Seminar Ib

1st year, 2nd cycle

Drops impacting on the surface

Author: Matic KnapMenthor: prof. dr. Rudolf Podgornik

Ljubljana, March 2015

Abstract

In our seminar we present the dynamics of drops impacting on a solid surface.

Firstly we describe the essential role of the air trapped between the impacting

drop and surface. Further on we focus on boundary conditions for drop splash-

ing. At the end we conclude that air film trapped between drop and surface

does not have impact directly on boundary conditions but it plays essential role

in dynamics.

Contents

1 Introduction 1

2 Skating on a film of air 2

3 Critical Impact Speed for Drop Splashing 6

4 Conclusion 10

5 Bibliography 11

1 Introduction

There’s no doubt that you have been outside in the rain watching little drops of watersplashing on the lake or a car window. Drops impacting on a surface is a well knownphenomena but quite complex to fully understand. Since 1908 when Worthington [1]started to study drops impacting on surface he wanted to attract researchers fromother fields, not only to explain phenomena, but also because of its importance intehnological applications. Most stunning and beautiful splashing paterns that occurare explained by rapid impact followed by shockwave as the fluid bounces back thesurface, seen on Figure 1a.

(a) deep pool splash (b) flat surface splash

Figure 1: In figure 1a is shown splash on deep pool of water, caused by water drop.Figure 1b is splash on flat solid surface.

As previously mentioned splashes are not just beautiful to observe but also presentin tehnological and scientific fields:

• drop impact at spray cooling, fuel injection or spray painting

• thin coatings on surfaces

• erosion in steam turbines caused by high-speed impacts with solids

• splashed pesticide droplets can be blown away by wind and pollute neighboringareas.

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• splashing raidrops provide mechanism for dispersal of fungus spores

• in forensics the study of patterns of imacted blood drops can be significant inreconstructing crimes.

Droplets impacting on a surfaces have two major point of view, impacting a deeppool of liquid or imapcting a solid surface. Those two approches are divided in twocategories:

• those where velocity is above treshold for splash occurance.

• Those below treshold.

Studying droplets impacting on a deep pool of liquid is more complex than dropletsfalling on flat solid surface, because of deformation of surface before and after impact.That’s why in our seminar we decided to foucus on droplets falling on a flat solidsurface. We describe the essential role of the air trapped between the impacting dropand the surface. Later on we foucus on critical impact speed where splashing occur.

2 Skating on a film of air

Dynamics of free drops failing from height on surface well known trough full path,but when it come close to impact there are still some missing explanations of phe-nomena that occurs. Before the contact of droplet with surface can occur, the dropmust first drain the air separating it from the surface. Recent theoretical calculationsuggest that, even at moderate impact velocities the air fails to drain and is insteadcompressed, deforming and flattening the bottom of the drop while serving as a thincushion of air a few tens of nanomenters thick to lubricate spread of the drop andleading to eventual formation of a trapped buble of air within the drop [4]. Withnew experimental methods we can measure contact dynamics of a drop impacting ona dry glass surface [3]. We directly observe a thin film of air that initially separatesthe liquid from the surface enabling muche more rapid lateral spreading of the dropproviding strinking confirmation of the theoretical predictions. To vizualize the dy-namics of contact we take a cut trough recorded image at the location shown by thedashed red line in Figure 2a. Then convert the measured intensity to separation, andplot the time evolution, using color to denote the height as shown in Figure 2b.

2

TA STAVEK NI POPOLN.

MUCH

Figure 2: The behavior of the thin air film separating the impacting drop from thesurface. Figure 2a: six snapshots of a drop, released from H “ 1m illustrating thefilm of air and the impact dynamics. Figure 2b: the impact dynamics along the cutshown by the dashed line in Figure 2a(ii). The height is indicated by the color. Thearrow indicates one of the bubbles that remains trapped in the liquid [2].

