EXPERIMENTAL INVESTIGATION OF A SINGLE DROPLET IMPACT ONTO A SESSILE DROP

Atomization and Sprays, 20(10):909–922, 2010

EXPERIMENTAL INVESTIGATION OF A SINGLEDROPLET IMPACT ONTO A SESSILE DROP

N. Nikolopoulos,1 G. Strotos,1 K. S. Nikas,1,∗ M. Gavaises,2

A. Theodorakakos,3 M. Marengo,4 & G. E. Cossali4

1Technological Education Institute of Piraeus, Department of Mechanical Engineering, Fluid Me-chanics Laboratory, 250 Thivon & P. Ralli St., Aigaleo 12244, Greece

2City University London, School of Engineering and Mathematical Sciences, NorthamptonSquare, EC1V 0HB, London, United Kingdom

3Fluid Research Co., 49 Laskareos Str, 11472, Athens, Greece4Department of Industrial Engineering, University of Bergamo, viale Marconi 5, 24044 Dalmine,Italy

∗Address all correspondence to K. S. Nikas E-mail: [email protected]

Original Manuscript Submitted: 19/10/2010; Final Draft Received: 08/01/2011

This paper presents an experimental study of the impact of a single water droplet onto a stationary liquid bulk built bythe previous impact of one or two droplets. The experiments were performed with two different film thicknesses, threedifferent Weber (We) numbers, and two surface contact angles. In both cases we have hydrophilic conditions, which donot allow for a good investigation of this parameter. The morphology of the drop impact was studied using a charge-coupled device (CCD) camera, and the corresponding qualitative and quantitative characteristics regarding the timeevolution of the phenomena, such as the diameter and height of the evolving crown, were obtained by image analysis.Analysis of the experimental data evidences that the phenomenon has a strong similarity to the impact of a single drop onshallow films, although the effect of the surface wetting characteristics plays, in this case, a negligible role. The regimesof deposition and splashing are identified as a function of the We number and the maximum thickness of the steady film,which is affected by the surface wettability properties.

KEY WORDS: drop impact, wettability, splash, liquid film

1. INTRODUCTION

Droplet impact onto solid and liquid surfaces has beenstudied extensively over the last decades, both experimen-tally and numerically. This phenomenon has great rele-vance in many engineering applications, such as surfacecooling or coating, spray injection in internal combustionengines, inkjet printing, etc. Therefore, it is consideredessential to have an in-depth understanding of the govern-ing physics of droplet impingement on dry and wet wallsin order to have better control of those process charac-

teristics that may significantly affect the performance ofthe applications. Nevertheless, little information regard-ing the chain droplet impingement onto solid surfaces ex-ists in recent literature, despite the fact that sequentialdroplet impingement is one of the most typical of dropletinteractions (Kang et al., 1994, 1995; Ghafouri-Azar etal., 2003; Fujimoto et al., 2001).

Droplet impingement onto solid surfaces is a complexphysical process, encompassing fluid dynamics, physics,and interfacial chemistry. Therefore parameters such as adroplet’s impact velocity and diameter, liquid viscosity,

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910 Nikolopoulos et al.

surface tension coefficient coupled with hydrophobicity,and temperature of substrates have been examined to eval-uate their effects on the evolution of the spreading process(Zhang and Basaran, 1997; Gu, 2001; Cooper-White etal., 2002; Fujimatsu et al., 2003; Yu et al., 2007; Werneret al., 2007; Wang et al., 2009; Wachters and Westerling,1966; Hatta et al., 1995, 1997).

Yarin and Weiss (1995), using a chain droplet genera-tor, observed the impingement process of varying dropletdiameters between 70 and 340µm, while the impactingdroplet velocity was varied up to 30 m/s. They derived asplashing threshold rule, dependent on impact frequencyand velocity but independent of droplet size and ambi-ent pressure. Furthermore, they formulated a correlationfor the secondary droplet size distribution. This seriesof experiments may be regarded as the impact of a sin-gle droplet onto a partially wetted surface. Cossali et al.(1997, 1999, 2004) identified a deposition: the splashinglimit after impact, using single droplets with a diame-ter between 2 and 5.55 mm and a maximum velocity ofaround 6.5 m/s. This threshold between the depositionand splashing regime was identified by the formation of acrown with emerging jets at its rim, which at subsequenttimes break up into secondary droplets.

Most of the studies were focused on the characteris-tics of splashing and its transition mechanisms beyondthe regime of total absorption, which occurs when theimpact is sufficiently harsh (Worthington, 1908; Hobbsand Kezweeny, 1967; Rodriguez and Mesler, 1985; Van-der Wal et al., 2006). Under these conditions a crown isformed, and a multitude of secondary droplets is furtherdisintegrated from the rim after a corrugated shape is gen-erated by the capillary instability. This is followed by theejection of a central column known as a Rayleigh or Wor-thington jet (Shin and McMahon, 1990).

Comprehensive reviews of single-drop studies can befound in Rein (1993) and Yarin (2006). The collision ofa single drop with a liquid layer can result in sticking,rebound, deposition, and splash (Stanton and Rutland,1998), depending on the impact parameters. In Cossali(1997) and Rioboo (2001), two types of splash were ob-served: prompt splash, immediately after drop impact,and corona splash. The formation of a cavity and theemergence of a central fingerlike jet was also observedin the case of high-velocity impact into a deep pool oronto a relatively thick film (Oguz and Prosperetti, 1990).Numerous studies of the drop impact onto a liquid filmare focused on the experimental investigation and mod-eling of the splash threshold, crown propagation, obser-vations of the rim fingering, and the parameters of the

secondary drops (Levin and Hobbs, 1971; Macklin andMetaxas, 1976; Okawa et al., 2006). The studies are de-voted to the investigation of drop impact into relativelythin (Wang and Chen, 2000), moderate (Cossali et al.,1997, 1999; Vander et al., 2006), and comparable withthe initial drop diameter or thick liquid layers (Prosperettiand Oguz, 1993).

Naber and Farell (1993) characterized the secondaryatomization of droplets of various liquids with the samedroplet size range but in the case of impacts onto heatedsubstrates, examining the effect of heat transfer on theevolution of the phenomenon. This investigation was laterextended by Bernardin (1999) and Anders et al. (1990) forethanol droplets, both for low wall temperatures, leadingmainly to deposition, and for high temperatures above theLeidenfrost limit, promoting the rebound of the incom-ing droplet over the substrate. Similar experiments can befound in Richter et al. (2003, 2004, 2005), where succes-sive impingement processes were examined under atmo-spheric conditions.

Roisman et al. (2006), using a phase Doppler instru-ment, measured the drop size and velocity of secondarydroplets ejected from the protruding jets at the crown’srim during splash. Based on these experimental series,they introduced a semiempirical model for the secondaryspray, which was validated using a numerical simulationof spray transport based on the Euler-Lagrange approach.Moreira et al. (2007), applying a phase Doppler analysis(PDA) technique, extended the aforementioned investiga-tion to gather information about the size and velocity ofthe secondary droplets as a function of incidence angleand heat-transfer regimes in the case of isooctane and wa-ter droplets impinging on hot surfaces.

Finally, Muller et al. (2007) analyzed droplet wall in-teraction under realistic conditions for deionized (DI) en-gines, with elevated pressures up to 3 bar, and for morerealistic droplet sizes around 50µm. They presented de-tailed measurements of the induced flow field and sizerange of the evolving secondary droplets as a functionof wall temperature after splashing, while Rahimi andWeihs (2009) experimentally investigated the impact ofmagneto-rheological droplets onto a smooth surface forthe first time. In addition, the dimensionless time to reachthe maximum spreading diameter (Dmax) of a droplet af-ter its impact onto a solid substrate is not a constant, asused in several previous investigations, but ranges from0.6 to 2.99, depending on the Weber (We) number andcontact angle.

Drop retraction fromDmax depends on We as wellas on the wetting angle. For hydrophilic surfaces, the re-

Atomization and Sprays

Experimental Investigation of a single Droplet Impact 911

traction height decreases as We increases. In contrast, forhydrophobic surfaces the retraction height increases withWe, and rebounding occurs when We exceeds a criticalvalue. Mao et al.’s (1997) rebounding model correctlypredicts rebounding in the present experiments for mostcases.

The present study is focused on the experimental in-vestigation of a finite chain of water droplets (two andthree droplets in a row) impinging on a solid wall un-der atmospheric conditions, where the impacting dropletvelocity, the height of the pre-existing liquid film, andthe wettability characteristics of the solid substrate werethe controlling parameters. The main difference with themost common experiments of impact on uniform liquidfilms is related to the influence of wall wettability on thisphenomenon. Such an influence is evidenced both on thethickness of the film formed by the previously impacteddrops and on the dynamic contact angle, which may in-fluence the movement of the contact line after the impactof the last drop.

2. EXPERIMENTAL SETUP

2.1 Apparatus

The experimental rig used for the experiments is shownin Fig. 1. It comprises a drop generator made by a sus-pended needle positioned at a distance ranging between150 and 440 mm from the solid wall and connected with

Trigger Delayer

Laser He-Ne

Strobolamp

Colour PCO SensiCam, 1280x1024 pixels

Aluminium/glass plate

Oscilloscope

Tank Drop generator

Photodiodes

`

FIG. 1: Drop impact test rig.

a small water tank. The detachment of single droplets isachieved by a small pressure pulse produced by the open-ing of a solenoid valve. The droplet diameter was mea-sured after enlarging the falling droplet’s pictures at var-ious time instants. The mean value was equal toDo =4.02 mm, with a repeatability better than 6%, while theimpact velocities ranged between 1.7 and 2.8 m/s. Twodifferent solid plates were used, of either aluminum orglass composition, in order to analyze the effect of wetta-bility on the evolution of the phenomenon. In all cases,distilled water at approximately 20◦C was used, whilethe average ambient pressure and temperature were equalto the atmospheric (1.015) bar and 23◦C, respectively. Acharge-coupled device (CCD) camera (Colour PCO Sen-siCam, 1280× 1024 pixels) and a short duration flashlamp (Strobe) were used in order to capture the evolu-tion of the phenomenon at various time instants beforeand after impact. The flash duration was equal to about10 µs, and the time of image shooting was controlled bya delay circuit, which was triggered by the obscuring of alaser beam, caused by the passage of the falling droplet.The chosen configuration allowed for measuring the maindimensions of the lamella’s crown (diameter and height)from the side view. Calibration for size measurements wasobtained from the image analysis of pictures of known di-mension bodies (needles).

The image analysis was performed using commercialsoftware (Image ProPlus), while the coordinates of thepoints of interest were taken manually, since the complex-ity of the liquid shapes rendered the development of sub-routines for the tracking of the specific points inadvisable.The accuracy of the measured dimensions is estimated toabout three pixels, equal to 69µm.

Despite the good repeatability, the impact process can-not be regarded as deterministic. In order to collect a sta-tistically significant sample of data for each time step, weconducted the same experiment at least 30 times so that areconstruction of a sort of representative shape evolutionwith time would be available.

2.2 Free-Falling Droplet’s Shape and ImpactVelocity

After the detachment from the needle tip, the dropletfollows a free falling body motion. By monitoring thedroplet’s shape just before impacting the solid substrate,it was found that the droplet had an ellipsoid shape ratherthan a spherical one, with the mean diameter in the hori-zontal direction being equal to aroundDh,avg = 4.20 mmand the vertical one equal toDv,avg = 3.69 mm, as shown

Volume 20, Number 10, 2010


in Fig. 2a. This figure also evidences the difference in theshape of the residual liquid bulk when a droplet impactsan aluminum or glass substrate. In the latter case, the liq-uid spreads more due to the smaller wetting contact angle.The variation of these two dimensions is less than 3%,and the mean initial diameter of the droplet is calculatedby the commonly used expression

D0 = (D2h,avgDv,avg)1/3 (1)

In order to measure the impact velocity at the start of eachexperimental series, droplet images before impingementwere recorded (see Fig. 2b). The edge locations of the topand bottom points of the same droplet (two points each)were plotted against the exposure time of each frame, andthe results were fitted by the time gradient of distance(dx/dt) in order to calculate the velocity of the impactingdroplet just before its first contact with the solid substrate.This procedure was repeated for around 15 pictures foreach case, examined, and the mean value of the impact-ing velocity was then calculated. The standard deviation

of the velocity was found to be less than 0.01 m/s for allcases examined.

3. RESULTS AND DISCUSSION

The main emphasis of the present paper is the temporalevolution of the interaction of an impacting droplet onto aresidual liquid bulk, generated by the previous impact ofone or two droplets of the same size. Hereafter the staticliquid bulk on the solid wall is called first droplet, and theincoming droplets are called second or third droplet. Aftereach impact, the surface is cleaned carefully by air soot,so that the first droplets always impinge on a dry solidsurface, maintaining a good repeatability of the wettingeffects. The experimental campaign includes the paramet-ric study of two different solid substrates (aluminum andglass), three different impacting velocities ranging from1.7 to 2.65 m/s, and different maximum thicknessesδ

of stationary droplets between 0.12 and 0.33D0. In allcases examined, the droplet diameter was kept constant;

Dvavg

Dhavg

hAlum> hGlass

hGlass

Drb,Alum < Drb,Glass

(a)

(b)FIG. 2: (a) Definition of droplet diameters and stationary liquid bulk maximum thickness in the case of an impacton aluminum and glass surfaces, and (b) droplet sequential images before impact. The case is represented among themost deformed drop conditions.



therefore, the effect of droplet size on the evolution of thephenomenon is not included. The image analysis allowedidentification of at least two different regimes: (a) a de-position regime, with the formation of a crown but notfollowed by a breakup, and (b) a splashing regime char-acterized by the crown’s breakup into numerous satellitedroplets. Reasonably, an additional regime should existfor low impact velocities, where the impacting droplet de-posits on the stationary liquid bulk, without the formationof any crown. This also occurs in the case of an impact-ing droplet onto a thin liquid film (Rioboo et al., 2003),but this regime was not investigated experimentally in thepresent paper.

The main parameters affecting chain droplet impinge-ment with long time intervals between subsequent dropsare the droplet diameterDo and the impact velocityUo;other significant parameters are the densityρ and viscos-ity µ of the liquid and gas phases, and the liquid surfacetension coefficientσ. These variables are grouped in di-mensionless parameters, namely, the We and the Ohne-sorge (Oh) numbers.

We =ρliqD0U

20

σ, Re =

ρliqD0U0

µliq,

Oh =µliq

(σρliqD0)1/2(2)

In the present investigation the Oh number for all casesexamined is equal to 0.00187. In addition, this phenom-ena is significantly governed by the wettability of the gas–liquid–solid interface, represented here by the static con-tact angleθwet, since this parameter significantly affectsthe maximum thickness,δ, of the stationary liquid bulk.The static apparent contact angle for water on the alu-minum surface was measured to be equal to 60◦, whilefor the glass plate it was equal to 10◦. Table 1 reports theexperimental conditions, in nondimensional form, for allthe performed experiments.

The scope of the present paper is to describe the evo-lution of the phenomenon during its advancing phase andnot during its receding one, since during the initial stagesof the phenomenon, inertial forces override the surfacetension forces, and the axisymmetric character of the phe-nomenon around the impact axis is valid. As a result, theeffect of the We number characterizing the impact in con-junction with the number of impacting droplets can beinvestigated, and valuable information as far as the gov-erning physics of the phenomenon can be found. Fig-ures 3a–3c report the time evolution of the impact of a sin-gle droplet onto a stationary liquid bulk formed after the

t=1.44 ms

t=2.64 ms

t=3.84 ms

t=5.04 ms

t=6.24 ms

t=7.44 ms

t=8.64 ms

t=9.84 ms

t=11.04 ms

t=12.24 ms

t=13.44 ms

t=0.1 ms

t=0.55 ms

t=0.825 ms

t=1.1 ms

t=1.375 ms

t=1.925 ms

t=2.475 ms

t=4.125 ms

t=6.325 ms

t=8.525 ms

t=0.55 ms

t=1.1 ms

t=1.65 ms

t=2.2 ms

t=2.75 ms

t=3.3 ms

t=3.85 ms

t=8.8 ms

t=11 ms

t=13.2 ms

(a) (b) (c)

FIG. 3: (a) Time evolution of case A (two droplets), (b)of case B (two droplets), and (c) of case C (two droplets).

impact of one droplet on the aluminum substrate [casesA, B, and C (two droplets)]. Figures 4a–4c report thecorresponding time evolution of the droplet impact ontothe liquid bulk formed by one droplet for the case of theglass substrate [cases D, E, and F (two droplets)]. Finally,Figs. 5a–5c and Figs. 6a–6c report similar results for thecase of the residual liquid bulk made by two droplets.The time evolution of the phenomenon can be describedby four main phases, whose characteristics can be distin-guished from the relevant images. As shown in Fig. 3a,the droplets merge at once after their first contact, and thelamella’s rim moves toward the liquid–solid contact linethat, after being reached by the lamella, starts to move



TABLE 1: Examined cases–the number of droplets refers to those used to build the residual liquid bulk.Case D0 (mm) U0 We Re Plate δ∆ h/D0 ACA (◦)

Two droplets

A 4.02 1.7 161 6800 Aluminum 0.215 60B 4.02 2.5 348 10000 Aluminum 0.23 60C 4.02 2.6 377 10400 Aluminum 0.234 60D 4.02 2.2 270 8800 Glass 0.130 10E 4.02 2.4 321 9600 Glass 0.146 10F 4.02 2.65 391 10600 Glass 0.120 10

Three droplets

A 4.02 1.7 161 6800 Aluminum 0.329 60B 4.02 2.5 348 10000 Aluminum 0.277 60C 4.02 2.6 377 10400 Aluminum 0.266 60D 4.02 2.2 270 8800 Glass 0.140 10E 4.02 2.4 321 9600 Glass 0.180 10F 4.02 2.65 391 10600 Glass 0.162 10

(a) (b) (c)

t=0.253 ms

t=0.693 ms

t=1.243 ms

t=1.793 ms

t=2.343 ms

t=2.893 ms

t=3.443 ms

t=3.993 ms

t=5.093 ms

t=6.193 ms

t=7.293 ms

t=8.393 ms

t=9.493 ms

t=0.36 ms

t=0.81 ms

t=1.26 ms

t=1.71 ms

t=2.16 ms

t=3.06 ms

t=3.96 ms

t=4.86 ms

t=5.76 ms

t=6.66 ms

t=7.56 ms

t=8.46 ms

t=0.25 ms

t=0.75 ms

t=1.25 ms

t=1.75 ms

t=2.25 ms

t=2.75 ms

t=3.25 ms

t=3.75 ms

t=4.25 ms

t=5.25 ms

t=6.25 ms

t=7.25 ms

t=9.25 ms

(a) (b) (c)

FIG. 4: (a) Time evolution of case D (two droplets), (b)case E (two droplets), and (c) of case F (two droplets).

t=1.2 ms

t=2.2 ms

t=3.2 ms

t=4.2 ms

t=5.2 ms

t=6.2 ms

t=7.2 ms

t=8.2 ms

t=9.2 ms

t=10.2 ms

t=11.2 ms

t=0.55 ms

t=1.05 ms

t=1.3 ms

t=1.55 ms

t=1.8 ms

t=2.3 ms

t=2.8 ms

t=3.3 ms

t=4.3 ms

t=6.3 ms

t=8.3 ms

t=0.6 ms

t=1.1 ms

t=1.6 ms

t=2.1 ms

t=2.6 ms

t=3.1 ms

t=3.6 ms

t=4.1 ms

t=6.1 ms

t=8.1 ms

t=10.1 ms

t=12.1 ms

(a) (b) (c)

FIG. 5: (a) Time evolution of case A (three droplets),(b) of case B (three droplets), and (c) of case C (threedroplets).

radially (compare the picture at 2.64 ms and 13.44 ms).During this spreading phase, the crown evolves radiallyand vertically. At subsequent times this crown evolves, as



t=0.4 ms

t=0.9 ms

t=1.4 ms

t=2.9 ms

t=3.4 ms

t=3.9 ms

t=4.4 ms

t=5.4 ms

t=6.4 ms

t=7.4 ms

t=9.4 ms

t=0.35 ms

t=0.75 ms

t=1.25 ms

t=1.75 ms

t=2.25 ms

t=2.75 ms

t=3.25 ms

t=3.75 ms

t=4.75 ms

t=5.75 ms

t=6.75 ms

t=7.75 ms

t=8.75 ms

t=0.5 ms

t=1.0 ms

t=1.5 ms

t=2.0 ms

t=23.5 ms

t=3.5 ms

t=4.5 ms

t=5.5 ms

t=6.5 ms

t=7.5 ms

t=8.5 ms

t=9.5 ms

(a) (b) (c)

FIG. 6: (a) Time evolution of case D (three droplets), (b)case E (three droplets), and (c) case F (three droplets).

in the case of a droplet impacting onto a liquid film (Cos-sali et al., 1997, 1999, 2004). Dependent on the initialkinetic energy (impact velocity) and surface energy of theimpacting droplet, this crown may either start collapsingwithout (deposition regime, Fig. 3a,t = 7.44–13.44 ms) orwith (splashing regime, Fig. 3c,t = 2.75–13.2 ms) jettingphenomena and secondary droplets.

During the final collapse, the surface energy is trans-formed back to kinetic energy under the effect of surfacetension forces and gravity, and the crown begins to recedetoward the solid substrate. For case A (low We, equal to161) the formation of rim instabilities is observed, but no

jets are produced because the kinetic energy is not highenough (Fig. 3a,t = 6.24 ms). By increasing the We num-ber of the impacting droplets, the detachment of satellitedroplets becomes more evident, even at the first stages ofimpact. This phenomenon shows similar characteristics tothe ”prompt” splash phase occurring in the case of a sin-gle droplet onto shallow films (Cossali et al., 1997, 1999,2004). The remaining lamella later collapses in all cases.

3.1 Crown’s Diameter

After its formation, the crown diameter and height start toincrease. The definition of these two characteristic dimen-sions is ambiguous. From Fig. 7 it is easy to see that thecrown is not cylindrical, and the diameter of the surfaceof the expanding lamella varies with the distance from thesubstrate.

To take into consideration this uncertainty and to eval-uate the effect of wettability of the evolution of the phe-nomenon, two diameters were defined and measured; thewetting diameterDlb, measured at the lamella’s base incontact with the substrate, and the upper diameter mea-sured at the crown’s top,Dlr (see Fig. 7). In any case,these diameters cannot be defined precisely, since thereis no clear definition in recent literature for the relevantin-between points defining each of the measured lengths.Moreover, since these points were measured manually,the reading error from the operator should also be takeninto account. This was always found to be less than 6%by comparing the results from two different users. Dueto the aforementioned facts, one should consider an ex-pected uncertainty of the measured lengths ranging from5 to 10% of the presented mean values when referring tosimilar numerical or experimental results.

In order to compare the experimental data and to jus-tify the effect of the parameters investigated in this pa-

FIG. 7: Definition of the main dimensions characterizingthe evolving crown’s formation.



per (We number, maximum thickness of the stationaryliquid bulk, and wettability), the dimensionless timeτ =t× U0/D0, and the dimensionless lamella’s lengthsrlb =Rlb/D0, rlr = Rlr/D0, andH/D0 are used.

Figures 8 and 9 present the time evolution of the di-mensionless lamella’s base and its rim radius for all thecases examined. Regarding the wetting crown’s diameter,its temporal evolution begins only when the expandinglamella reaches the contact line (see Fig. 9). This time in-stant significantly depends on the initial diameter of thestationary droplet, which varies with We number and thecontact angle. As far as the upper diameter is concerned,the results are far more interesting. Following Yarin andWeiss’s (1995) approximation for the case of an impact-ing droplet onto a film, the evolution of the crown sizeshould follow the relation depicted by Eq. (3):

Rlr/Do = C(τ− τo)n (3)

with n being equal to 0.5, andC depending only on thedimensionless film thickness asC = (2/3δ)0.25.

Here τo is the time it takes the propagation waveto reach the surface of the film after the impact of thedroplet. In our case, this value can be considered almostequal to zero, since the time elapse between this time andthe time the impacting droplet comes into contact with theone on the surface for the first time is very small. Such arelationship is not expected to describe the whole evolu-tion of the crown, since, as also stated by Cossali et al.(2004), the fitting of all the experimental data for the caseof an impinging droplet onto a liquid film was unsuccess-ful. Following the same approach, the exponentn is con-sidered to be independent of both drop velocity and liquid

FIG. 8: Temporal evolution of dimensionless base radiusRlb/Do for all cases examined.



FIG. 9: Temporal evolution of dimensionless rim radiusRlr/Do for all cases examined.

bulk thickness, while the constantC shows a dependenceon the drop impact Weber number and initial maximumdimensionless liquid bulk thicknessδ(h/D0, h = liquidbulk thickness).

Table 2 reports the values ofC andn for each case ex-amined, as well as the value ofC when the exponentnis equal to 0.5. These values were extrapolated from theexperimental data, after following a best fitting procedure

TABLE 2: Coefficients C and n for the experimental cases. The theoretic coefficient Cth is also presented, whilenth = 0.5.

Two droplets Three dropletsCase Plate We h/D0 τ0 Cth Cexp nexp h/D0 τ0 Cth Cexp nexp

A Al 160.9 0.215 0.230 1.327 1.049 0.470 0.329 ∼ 0 1.193 1.039 0.485B Al 348.0 0.230 ∼ 0 1.305 1.314 0.403 0.277 0.235 1.246 1.223 0.405C Al 376.4 0.234 ∼ 0 1.299 1.045 0.524 0.266 ∼ 0 1.258 1.109 0.502D Glass 269.5 0.130 0.014 1.505 1.196 0.471 0.140 ∼ 0 1.477 1.065 0.488E Glass 320.8 0.146 0.012 1.462 1.314 0.436 0.180 0.075 1.387 1.308 0.398F Glass 391.1 0.120 ∼ 0 1.535 1.072 0.544 0.162 ∼ 0 1.424 1.029 0.586



of the experimental data for the time evolution of the di-mensionless lamella’s radius at its bottomRl,b and at itsrim Rl,r.

From Table 2 it is deduced that the argument of Yarinand Weiss (1995) and Cossali et al. (2004) that the timeevolution of the dimensionless lamella’s radius at its rimcan be fitted by an exponentn equal to 0.5 in the caseof a single droplet impingement onto a liquid film is alsovalidated in the present case. The coefficientn is closeto the theoretic value of 0.5, with a mean value equal to0.476± 0.06, and it seems that there is no dependencyon the sessile droplet’s thickness, or on the We and Frnumbers. The mean value ofτ0 is equal to 0.047.

CoefficientC shows a large deviation from the theoret-ical values for a film. In order to identify the dimension-less parameters on which the value ofC is dependent on,we assumed this to be a function of We and Fr numbers,as well as of the geometry of the deposited sessile droplet.The latter is taken into consideration by defining the ra-tio of the droplet thickness divided by the initial contactdiameter of the sessile droplet, i.e.,

w =h

DLB,0(4)

This ratio is a measure of the wettability of the ses-sile droplet, and for a spherical cap this can be approx-imated by the value of the wetting contact angle, i.e.,w = 1/2 tan (θ/2). Keeping the aforementioned com-ments in mind, the following relationship was examined:

C

Cfilm,th= [f (We,Frh)]w (5)

In terms of a physical interpretation, this type of cor-relation indicates that for a fully wetted surface withDLB,0 → ∞, w → 0, and from Eq. (5) it is concludedthat C = Cfilm,th. Using a multiple regression analysis,it has been proven that the coefficientC depends on boththe We and Fr numbers. A simplified expression is given,

which relates the coefficientC of Eq. (3) with the theo-retical values of coefficientC for the impact on a liquidfilm:

Cexp = Cfilm,th

[4.09e− 3×We9.29Fr−7.73

h

]w(6)

The mean deviation of the calculated results in compari-son with the corresponding data was approximately 9.9%,which is relatively satisfactory since only 12 experimen-tal data were available. The exact values of the calculatedresults are listed in Table 3.

The initial values ofRlb for the case of glass substrateand its rate of increase with time are greater than for thecase of aluminum substrate. This observation reveals theeffect of wettability on the evolution of the phenomenon,as the lamella spreads more easily on hydrophilic sur-faces. Moreover, the We number does not seem to mono-tonically affect the spreading of lamella on the substrate,since the detachment of satellite droplets, especially inthe case of the two higher We numbers, may significantlyaffect the initial values ofRlb.

A similar expression to Eq. (3) is expected to hold forthe base radius. The main difference is thatτ0 should begreater than zero, since there is a time delay for the de-velopment of the phenomenon, and atτ = τ0 the radiusshould be equal to the initial radius:

RLB

D0=

RLB,0

D0+ C(τ−τ0)

n , τ > τ0 (7)

The results obtained from the regression analysis arelisted in Table 4. An effort was made to relate the cor-responding coefficients of Eq. (3) for the rim radius withthe corresponding coefficients of Eq. (7) for the base ra-dius. It is concluded thatCbase = 0.24rim independent ofWe, Fr, andδ, while for the coefficientn there was a largescattering and an important dependency onδ was found,i.e., nbase/nrim = 0.465 × δ−0.667. Figure 10 presentsthese results.

TABLE 3: Parameters affecting the coefficients CTwo droplets Three droplets

Case We δ Frh w Ccalc Cexp δ Frh w Ccalc Cexp

A 160.9 0.215 340.5 0.091 0.977 1.049 0.329 222.5 0.121 1.183 1.039B 348.0 0.230 688.4 0.105 1.099 1.314 0.277 571.6 0.114 1.219 1.223C 376.4 0.234 731.8 0.099 1.135 1.045 0.266 643.8 0.109 1.213 1.109D 269.5 0.130 943.1 0.040 1.165 1.196 0.140 875.8 0.035 1.204 1.065E 320.8 0.146 999.4 0.057 1.083 1.314 0.180 810.6 0.061 1.109 1.308F 391.1 0.120 1482.5 0.046 1.140 1.072 0.162 1098.1 0.057 1.122 1.029



TABLE 4: Coefficients C and n for the experimental cases for the base radiusTwo droplets Three droplets

Case Plate We h/D0 τ0 Cexp nexp h/D0 τ0 Cexp nexp

A Al 160.9 0.215 1.200 0.251 0.708 0.329 1.800 0.215 0.350B Al 348.0 0.230 0.850 0.303 0.745 0.277 2.029 0.252 0.598C Al 376.4 0.234 1.358 0.368 0.550 0.266 1.668 0.279 0.429D Glass 269.5 0.130 1.884 0.206 0.998 0.140 0.570 0.048 0.917E Glass 320.8 0.146 1.239 0.418 0.733 0.180 2.209 0.292 0.635F Glass 391.1 0.120 1.483 0.448 0.738 0.162 1.626 0.235 0.825

FIG. 10: Dependence ofCbase/Crim andnbase/nrim onδ = (h/Do) for the cases examined.

3.2 Crown’s Height

The time evolution of the crown height (H) nondimen-sionalized with the initial droplet’s diameterDo is de-picted in Fig. 11 for all cases examined. The crown’sheight always reaches a maximum valueHmax after acertain timeτmax from impact. The maximum crown’sheight increases with increasing We number of the im-pacting drop. Apparently, the contact angle does not sig-nificantly influence this parameter. BothHmax andτmax

show a reasonable dependence of the initial impact veloc-ity Uo, but they are almost independent ofδ (see Fig. 12).This is also found in the case of a droplet impact ontoa film (Cossali et. al., 2004), which is indicative of thesimilar governing physics between the two phenomenon.

From Fig. 12, following a best-fitting procedure, thedimensionless maximum height the crown reaches canbe approximated by the equation given asHmax/Do =0.4088 – 0.0626, with the corresponding time asτmax =1.4502 – 0.0855 for the range of parameters investigatedin the present paper.

3.3 Creation of Jets

After the initial steps of crown’s generation, along itsexpansion two kinds of perturbation waves are induced

on its surface, namely, longitudinal and azimuthal (seeFig. 13a,t = 2.75 ms), as in the case of a droplet imping-ing onto liquid films (Cossali et al., 1997). The longitudi-nal waves propagate vertically in the liquid sheet, and theazimuthal are visible in the form of periodical swallowingalong the cylinder. At the crown’s top, a rim is generatedand then cusps are formed, which under the simultane-ous effect of the aforementioned waves, form the root ofthe jets’ ejection. These jets break up into a number ofsatellite droplets, possibly through a Rayleigh process. InFigs. 13a and 13b, primary and secondary break-up stagesfrom the same jet are evident.

Despite the detachment of satellite droplets, the crowncontinues to develop until it reaches its maximum defor-mation, and under the effect of surface tension forces andgravity it begins to collapse toward the solid plate. Duringits collapse, some bigger droplets at the crown’s peripheryare formed and detach until its final collapse (Fig. 13b,t= 6.75 ms).

4. CONCLUSIONS

This paper reports the main findings related to the crownand base radius development after the impingement of adroplet onto a stationary liquid bulk, formed by the pre-vious impact of one or two droplets with the same im-pact parameters. The experimental analysis of the mea-surements revealed that the governing physics of this typeof phenomenon is quite similar to the one controlling theimpact of a droplet onto shallow liquid films, althoughthe influence of the wall wettability introduces some im-portant differences in terms of the shape of the aforemen-tioned liquid bulk. The crown rim diameter follows a lawsimilar to the case of a droplet impingement onto a uni-form film. Following analysis of the controlling param-eters describing the temporal evolution of this variable,the roles of We and Oh numbers, as well as of the wet-ting contact angle, were identified, providing analyticalexpressions. These expressions are based on the proposed



FIG. 11: Temporal evolution of dimensionless crown’s heightH/Do for all cases examined.

1.6 2 2.4 2.8

U0 (m/s)

0

1

2

3

4

5

Hm

ax/D

0

,

� max

Hmax/D0

tmax

FIG. 12: Dependence ofHmax andτmax onUo andδ.



(a)

(b)

FIG. 13: (a) Longitudinal and azimuthal waves on thesurface of the expanding crown,t = 2.75 ms, and (b) satel-lite droplet detachment during the crown’s collapse,t =6.75 ms (case E, three droplets).

correlation by Yarin and Weiss for the case of a dropletimpacting onto a shallow film. Regarding the wetting baseradius, the coefficientC was found to be related to thecorresponding coefficient for the rim temporal evolution,while the exponentn was found to be a function both ofthe corresponding exponentn of the rim and of the di-mensionless bulk liquid thicknessδ.

The maximum crown’s height and the correspondingtime that this occurs increase with We number and do notappear to depend significantly on the maximum thickness.Gravitational forces might not have a significant influenceon the evolution of the phenomenon for the range of pa-rameters investigated in the present paper. In the case ofdroplet impact onto a shallow film, two types of pertur-bation waves are induced on the lamella surface, namely,longitudinal and azimuthal. The investigation of this com-plicated phenomenon should be continued, taking into ac-count the effect of liquid viscosity for a wider range of Wenumbers and film thicknesses, in order to better under-stand the complicated governing physics and to build upreliable correlations which can be used in spray modelssimulating such interactions.

ACKNOWLEDGMENTS

Nikolopoulos Nikos is grateful for a Marie Curie Fellow-ship (2003).

REFERENCES

Anders, K., Roth, N., and Frohn, A., Operation characteristicsof vibrating-orifice generators as a calibration standard forsizing methods and for the study of basic phenomena,Proc.2nd Int. