The critical impact speed for the splash of a drop

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by  Guillaume Riboux

The critical impact speed for the splash of a drop
Guillaume Riboux & José Manuel Gordillo
A´rea de Mec´anica de Fluidos, Departamento de Ingener´ıa Aeroespacial y Mec´anica de Fluidos,
Universidad de Sevilla, Avenida de los Descubrimientos s/n 41092, Sevilla, Spain.
(Dated: March 31, 2014)
Makinguseofexperimentalandtheoreticalconsiderations,inthisLetterwededuceacriterionto
determinethecriticalvelocityforwhichadropimpactingasmooth drysurface,eitherspreadsover
thesubstrateordisintegratesintosmallerdroplets. Thederivedequation,whichexpressesthesplash
thresholdvelocityasafunctionofthematerialpropertiesofthetwofluidsinvolved,thedropradius
4
and themean free path of themolecules composing thesurrounding gaseous atmosphere, has been
1
thoroughlyvalidated experimentally at normalatmospheric conditionsusingeight differentliquids,
0
with viscosities ranging from µ = 3 10−4 to µ = 10−2 Pas and interfacial tension coefficients
2
varying between σ =17 and σ = 72 m×Nm−1. Our prediction·s are also in fair agreement with the
r measured critical speed of drops impacti·ng in different gases at reduced pressures given by Xu et
a
M al[1].

8
2 Thecollisionofadropagainstasolidsurfaceisubiqui- a T=0 e
tousinNatureandispresentinamyriadoftechnological

] and scientific fields comprising ink-jet printing, combus-
n b T=50µs f
tion or surface coating [1–3]. Given the physical prop-
y
d erties of both the liquid and the gas, the atmospheric c g T=160µs
- pressure[1],thesizeofthedropandthephysicochemical
u
properties of the substrate [4], experience reveals that d T=928µs h T=580µs
l
f there exists a critical impact velocity below which the
.
s liquid simply spreads over the surface and above which
c the original liquid volume fragments into tiny droplets i T=210µs j T=260µs
i
s
violently ejected outwards, creating what is known as a
y
h splash (see Fig. 1). In spite of the number of advances k Ha Ht V
p on the subject [1, 5–14], a precise descriptionof the crit- t
[ ical conditions leading to drop splashing, is still lacking A(T) Rt
[15,16]. Indeed,thewellknownequationdeducedtwenty
3
v yearsagobyMundoet al [5,11,17],aswellasmanyem-
FIG. 1. (a)–(h): Sequence of events after the impact of an
3 piricalcorrelations[14], provideexpressionsfor the criti-
ethanoldropletofradiusR=1.04mmfortwodifferentimpact
4 calvelocitywhichdependonlyonthematerialproperties velocities, V=1.29ms−1 (1(a)–(d), left) and V=2.28ms−1
9
of the liquid, and do not take into account that splash-
6 (1(e)–(h),right). Thedropsimplyspreadsoverthesubstrate
. ing is largely affected by the gaseous atmosphere [1]. In forthesmallervalueofV butbreaksintotinydrops,violently
1
this Letter, we provide a theoretical framework which is ejected outwards, for the larger impact velocity. Note from
0
consistentwithourownexperimentaldataandalsowith images 1(f)–(g) that the lamella dewets the substrate before
4
1 all previous findings. cevreeanttisngrepthreesesnptlaesdhindeFpiigcst.ed1(iin)–F(ji)g,.fo1r(wh)h.ichThVe=s2eq.0u1emncse−o1f,
:
v Toelucidatethepreciseconditionsunderwhichadrop illustrates the existence of an intermediate range of impact
i velocitiesforwhichthelamellafirstlydewetsthesolidtocon-
X hitting asolidsurfacesplashesornot,weperformexper-
tactitagainasaconsequenceoftheradialgrowthoftheedge
iments with millimetric drops of radii R formed quasi-
r of the lamella. The splash threshold velocity corresponding
a staticallyatnormalatmosphericconditions. Eightdiffer- to these experiments is V=2.19ms−1. The times in 1(a)–
ent liquids are slowly injected through hypodermic nee- (c) are identical to those corresponding to images 1(e)–(g).
dles of different diameters. Drops generated in this way The sketch in Fig. 1(k) illustrates the definition of the main
are spherical and fall under the action of gravity onto a variables used along the paper.
dry glassslide, with a composition suchthat the liquids,
whosephysicalpropertiesarelistedinTable1ofthesup- dicularly to each other.
plementary material section, partially wet the substrate Figure1showsthedetailedsequenceofeventsrecorded
withastaticcontactangle 20o. TheimpactspeedV is from the instant T=0 at which the drop first contacts
∼
varied by fixing the vertical distance between the exit of the solid. These images reveal that, initially, the drop
the needles and the impactor. To simultaneously record deforms axisymmetrically, with A(T) the radius of the
the impact process fromthe side with two different opti- circularwettedarea[Figs. 1(a)–(b)]andthatanairbub-
cal magnifications and acquisition rates, two high speed ble is entrapped at the center of the drop [3, 18]; how-
cameras focusing the impact region are placed perpen- ever, the presence of this tiny bubble does not affect the
2
splash process. Figure 1(c) illustrates that for T T , a 1.5
e
thin sheet of liquid starts to be expelled from the≥radial 1.5 χ
1
√3
position where the drop contacts the solid, i.e. A(T ),
e 0.5 rt(t)
with Te the ejection time. Of special relevance for the r 1 100 W20e0 300
purposes of this study is to observe the change of trajec-
a(t)
tory experienced by the edge of the sheet as the impact
0.5
velocity increases. Indeed, for the smallest values of V
[Figs. 1(a)–(d)], the lamella spreads tangentially along
0
the solid but, for a range of larger impact velocities, the 0 0.t05 0.1 0t.15 0.2 0.25 0.3
e
liquidinitiallydewetsthesubstrateandcontactsthesub-
strate again [Figs. 1(i)–(j)]. For even higher values of
FIG. 2. Experimental radius of the wetted area compared
V, the front of the lamella dewets the solid [Figs. 1(f)–
with a=√3t (solid line) for We=98, Re=3462 (water drop).
