Ex2: Splash !Lorsqu’une goutte impacte sur un film liquide, ou lorsqu’une vague déferle, il peut se former un splash correspondant à l’éjection de deux feuilles liquides (en 2D).Une modélisation simple de ces deux situations a été proposée par Howell Peregrine et peut se schématiser comme suit :

ou encore sous la forme plus générale ci-dessous : une feuille liquide (vitesse u1, épaisseur h1, angle avec l’horizontale A1)impactant un film au repos d’épaisseur d0 forme deux feuilles liquides dont les caractéristiques sont représentées ci-dessous.En exprimant des relations de conservation simples, écrire un jeu d’équations permettant de déduire A2, V2, u2, h2, A3, u3, h3, V3, v1, d1, v2, d2 à partir des données initiales. On ne cherchera pas à résoudre le système d’équations.

Ex1: Origine des gouttes satellites. Proposer un scénario expliquant l’origine des gouttes satellites. Peut-on les éviter ?

Ecoulements multiphasiques TD7: Singularités en mécanique des fluidesUMPC. NSF16. 2009-2010 Jérôme Hoepffner & Arnaud Antkowiak

Gauche : fragmentation d’une colonne d’huile d’olive dans de l’alcool reportée par Plateau (1849). Bas : Simulation de la rupture d’un jet liquide (Eggers & Dupont, 1994). Droite :

Tailles de gouttes résultant de la rupture d’un jet (Rutland & Jameson, 1970)

open problem. Likewise, the reason for the appearance

of the exponent ��2/3, which seems to be selected over

some range of scales, has not been understood on a

deeper level.

C. Satellite drops

The understanding of satellite drops and their possible

control has been the driving force behind much of the

research on drop formation. It seems they were first ob-

served by Plateau (1849) in his experiments with an un-

stable cylinder of viscous fluid suspended in another (see

Fig. 37).

In air, where drops are not held in place by the outer

fluid, they are much harder to observe, but still did not

escape the attentive eyes of Guthrie (1864). He noticed

tiny drops being thrown upward towards the nozzle in a

dripping-faucet experiment and conjectured that they

resulted from the profile’s not being spherical at the

point of breakup. Rayleigh (1882), using stroboscopic

methods, saw small drops appearing between main

drops in jet experiments and called them ‘‘spherules.’’

For engineering purposes, jets are of particular impor-

tance, the ink-jet printer playing the role of a prototypi-

cal application (Pimbley and Lee, 1977; Chaudhary and

Maxworthy, 1980b). As the drops separate from the ink

jet, they are individually charged and deflected onto

their target by an electric field. The smaller satellites are

deflected differently, producing an undesirable ‘‘halo’’

around the intended point. So the goal is to avoid satel-

lite formation altogether or to avoid having the satellites

charged.

Under normal circumstances, satellite drops are a uni-

versal feature of breakup, quite independent of the ex-

perimental configuration. Qualitatively, they come

about through two properties of the pinching singularity:

(i) The singularity is localized, producing pinchoff at a

point. Since only a small amount of fluid is involved, it

acts on time scales much shorter than the growth of dis-

turbances on the jet, once it sets in.

(ii) The singularity is asymmetrical, so the only way it

can be matched onto an outer solution is by pinching

next to a drop, the steep part forming on the side of the

drop, the flat part merging into the flat neck.

Hence the neck is pinched off from either side and

eventually contracts into a satellite drop. By looking

back at the sequence of pictures in Fig. 6, one can see

this very clearly. Although the neck is initially thinnest

in its middle section, localized pinching sets in only at

the time of the last picture of the first row, after a well-

separated drop has begun to form. From this point to

breakup in the next picture the outer part of the solution

has hardly moved, illustrating the separation of time

scales. Before the neck can retract, it pinches on its up-

per end, and thus a satellite is formed. The same argu-

ments of course apply to jet decay, so within one period

two drops are produced, a main drop with volume Vdand a satellite drop with a much smaller volume Vs . At

long wavelengths the neck becomes very long and is sub-

ject to secondary breakings, which we shall disregard

here. We are going to focus on flow at low viscosities,

but similar mechanisms are present at higher viscosities,

as can be seen in the pictures of ‘‘micro-threading’’ (Fig.

30).