As seen on the Figure 2b the first 500µm clearly show the formation of the layerof air as the drop spreads before the liquid contacts the surface. The liquid does notspread inward, as seen by the boundaries of the thin film, denoted by the centralregion. This reflects the pocket of air which ultimately becomes a bubble trappedin the drop. While the layer of air is clearly responsible for decelerating the drop, itcannot retain the separation of the fluid and surface indefinetly. Ultimately, the thinfilm of air becomes unstable and contact occurs. Initially, two small dark spots appearin the film when the liquid fully contacts the surface, as shown on Figure 2b. As thesespots grow, other spots appear, as the film of air breaks down. These liquid wettingfronts spread rapidly, wetting the surface at a velocity of « 1.5m{s, comparable tothat of the liquid spreading on the thin film of air. Interestingly, there is a thin lineof air at the front of the spreading fluid where the air film becomes thicker as the airis pushed by the advancing wetting front, as shown by the white region leading theedge of the black wetting front.

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(a) (b)

Figure 3: The initial dynamics of the wetting. Figure 3a represents peak velocitiesfor the outward (blue dots) and inward (red dots) fronts. Blue line is theoreticallypredicted initial outward spreading velocity [2]. The dashed line indicates the tresholdheight above which splash is observed. Figure 3b intensity of the reflected light directlyunderneath the thin air film at a location marked by a red spot in the inset for height21cm. Closeup of low intensity region. The image shown is a direct visualzation ofthe thin film of air separating the liquid from the surface prior to contact [2].

To explore the initial dynamics of the wetting associated with the rupture or break-down of the air cushion, they numerically calculated local instantaneous velocity andplot its magnitude as a function of radial position r. The inward-moving velocityis constant, propagating at approximately 1.3m{s. The outward-moving velocity de-creases as 1{r, and can exhibit remarkably high values, as large as 70m{s, as shown inFigure 4b. Suprisingly, the velocity of the inward-moving front is independed of heightH. The maximum velocity of the outward moving front increases strongly with H, asshown on Figure 3a. Moreover, the maximum velocity of the outward-moving frontis nearly an order of magnitude greather than the capillary velocity for isopropanol.When contact line advances, it must flow on very small scales to maintain contactwith interface. Flow on these small scales is dominated by viscous dissipation andthus, the propagation rates are limited by the liquid capillary velociy. By contrast thevelocities measured here are much larger. This suggest that the fluid is not in contactwith the surface but is instead spreading on a thin film of air. Thus, the very earlyviscous dissipation is in the gas as it is squeezed out from under the liquid that wetsthe surface at an acceleratd rate. Indeed such high velocities are predicted theoreti-cally [2]. But only with explicit assumption that the spreading occurs over a film ofair, as indicated by the excellent agreement between the calculated behavior, shownby the solid line, and the data in Figure 3a. The intensity initially drops rapidly,corresponding to the passage of the liquid over the area sampled by the photodiodex- the steep slope of the intensity drop is indicative of the very high speed at which theliquid spreads. However, the intensity does not drop all the way to zero, but insteadlevels off, reaching a plateau at a value I{I0 « 0.1, where I0 is normalization intensitybefore drop hits the interface. This intensity decreases to zero after approximately

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5µs as shown in enlarged data set in Figure 3b and depends on height at which dropletwas dropped. This plateau directly reflects the existence of the thin film of air thatseperates the liquid from the surface. The nature of the final decay of this plateaudiffers from experiment to experiment.

Figure 4: The initial dynamics of the wetting. Figure 4a: R0, as a function of height(inset) same as main figure in log scale. Figure 4b: The inward (solid circles) andoutward (open circles) velocity of the spreading liquid for different heights [2].

This reflects the specific dynamics of the dewetting of the air film, which can varydue to the specific spinodal decomposition that occurs in each case. These measure-ments directly confirm the spreading of the liquid on a thin film of air of order 10nmthick. This is trailed closely by a wetting front that rapidly expands due to the break-down of the air film. Experimental results seen on Figure 3b directly demonstratesthe existence of a thin film of air over which the liquid spreads. This provides strikingconfirmation of the theoretical prediction [4, 5] that a drop is skating on thin film ofair until it breaks down. In addition those reults reveal that qualitatively new phe-nomena occur as the thin film of air becomes unstable simultaneously breaking downat many discrete locations leading to wetting patches that grow and coalscence tofully wet the surface. Similar dynamics has aslo been reported to occur when a sheetof fluid is ejected as a drops splash after high velocity impact [6, 7] For a perfectlywetting fluid such as isopropanol on a glass, a thin film of air behaves as does a poorsolvent - it cannot remain stable and van der Waals forces wil cause it to dewet thesurface trough a nucleation or spinodal-like process.