(g)]anddropsarefinallyejectedradiallyoutwards[Figs.
Theradial position of theejecta sheet, r , isalso represented
t
1(g)–(h)]inawaysimilartothe experimentsreportedin fortimest>t ,witht theejection time. Theinitialvelocity
e e
[19–21]. Therefore, the analysis of the images in Fig. 1 of the ejected sheet coincides with that of the radius of the
reveals that, for a splash of the type illustrated in Figs. wetted area at t=t since both r (t) and a(t) are tangent
e t
1(e)–(h)totakeplace,twoconditionsneedtobe fulfilled to each other at the ejection time. The inset represents the
simultaneously: the liquid must dewet the solid and the ratio χ/√3 1, where χ is the coefficient obtained from the
≃
bestfitofafunctionofthetypea=χ√ttotheexperimentally
verticalvelocityimpartedtothefrontpartofthelamella
measuredradiusa(t),foralargerangeofWebernumbersand
needs to be large enough to avoid the liquid to contact two different liquids: water ( : µ 0.9cP, σ 67.5mNm−1)
the solid again. To obtain a splash criterion, it is essen- • ≃ ≃
and a silicon oil ( ) of viscosity µ=10cP and surface tension
tial to observe from Figs. 1(i)–(j) and the movies in the σ 19.5mNm−1. •The ratio χ/√3 1 also for therest of the
≃ ≃
supplementary material section, that the rewetting is a fluidsinvestigated.
consequenceoftheradialgrowthoftherimthickness,H ,
t
caused by capillary retraction. Along the Letter, times, aframeofreferencemovingatavelocitya˙ (seefigure1k),
velocities andpressuresaremade dimensionless using R, yieldsthatfluidparticlesareejectedfroma(t)ataspeed
V,R/V, ρV2 ascharacteristiclength,velocity,time and relative to that of the ground given by va=2a˙= 3/t.
pressure, with ρ the liquid density and lower-caseletters Moreover, since the flux of tangential momentum per
denoting dimensionless variables; the subscript g will be unit length is ρV2Ra(t) [21],[26], we thus conpclude
∝
used to denote gas quantities. Next, the instant Te at thatthe heightha ofthe lamellaatthe intersectionwith
which the lamella is ejected, as well as its initial height the spreading drop, i.e. at the radial position r=a(t), is
and velocity, Ht and Vt respectively (see figure 1k), will ρV2Ra˙2ha ∝ ρV2Ra ⇒ ha ∝ a/a˙2 ∝ t3/2, with dots
be calculated. denoting time derivatives [27–29].
Splashing occurs when the values of both the Weber Figure 2 shows that the measured radius of the wet-
and Reynolds numbers are such that We=ρV2R/σ 1, ted area perfectly matches a=√3t for all the different
≫
Re=ρV R/µ 1, with the dimensionless numbers We impact events and fluids considered, a result that fully
≫
andRemeasuringtherelativeimportanceofinertialand validates our potential flow calculation. However, while
surface tension stresses (We) and inertial and viscous ourexperimentalevidenceindicatesthattheejectasheet
stresses (Re). Consequently, during the characteristic is only produced for t te, the potential flow approach
≥
impact time, R/V, viscous effects are confined to thin predictsthegenerationofalamellafort 0ofvanishingly
boundary layers of typical width RRe−1/2 R [22] small thickness ha t3/2 with fluid velo≥city diverging as
a fact suggesting that the use of p∼otential flow≪theory va t−1/2. ∝
∝
[23, 24], which neglects liquid viscosity,is appropriate to To understand the differences between potential flow
describe the liquid flow at the scale of the liquid drop. results and observations note first that, in analogy with
For the sake of clarity, we report here only the main re- the cases of bubbles bursting at a free interface [30] and
sults of the analysis, being further details provided in Worthington jets [31, 32], the fluid feeding the lamella
the Supplementary Material Section, where using Wag- comes from a region where shear stresses are negligible,
ner’stheory [23]we deduce thatthe radiusofthe wetted namely, a very narrow boundary straddling the drop’s
region evolves in time as a(t)=√3t [25]. Potential flow interface (dark shaded regionin panel 1k), and not from
theoryalsopredictsthat,asaconsequenceofthesudden the boundary layer growing from the stagnation point
inertial deceleration of the liquid when it hits the wall, located at the axis of symmetry.
a flux of momentum is directed tangentially along the Fastfluidparticlesenteringtheliquidsheetarerapidly
substrate [21], giving rise to the ejection of a fast liquid deceleratedwithinthelamelladuethecombinedactionof
sheet, like the one depicted in Figs. 1(c) and 1(f)–(g). boththeviscousshearstressesdiffusingfromthewalland
The application of the Euler–Bernoulli equation at the capillary pressure. The characteristic thickness δ of the
drop’s interface, where the pressure remains constant,in region affected by viscous stresses at a distance h 1
a
∼ ≪
3
a Dv/Dt=a¨ t−3/2 is
0 ∝ e
−4/3 2.9
−0.5 71.13.4 c1Re−1te−1/2+Re−2Oh−2 =a¨h2t =c2t3e/2, (1)
log(te) 3775..13 where we set c1 = √3/2 for simplicity and c = 1.1 ac-
−1 −1/2 counts for the proportionality constant in ht ha t3e/2.
∝ ∝
Figure3a,illustrates that equation(1) verywellapprox-
−1.5
imates the experimental results. Experiments also vali-
date the high–Oh and low–Oh limits of Eq. (1), respec-
−12. 5 2 lo2.5g(Re)3 3.5 4 tively given by Re−1te−1/2 t3/2 te Re−1/2 and
b Re−2Oh−2 t3/2 t R∝e−4/3O⇒h−4/3∝. In addition,
e e
1 Fig. 3b reve∝als that⇒, v ∝t−1/2 and h t3/2, providing
t e t e
0.8 ∝ ∝
further support to our theory.
0.6 Now, following the ideas in [4, 15], we represent in
log(vt)0.4 log(ht) the inset of Fig. 4(b) the capillary number, Ca∗, de-
−1
0.2 −2 3/2 −1/2 fiDnueedzeutsianlg[4]t,hCeav∗aliusecoonfsVtatnattfotrhethsepclaassheotfrawnestittiinogn.suIrn-
0
−3
−1.5 −1 −0.5 faces and low viscosity liquids because, in their case, the
−0.2 log(te) splashanddewettingtransitionscoincide. However,since
−2 −1.5 −1 −0.5
log(te) dewetting is a necessary but not sufficient condition for
drop splashing in our case (see Figs. 1 i–j), we find that
∗
i) Ca varies appreciably with the liquid properties and
FIG. 3. (a) Comparison between the experimentally mea- ii) Ca∗ is larger than the critical capillary number Ca∗
sured valueof theejection time, and theone calculated solv- d
abovewhichthe liquiddewetsthe substrate. Indeed, the
ing equation (1). Each of the solid lines represent the differ-
type oflubricationequationsin[33]representingastatic
ent theoretical results for different Ohnesorge numbers. The
force balance in the direction tangent to the wall, are in-
values of 1000 Oh, are represented in the legend. The ex-
perimentally d×etermined ejection time is very well approxi- tegratedtodetermineCa∗ (seetheSupplementaryMate-
d
mated by the solution of equation (1) and clearly follow the rial for details) and the result, particularized for µ=5cP
low and high Ohnesorge number limits reflected by equation andillustratedinthe insetofFig. 4(b),revealsthatCa∗
(1), te Re−4/3Oh−4/3 and te Re−1/2. (b) The mea- is well above Ca∗.
suredex∝perimentaldatafollow our∝prediction vt t−e1/2 and Thus, to detedrmine the critical speed of an impact-
ht te3/2. We find that ht 2.8ha, with ha =∝√12/πt3e/2 ing drop, we note first that splashing occurs as a con-
∝ ≃
deduced in the supplementarymaterial section
sequence of the vertical lift force ℓ, imparted by the
gas on the edge of the liquid sheet. The lift force
downstreamthejetroot(seeFig. 1k),whichistheregion
results from the addition of two contributions: the
wherethejetmeetsthedrop,isδ/h 1(seeSupplemen-
a∼ lubrication force KlµgVt and the suction force
tary Materials Section). The deceleration, provokes the K ρ V2H . The f∼ormer is exerted at the wedge forme∼d
fluid to accumulate at the edge of the liquid sheet and, u g t t
between the substrate and the edge of the lamella and
consequently, h >h (see Fig. 1k).
t a the latter, at the top part of it (see Fig. 4a). In
To determine t , note from Fig. 2 that, although the
e the Supplementary Material Section we show that K
l
velocitiesoffluid particlesenteringthe jetarev =2a˙ > 2 ≃
a (6/tan (α))[ln(19.2λ/H ) ln(1+19.2λ/H )], with
t t
a˙, both a(t) and r are tangent to each other at the − −
t λthemeanfreepathofgasmolecules,αthewedgeangle
instant of ejection, namely, v = a˙ at t = t . Thus,
t e which, for the case of partially wetting solids considered
since the lamella can only be ejected if its tip advances
here does not seem to significatively depend on liquid
faster than the radius of the wetted area, the condition
viscosity[34] and K 0.3. Since the local gas Reynolds
for sheet ejection is Dv/Dt = ∂p/∂x+Re−1 2v a¨ u≃
number based on H and V is O(10), both the vis-
− ∇ ≥ t t
at t=t . Here, Dv/Dt<0 is the dimensionless acceler- ∼
e cous and inertial contributions to the lift force need to
ation of the material points in the sheet given by the
be taken into account (see the Supplementary Material
momentum equation, p denotes pressure and x mea-
fordetails). Therefore,theverticalforcebalanceperunit
sures the distance from the jet root. To determine t ,
e length, when applied at the edge of the lamella, reads
since δ∼ht, ∇2v∼a˙/h2t ∝ te−1/2/h2t; moreover, the in- ρHt2V˙v ∼ ℓ with Vv the vertical velocity, whose char-
crement of pressure experienced by fluid particles flow- acteristic value at the instant when the liquid front has
ing into the edge of the lamella is the capillary pressure raised a distance H above the substrate is given by
t
We−1/h and thus, ∂p/∂x Re−2Oh−2/h2 with ∆x h ∼
and Oh t= µ/√ρRσ = √∼We/Re the Ohtnesorge n∼umt- V ℓ/(ρH ) where ℓ=K µ V +K ρ V2H .
v ∼ t l g t u g t t
ber. Therefore, the critical condition for sheet ejection (2)
p
4
(seFeorFtigh.ee1dig-je),ofVtheneshedeesttnootbetolacorgnetractthtahnesR˙ol.idaHgearien, a Kuρ−gVt2Ht
v c
Rc is the rim radius of curvature which, once the liquid Vt
dewets the substrate, grows in time as a consequence of α + KlµgVt
capillary retraction. Naming Vr the well known Taylor– b Vt
Culick velocity given by the momentum balance 2σ =
ρVr2Ht, 0.5 Ca∗4 Ca∗d(5cP) 00..68
0.4 2 0.4
Vr = 2σ/ρHt (3) β
0.3 0− 2.5 −2 −1.5 −10.2−2.6−2.4−2.2−2−1.8
and inserting this resuplt into the mass balance log(Oh) log(Oh)
2 0.2
dR /dT V H , one readily obtains that, at the instant
c ∼ r t
when the liquid separates from the substrate, for which 0.1
Palaciosetal.