The total volume of both main drop and satellite is

controlled by mass conservation. That is Vd�Vs is equal

to the volume of fluid leaving the orifice during one pe-

riod, and thus

FIG. 36. The similarity function (169) for the inviscid singular-

ity of the lubrication equations. Theory is compared with a

numerical simulation at Re�5�109, at a time when the maxi-

mum slope has reached 104. FIG. 37. The final stages of the breakup of a column of olive

oil in a mixture of alcohol and water, as observed by Plateau

(1849).

911Jens Eggers: Nonlinear dynamics and breakup of free-surface flows

Rev. Mod. Phys., Vol. 69, No. 3, July 1997

The velocity of the satellite, however, can vary greatlyowing to its small mass. Its value will depend on thedetails of the breakup. The most extensive experimentalinvestigation of satellite formation has been performedby Chaudhary and Maxworthy (1980a,1980b). Figure39(a)–(c) shows their photographs of jets with varyingdisturbance amplitudes. At low driving [Fig. 39(a)], theneck breaks first on its front, while at the highest driving[Fig. 39(c)] it breaks on its rear. The latter case meansthat drop and satellite detach as a single unit, and onlylater do the two separate. At intermediate driving am-plitudes, there is a transition region where the pinchingbecomes more and more symmetric, causing thebreakup in front and behind the neck to occur almostsimultaneously, as can be seen in Fig. 39(b). Thisgradual transition can best be characterized by the timedistance t i between the two breakings, which has beencalled the ‘‘satellite interaction time’’ by Pimbley andLee (1977). Its significance lies in the fact that it is ameasure of the momentum transferred between the twobreakings. In case (a), there is a strong backward pull onthe satellite, causing it to rear merge, while in (c) itmerges with the drop leading it. If t i is very small, as in(b), the momentum transfer is also small and the totalmomentum of the satellite is given mostly by the veloc-ity distribution at the first breaking. Accordingly, it mayeither rear merge or forward merge, as in Fig. 39(b), anda prediction of the merging characteristics is difficult.

The focus of Pimbley and Lee’s (1977) andChaudhary and Maxworthy’s (1980b) work is on the de-pendence of the breakup characteristics on the reducedwave number x and on the driving amplitude. Their re-sults agree only qualitatively in that (i) there is a transi-tion from rear merging to forward merging as the driv-ing amplitude is increased, corresponding to a transition

from separation in front of the neck to separation on itsrear, and (ii) the critical amplitude that corresponds tothe no-merge condition increases with wave number.Left out of their investigation was the Weber number,which has been shown by Vasallo and Ashgriz (1991) tohave a significant effect as well.

No accepted explanation exists of the observed tran-sition from front to rear pinching. It was suggested byPimbley and Lee that the transition is connected to thegrowth of the second harmonic component or octaveabove the fundamental input, which they also tried toconfirm within a second-order perturbation calculation.Indeed, if a component with a shorter wavelength hashad a chance to grow sufficiently with respect to thefundamental, which will be the case at low driving, it willproduce an additional growth of surface perturbationsacross the neck. Thus the neck will break first on itsfront.

This idea was tested within the one-dimensionalmodel (79), (80), (82) by Eggers (1995c). The jet wasbroken as described in more detail by Brenner et al.(1997) whenever the radius reached a minimum heightof 10�3r0. After a number of breakings (typically 50–100), the jet reached a stationary state in which the en-tire breaking process had the same periodicity T�1/f asthe driving. This was the first time such a stationary statehad been established in a numerical simulation. Figure40 shows stationary jet breakup at a reduced wave num-ber x=0.4312 and at weak driving ��0.04, which led tobreakup 16 wavelengths away from the nozzle, similar toFig. 39(a). The Reynolds and Weber numbers were thesame as in the experiment performed by Chaudhary andMaxworthy. The neck formed symmetrically, causing itto pinch at both ends at almost the same time. In dis-agreement with experiment, the simulated breakup re-mained almost identical as the driving amplitude wasincreased. This was because the amplitude of harmonicsgenerated by the nonlinear interaction was very small,so it could not affect the breakup.

