omeadrawba k.

When

Oh

1

and

Oh

2

aresimilar,theadditional me ha-nismsbalan ethemselvesandthe apillarywavesremain

the only dominant fa tor. This ould explain the

out-growthofthepartial oales en eregiononthesolidline

ofthe

Oh

1

− Oh

2

map(Fig. 5).

The main trend of movement during a oales en e is

apowerful rotation,as it was shown in the PIV

exper-iments of Mohamed-Kassim et al. [15℄, as well in the

numeri alsimulationsofBlan hetteetal. [9℄andYueet

al. [14℄. Thisnominalmovementiss hemati ally

repre-sentedwithsolidarrowsinFig. 14.

The uid motion is due to the onversionof

interfa- ial energyinto kineti energy. As itwas shown in

se -tionIVC,thekineti energysupplyrateis onstant,and

does not greatly depend on vis osities. This energy is

distributedin ea huidsu has thenormalvelo ityhas

to be ontinuous at the interfa e. When vis ositiesare

negligible,itgivesrisetoan'idealuid'movement.

Inpresen eofvis osity,theregionsofhighvelo ity

gra-dients(visibleonthenumeri alsimulationsofBlan hette

[9℄) arediusing momentum and energy. Sin e the

mo-tionisimposed bytheinterfa e,thehigherthevis osity

is,thebiggerthemassofuidisdispla ed. Inuid1,this

additional movement (represented by dotted arrows in

Fig. 14)tendstoa eleratetheemptyingofthedroplet.

Note that other movements in uid 1 an be enhan ed

byvis ousdiusionwhen

Oh

1

isin reased,espe iallythe ollapseofthenal olumnas awhole.

Inuid2,su haindu edmovementenhan esthe

hor-izontal ollapse. This ould explain the amazingly thin

aspe tofthe olumn-shapedinterfa eonthelateststages

of oales en es when

Oh

2

is high (see snapshotsin Fig. 2( )and2(d)). When

Oh

2

is intermediate,the horizon-tal ollapse is onned to regions lower than the

equa-tor of the initial droplet, and this ensure a partial

o-ales en e (dashed arrows in Fig. 14, snapshot in Fig.

2( )). When

Oh

2

in reases,thevis ousdiusionentrains uidathigherlatitudes,andtheverti alextensionofthe

horizontal ollapse is bigger: oales en e be omestotal

(dashdotarrowsin Fig. 14,snapshotinFig. 2(d)). The

boundary between these two opposite inuen es is

ap-proximatelyrea hed when energy is diused on as ale

oftheorderof

R

i

,theinitial radiusofthedroplet. The riti alOhnesorgeinuid2wasfoundtobeabout0.32.

This means that during the oales en e (

≃ 1.5τ

σ

), the energyisspreadonas aleroughlyequalto

0.5R

i

,whi h is oherentwiththepreviousassumptions.

FIG.12: (Coloronline)Highvis osityinuid1-Capillarywavesandtotal oales e n e(

Bo

= 0.10

,

Oh

1

= 0.020

,

Oh

2

= 0.006

)

FIG. 13: (Color online) High vis osity in uid 2 - No apillary waves and partial oales e n e (

Bo

= 0.03

,

Oh

1

= 0.003

,

Oh

2

= 0.16

)

FIG.14: (Coloronline)Hypotheti s hemati motiontrends

during the oales en e. Thesolid arrows orrespond to the

nominal movement; the dotted arrows to the movement

in-du ed by a high vis osity of uid 1; the dashed arrows to

themovementindu edbyanintermediatevis osityofuid2;

andthe dashdotarrowsto themovementindu edbya high

vis osityofuid2.

Further investigation is needed (for example a

om-plete P.I.V. study) in order to onrm the existen e of

theseme hanisms,andtoassessaboutthepre iseimpa t

ofvis ositiesontheow.

VI. CONCLUSIONS

Partial oales en eonlydependsonfourdimensionless

parameters,theBondnumber(gravity/surfa etension),

theOhnesorgenumbersin bothuids (vis osity/surfa e

tension),andthedensityrelativedieren e. Thepartial

oales en e ano uronlywhentheBondandthe

Ohne-sorgenumbersare below riti alvalues

Bo

c

,

Oh

1c

, and

Oh

2c

,sin esurfa etensionhasto betheonlydominant for e. Otherwise,the oales en eistotal.