R H /2, R˙ V . Hence, the splash threshold condi-
tiocn≃reatds β =cV∼/Vr (ℓ/σ)1/2 O(1), with V andV 0 −2.5 −2 −1.5 −1
v r ∝ ∼ v r c
respectively given in Eqs. (2)–(3). Using the expression 6
for the lift force ℓ in Eq. (2), the splash condition can 5 Palaciosetal. 0.3
thus be expressed as β0.2
4 0.1
K µ V +K ρ V2H 1/2 V
β = l g t u g t t O(1). (4) 3 1 2 3 4 5 6 7
σ ∼ [m/s] V [m/s]
(cid:18) (cid:19) 2
To check the validity of Eq. (4), Eq. (1) is used first
1
to calculate the ejectiontime t correspondingto the
e,crit
values of We, Re, ρ/ρg and µ/µg for which the splash 0 −2.5 −2 −1.5 −1
transition is experimentally observed. Once t is log(Oh)
e,crit
known, V = √3/2V t−1/2, H = 2.8R√12/πt3/2 and
t e,crit t e,crit
β isdeterminedthroughEq. (4). Fig. 4(b)demonstrates
FIG. 4. (a) The total lift force arises as the addition of the
thatthesplashthresholdischaracterizedbyanearlycon- forcein thewedgeof angleαandtheforceat thetop partof
stantvalueofβ,independentofthetypeofliquidconsid- thelamella. Theplus/minussigns indicatetheregions where
ered, as predicted by Eq. (4). The open symbols in Fig. the gauge pressure is positive/negative respectively. (b) Val-
4(c), representing the splash threshold velocities calcu- ues of the function β calculated through Eq. (4) for the dif-
ferentliquidsanddropdiametersinvestigated. Experimental
latedsolvingequations(1)and(4)forβ 0.14,arefairly
≃ data from [14] have also been included. The inset represents
close to those measured experimentally, a fact further thevaluesofCa∗=(µV/σ)√3/2t−1/2,witht thesolution
supporting our theory. Interestingly enough,the inset in e,crit e,crit
of equation (1) for the experimental splash threshold veloc-
Fig. 4(c) shows that the splash threshold corresponding
ity given in Fig. 4(c). (c) The open symbols represent the
to all the experimental data in Xu et al[1], where the splash threshold velocities predicted by the solution of equa-
critical speed of drops of different liquids falling within tions (1) and (4) with β 0.14. The inset shows that the
≃
several gases and different pressures is investigated, is splashthresholdvelocityV correspondingtotheexperiments
also characterizedby β 0.14. in Xu et al [1] is also characterized by β 0.14. Our own
≃ data, as well as those in [14] are also inclu≃ded in the inset.
Since the lift forceℓ is dominatedby the termK µ V
l g t
The symbols in black have the same meaning as in Xu et al
(see the Supplementary Material for details) and K is
l [1].
approximatelyconstantbecauseofitslogarithmicdepen-
denceonthephysicalparameters,thesplashcriterion(4)
explanation to the well-known correlation by Mundo et
at normal atmospheric conditions can be approximated
al [5, 14, 15] and to the interesting finding in [35] that
by (µ V/σ) t−1/2 C, with t the solution of equa-
g e,crit ∝ e,crit the splash threshold condition cannot be solely charac-
tion (1) and C a constant. Due to the fact that t
e,crit terized in terms of Oh and Re. Let us emphasize that
Re−4/3Oh−4/3(seeFig. 3a)inthelow–Ohlimit,thepr∝e-
these power law expressions are simply approximations
viousapproximatecriterionresultsinOh Oh5/3Re5/3
g to our theory, expressed by equations (1) and (4).
ReOh8/5 5/3(µ/µ )5/3 C′, where Oh = µ /√ρRσ≡. To conclude, we have deduced a criterion expressing
g g g g
∝
I(cid:16)n the hig(cid:17)h–Oh limit, t Re−1/2 (see Fig. 3a) the splash threshold velocity of a drop impacting on a
e,crit ∝ smooth, dry surface as a function of the liquid density
and, in this case, the approximate splash criterion is
OhgOhRe5/4 ReOh8g/5 5/4(µ/µg) C′′. These aannddvviissccoossiittyy,athnedmthilelimnaentroimcdetrroipcrmaedaiuns,frteheepgaatshdoenfsgiatys
≡ ∝
power law express(cid:16)ions are e(cid:17)xperimentally validated in molecules.
the Supplementary Material Section, giving a physical The authors wish to express their most sincere grati-
5
tude to Professor Alexander Korobkin for useful sugges-
tions, discussions, for providing them with many rele-
vantreferencesonthe subjectandfor his kindinvitation
to JMG to participate in the seminar Mathematics of
Splashing, Edinburgh, June 2013. Useful comments by
Alejandro Sevilla, Javier Rodr´ıguez-Rodr´ıguez and De-
varaj van der Meer are also very much acknowledged.
We are also grateful to Alonso Fernández for providing
uswiththenumericalvaluesofK asafunctionofthelo-
u
calgasReynoldsnumber. Thisworkhasbeensupported
by the Spanish MINECO under ProjectDPI2011-28356-
C03-01, which has been partly financed through Euro-
pean funds.
6
SUPPLEMENTARY MATERIAL
TABLEI.Physicalpropertiesofthedifferentfluidsused,drop
radius and thecorresponding Ohnesorge numbers.