This situation changes if a small component of a har-monic frequency is added to the driving. The existenceof higher harmonic components in the driving by real-world nozzles has been recognized and measured before(Taub, 1976; Chaudhary and Maxworthy, 1980b; Torpey,1988). They are produced by nonlinearities of the pi-ezocrystal and by the fact that the velocity profile com-ing out of the nozzle is not constant, but is more or lessparabolic, depending on the length of the nozzle. This

FIG. 39. Photographs of satellite formation at three differentdriving amplitudes, increasing from (a) to (c), at a reducedwave number x=0.4312 (Chaudhary and Maxworthy, 1980b).In (a) the satellite breaks first at its front and then at its rear.In (b) the situation is the same, but the time in between thetwo events is much shorter. In (c) the satellite breaks first at itsrear.

FIG. 40. Simulation of satellite formation in the stationarystate of a decaying jet. The dimensionless wave number isx�0.4312, the Reynolds number Re=39, and the Weber num-ber �2�239. The satellite first separates at its front, but almostsimultaneously at its rear.

913Jens Eggers: Nonlinear dynamics and breakup of free-surface flows

Rev. Mod. Phys., Vol. 69, No. 3, July 1997

� rd

r0� 3

�� rs

r0� 3

�3�

2x. (170)

Here rd and rs are the equilibrium radii of drop andsatellite, respectively. Measurements of rd and rs wereperformed by Rutland and Jameson (1970) andLafrance (1975). As shown in Fig. 38, the volume of thesatellite drop decreases disproportionally as x is in-creased. Very close to x�1 measurements become lesscontrolled as one is approaching the regime of randombreakup. For x�0.2, satellite and main drops becomesimilar in size. Drop sizes are very robust, as no signifi-cant dependence on the perturbation amplitude or theWeber number has been reported. This statement is am-plified by the fact that simulations with idealized bound-ary conditions and even perturbation theory have beenquite successful in predicting the correct drop sizes. Thefirst comparison was carried out by Rutland and Jame-son (1970), using Yuen’s (1968) theory. The third-ordertheory was applied up to the point where the local ra-dius first goes to zero—way beyond the range of appli-cability of the theory. The other breaking point was ex-trapolated from this profile. Mass conservation [Eq.(170)] was usually valid to within 10%. The fact that thiscrude theory gave qualitatively correct results indicatesthat the drop volume is set early in the evolution of thesurface. Once some local minimum is produced, pinch-

ing sets in before the drop volumes can change appre-ciably. Thus only the position of the initial indentationsare needed. An even better agreement between pertur-bation theory and experiment was reported by Lafrance(1975). However, as we discussed at length in Sec. IV.C,this solution is erroneous and we feel unable to judgethe significance of this agreement.

Simulations have been even more successful in pre-dicting drop volumes. Mansour and Lundgren (1990)employed a simulation of inviscid, irrotational flow,while Ashgriz and Mashayek (1995) simulated theNavier-Stokes equation at Re=200. The results of thelatter group are included as solid lines in Fig. 38 and canbe seen to agree with experiment to within experimentalscatter. Two significant simplifications were made in thesimulations: First, the perturbations were uniform anddid not grow in space. Second, the simulation did notrepresent the stationary state of a decaying jet, whosedownstream side has a free-boundary condition. Bothsimplifications considerably influenced the profiles, buthad little effect on the total volume within a neck. Notealso that Eq. (170) was obeyed exactly by the fully non-linear calculation.

Apart from the volumes Vd and Vs , the most relevantparameters are the velocities vd and vs of drop and sat-ellite. They determine whether the satellite will mergewith the drop leading it (‘‘forward merge,’’ vs�vd) orwith the one following it (‘‘rear merge,’’ vs�vd). In thelimiting case vs�vd (‘‘no merge’’ or ‘‘infinity condi-tion’’), the satellite will not merge at all, which is par-ticularly undesirable, as satellite drops will persist fardownstream. Analogous to Eq. (170), there is an exactrelation involving vd and vs , which represents momen-tum conservation. In it, the momentum �(vdVd�vsVs)carried by the drop and the satellite equals the momen-tum flux of the unperturbed jet, integrated over one pe-riod. In addition to the momentum carried by the fluid,there is a contribution coming from surface tension,which pulls the fluid back towards the nozzle. In the caseof an unperturbed jet, this is just the excess pressure�p���/r0, integrated over the cross section of the jet.Here we have assumed that the initial perturbations onthe jet are small. This leads to the second exact balance,

v j�rd3vd�rs

3vs��3�

2x � v j2r0

3��

�r0

2� . (171)

As expected, the main drop will be slowed down relativeto the jet velocity. In fact, neglecting the contributionfrom the satellite drop, one finds from Eqs. (170) and(171) that

vd�v j��

�

1r0v j

.

