Ex1: Thermocapillarité.Expliquer pourquoi une goutte (une bulle) se déplace toujours vers les régions chaudes.

Ecoulements multiphasiques TD8: ÉbullitionUMPC. NSF16. 2009-2010 Jérôme Hoepffner & Arnaud Antkowiak

CHAUDFROID

1r2

∂

∂r

�r2u�

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ρ�

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�∂2u�

∂r2+

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R

Ex2: Équation de Rayleigh-PlessetOn cherche à décrire la croissance (ou même les oscillations) d’une bulle sphérique de rayon R(t). On suppose que les mouvements du liquide environnant sont purement radiaux, et on suppose que la vitesse du liquide au voisinage de l’interface peut être assimilée à celle de l’interface (on n’interdit pas pour autant les transferts de masse).On rappelle les équations de conservation de la masse et de la quantité de mouvement dans le liquide :

où la vitesse u_l décroît vers 0 à l’infini, et la pression p_l tend vers p_∞ dans cette même limite.

1/ Donner la vitesse du liquide en r=R en fonction de R, ainsi que l’écart de pression p_v-p_l entre la vapeur et le liquide. On négligera ici le phénomène de recul de vapeur. La valeur de la contrainte visqueuse radiale est :

2/ Donner la structure du champ de vitesse, à une constante multiplicative A(t) près. Que vaut A(t) en fonction de R ?

3/ Injecter cette expression dans l’équation de conservation de la quantité de mouvement.

4/ En déduire la valeur de la pression p_l en un point r du liquide, en fonction de p_∞, rho_l, r et R.

5/ Montrer que la pression dans la bulle est donnée par l’équation suivante, dite de Rayleigh-Plesset :

Ex4: Dégazage catastrophique : un volcan à la maison

Lorsqu’on plonge une poignée de Mentos dans une bouteille de Coca-Cola, on observe une éruption impressionnante (faire la manip chez soi -ou chez un ami- ou la regarder sur youtube).

1) Proposer une explication du phénomène.

2) L’expérience marche beaucoup mieux avec les Mentos parfumés à la menthe qu’avec n’importe quel autre parfum. Pourquoi ?

Ex3: L’effet Leidenfrost.

1) Décrire en un paragraphe l’expérience du film.

2) Dans une seconde expérience, on dépose une grosse goutte d’eau sur une plaque chauffée (fig. 2) et on reporte le ‘temps de vie’ de la goutte en fonction de la température (fig. 1). Commenter.

3) Ces deux phénomènes sont-ils liés ? Proposer une interprétation.

Immersion d’une boule de cuivre surchauffée

dans de l’eau.

Leidenfrost dropsAnne-Laure BianceLaboratoire de Physique de la Matiere Condensee, UMR 7125 du CNRS, College de France,75231 Paris Cedex 05, France

Christophe ClanetInstitut de Recherche sur les Phenomenes Hors Equilibre, UMR 6594 du CNRS, BP 146,13384 Marseille Cedex, France

David QuereLaboratoire de Physique de la Matiere Condensee, UMR 7125 du CNRS, College de France,75231 Paris Cedex 05, France

!Received 30 September 2002; accepted 6 March 2003; published 5 May 2003"

A Leidenfrost drop forms when a volatile liquid is brought in contact with a very hot solid. Then,a vapor film comes in between the solid and the drop, giving to the latter the appearance of a liquidpearl. After a brief description of the shape of a Leidenfrost drop, we show that its size cannotexceed a certain value. Then, we describe the characteristics of the vapor layer on which it floats.We show how it is related to the drop size, and how both vary with time, as evaporation takes place.We finally deduce scaling laws for the lifetime of these drops. © 2003 American Institute ofPhysics. #DOI: 10.1063/1.1572161$

I. INTRODUCTION

When a drop of liquid is deposited on a hot solid, oftemperature around the boiling temperature of the liquid, thedrop boils and quickly vanishes. But if the solid temperatureis much higher than the boiling point, the drop is not any-more in contact with the solid, but levitates above its ownvapor. Because of the insulating properties of the film, theevaporation is rather slow: a millimetric droplet of water ona metallic surface at 200 °C is observed to float for more thana whole minute. In addition, the absence of contact betweenthe liquid and the solid prevents the nucleation of bubbles, sothat the drop does not boil but just quietly evaporates. Suchfloating drops are called Leidenfrost drops, after the name ofthe German physician who first reported the phenomenonaround 1750.1

As an example, we display in Fig. 1 the lifetime % of awater drop !radius R!1 mm) deposited on a duraluminplate, as a function of the plate temperature T. Below 100 °C,% decreases to reach typically 200 ms at 100 °C. At this point,the drop is boiling at once after touching the surface. Whenheating the plate between 100 and 150 °C, the droplet life-time dramatically increases, typically by a factor 500, whichcan be associated with the formation of an insulating vaporlayer below the drop. This sharp maximum defines theLeidenfrost temperature.2,3 At larger temperatures, % slowlydecreases, passing from 100 s at 150 °C to 40 s at 350 °C.