Experiments
ρ σ µ R ℓ Oh 103
σ
Since the liquids with physical properties given in ta- (kg/m3) (mN/m) (cP) (mm) (mm) (×-)
ble 1 are injected quasi-statically, the radius R of the (a) N 789 24.0 0.3 1.03 1.76 2.4
drops generated are approximately given by R/D (b) 1000 71.8 0.95 1.96 2.71 2.5
(ℓ /D )2/3, with D the diameter of the injection tucb∝e, •(cid:7) 1000 71.8 0.95 1.74 2.71 2.7
ℓ σ= (cρg/σ)−1/2 thce capillary length, ρ the liquid den- ◮ 1000 71.8 0.95 1.63 2.71 2.8
σ 1000 67.5 0.9 1.45 2.62 2.9
sity, g the acceleration of gravity and σ the interfacial (c) ⋆• 791 23.5 0.6 1.53 1.74 3.5
tension coefficient. The acquisition rate of the camera ⋆ 791 23.5 0.6 1.05 1.74 4.2
used to describe the overall impact process varied be- (d) (cid:4) 789 22.6 1.0 1.53 1.71 6.1
tween 17241 and 29197 frames per second and the reso- (cid:4) 789 22.6 1.0 1.04 1.71 7.3
(e) ◭ 854 17.2 1.3 1.34 1.43 9.1
lution in microns of the captured images varied between
◭ 854 17.2 1.3 0.86 1.43 11.4
16.60 to 31.91 microns/pixel. The high speed camera (f) H 875 17.8 1.7 1.37 1.44 12.0
used to record the impact details was operated between H 875 17.8 1.7 0.92 1.44 14.7
105 and 641509frames per second, providingspatial res- (g) (cid:7) 913 18.6 4.6 1.32 1.44 30.5
olutions ranging from 4.23 to 14.00 microns/pixel. (cid:7) 913 18.6 4.6 0.89 1.44 37.1
The characteristic thickness δ √νt of the region (h) 1000 19.5 10.0 1.32 1.41 62.2
affected by viscous stresses at a di∼stance h 1 down- • 1000 19.5 10.0 0.90 1.41 75.3
a •
∼ ≪
stream the jet root (see Fig. 1k), which is the region (a) Acetone, (b) Water, (c) Methanol, (d) Ethanol, (e)
where the jet meets the drop, with t h /v is Decamethyltetrasiloxane, (f) Dodecamethylpentasiloxane,
a a
∼ (g)Poly(Dimethylsiloxane) and (h) 10 cP Silicone Oil.
δ Re−1/2(h v )−1/2 (Ret)−1/2 . (5) Oh=√We/Re=µ/√ρRσ.
a a
h ∝ ∼
a
For all our experimental data, δ/h 1 (see Fig. 5), and
thus the lamella is decelerated byav∼iscous shear stresses 4
[10] at a tiny distance h from the jet root and also 2.9
a
bythecapillarypressure∼. Thedecelerationalsoprovokes 3 7.3
11.4
the fluid to accumulate at the edge of the liquid sheet δ
37.1
and,consequently,h >h (seeFig. 1k). Theanalysisof
t a
theexperimentaldatadepictedinFig. 5revealsthatthe h 2 75.3
t
characteristic boundary layer thickness based on the ex-
perimentally measured values of the velocity and height 1
oftheedgeofthelamella,isalwaysclosetothethickness
of the lamella.
0
1. 5 2 2.5 3 3.5 4
Re
log( )
Determination of a(t)
Due to the fact that Re = ρV R/µ 1, the veloc- FIG. 5. Experimentally obtained values of the ratio
ity field sufficiently far from the wall ca≫n be expressed δ/ht 5(Rete)−1/2 for some of the fluids used in this study.
∼
in terms of a velocity potential u= φ(r,z,t) which, by Theprefactorof5usedintheestimationofδismotivatedby
virtue of the continuity equation, ∇ u=0, verifies the the factor in the thickness of boundary layers growing with
Laplace equation 2φ=0. To satis∇fy·the impermeabil- zeropressuregradient[22]. Thevaluesinthelegendrepresent
ity boundary cond∇ition u k=0 at the circular region of 1000×Oh.
·
radius a(t) formed by the intersection of the drop with
the substrate [see Fig. 6], the flow field within the drop
can be expressed as the addition of two velocity fields: wall. In the previous expression, we have taken into ac-
the one associated with the impact velocity ( k) plus count the fact that, since We 1, the contribution of
u′= φ′, which verifies the Laplace equation s−ubjected the capillary pressure σ/R has≫been neglected with re-
to t∇he the condition at z=0, k u′=1 for r a(t) and spect to ρV2. Now note that the boundary condition
· ≤
to the Euler–Bernoulli equation at the drop interface at the free interface can be further simplified due to the
r>a(t), z=z (r,t), ∂φ′/∂t + u2/2 1/2 with z the fact that, during the initial instants, the radius of the
d d
≃
vertical height of the drop interface with respect to the wetted area verifies the condition a(t 1) 1 and
≪ ≪
7
thus, since at t=0, z =r2/2, the position of the free in-
d
terface can be approximated for t 1 as z 0 with
d
errors ∼ O(a2) ∼ O(t). Moreover,′f≪or t ≪ 1, th≃e Euler– n Σc Ur
Bernoulli equation simplifies to ∂φ/∂t 0 since the lo-
≃
cal acceleration term dominates over the convective one
[t2h1e,fr2e4e],inimteprflaycine,gwthhiacthtisheapEpurolexri–mBaetrenlyoulollciaetqeudaattiozn=a0t, Ur Rc Σd n
simplifies to φ′=0 with errors of the order of O(t) 1 k θ θo Ha
due to the fact that, at t = 0, φ′ = 0 at the free≪in- A˙ r
2 ′ Σ
terface. Consequently,the analyticalsolutionof φ=0 s
subjected to the boundary conditions at z=0, k∇ u′=1 A(t)
′ ′ ·
for r a(t), φ=0 for r>a(t) and to φ=0 for z
≤ ∇ →∞
leads to the following expression for the normal velocity FIG. 6. Definition of the control volume used to determine
at z=0 and r>a(t) (see [36]): the thickness Ha of the ejected sheet, which is defined in a
frame of reference moving at thevelocity A˙.