This result was also obtained by Schneider et al. (1967),except that they had an erroneous factor of two in frontof the second term on the right. If this is corrected,agreement with their measurements of the drop velocityimproves substantially.

FIG. 38. Radius of main and satellite drop as a function of thereduced wave number. Experimental data by Rutland andJameson (1970, solid symbols) and Lafrance (1975, open sym-bols). The lines represent the radius of the main and the sat-ellite drop according to the simulations by Ashgriz andMashayek (1995).

912 Jens Eggers: Nonlinear dynamics and breakup of free-surface flows

Rev. Mod. Phys., Vol. 69, No. 3, July 1997

C2 Cover Illustration

Figure 1. A sequence of milk drops falling on a solid surface.

Figure 2. Impact of propanol drop on the same fluid.

In this paper we illustrate that the regular crown structure found by Edgerton is a rareexception, observed only in a narrow range of parameters. As we show in more detail below,the formation of an Edgerton milk crown results from an intricate sequence of events. First,a cylindrical fluid sheet shoots out from the layer, forming a fat rim. Second, cylindricalsymmetry is broken by a linear instability of the rim, which selects a preferred wavelength atwhich the thickness of the rim varies periodically. Third, in a nonlinear stage of the originalinstability, the rim develops tips, from which jets emanate. Fourth, drops pinch off from theend of each jet. Irrespective of its regularity, we herein call the generation of droplets by thismechanism a crown splash.

In practice, a whole range of other phenomena can take place as one moves outside the lightshaded region of figure 3. In particular, there are at least two separate mechanisms by whichsheets form upon impact, and the end products of the impact event are strongly influencedby whether these sheets decay separately, commingle or interact. The symmetry breakinginstability of each sheet may occur by one or more mechanisms, leading to noisy structures ingeneral.

C6 Cover Illustration

Figure 6. Left: an experiment corresponding to the diagram on the right. The parameters areRe = 676, We = 428, 775 µs after impact. Corresponds to • in figure 3. Right: a diagram of themechanism for sheet formation as proposed in [2]. Mass and momentum conservation determinethe velocity and angle of the upward moving sheet, which usually rises at a steep angle.

Figure 7. An ejecta sheet ‘riding’ on a Peregrine sheet. The former leads to the formation of tinydroplets that precede the much larger Peregrine sheet. Re = 1354, We = 874, 580 µs after impact.Corresponds to ! in figure 3.

3.2. Peregrine sheet

After the drop has penetrated the layer, a sheet of liquid travels outwards over the solid surface.It collides head on with the as-yet unmixed fluid in the preexisting layer. This mechanism,first described by Peregrine [2], is illustrated in figure 6. Elementary considerations of massand momentum balance show that, given the thicknesses and velocities of the two incomingsheets (in the frame of reference of the moving base), that the speed, thickness and ejectionangle of the resulting sheet can be calculated. An experimental example is shown in figure 6.Peregrine sheets form for We > 40, and are disordered from the earliest observable times forRe > 1300. These features are, respectively, denoted by dashed horizontal and vertical linesin figure 3.

3.3. Combination of two sheets

This qualitative breakdown of the splash is summarized in figure 3. For most of the parameterregime of our experiments, the two sheets described appear together in any given impact event.We are aware of three different fates of the ejecta sheet.

(i) The ejecta sheet becomes the leading edge of the Peregrine sheet, as the latter growsunderneath the former. In figure 7 we show a Peregrine sheet, whose rim has not yetbroken up. Yet a number of very small drops precede the rim. These form from ahorizontal sheet attached to the rim at early times.

62 D. H . Peregrine

f- ---+ \ \ \ \ \ ~ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ \

“2, d, 0 1 , di

FIGURE 1. One-dimensional model of the splash of a ‘moving waterfall ’ onto a thin layer of water, defining depths, d,, thicknesses, h,, velocities, ui, Vj, and angles A i .

