Dynamics of a bouncing droplet onto a vertically vibrated interface

T. Gilet, 1,

∗

D. Terwagne, 1 N. Vandewalle, 1 and S. Dorbolo 1,

†

1

GRASP, Physi s Department B5a,

University of Liège, B-4000 Liège, Belgium

(Dated: April28, 2008)

Low vis osity (

<

100 St)sili on oil droplets are pla ed ona highvis osity (1000 St)oil bath that vibrates verti ally. The vis osity dieren e ensures that the droplet is more deformedthan

thebathinterfa e. Dropletsboun eperiodi allyonthebathwhenthea elerationofitssinusoidal

motion is larger thana threshold value. Thethreshold is minimumfor aparti ular frequen y of

ex itation:dropletandbathmotionsareinresonan e. Theboun ingdroplethasbeenmodelledby

onsideringthedeformationofthedropletandthelubri ationfor eexertedbytheairlayerbetween

thedropletandthebath. Thresholdvaluesarepredi tedandfoundtobeingoodagreementwith

ourmeasurements.

PACSnumbers: 68.15.+e,47.55.Dz,68.03.Cd

Keywords: Dropletphysi s,Vibratedinterfa e,Boun ing

Themanipulationofindividualdropletsbe omes

pro-gressively more important in mi rouidi s, as it is a

promising alternative to uid displa ement in

mi ro- hannels[1,2℄. Adropletmaybe onsideredasa

mi ro-s ale hemi al rea tor with a high e ien y [3℄ or as a

variablefo usopti allens[4℄. Whenadropletislaidon

aliquid bath, its oales en e with the bath oftentakes

a short time sin e the air layer separating the droplet

from the bath has to be drained out. This drainage

maybedelayedbyverti allyvibratingthebath [5℄: the

droplet boun es periodi ally without oales ing. With

this experiment, adropletmaybemanipulated without

any onta t with a solid element, whi h minimizes the

hemi al ontamination. Themanipulation ofboun ing

dropletsisstraightforward: dropletsmovespontaneously

by intera ting with thewavethey produ eon the bath

at ea h impa t[6℄. By using this wave, theyprobethe

surroundingsanddete tthepresen eofotherdropletsor

solidobsta les,theymaybeguided[7℄. Severaldroplets

on the same bath intera t together and experien e

or-bitalmotions[6℄, orform 2D rystallinelatti es[8℄.

Fi-nally,partial oales en eallowslowvis ositydropletsto

be emptied step-by-step. This emptying as ade stops

whendropletsareabletoboun eperiodi ally[9℄.

Couderetal.[5℄investigatedtheboun ingofan

homo-geneoussystem: thedropletandthebatharemadewith

the samevis ous oil(500 St). The verti al position of

thevibratedbathis givenby

A cos(2πf t)

,where

A

and

f

are the for ingamplitude and frequen yrespe tively. Theredu eda eleration

Γ

is denedas

Γ = 4π

2

_Af

2

_/g

.

Periodi boun ing is observed when

Γ

is higher than a riti alvalue

Γ

C

,thethresholdforboun ing. Couderet al. observedthat

Γ

C

−

1 ∼ f

2

, andexplained this

s al-ing by balan ing thegravity, theinertialfor es and the

lubri ationfor eexertedonthedropletbythesqueezed

∗

Ele troni address: Tristan. Giletulg.a .be

†

URL:http://www.grasp.ulg.a .be

airlayer. Deformationsofboththedropletandthebath

werenot onsidered.

Boun ingme hanismsofliquidobje tshavebeen

stud-iedinawiderangeof ongurations: adropletboun ing

ona hydrophobi solid surfa e[10℄ or on an horizontal

wall immersed into an immis ible liquid [11℄, a bubble

boun ing on a water/air interfa e [12℄... As in elasti

solids, the deformation of those liquid obje ts is often

the key ingredient that ensures the boun ing property,

due to surfa e tension. Deformations may be

signi- antly damped by vis ous ee ts when the Ohnesorge

number

Oh = νpρ/σR

is largerthanunity, where

R

is thedropletradiusand

ρ

,

ν

and

σ

arethedensity,the vis- osityand thesurfa etensionof theliquid respe tively.

IntheexperimentofCouder,

Oh ∼ 4

,whi hmeansthat deformationsmaybenegle ted.

In this letter, we investigate theboun ing oflow

vis- osity droplets, for whi h the deformationis important

(

Oh ≪ 1

). The s aling law proposed by Couder [5℄ for thethresholda eleration is notvalidanymore : values

of

Γ

C

belowunityhavebeenobserved[7,9℄. Inorderto fo usonthedropletdeformation,weanalyzethe

boun -ing of low vis osity droplets (1.5 to 100 St) on a high

vis ositybath(1000 St) : thesystemisinhomogeneous

andthedeformationofthebathsurfa eismu h smaller

thanthedropletdeformation. First,wemeasure

Γ

C

for various

ν

and

f

with

R

xed. Then,amodelthat in or-poratesboththedropletdeformationandthelubri ation

for eisdeveloped.

A ontainer lled with 1000 Stsili onoil is xedon

averti allyvibratingele tromagneti shaker. Byusinga

syringe,dropletsofradius

R =

0.765mmwithvis osities of1.5, 10,50and 100 Stare pla edonthebath.

Mea-suredthresholda elerations

Γ

C

areshowninFig.1. The thresholdisdeterminedbyusingtwodierentproto ols.

First,thedropletis reatedwhenthea elerationis

su- ientlyhighforboun ingtoo ur.Thefor ingamplitude

isthenslowlyde reased(

f

xed)untilthedropletstops boun ingand oales eswiththebath(

•

inFig.1). Then, startingfromzero,

Γ

isin reased. Afterea hin rement,

adroplet is laid onthe surfa eof the bath. Whenthis

droplet boun es, the threshold is rea hed (

N

in Fig.1). An hysteresis, i.e. a dieren e in

Γ

C

dedu ed by both methods,isobservedfor

ν = 1.5

St(Fig.1(a)). Forhigh vis ositydroplets(Fig.1(d)),

Γ

C

> 1

,and

∂Γ

C

/∂f > 0

as predi tedin[5℄. Forlowerdropletvis osities,the

thresh-old may be lower than 1, and there is a minimum in

the

Γ

C

(f )

urve. Athigh frequen y, this urvestrongly in reaseswith

f

.

The following model is proposed in order to des ribe

the

Γ

C

dependan e on the for ingfrequen y

f

and the dropletvis osity

ν

. Theow is assumedto be axisym-metri andthemotionof thedropletmass enter(mass

M

) onned to a verti al axis. The droplet boun ing is modelled with two s alar ordinary dierential

equa-tionsdes ribingtheverti alposition

x

c

ofthemass en-terandtheverti aldeformation

η

ofthedroplet(Fig.2) respe tively. The bath deformation is negle ted.

Dur-ingitsight,thedropletexperien esanapparentgravity

M g(Γ cos 2πf t − 1)

in a frame moving with the bath. Moreover,thedropletisstressedbythesurroundingair,

resulting in a verti al for e

F

. The inuen e of air is negligibleonthedropletmovement,ex eptwhenthereis

athinairlayerbetweenthedropletandthebathsurfa e.

Then,

F

anbeestimatedbylubri ationtheory[13℄,and dependsonthethi kness

h

ofthelmanditsrateof de- rease

˙h

. Movementsinsidethedropletalsohavea signif-i antinuen e on theairlmdrainage. This latter an

bemodelledtoleadingorderbyaPoiseuilleowbetween

twoparallelplanarinterfa es. Thebottominterfa eisat

rest (the bath is stati ), while the upper moves with a

verti al velo ity equal to

˙h

and an horizontal velo ity proportional to

˙ηr/R

, where

r

is the radial ylindri al oordinate, and

R

theradius of theunstrained droplet. Therefore

F = c

1

µ

a

R

4



c

2

˙η

h

2

_R

−

˙h

h

3



(1)

where

c

1

,

c

2

arepositive onstants,and

µ

a

isthedynami vis osityof theair. A ordingtothelubri ationtheory,

c

1

= 3π/2

. The parameter

c

2

represents the inuen e oftheowinside thedropletontheowin theairlm.