2 a a
k·u=−1− π √r2 a2 −arcsin r . (6) perimeterofthewettedarea,namelyr=a(te)=√3te,will
(cid:20) − (cid:16) (cid:17)(cid:21) be analyzed in a frame of reference moving at a velocity
Now, the equation that determines a(t) is deduced by da/dt. In this frameofreference,the velocityfieldU at
r
means of the so-called Wagner condition [23, 24], which the surface Σ sketched in Fig. 6 is given by
c
is nothing but the time integralof the kinematic bound-
U = W r−1/2(sin(θ/2)e +cos(θ/2)e ) A˙e (10)
ary condition obtained using the velocity field given by r r θ x
− −
equation (6), namely,
whererisnowusedtoindicatethedistancefromr=aand
R A 2 2V T κ(τ)dτ the expression for W=√2aV/π is obtained taking the
VT limit r a once the variable r in equation (6) is replaced
2 (cid:18)R(cid:19) − − π "Z0 A(T)2−κ(τ)2 (7) by r+≪a.
T κ(τ) p It will be shown below that the flow in the control
arcsin dτ =0, surface Σ sketched in Fig. 6, composed by the arc of a
−Z0 (cid:18)A(T)(cid:19) # circle with radius r namely, Σ , the drop surface Σ as
c c d
well as by the exit surface Σ , is quasi steady. Thus, ne-
with κ(τ) the radius of the wetted area for times τ < t. s
glectingcapillarityandviscosityandapplyingthesteady
Equation (7) simply establishes that the wetted radius
Euler–Bernoulli equation, P +ρU2/2=ρA˙2/2 along the
A(T) is fixed by the instant of time at which a point r
constant pressure streamline Σ with P the liquid pres-
on the drop interface with initial coordinates R=A(T), d
Z =R(A(T)/R)2/2 reaches Z=0. The dimensionless sure, yields that the liquid velocity at the exit surface
d Σ in Fig. 6 is uniform and equal to A˙e in the mov-
version of (7) then reads, s x
ing frame of reference. Indeed, by virtue of equation
a2−2t− π4 0a √a2κ κ2 −arcsin κa ddκτdκ=0, a(1p0p)r,oxthimeamteoddublyusA˙oifntthheelliiqmuiidt rvcelociaty3 ont3Σ/2d acnadn,bine
Z (cid:20) − (cid:16) (cid:17)(cid:21) (8) addition,duetothefactthatthestr≫eamlin∝esareparallel
where dτ=(dτ/dκ)dκ. The integral equation (8), which in the liquid sheet, the pressure drop across the lamella
expresses t as a function of a, can be solved by can be neglected. Now note that the integral balances
noticing that equation (8) possesses a solution of the of mass and momentum applied to the control surface
type dt/da=Ca. Performing the change of variables Σ=Σ Σ Σ sketched in Fig. 6, provide a couple of
c d s
κ/a=sinλ, equation (8) then reads equations to express h as a function of a. Indeed, using
a
S S
the expression for the relative velocity given in equation
2 4 2 π/2 2 (10), the mass balance yields
a 2t Ca sin λ λsinλcosλ dλ=0 (9)
− −π Z0 − π
and,thus,t=a2(1 C/(cid:0)2)/2;therefore,since(cid:1)weassumed A˙Ha = Wrc−1/2sin(θ/2)+A˙cosθ Rcdθ, (11)
that dt/da=Ca, w−e conclude that C=2/3 and, conse- Zθ0 (cid:16) (cid:17)
quently, a=√3t. which, in dimensionless form, reads
a˙ha =2wrc1/2−a˙rcθ0, (12)
Sheet ejection with θ0 1 due to the fact that the drop interface is
≪
nearly tangent to the impacting wall [see Fig. 6]. Us-
The jet ejection process depicted in Fig. 1(c) of the ing the expressionfor the relativevelocitygiveninequa-
main text, which takes place from t=t on from the tion (10), as well as the steady Euler–Bernoulliequation
e
8
140 0.5 ✻H0 Ca∗ 1
ℓ
120 0.3 g
0.8
Z 100 0.1
80 0.6
ℓ −0.15 −0.1 −0.05 0 0.05
g 60 (X−Xc)/ℓg ∗
Cad 0.4
40
1cP
20 10cP 0.2
❄
−15 0 −125 −100 −75 −50 −25 0
0
ℓ 0 0.01 0.02 0.03 0.04 0.05 0.06
X/
g h /2
t
FIG.7. Solutionofequation(22)foravalueofthecurvature
at the tip of the lamella dγ/ds=-100 and the corresponding FIG. 8. Solid points represent the critical capillary num-
criticalcapillarynumbers. Inthecaseoftheliquidofviscosity ber calculated from the numerical solution of equation (22)
µ=10cP,Ca∗=0.55,andinthecaseoftheliquidwithµ=1cP, for different values of [dγ/ds(γ = π/2)]−1 and µ=5cP.
d
Ca∗=0.35. The solid curve represents parametrical representation of
d (µV/σ)√3/2t−1/2 vs h /2 1.5h =√12/(2π)t3/2 (see Fig.
e t a e
≃
3bofthemaintext)andthecrossingbetweenbothfunctions
determinethe theoretical capillary numberfor which theliq-
P +ρU2/2=ρA˙2/2 to calculate pressure on Σ , the mo- uid dewets thesolid.