This model is complete once relations are found between the three streams, a t the impact point and the projection points. These relations a,re readily found once it is noted that a t each of these points the configuration is similar to that of the sym- metrical impact of two plane jets (e.g. see Milne-Thomson 1960, § 11.42, interestingly it is noted in 3 1 I .40 that only the symmetric solution is determinate). I n the reference frame moving with the impact or projection point the flow in that neighbourhood is steady. Bernoulli’s theorem holds along stream lines and with our neglect of gravity reduces to constant velocity on surface streamlines. That is

and u3 = v,- v, = v3. The flow into and away from these points must conserve mass and this reduces to

h, = d, + d,, (4)

h, = dl+d,, (5)

and h, = d , + do. (6)

Similarly the rate of flow of horizontal momentum is conserved, which simplifies to

-h,cosA, = d1--d, (8)

and

LETTERS

Making a splash with water repellency

CYRIL DUEZ1, CHRISTOPHE YBERT1, CHRISTOPHE CLANET2 AND LYDERIC BOCQUET1*1Laboratoire PMCN, Universite Lyon 1, UMR CNRS 5586, 69622 Villeurbanne, France2IRPHE, UMR CNRS 6594, 13384 Marseille, France*e-mail: lyderic.bocquet@univ-lyon1.fr

Published online: 25 February 2007; doi:10.1038/nphys545

A splash is usually heard when a solid body enters water at largevelocity. This phenomenon originates from the formation of anair cavity during the impact. The classical view of impacts on freesurfaces relies solely on fluid inertia; therefore, surface propertiesand viscous effects should be negligible at sufficiently largevelocities. In strong contrast to this large-scale hydrodynamicviewpoint, we demonstrate here that the wettability of theimpacting body is a key factor in determining the degree ofsplashing. This unforeseen fact is further embodied in thedependence of the threshold velocity for air entrainment on thecontact angle of the impacting body, as well as on the ratiobetween surface tension and fluid viscosity, thereby defininga critical capillary velocity. As a paradigm, superhydrophobicimpactors make a big splash for any impact velocity.

The first systematic study of splashes was published more thanone century ago by Worthington1. In this pioneering work, heused high-speed photography to examine impacts of drops andsolid bodies on a liquid surface, with beautiful illustrations ofsplashes. (For impact of solid spheres on a liquid, it is interestingto mention that Worthington did observe different splashingbehaviours, referred to as ‘rough’ or ‘smooth’, which are similar tothe wettability effect discussed in the present paper.) In recent years,there has been a resurgence of interest in the physics of impact, duein particular to the development of rapid video imaging. And newperspectives have emerged, showing that unforeseen mechanismsplay a central role in impact: to cite a few, the inhibition ofdroplet rebound by adding tiny amounts of polymers2, the complexdeformation dynamics of a rebounding drop3 and the unexpectedrole of ambient air on drop splashing4,5. Here, we consider thesituation of a solid body impacting a gas–liquid interface. Thissituation is obviously relevant for many naval applications, suchas ship slamming and air to sea weapons, and for any industrialcoating process that involves the dipping of a solid object in a liquidbath (where air entrainment is to be avoided). The traditionaldescription of an impact of a solid body on a free interfacefollows the work of von Karman and Wagner6,7, in which viscosity,surface tension and compressibility effects are neglected8,9. Thisidealized framework is formally justified by the fact that inthe situations relevant to impacts, the Reynolds Re and WeberWe numbers, quantifying the role of inertia versus respectivelyviscous and capillary effects, are very large. This is preciselythe regime of interest in this study: Re = ρU a/µL ∼> 104–105

and We = ρU 2a/γLV ∼> 103–104 (for impacting body diameter a,velocity U , liquid density ρ, liquid viscosity µL and liquid–vapoursurface tension γLV). Accordingly, capillarity and viscosity are notexpected to play any role in the impact and can be ignored inthis description.

a b

c d

125 ms

Figure 1 Visual and audio recordings of impact. a,b, Photographs of the impactof two spheres differing only in wettability by a nanometric coating on their surface:impact of a perfectly wetting sphere, with static contact angle θ0 " 15◦ (a); impactof a hydrophobic sphere with static contact angle θ0 " 100◦ (b). The impact velocitywas 5.0 m s−1 in both cases, corresponding to a 1.25 m height drop. Thephotographs were taken 15.5 ms (a) and 15.0 ms (b) after initial impact.c,d, Time-dependent audio recordings of the impacts, as measured by a microphone∼10 cm from the impact point, for a hydrophilic (c) and a hydrophobic (d) sphere.The signal is proportional to the acoustic pressure emitted during the impact. Theunits on the vertical scale are arbitrary (but identical). A big ‘splash’ is evident for thehydrophobic sphere, whereas a tiny ‘plop’ is heard for the hydrophilic sphere. Thesound is associated with the rapid closure of the cavity (not shown).