An experimental work on 150 oales en e eventshas

beenmadeinordertostudytheimpa tofvis osityand

gravity on the oales en e pro ess. The droplets were

lmedand theimages post-pro essed in order to

deter-minethedropletradii,the apillarywaves,butalso

quan-titiessu hasthepotentialsurfa eenergyalongany

oa-les en epro ess. Surprisingly,itseemsthatthis surfa e

energyis onvertedintokineti energyata onstantrate,

thatdoesnotdepend onthe oales en eout ome. This

observationhasnoexplanationuntil now.

The

Ψ

law, whi h represents the ratio between the daughterandthemotherdropletradii,hasbeen

investi-gatedas a fun tion of

Bo

,

Oh

1

, and

Oh

2

. When these numbersaresmallenough,

Ψ

be omesequalto approxi-mately0.5. Gravityisknowntoa eleratetheemptying

of the droplet, while vis osity damps the waves during

theirpropagation. The riti alOhnesorgeforthe

partial-to-totaltransition is morethan one order ofmagnitude

higherin uid2thaninuid1.

Blan hetteandBigioni[9℄havearguedthatpartial

o-ales en ewasduetothewaves onvergen eonthetopof

thedroplet. Thepropagationandthe dampingof these

waveshas been investigated. The dominant modes are

10 ≤ l ≤ 12

. The damping of these waves is roughly the same in uid 1 and in uid 2. It annot explain

thehuge dieren e between riti al Ohnesorges in both

suspe tedtoenhan eor toavoidpartial oales en e.

Inappendix,apredi tionof as adefeatureswasmade

foragreatnumberofdierentuids. There ordis

prob-ablya11-step as ade,whenusinganinterfa ebetween

mer uryandanothernon-vis ousliquid. Su ha as ade

hasnotbeenobservedfornow. Itwouldimplytheuseof

advan edmi ros opyte hniquetoobservethelatestages

ofthe as ade.

Althoughpartial oales en ebeginstobeunderstood,

many questions remain unexplored. A P.I.V. study

shouldaddressalot oftheseproblems.

A knowledgments

T.G.benetsof aFRIAgrant(FNRS,Brussels); S.D.

is a resear h asso iate of FNRS (Brussels). This work

hasbeennan iallysupportedbyColgate-PalmoliveIn .

Theauthorswouldliketothankespe iallyDowCorning

S.A., Dr. G.Broze, Dr. H. Caps,Dr. M. Bawin, PhD

Student S. Gabriel,and S. Be o fortheir fruitfulhelp.

Thisworkhasbeenalsosupportedbythe ontra tARC

02/07-293.

APPENDIX A:NUMBER ANDSIZEOF

DAUGHTER DROPLETS

It is possible to estimate the maximum number of

daughter droplets for a ouple of uids starting from

their physi al properties (available in [27℄ and [28℄ for

instan e). The maximumdroplet size (limited by

grav-ity-Eq.(4))hasaradiusof theorderof

R

M

≃

s

σBo

c

2ρ

m

∆ρg

,

(A1)

while the minimum droplet radius (vis osity limited

-Eq.(5))isoftheorderof

R

m

≃ max

i=1,2

 ρ

m

ν

i

2

σOh

2

ci



.

(A2)

Supposing that

Ψ ≃ 0.5

during the whole as ade, the upper bound of the possibles numbers of steps in the

as ade

N

isgivenby

max(R

m1

, R

m2

)

R

M

≃ (0.5)

N

_.

(A3)

When

N = 1

,the oales en e istotal.

The apillarytime anbeestimatedforthelargestand

the smallestdroplet. Moreover, wehave to ensurethat

thistime isalwaysveryshort omparedto thedrainage

time (when the droplet stays at rest on a thin lm of

uid 2). When the uid 2 has a dynami al vis osity

signi antly smaller than the uid 1, the lm drainage

doesnot entrainthe uid 1(neither in thedroplet, nor

in the bulk phase). The Reynolds'model [6℄, based on

thelubri ationequation, is of appli ation. The lifetime

LT

isgivenby

LT ≈

3µ

2

R

4

i

4πg(ρ

1

− ρ

2

)h

2

c

,

(A4)

where

h

c

is the riti al lm thi kness when the lm breaks. Whendynami alvis ositiesaresimilar,thelm

drainageentrainsuid 1,and theReynolds'theory have

toberepla edbyIvanovandTraykov'sequation[29℄

LT ≈



_ρ

1

µ

1

R

2

i

32π

2

_g

2

_(ρ

1

− ρ

2

)

2

h

2

c



1/3

.

(A5)

This equation is used to provide a rude estimation of

the lifetime of the droplets in Table II. Note that the

drainagetimeishighlyinuen edbyseveralfa tors:

sur-fa tant[6℄,ele tri elds[30℄,vibrations[31℄,andinitial

onditionsamongothers.

Tables II and III givepredi ted orders of magnitude

of

R

M

,

R

m1

,

R

m2

,theasso iatedlifetimesand apillary times, andthe numberof steps

N

for dierent uids 1. Theuid2isair(A),water(W)ormer ury(M). When

uid2isheavierthanuid1,thedroplet rossesthe

in-terfa efrombottomtotop. Whenthesymbol*isused,

uids1and2areinverted(forexampleadropletof

mer- urysurroundedbybenzeneinsteadofadropletof

ben-zenesurrounded by mer ury). Theresultsin bold have

beenapproximately he kedexperimentally(byCharles

andMason[5℄,Leblan [6℄orourresults).

We an see in Tables II and III that, ontrary to

Thoroddsen's predi tions [3℄, the oales en e of a

mer- urydropletsurroundedbyairis donein only6stages.

Themaximumnumberof as adestepsisobtainedfora

mer urydropletsurroundedbyanon-vis ousliquid,su h

as a etone, benzene, heptane, hloroform or ether. No

morethan 11 stages will beobserved in this ase. The

re ord ould be broken by working with giant droplets

in mi rogravity. Note that a 11-stage as ade is really

hard to observewith atraditional opti al

instrumenta-tion, sin ethe smallestdroplets havearadius of about

0.6 mi rometers. Moreover, these droplets have a

life-time about0.5 millise onds and they probably oales e

inlessthan100nanose onds. The ontinuummedia

ap-proa h isusually valid. Indeed, whenworkingwith two

liquids, the mean free path is learly smaller than the

lastdroplet, about600nanometers. Andwhen working

withaliquidsurrounded byair, theairvis osityavoids

dropletsbelow10mi rometers.

Data are not in agreement with our predi tions

on- erning an aniline droplet surrounded by water. The

anilineexperien esasmu hasthreepartial oales en es,

whileourpredi tionsstateforatotal oales en e.

There-fore,the riti al Ohnesorge

Oh

1c

is higher than0.02 in this ase. Thereasonisstillunknownandneedsfurther

C2H4O2

1049

1.16

27.8

1.4

0.19

7.0

64

27

25

70

4

◦

C3H6O

787

0.39

23.5

1.5

0.14

7.2

6.3

25

9.4

17

5

390.0

1.5

0.022

7.7

6.9

1.4[−3]

0.62

25

7

0.097

0.56

0.017

0.50

0.056

11

C6H7N

1022

4.30

42.9

1.7

0.35

7.9

550

17

160

1.4[3]

1

6.1

8.5

120

1

12[3]

2.7[5]

5.1

410

19

1100

3400

3

C6H6

879

0.74

28.9

1.5

0.18

7.4

21

22

10

12

6

357

1.4

0.029

7.6

27

1.5[−3]

21

20

5

0.09

0.61

0.068

0.54

0.068

11

35.0

1.4

50

36

0.17

56

36

6

1.7

0.091

100

90

CS2

1263

0.30

32.3

1.4

0.12

7.0

4.3

28

9.2

21

5

CCl4

1594

0.61

27.0

1.1

0.12

6.4

10

39

12

41

4

45.0

0.43

19

27

0.18

22

24

6

0.37

72

0.067

37

100

5

CHCl3

1489

0.38

27.1

1.1

0.12

6.4

10

39

12

41

4

28.0

2.0

0.36

19

16

0.28

15

14

6

0.38

110

0.041

56

250

4

357.0

1.5

0.034

8.0

7.7

1.6[−3]

1.0

3.1

7

0.099

0.64

0.019

0.57

0.074

11

◦

C6H12

773

1.15

41.1

3.6

0.83

32

71

0.11

61

89

5

0.91

43

0.18

47

41

6

◦

₇₆₁

_0.65

_15.9

_1.2

_0.15

_6.6

₂₅

₃₆

₁₄

₃₄

₅

≃

30

0.58

31

0.15

28

30

6

0.078

56

76

◦

₈₃₀

_1.50

_17.6

_1.2

_0.20

_6.6

₁₃₀

₃₆

₄₅

₂₄₀

₃

≃

30

1.1

17

0.15

150

390

4

0.43

72

82

◦

₉₃₄

_10.0

_20.1

_1.2

_0.37

_6.7

₅₈₀₀

₃₅

₁₀₀₀

_6.8[4]

₁

≃

30

6.0

8100

0.16

7600

1.3[5]

1

20

140

92

◦

₉₆₀

_50.0

_20.8

_1.2

_0.64

_6.7

_1.4[5]

₃₅

_1.5[4]

_8.3[6]

₁

≃

30

17

2.0[5]

0.16

1.6[5]

1.7[7]

1

65

780

2100

TABLE II:Theoreti alpredi tionof as adesofpartial oales e n e eventsfordierent ouplesofuids. Fluid1isgivenintherst olumnwhileuid2is air(A),

water(W)ormer ury(M).Whenthesymbol*isused,uids1and2areinverted. Notation

a

[b]

orrespondsto

a ×

10

b

965

100

20.9

1.2

0.8

6.7

5.8[5]

35

4.8[4]

6.7[7]

1

≃

30

25

8.2[5]

0.16

5.5[5]

1.3[8]

1

65

2200

1.7[4]

C4H10O

714

0.34

17.0

1.3

0.12

6.8

6.1

30

10

24

5

379.0

1.5

0.019

7.6

5.4

1.4[−3]

0.47

1.7

8

0.095

0.57

0.014

0.51

0.059

11

10.0

1.6

0.26

18

25

0.54

16

36

6

0.47

210

0.062

120

920

2

C12H26

750

1.80

47.0

0.9

30

0.12

110

250

0.89

47

0.38

49

43

6

C2H6O

789

1.52

22.8

1.4

0.22

7.2

100

25

37

130

3

C4H8O2

925

0.49

23.6

1.4

0.14

6.9

12

28

11

21

5

CH2O2

1220

1.32

37.13

1.5

0.21

7.3

71

23

28

77

4

C3H8O3

1261

1185

63.4

1.9

2.4

8.2

3.5[7]

14

1.7[6]

2.1[10]

1

370.0

0.45

7.9

7.0[7]

1.5[−3]

5.9[5]

8.3[10]

1

0.099

0.61

1.8[5]

2400

1.0[7]

Hg

13546

0.11

486.1

1.6

0.099

7.6

0.46

20

5.3

10

6

◦

₁₃₅₃₄

_0.11

₄₇₁

_1.6

_0.098

_7.5

_0.46

₂₂

_5.6

₁₂

₆

CH4O

791

0.75

22.6

1.4

0.17

7.2

25

25

12

17

5

C7H16

684

0.72

379.0

1.5

0.024

7.6

25

1.4[−3]

1.6

17

5

0.095

0.57

0.062

0.51

0.059

11

51.0

0.51

25

22

0.10

17

13

7

0.73

0.054

38

34

C6H14

659

0.58

18.4

1.4

0.16

7.1

15

26

11

17

5

C8H18

703

0.78

21.8

1.5

0.18

7.3

24

23

12

15

5

51.0

3.5

0.56

27

25

0.10

21

16

7

0.77

42

0.063

40

35

6

C7H8

867

0.67

28.4

1.5

0.17

7.4

17

22

10

13

6

28.0

1.1

44

38

0.21

52

43

6

1.4

0.095

110

140

H2O

1000

1.00

72.8

2.3

0.26

9.0

17

9.8

10

6.0

7

7.2[−3]

3900

0.043

10

2.0[4]

1

◦

_0.89

_72.0

_2.3

_0.25

_9.0

₁₄

₁₁

_8.3

_4.3

₇

TABLEIII:Theoreti alpredi tionof as adesofpartial oales e n e eventsfor dierent ouplesofuids. Fluid1isgivenintherst olumnwhileuid2isair(A),

water(W)ormer ury(M).Whenthesymbol*isused,uids1and2areinverted. Notation

a

[b]

orrespondsto

a ×

10

b

[1℄ M. Wu, T. Cubaud, and C-M. Ho, Phys. Fluids 16

(2004).

[2℄ H.A. Stone, A.D. Stroo k, and A. Ajdari, Annu. Rev.

FluidMe h.36 ,381(2004).

[3℄ S.Thoroddsen,Nat.Phys.2 ,223(2006).

[4℄ N.Vandewalle,D.Terwagne,K.Mulleners,T.Gilet,and

S.Dorbolo,Phys.Fluids18 ,091106(2006).

[5℄ G.E. Charles and S.G. Mason, J. Colloid S i. 15 , 105

(1960).

[6℄ Y.Leblan ,Ph.D.thesis,UniversitéParisVII(1993).

[7℄ S.T.ThoroddsenandK.Takehara,Phys.Fluids12 ,1265

(2000).

[8℄ P.PikhitsaandA.Tsargorodskaya,ColloidsSurf.,A167 ,

287(2000).

[9℄ F.Blan hetteandT.P.Bigioni,Nat.Phys.2 ,1(2006).

[10℄ E.M. Honey and H.P. Kavehpour, Phys. Rev. E 73 ,

027301 (2006).

[11℄ H.AryafarandH.P.Kavehpour,Phys.Fluids18 ,072105

(2006).

[12℄ X. Chen, S. Mandre, and J.J. Feng, Phys. Fluids 18 ,

051705 (2006).

[13℄ X. Chen, S. Mandre, and J.J. Feng, Phys. Fluids 18 ,

092103 (2006).

[14℄ P.Yue,C.Zhou,andJ.J.Feng,Phys.Fluids18 ,102102

(2006).

[15℄ Z. Mohamed-Kassim and E.K. Longmire, Phys. Fluids

16 ,2170(2004).

[16℄ S.T. Thoroddsen,K.Takehara,andT.G.Etoh,J.Fluid

Me h.527 ,85(2005).

[17℄ B.S.Dooley,A.E.Warn ke,M.Gharib,andG.

Tryggva-son,Exp.Fluids22 ,369(1997).

[18℄ S.Hartland,Trans.Inst.Chem.Eng.45(1967).

[19℄ A.F. Jones and S.D.R. Wilson, J.Fluid Me h.87 , 263

(1978).

[20℄ J. Eggers, J.R.Lister, and H.A. Stone, J. Fluid Me h.

401 ,293(1999).

[21℄ A.Men ha a-Ro ha,A.Martínez-Dávalos,R.Núñez,S.

Popinet,andS.Zaleski,Phys.Rev.E63 ,046309(2001).

[22℄ C.Hanson and A.H. Brown, Solvent Extra tion

Chem-istry522(1967).

[23℄ A.V.Anilkumar,C.P.Lee,andT.G.Wang,Phys.Fluids

A3 ,2587 (1991).

[24℄ P.N.ShankarandM.Kumar,Phys.Fluids7 ,737(1995).

[25℄ R.W.Cresswell andB.R.Morton, Phys.Fluids 7 , 1363

(1995).

[26℄ L.LandauandE.Lif hitz,Courseontheoreti alphysi s,

6-Fluidme hani s (AddisonWesley,1959).

[27℄ G.W.C. Kaye and T.H. Laby, Tables of physi al and

hemi al onstants, and some mathemati al fun tions

(Longman,1973),14thed.

[28℄ D. Lide, Handbook of Chemistry and Physi s (CRC

Press,2005),83rded.

[29℄ I.B. Ivanov and T.T. Traykov, Int. J. Multiphase Flow

2 ,397(1976).

[30℄ J.S. Eow and M. Ghadiri, Colloids Surf., A 215 , 101

(2003).

[31℄ Y. Couder, E. Fort, C.H. Gautier, and A. Boudaoud,

Phys.Rev.Lett.94 ,177801(2005).

[32℄ The gravitational energy is negligible sin e the Bond

numberissmallerthanunity.

[33℄ A similar development is made inthe Fluid Me hani s

ourseofLandauandLif hitz[26℄.

[34℄ When

l ≥

8

, the approximation of Eq.(14) by Eq.(16) leadstoarelativeerrorsmallerthan10%,nomatterthe

relativedieren eindensity.

[35℄ The 'damping'due to anin rease of the Bondnumber

is less relevant, sin e it is most probably the result of

a attening of the droplet instead of a real dissipative

me hanism.

[36℄ Sin etheReynoldsnumber(theinverseoftheOhnesorge

number)ishigh,thevorti ity anbesupposedtobe

on- entratedina thinboundarylayer. Thepotential

solu-tionisvalidelsewhereandthevis ousdissipation anbe