Dewetting dynamics are traditionally considered to be quite slow. However, forspinodal dewetting the rate of film breakup depends strongly on its thickness andalso on viscosity and may occur very rapidly - for example, a 10nm thick air filmwill remain stable for no longer than 1s [8]. Thus, rupturing occurs simultaneously atmany discrete locations. This leads to small wetting patches that grow and coalesce tofully cover the surface, thereby very rapidly following the advancing fluid front. Thisgives the appearance of a single contact line moving at the same velocity as the fluid,much faster than the calculated capillary velocity. Eventually, however, spinodal-like

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TO MORATE RAZLOZITI!

SPINODAL DEWETTING?

dewetting of the air film always leads to its breakup and complete contact of thesurface by the fluid. The rate at which contact occurs depends on the rate of thisspinodal-like process, which depends on the thickness of the air film. Initially, asheight is increased, the air film becomes thinner, and the breakup of the air filmoccurs more rapidly. Thus, even though the rate of initial drop spreading increaseswith height, the length over which the drop skates on the air film decreases. However,as height increases still further, the thickness of the air film saturates, and hence therate of breakup also saturates.The rate of initial spreading of the drop continues toincrease with height. Thus, the drop always can skate over the film of air, even asheight continues to increase. Interestingly, this skating on the film of air can persist,even until height increases enough that a sheet of fluid is ejected near the expandingrim, and a splash is produced. But unfortunately such experiments cannot be done,because of to small temporal resolution as the drop contact the surface. Those aretechnological restrictions which will be proably resolved in next years. Instead ofknowing the whole dynamics we rather focus on boundary conditions for splash tooccur.

3 Critical Impact Speed for Drop Splashing

To elucidate the precise conditions under which a drop hitting a solid surface splashesor not we need to generate spherical droplets which fall under gravity onto dry glasssurface. Initially the drop deforms axisymmetrically with air bubble entraped at thecenter of the drop. As previously said that air trapped between surface and dropis crutial for contact dynamics. For high velocity impacts is proposed that trappedair does not affect splash boundary conditions directly. With that estimation we canobserve only border of wetted area without considering what is the reason for wettingor dewetting.In this study are relevant observants changes of trajectory experienced by the edge ofthe sheet as the impact velocity increases. Indeed for the smallest values of verticalvelocity, the lamella spreads tangentially along the solid but, for a range of largerimpact velocities, the liquid initially dewets the subtrate and contacts the subtrateagain. For even higher values of vertical velocity, the front lamella dewets the solidand drops are finally ejected radially outwards.For splash to occur two conditions need to be fulfilled simultaneously: the liquid mustdewet the solid and the vertical velocity imparted to the front part of the lamellaneeds to be large enough to avoid the liquid to contact the solid again. Rewettingis a consequence of the radial growth of the liquid sheet edge, caused by capillaryretraction.

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Figure 5: Droplet failing from treshold heights in different preessures. Splash issuppresed with lowering the pressure.

Splashing occurs when the values of both the Webber and Reynolds numbers aresuch that [9]

We “ ⇢V2R

�°° 1, Re “ ⇢V R

µ°° 1. (1)

Where V -velocity, ⇢-liquid density, R-length, �-surface tension and µ-dynamic vis-cosity are normalized characteristic dimensionless variables. Therefore We and Reare dimensionless numbers measuring the relative importance of intertial and surfacetension stresses and intertial and viscous stresses. Consequently, during the charach-teristic impact time (RV ) viscous effects are confined to thin boundary layers of typicalwidth « R?

Re. A fact suggesting that theuse of potential flow theory [10], which ne-

glects liquid viscosity, is appropriate to describe the liquid flow at the scale of theliquid drop. Potential flow theory predicts that, as a consequence of the sudden in-terital deceleration of the liquid when it hits the wall, a flux of momentum is directedtangentially along the substrate, giving rise to the ejection of a fast liquid sheet. Theapplication of the Euler-Bernoulli equation at the drop’s interface, where the pressureremains constant, in a frame of reference yields that fluid praticles are ejected fromaptq at speed relative to that of the ground given by va “ 2 9a “

a3{t where aptq is

time evolving radius of the wetted area extracted from Wagner’s theory [10].

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(a) (b)

Figure 6: Figure 6a: experimental radius of the wetted area compared with a “?3t

(thin solid line) for We “ 98, Re “ 3462. The experimental radial position of theejecta sheet, rt, is also represented for times later than te the injection time. Theinset represents the ratio ⇠?

3« 1, where ⇠ is the coefficient obtained from the best fit

of a function a “ ⇠?t to the experimentally measured radius aptq for a large range of

Webber numbers, for water and silicon siicon oil. Figure 6b : The sketch illustratesthe definition of the main variables used in explanation of boundary conditions forsplash [9].

Moreover, since the flux of tangential momentum per unit length is proportionalto ⇢V 2Raptq we thus conclude that height of lamella at the intersection with thespreading drop at r “ aptq is proportional to

?t3. Figure 6a shows that the measured

radus of the wetted area perfectly matches aptq “?3t for different events and fluids

which fully validates potential flow calculation. Understanding difference betweenpotential flow results and observation takes into account analogy with the cases ofbubbles bursting at a free interface and Worthington jets [11]. The fluid feeding thelamella comes from a region where shear stresses are negligible. Namely a very narrowboundary straddling the drop’s interface and not from the boundary layer growingfrom stagnation point located at the axis of symetry. Those interface can be seen onFigure 6b. dark shaded region. High velocity particles entering the liquid sheet arerapidly decelerated within the lamella due to the combined action of both the viscousshear stresses diffusing from the wall and capillary pressure. Dimensionless thicknes� of the region affected by viscous stresses at a distance ha downstream of the jet rootwhich is the region where the jet meets the drop, is �ha « 1. The deceleration, provokesthe fluid to accumulate at the edge of the liquid sheet and, consequently ht ° ha asseen on Figure 6b. To determine te, note from Figure 6a that, although the velocitiesof fluid particles entering the jet are va “ 2 9a, both aptq and rt are tangent to eachother at the instant of ejection, namely vt “ 9a at t “ te. Thus, since the lamellacan only be ejected if its tip advances faster than the radius of the wetted area, thecondition for sheet ejection is [9]

dv

dt“ ´ BpBx ` Re

´1r2v (2)

here dv{dt † 0 is the dimensionless acceleration of the material points in the sheet

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(a) (b)

Figure 7: 7a: Experimentally measured value of the ejection time, and the one calcu-lated solving equation (2). Each of the solid lines represents the different theoreticalresults for different Ohnesorge numbers. The values of 1000Oh, are represented inthe legend. The experimentally determined ejection time is very well approximatedby the solution of equation (2) and clearly follow the low and high Ohnesorge number(Oh)limits of this equation deduced in the main text. Figure 7b : the total lift forcearises as the addition of the force in the wedge of angle ↵ and the force at the toppart of the lamella. The plus and minus signs indicate the regions where the gaugepressure is positive or negative [9].

given by the momentum equation, p denotes pressure, and x measures the distancefrom the jet root. Moreover, the increment of pressure experienced by fluid particlesflowing into the edge of the lamella is the capillary pressure. Therefore the criticalcondition for sheet ejection dvdt 9

ate3 reads [9]:

c1Re´1?te´1 ` Re´2Oh´2 “ :aht2 “ c2

ate3 (3)

Figure 7a illustrates that equation (2) very well approximates the experimental results.Experiments also validates the high determination of critical speed of an impactingdrop. Firstly we note that splashing occurs as a consequence of the vertical lift forcef , imparted by the gas on the edge of the liquid sheet. The lift force results from theaddition of two contributions:

• the lubrication force - KlµgVt

• the suction force - Ku⇢gV 2t Ht

The lubrication force is exerted at the wedge formed between the substrate and theedge of the lamella, and the latter, at the top part of it, seen on Figure 7b. Basedon reference [9] Kl in the case of partially wetting solids does not seem to significatlydepend on liquid viscosity. Since the local gas Reynolds numbers based on Ht andVt is around ten, both the viscous and interital contributions to the lift force needto be taken into account. Therefore, the vertical force balance per unit length, whenapplied at the edge of the lamella with vertical velocity Vv whose characteristic valueat the instant when liquid front has rised a distance „ Ht above the substrate is given

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(a) (b)

Figure 8: Figure 8a : values of the function � calculated through Eq. (4) for thedifferent liquids and drop diameters investigated. The inset shows that the splashthreshold velocity V corresponding to the experiments in Xu, Zhang, and Nagel [12](black symbols) is also characterized by � « 0.14 as also those from eperiment in [9].Figure 8b: the measured experimental data follow our predictions for both vt and ht[9]

by

Vv «d

f

⇢Ht(4)

For the edge of the sheet not to contact the solid again Vv needs to be larger thanrim raduis of curvature which, once the liquid dewets the substrate, grows in time asa consequence of capillary retraction. Hence, the splash treshold condition reads

� “ VvVr

9c

f

�« Op1q. (5)

Figure 8a demonstrates that the splash treshold is characterized by a nearly constantvalue of �, independed of the type of liquid considered, as predicted by equationEq. (4) for � in [9]. Interestingly enough, the inset in Figure 8a shows that thesplash treshold correrspoding to all the experimental data in [12], where the ciriticalspeed of drops of different liquids falling within several gases and different pressuresis investigated, is also charachterized by � “ 0.14.

4 Conclusion

We have overviewed the phenomena of drop impacting on surface which is not fullyunderstood. Furthermore we presented phenomena of drop skating on thin layer ofair which has not been directly observed until recentlt. It’s still not clear what isthe importance of that air layer and if it’s crutial for bubble or splash developement.No matter what the team from has prove that boundary conditions for splashig isindepended of that air layer. Paramters that are crutial for splashes to occur on

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dry smooth surfaces are vertical velocity, liquid and gas density and viscosity, dropradius, the interfacial tension coefficient and the free path of gas molecules. We canspeculate that air layer is indirectly connected to those parameters and is crutial forfurther developement of drop at impact but without simulations and experiemntswe can not conclude. We can conclude that those paramteres are crutial for dropsplashing which has significant importance in wide pallete of applications. Such asink-jet printing, turbine corrosion, or even to determine minimal plant placement toprevent disease spreading with splashing raindrops.

5 Bibliography

1 Worthington A.M., A Study of Splashes., Longmans, Green, and Co. (1908).

2 Kolinski J.M., Rubinstein S.M., Mandre S., Brenner M.P., Weitz D.A., MahadevanL., Physical Review Letters Phys. 108, 074503 (2012).

3 Rubinstein S.M., Cohen G., and Fineberg J., Nature (London) 430, 1005 (2004).

4 Mani M., Mandre S., and Brenner M.P., J. Fluid Mech., 647, 163 (2010).

5 Mandre S., Mani M., and Brenner M.P., Phys. Rev. Lett., 102, 134502 (2009).

6 Driscoll M.M, Stevens C.S., and Nagel S.R., Phys. Rev. E, 82, 036302 (2010).

7 Thoroddsen S.T., Takehara K., and Etoh T.G., Phys. Fluids., 22, 051701 (2010).

8 De Gennes P.G, Brochard-Wyart F. , and Quere D., Capillarity and Wetting Phe-nomena: Drops, Bubbles, Pearls, Waves, (Springer Verlag, New York, 2004).

9 Riboux G. and Gordillo J.M., Phys. Rev. Lett., 113, 024507 (2014).

10 Wagner H., Angew. Z., Math. Mech. 12, 193 (1932).

11 Gekle S. and Gordillo J.M., J. Fluid Mech. 663, 293 (2010).

12 Xu L., Zhang W.W., and Nagel S.R., Phys. Rev. Lett. 94, 184505 (2005).

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