Our experimental results contrast with this simple picture. Asillustrated in Fig. 1, two spheres that differ only by a nanometriccoating that modifies wettability exhibit very different impactbehaviour: a huge air cavity is entrained for the hydrophobicsphere, whereas no such behaviour is observed for the hydrophilicsphere. However, apart from the static contact angle (θ0 " 15◦

versus θ0 "100◦), the spheres are identical in terms of bulk material(glass), diameter, very low surface roughness and impact velocity(U = 5 m s−1). Moreover, during the experiment, a ‘splash’ is heardfor the hydrophobic sphere, whereas only a tiny ‘plop’ is producedby the hydrophilic sphere, as shown in Fig. 1c,d.

180 nature physics VOL 3 MARCH 2007 www.nature.com/naturephysics

!"#$#%&'()***) )+,-,./0*12+/234*56

Ex5: Plouf ! (?)Dans l’expérience ci-dessous (Duez et al., Nature Physics 2007), des expériences de lâcher de billes dans l’eau sont conduites. À gauche, la bille a un revêtement hydrophile alors qu’à droite le revêtement est hydrophobe.1) Décrire ces expriences en un paragraphe2) Proposer une interprétation du phénomène

Ex3: Gouttes affamées. 1) Décrire en un paragraphe l’expérience ci-dessus.2) Proposer une interprétation du phénomène

Rupture d’un jet d’une solution de polymère (Clasen et al., 2009)

6 C. Clasen, J. Bico, V. M. Entov and G. H. McKinley

1 mm

Figure 1. The ‘gobbling’ phenomenon: a large terminal drop periodically develops at the endof a thin jet of a viscoelastic fluid (100 ppm PAA solution, Q = 39.7 mm3 s−1, Ri = 0.125 mm).A sequence of video images is shown; the time interval between consecutive images is 6 ms.

or hastily in large pieces; gulp’: American College Standard Reference Dictionary)of a chain of tiny beads by a greedy terminal drop, until it is ‘sated’ or ‘saturated’and falls off. In some cases, the terminal drop can ‘swallow’ up to several scores ofbeads before detachment. A movie of the gobbling phenomenon can be found athttp://web.mit.edu/clasen/Public/gobbling.avi.

The ‘gobbling’ phenomenon is specific to macromolecular solutions and is neverobserved in experiments with jets of pure water or other Newtonian fluids. However,even minute amounts of polymeric additive bring it into existence. If one focuses onthe central axial column, it is clear that the primary role of the polymeric additive isto stabilize the later stages of the capillary thinning process and severely retard theinertial breakup of the fluid column (Christanti & Walker 2001; Amarouchene et al.2001; Tirtaatmadja, McKinley & Cooper-White 2006). This stabilization then enablesus to image the temporal evolution and axial development of a beads-on-a-stringmorphology along the jet. Somewhat analogous bead dynamics can be seen even withNewtonian fluids when a thin annular film of viscous fluid is coated on a solid fibre(as described originally by Boys 1912 and studied in detail by Kliakhandler, Davis& Bankoff 2001; Craster, Matar & Papageorgiou 2005 and references therein). Inthe present case the rigid central fibre is replaced by the highly elongated polymermolecules in the thin viscoelastic ligaments connecting the drops .

The dramatic effects of dilute amounts of high-molecular-weight additives onthe breakup of aqueous fluid filaments is well known and has been extensivelystudied since the pioneering work of Middleman (1965) and Goldin et al. (1969).The hydrodynamic consequences of small amounts of polymeric additives can berationalized in terms of the unravelling and extension of the initially coiled polymericmolecules by strong extensional flows (Entov & Yarin 1984; Bazilevskii et al. 1990b;Anna & McKinley 2001; Clasen et al. 2006b). In the case of steady jets issuing froma nozzle at high flow rates, significant elastic stresses can be generated (even fordilute polymer solutions) which affect the breakup length of the jet and the ensuingdroplet size distribution (Bousfield et al. 1986; Christanti & Walker 2001). In thecase of dripping from a faucet at very low flow rates, the presence of even diluteconcentrations of polymer can dramatically extend the time to pinch off and inhibitthe existence of satellite droplets (Amarouchene et al. 2001; Tirtaatmadja et al. 2006;Sattler, Wagner & Eggers 2008). In each case, the large elongational viscosity of the

size of the geometric length scales of the system. We arewell above that regime in these experiments. We alsomeasured that the surface tension of the liquid [12] doesnot vary with pressure in our experimental regime. Webelieve that our splashes are not caused by air entrapmentnear the drop center on impact because we do not observeany air bubbles and the splash occurs at the expanding rim[13,14]. In order to understand air’s role better, we havevaried the composition of the gas. The inset of Fig. 2(b)shows the threshold pressure versus impact velocity forfour different gases; the values of PT are displaced fromeach other but the trends in the data have the same quali-tative shape. We note that the four gases used have similarviscosities (varying only from 15:3 !Pa s for SF6 to25:6 !Pa s for Kr) but have very different molecularweights, MG (MHe ! 4, Mair!29, MKr ! 83:8, and MSF6 !146 Daltons) [15]. We have tried to scale the differentcurves on top of one another and found that the best datacollapse, in the region with impact velocities greater thanV", is obtained by plotting #MG=Mair$0:5PT versus V0. Theresult is shown in the main panel of Fig. 2(b).

Our analysis concentrates entirely on the regime withV0 >V". We consider two contributions to the stress on theexpanding liquid layer after impact: the first, !G, is due tothe restraining pressure of the gas on the spreading liquid,which acts to destabilize the advancing front and deflect itupward; the second, !L, is due to the surface tension of the

liquid, which favors keeping the liquid layer intact afterimpact. When the two stresses become comparable, weexpect the spreading liquid to become unstable and tobreak up into droplets.

An estimate of !G should include air compressibility,either because the drop spreads out suddenly and rapidlyafter impact, causing a weak shock in the air [16] orbecause the characteristic length scale of air flow nearthe spreading liquid rim is small [17]. Under such a cir-cumstance the stress from gas is proportional to the gasdensity, "G, the speed of sound in the gas, CG, and theexpanding velocity of the liquid layer on the substrate, Ve:

!G % #"G$#CG$#Ve$ %PMG

kBT

!!!!!!!!!!!!

#kBTMG

s!!!!!!!!!

RV0

2t

s

: (1)

Here # is the adiabatic constant of the gas, T is thetemperature, kB is Boltzmann’s constant, R is the initialradius of the drop, and t is the time measured from theinstant of impact [18].

In order to estimate !L, we consider the surface tensionpressure near the front of the advancing liquid. This isgiven by the surface tension coefficient, $, divided bythe thickness of the layer, d. The thickness d is assumedto be the boundary layer thickness which is controlled bythe diffusion of vorticity from the solid substrate [19].Thus,

FIG. 1. Photographs of a liquid drop hitting a smooth dry substrate. A 3:4& 0:1 mm diameter alcohol drop hits a smooth glasssubstrate at impact velocity V0 ! 3:74& 0:02 m=s in the presence of different background pressures of air. Each row shows the drop at4 times. The first frame shows the drop just as it is about to hit the substrate. The next three frames in each row show the evolution ofthe drop at 0.276, at 0.552, and at 2.484 ms after impact. In the top row, with the air at 100 kPa (atmospheric pressure), the dropsplashes. In the second row, with the air just slightly above the threshold pressure, PT ! 38:4 kPa, the drop emits only a few droplets.In the third row, at a pressure of 30.0 kPa, no droplets are emitted and no splashing occurs. However, there is an undulation in thethickness of the rim. In the fourth row, taken at 17.2 kPa, there is no splashing and no apparent undulations in the rim of the drop.

PRL 94, 184505 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending13 MAY 2005

184505-2

Ex4: Splash ? 1) Décrire en un paragraphe l’expérience ci-dessous.2) Proposer une interprétation du phénomène

Impact de goutte à différentes pressions ambiantes (Xu et al., 2005)