It annot beestimated by simplearguments. Newton's

se ondlawappliedto thedropletiswrittenas

M

d

2

_x

c

dt

2

= M g(Γ cos 2πf t − 1) + F

(2)

For pra ti alpurposes,weuse

h = x

c

−

R − η

insteadof

x

c

. Theevolutionof

η

ispres ribedbyanenergybalan e intheframeofthemass enterofthedroplet:

d(K + E)

dt

= −P

d

−

P

f

(3)

where

K

is the kineti energy of the motion inside the droplet,

E

istheinterfa ialenergyand

P

d

isthevis ous dissipative power inside the droplet. The power

devel-oped by

F

, alled

P

f

, is supposed to be equalto

c

6

˙ηF

.

0

20

40

60

80

100

0

1

2

3

4

5

f [Hz]

Γ

C

(f,

ν

=1.5cSt)

(a)

0

20

40

60

80

100

0

1

2

3

4

5

f [Hz]

Γ

C

(f,

ν

=10cSt)

(b)

0

20

40

60

80

100

0

1

2

3

4

5

6

7

f [Hz]

Γ

C

(f,

ν

=50cSt)

( )

0

20

40

60

80

100

0

1

2

3

4

5

6

7

f [Hz]

Γ

C

(f,

ν

=100cSt)

(d)

FIG.1: A eleration threshold

Γ

C

for various vis osities of the droplet: (a)

ν = 1.5

St, (b)

ν = 10

St, ( )

ν = 50

St and(d)

ν = 100

St. For

ν = 1.5

St, thresholdsare dier-enta ordingtowhetherthea elerationisin reased (

N

)or de reased(

•

). Thesolidline orrespondstothemodel predi -tion(Eq.9), with oe ientsgiveninTable I. Thedash-dot

line is a t by the s aling of Couder (

Γ

C

− 1 ∼ f

2

). The

verti al dashed line enhan es the resonan e frequen y

ω

res

des ribedinthemodel. Error bars orrespondtothesize of

FIG. 2: Geometri al variables neededto model a boun ing

dropletof undeformedradius

R

:

x

c

is thedistan e between thedroplet enterofmassandthebath,

h

istheminimumair lmthi knessand

η

istheverti aldropletdeformationabout theaxisofsymmetry.

Law Mode2 Mode3 Fit

K = c

3

M ˙η

2

/2

c

3

= 3

/10

c

3

= 1

/7

c

3

= 0

.1

E = c

4

ση

2

/2

c

4

= 16

π/5 c

4

= 40

π/7 c

4

= 10

P

d

=

c

5

νM ˙η

2

/R

2

c

5

= 3

c

5

= 4

c

5

= 3

.3

TABLE I: Constitutive laws for the energy balan e of the

dropletdeformation. Se ond andthird olumnsare

theoret-i al oe ients

c

i

for modes 2 (spheroid)and 3, while the fourth olumn orrespondstothebesttofEq.(9)on

exper-imentaldata.

A onvenientwaytoestimate

K

,

E

and

P

d

asafun tion of

η

refers to the potential ow related to innitesimal apillarywavesatthesurfa eofadroplet[14℄(seeTable

I). The deformation

η

measured experimentally is less than 10% of the initial radius, whi h validates the

lin-ear approa h [15℄. Wesupposethat only themode2 is

ex ited by the boun ing sin ehigher modeshave mu h

higherresonan efrequen ies. Thewhole systemis

writ-tenindimensionlessformbyusing

R

asalengths aleand the apillary time

τ

σ

=

pM/σ

as a time s ale. More-over,Eq.(3)isrepla edby

c

6

timesEq.(2)plus

1/ ˙η

times Eq.(3),in ordertoremovethelubri ationterm.







¨

h + ¨

η = Bo(Γ cos ωt − 1) + c

1

_4πνρ

3µ

a

Oh



c

2

_h

η

˙

2

−

˙h

h

3



(c

3

+ c

6

)¨

η + c

5

Oh ˙η + c

4

η = c

6

Bo(Γ cos ωt − 1) − c

6

h

¨

(4) where

Bo =

M g

σR

is theBond numberand

ω = 2πf τ

σ

is theredu edfrequen y.

Terwagne etal. [16℄observedthedynami s oftheair

lm lo ated between the droplet and the bath using a

mono hromati light: on entri fringes of interferen e

appearwhentheairlmissqueezed. Whenthedroplet

boun es,the motionof thefringes is perfe tly periodi :

noattenuationorphasedrifttakepla eandtheboun ing

is stationary. Ontheother hand,thenumberof fringes

de reases when the droplet does not boun e: the lm

thins. Theperiodi ityofthefringesmotionsuggests

pe-riodi solutions from Eq.(4). Conditions for su h

solu-T = 2π/ω

. Under theassumption of periodi ity, many termsvanish,giving

(

−

R

T

0

ηdt =

c

6

c

4

BoT

R

T

0

˙

η

h

2

dt =

4π

3c

1

c

2

νρ

µ

a

BoT

Oh

(5)

Termsontheright-handsidearealwaysstri tlypositive.

A ordingtotherstrelation,ame hanismofpotential

energystorage(here,thedropletdeformation)shouldbe

taken into a ount (

η 6= 0

). The droplet has to spend moretime in anoblate state (

η < 0

) thanin aprolate state(

η > 0

). A ordingtothese ondequation,internal movements in the liquid phase, relatedto the

deforma-tion rate, must have a signi ant inuen e on the lm

drainageandtheresultinglubri ationfor e. Moreover,a

signi antphaseshiftbetweentheminimumlm

thi k-ness and the maximum ompression must be observed.

Indeed,

R

T

0

˙ηdt = 0

, while

1/h

2

is stri tly positive and

vanisheswhenthelm thi kens. Tohaveapositiveleft

handside in these ond equation, weexpe tthelm to

be the thinnest when the droplet begins to re over its

spheri alshape (

˙η > 0

). All these required onditions show us that this model is minimal: if the model does

nottakeintoa ountallabovelisted onditions,its

pre-di tion fails and no periodi boun ingsolutions an be

found.

The a eleration threshold

Γ

C

required for periodi boun ingmaybeestimatedstartingfromEq.(4). When

Γ < Γ

C

,thedropletdoesnotboun e,theairlmremains thinand

¨

h ≪ Bo

. These ondequationinEq.(4)doesnot depend on

h

anymore. Thedropletbehavesas asimple for edos illator,i.e.

η = c

6

BoB(ω)Γ cos(ωt + φ) −

c

6

Bo

c

4

,

where

B(ω)

and

φ

aretriviallyobtained. Theresonan e frequen yrelatedtothisos illatoris givenby:

ω

res

2

=

c

4

c

3

+ c

6



1 −

c

4

c

3

+ c

6

Oh

2

2



(6)

Tond

h

withtherstequationofEq.(4),itis onvenient to dene the amplitude

H(t)

of the thi knessvariation as

h(t) = H(t)e

c

2

c

6

BoB(ω)Γ cos(ωt+φ)

. Cal ulationsyield

3c

1

µ

a

Oh

4πρνBo

˙

H

H

3

=



BΓ



(c

4

−

c

3

ω

2

) cos(ωt + φ)

−c

5

Ohω sin(ωt + φ)



−

1



e

2c

2

c

6

BoBΓ cos(ωt+φ)

(7)

Byintegratingthisequationover

n

periods(

T = 2π/ω

), weobtain:

H

nT

=



H

0

−2

−

8πρνBo

3c

1

µ

a

Oh

CnT



−1/2

(8) where

 C = (c

4

−

c

3

ω

2

)BΓI

1

(2c

2

c

6

BoBΓ) − I

0

(2c

2

c

6

BoBΓ)