The existence and the characterization of the Leidenfrosttemperature has been widely investigated.2–4 It depends onthe solid roughness,5 on the purity of the liquid6 !which canalso affect the lifetime of the drop7", and even on the way theliquid is deposited.8 We focus here on other aspects of theLeidenfrost phenomenon, such as the shape of the drops,their ability to evaporate, and the characteristics of the vaporlayer.

II. DROPS SHAPES AND STABILITY

These levitating drops can be considered as nonwetting.We call contact the region where the drop interface is paral-lel to the solid surface. If the drop radius R is smaller thanthe capillary length a (a!!&/'g , denoting the liquid sur-face tension and density as & and '", the drop is nearlyspherical, except at the bottom where it is flattened. In thislimit, Mahadevan and Pomeau showed that the size ( of thecontact is given by a balance between gravity and surfacetension.9 Denoting ) as the lowering of the center of mass,this balance can dimensionally be written: &)*'gR3. To-gether with the geometric Hertz relation (*!)R , thisyields9

(*R2/a . !1"

This relation was checked experimentally with nonwettingliquid marbles.10 Drops larger than the capillary length form

FIG. 1. Lifetime % of a millimetric water droplet of radius R!1 mm, as afunction of the temperature T of the Duralumin plate on which it is depos-ited.

PHYSICS OF FLUIDS VOLUME 15, NUMBER 6 JUNE 2003

16321070-6631/2003/15(6)/1632/6/$20.00 © 2003 American Institute of Physics

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puddles flattened by gravity, as it can be observed in Fig. 2,and the contact becomes of the order of the puddle radius(!"R).

The thickness h of this puddle is given by balancing thesurface tension #2$, per unit length, taking into account theupper and the lower interface% with the hydrostatic force(&gh2/2, also written per unit length%. This yields

h!2a . #2%

The temperature inside the water drop was measured andfound to be constant and equal to 99"1 °C. This implies adensity &!960 kg/m3 and a surface tension $!59 mN/m,and thus a capillary length a!2.5mm. For Leidenfrostpuddles such as in Fig. 2 or larger, we measuredh!5.1mm, in good agreement with Eq. #2%.

Up to now, the shape of these static drops was found tobe characteristic of a situation of nonwetting, close to whatcan be obtained on superhydrophobic solids. But as an origi-nal property, it is observed that the radius R #and thus thevolume 2'R2a) of a Leidenfrost drop is bounded, by avalue of the order of 1 cm #corresponding to about 1 cm3 forthe volume%. If it is larger, a bubble of vapor #or possiblyseveral ones, for very large puddles% rises at the center of thedrop and bursts when reaching the upper interface, as re-ported in Fig. 3.

We interpret this effect as a Rayleigh–Taylorinstability11 of the lower interface. The vapor film tends torise because of Archimedes’ thrust, but this implies a defor-mation of the lower interface, which the surface tension op-poses. Thus, we expect the maximum size of the drop toscale as a, the capillary length. Classically,12 the instabilitythreshold can be determined by looking at the evolutionof a small sinusoidal perturbation of the lower interface

z!((1#cos kr), with (k$1 and r the radial coordinate. Thesmallest cost in surface energy being achieved for a singlebump centered in r!0, we choose a wave vector k!'/R .Considering capillary and gravitational effects, the differenceof pressure )P between the center and the edge of the dropfor two points A and B at the same level is )P!PA%PB!2&g(*1%3(ak)2/2+ . The perturbation increases for posi-tive values of )P and is stabilized for negative ones. Thethreshold of the instability is thus for )P!0, which leads toa critical radius Rc!3.84a . Using Eq. #2%, we can expressthis quantity as a function of the puddle height:

Rc!1.92h . #3%

Figure 4 shows the largest radius Rc observed withoutbubbles as a function of the puddle height, also measured.Different liquids were used in order to vary the capillarylength, and thus the height. In particular, the thinner puddleswere obtained with liquid nitrogen and oxygen. The variationis indeed linear, and the slope found to be 2, in close agree-ment with Eq. #3%.

III. THE VAPOR LAYER: STATIONARY STATES

We now investigate the characteristics of the vapor layersupporting the drop. In order to measure its thickness, weused the diffraction of a He–Ne laser beam by the slot madeby the interval between the liquid and the solid #Fig. 5%. Werecorded the diffraction pattern #the distance X between twomaxima is about 1 cm on the screen and three to ten maximacan be observed%, and thus could deduce the film thickness e,which was found to be in the range 10–100 ,m.

FIG. 2. Large water droplet deposited on a silicon surface at 200 °C.

FIG. 4. Largest possible radius Rc of a Leidenfrost puddle without bubbling,as a function of its height h. The data are obtained with different fluids *#!%liquid nitrogen, #!% acetone, #&% ethanol, #"% water–ethanol mixtures ofvarious compositions, ##% water+ deposited on a duralumin plate at T!300 °C. For drops of radius R larger than Rc , bubbles such as photo-graphed in Fig. 3 are observed.

FIG. 3. Large puddles of water on a slightly concaveDuralumin plate at 300 °C, seen from above. Accordingto the puddle size, one or several bubbles rise and burstat the upper surface. The bars respectively indicate 0.5and 1 cm.

1633Phys. Fluids, Vol. 15, No. 6, June 2003 Leidenfrost drops

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Éruption de Coca-Cola Influence du soda sur l’éruption