r c
mentum balance equation
π π
ρe U (U n)R dθ+ρA˙2H = Pe ( n) R dθ, Dewetting
x r r c a x c
· · · −
Zθ0 Zθ0
(13)
with ex a unit vector tangent to the wall yields, Figures 1(e)–(g) in the main text illustrates that the
lamella needs to separatefromthe solid substratebefore
w2 breaking into drops under the action of capillary forces
− 2 π+2wa˙rc1/2−a˙2rcθ0+a˙2ha =0. (14) [15]. The dewetting process in our experiments can be
qualitatively described using the theory in [33], which
Substitutingtheresultinequation(12)intoequation(14) expressesavisco–capillaryforcebalanceprojectedinthe
leads to the expression for the jet thickness, direction tangent to the wall. Note that the tangential
force balance in [33] is different from the vertical force
πw2 a
h = = . (15) balance in the main text.
a 4a˙2 2πa˙2 The equations in [33] permit to determine, in an ap-
proximate manner, the critical velocity at which air is
Notethat,inthedeductionabove,wehaveassumedthat
entrained when a solid is plunged into a viscous liquid.
the flow is quasi steady, and this holds for values of R
c
In this case, the condition for the liquid sheet to detach
such that
from the solid substrate is the one determining the air
d ρU dω ρU (U n)dσ. (16) entrainment in an advancing contact line,
r r r
dT ZΩc ≪ZΣc · µV
t ∗
Since σ >Cad(qµg/µ,γ0,Ht/ℓg,l) , (19)
d ρU dω ρV2Ra¨r2 and with γ0 the static contact angle and ℓg,l the slip lengths
dT ZΩc r ∼ c (17) in the solid–gas or solid–liquidinterfaces, which are pro-
portional to the mean free path in the case of the gas
ZΣcρUr(Ur ·n)dσ ∼ρW2R∼ρV2Ra, annodte, ainlstohtehcaatse of the liquid, ℓl/ℓσ ∼10−5 [33]. In (19),
the radius r thus needs to satisfy the relationship
c H
q = , (20)
r2 a r t. (18) H +3ℓg
c ≪ a¨ → c ≪
with H the vertical distance of the free surface to the
Butr hasalsotosatisfytherelationshipr h t3/2, solid substrate, which is geometrically related to the an-
c c a
so it suffices to take r t5/4. ≫ ∼ gle formed by the free interface with the horizontal sub-
c
∼
9
a V
TABLE II.Physical properties of the different gases. ℓ t
λ0 µg ρg0
( 10−9 m) ( 105 Pas) (kgm−3)
× × ·
Helium 180 1.98 0.16
R
Air 65 1.85 1.18 c
Krypton 55 2.51 3.42 b
SF6 39 1.53 6.04
30
0.9
The tabulated values correspond to Tg0 =298.15 K, 20
pg0 =105 Pa. Therefore, for different valuesof thegas 0.7 Relocal
temperature Tg and pressure pg, λ=λ0(Tg/Tg0)(pg0/pg) K 10
and ρg =ρg0(Tg0/Tg)(pg/pg0). u 0.5 0
1 2 3 4 5 6 7
V [m/s]
strate, γ, as 0.3
dh 0.1
=sinγ. (21)
ds
0 5 10 15 20
Re
The equation for γ(s), with s the arclength, is given by local
the viscocapillarybalance projectedinthe directiontan-
gent to the wall [33]
FIG. 9. (a) Figure sketching the top part of the advancing
lamella. (b) Values of the local Reynolds number for the
d2γ 3Ca µ Bo
= f(γ,q g)+cos(γ) t , (22) experimental data considered in this study.
ds2 H(H +3ℓ) µ h2
l 0
with the function f in (22) given in [33],
Bot=ρ dVt/dt H02/σ and H0 Ht/2 the distance dγ/ds(γ = π/2). To determine the approximate experi-
of the t|ip of th|e lamella to the w≃all. The term involving mental value of Ca∗ for the case of siliconoil drops with
d
theBondnumberisincludedduetothefactthattheflow µ=4.6cP, the values of (µV/σ)√3/2t−1/2(Re,We) vs
e
isdescribedinanacceleratedframeofreference,namely, h /2,withh =3h =√12/πt3/2(Re,We)andt givenby
t t a e e
that moving at the velocity of the tip of the lamella
equation(3)ofthemaintextarerepresentedinthesame
with respect to the wall. Since the tip is decelerated
Fig. 8. The crossing between the two curves fixes the
bliymistuerdfaocuertceonmsipountaatnidonρs|tdoVBt/odt=|H1.0 ∼ σ/H0, we have critical capillary number Ca∗d≃0.42, which is far smaller
t than the critical capillary number at which splash is ex-
Themaindifferencebetweenourcalculationsandthose
perimentally observed (see the inset in Fig. 4(a) of the
in [33] are the boundary conditions that need to be sat-
main text). These results reinforce our starting hypoth-
isfied by equation (22). Indeed, in our case, equation
esisthat the liquid dewets the substratefor values ofthe
(22) has been solved fixing Ca and shooting from a po-
capillarynumbersmallerthanthoseforwhichthe splash
sition at the interface with γ=π/2, dγ/ds= 2/h and
− t transition is experimentally observed.
varying the height h0(γ = π/2) until γ=γ0 at the wall,
with the static contact angle fixed to γ0=π/6 in all the
calculations presented in this section. The value of the
Determination of K and K
capillary number is increased until the system (19)–(22) u l
fails to converge to the fixed value of γ0, a condition
that determines the value of the critical capillary num- The lift forces exerted by the relative gas flow in the
ber Ca∗. Figure 7 depicts the computed local shapes of regions sketched in Figs. 9–10, will be calculated next.
d
the tip of the lamella at the corresponding critical cap- The force in the quarter of a circle markedin red in Fig.
illary numbers for the same value of the initial curva- 9 has been computed numerically, using the comercial
ture, dγ/ds=100 and two liquid viscosities, µ=10cP code Fluent, for severalvalues of the gas Reynolds num-
| |
andµ=1cP, showingthat the angle ofthe wedge formed ber Relocal = ρgVtRc/µg, with Rc Ht the radius of
≃
between the edge of the lamella and the solid substrate curvatureofthefrontpartoftheadvancinglamella. The
varies only slightly with the viscosity ratio. Figure 8 result, depicted in Fig. (9)b, reveals that Ku 0.3 for
≃
illustrates the critical capillary number calculated solv- the range of local Reynolds numbers relevant to drop
ing equation (22) for µ=5cP and different values of the splashing, namely, 3<Relocal <102.
interfacial curvature dγ/ds(γ = π/2). Figure 8 also Tocalculatetheliftforceintheregionlocatedbetween
shows that the critical capillary number increases with thelamellaandthesubstrate,theshapeoftheadvancing
10
frame of reference, namely,
V
t H
q = U (S,Y)dY =0, (27)
g
0
Z
Vt+Ug Y with Ug given by equation (26) from which we obtain
H
0 dπ¯ h¯+2ℓ¯
g µ
S = , (28)
ds¯ h¯ h¯2+4¯h(ℓ¯ +ℓ¯ )+12ℓ¯ ℓ¯
µ g g µ
∂U
U =ℓ g where (cid:0) (cid:1)
g g ∂Y
6V µ
Pg = 2t g π¯g, S =Ls¯, H =H0h¯
tan (α) (29)
FIG.10. Figuresketchingtheflowinthelubricatingairlayer
located between the substrateand thelamella. ℓµ =H0ℓ¯µ and ℓg =H0ℓ¯g.
Thus, the vertical force per unit length exerted by
the pressure distribution obtained from the integration
front is approximatedto that of a wedge of angle α such of equation (28), is approximately given by
thattan(α)=H0/L,withH0 ∝Ht (see Fig. 10). Under 6µ V 1 µ V
the lubrication approximation, the general form of the F = g t π¯ ds¯= g t K , (30)
v 2 g 2 l
velocity field, tan (α) 0 tan (α)
Z
Y2 ∂P where
g
Ug(Y,S)=U0(S)+U1(S)Y + 2µg ∂S , (23) Kl =− 6/tan2(α) ×
(C2[a(cid:0)ln(1+a) (cid:1)alna]+C3[bln(1+b) blnb]) ,
needs to satisfy the boundary conditions at Y=0 and × − −
(31)
Y=H(S)=H0(1 S/L),
−
with
∂U
g
Y =0, Ug =ℓg∂Y and a= ℓ¯ +ℓ¯ +2 ℓ¯ ℓ¯ 2+ℓ¯ ℓ¯
Y =H(S), Ug =−Vt−ℓµ∂∂UYg . (24) b= (cid:0)ℓ¯gg+ℓ¯µµ(cid:1) 2q(cid:0)ℓ¯gg−ℓ¯µµ(cid:1)2+ℓ¯ggℓ¯µµ, (32)
− −
q
with ℓ 1.2λ the slip length of the gas[37], λ = and (cid:0) (cid:1) (cid:0) (cid:1)
g
kBTg/(√≃2πd2pg) the mean free path between gas 2ℓ¯µ 1 C1
molecules, k Boltzmann constant, T and p the gas C1 = , C2 = − , C3 = (C1+C2) . (33)
B g g ab b a −
temperature andpressurerespectively,d the effective di- −
ameterofgasmolecules(seeTableII)andℓµarisingfrom To simplify the expressionfor Kl, we set ℓ¯µ =0 in equa-
thecontinuityofshearstressesattheinterface,whichde- tion (28), yielding, for H0 =Ht/4
mands that, at Y=H(S), 2
K = (6/tan (α)) (ln[16ℓ /H ] ln[1+16ℓ /H ]) .
l g t g t
− −
∂U ∂V µ ∂U (34)
g g g
µg µ Vs Ht . (25) The theoretical results presented in Fig. 4 of the main
∂Y ≃ ∂Y → ∼ µ ∂Y
text have been obtained using equation (34) with α =
In (25), it has been assumed that ∂V/∂Y V /H , with 60o, independent on the type of fluid considered (as
s t
∼
V the liquid velocity at the interface in a frame ofrefer- suggested by the results in Fig. 7) and for a value of
s
ence moving at V and, thus, ℓ H µ /µ. Therefore, K = 0.3 (see figure 9). The angle α should be depen-
t µ t g u
∼
the gas velocity field that satisfies the boundary condi- dent on the wetting properties of the solid but, in the
tionsgivenin(24)expressedinaframeofreferencemov- case of our experiments, all the liquids partially wet the
ing at V is substrate with a similar contact angle 20o.
t
∼
To conclude, note that corrections to the mean free
Y ℓ H
U = V − µ− path as a consequence of the modification of both pres-
g t
− H +ℓg+ℓµ− sure and temperature along the coordinate s¯could have
H Y +ℓ H ∂P Y2 ∂P also been included in the formulation[37], but in order
g g g
ℓ + + .
−(cid:18)µg(cid:19)H +ℓg+ℓµ (cid:18) µ 2 (cid:19) ∂S 2µg ∂S to deduce an analytic expression for Kl and since the
(26) improvement in the results should not be significative,
we only report here the result corresponding to ℓ
g
Thepressuregradient∂P /∂S canbededucedimposing independent on s¯.
g
thatthenetflowrateperunitlengthiszerointhemoving