I

k

(x) =

1

_π

R

π

0

e

When

C < 0

, the averaged lm thi kness

H

de reases with timeand the dropletnally oales es. Conversely,

when

C > 0

,

H

divergesand the solutionis not longer valid. Thedroplettakeso,

¨

h

annotbenegle ted any-morein Eq.(4) andboun ingo urs. Thethreshold

a - eleration for boun ing

Γ

C

an thus be dened as the valueof

Γ

su h that

C = 0

. Thisequation hasone pos-itive solution when

c

4

−

c

3

ω

2

_{> 0}

, and no solution in

the other ase. There is a ut-o frequen y

ω

2

c

=

c

4

c

3

above whi h the model annot predi t boun ing (

C

is alwaysnegative). Thisfrequen y orrespondstothe

nat-ural resonan e of mode 2, when the droplet is dire tly

ex ited (i.e. not through the air lm dynami s). It is

always higherthan

ω

res

, related to the for ingthrough theairlmdynami s. Su h afrequen ywasalready

ob-served in [9℄. The urve

Γ

C

(ω)

tends asymptoti allyto a onstantvalue

> 1

when

ω → 0

. Moreover,when

Oh

is su ientlysmall, aminimum in

Γ

C

isobserved fora nite valueof

ω

, lowerthan

ω

res

sin e

∂(BΓ

C

)/∂ω > 0

when

ω < ω

c

. Therefore, nominimumisobservedwhen

ω

res

is omplex,i.e. when

Oh

2

_{> 2(c}

3

+ c

6

)/c

4

.

Inorder to ompare themodelpredi tions to the

ex-perimental data shown in Fig.1(a) to 1(d), a single t

hasbeenmadeon oe ients

c

2

,

c

3

,

c

5

and

c

6

(

c

1

isnot presentin Eq.(9) and

c

4

isxed to

10

). Obtainedvalue

(c

2

, c

3

, c

5

, c

6

) ≃ (25, 0.1, 3.3, 1)

are similar to the values estimated theoreti ally (TableI). The omparison with

experimentsis a eptable, bothqualitativelyand

quan-titatively. In parti ular, the minima for low vis osities

and thedivergen e for high frequen iesare reprodu ed.

Quantitativedis repan ies maybe due to the fa t that

only mode 2 is onsidered in the modelling. The

reso-nan e o urs for a redu ed frequen y

ω

res

< 3

as long as

Oh . 0.47

. The ut-o redu ed frequen y for the mode2is

ω

c

≃

10

. Foranoildropletwith

R =

0.765mm boun inginmode2,resonan eisobservedatamaximum

frequen yof51Hzwhenthevis osityislessthan32 St,

andthe ut-o frequen yofthis mode isabout165Hz.

Thisis onsistentwithourexperimentalobservations.

In on lusion, we have measureda eleration

thresh-oldsfor boun ing dropletson averti ally vibratedhigh

vis osity bath. The for ing frequen y and the droplet

vis ositywerevaried. Thereisa ut-ofrequen yabove

whi h the droplet annot use a mode 2deformation to

boun e. For low vis osity droplets, a minimum in the

Γ

C

(f )

urveisobserved,whi h orresponds tothe reso-nan eofthemode2. Inorder toexplainthese features,

a theoreti al model was developed, that in ludes both

thedeformation of the droplet and its inuen e on the

lmdrainage. Theseee tsarene essaryinorderto

ob-tainperiodi boun ingsolutionssu h as those observed

experimentally: themodel isthus minimal. Byvarying

fourofthesix onstitutive oe ientsofthemodel,itis

possibletottheexperimentaldata.

TG andSDthankFRIA/FNRS fornan ialsupport.

Ex hanges between laboratories have been nan ially

helped by the COST a tion P21. J.M.Aristo(MIT),

J.W.Bush(MIT),H.Caps(ULg),J.P.Le omte(Dow

Corn-ing)andA.Bourlioux(U.Montreal)area knowledgedfor

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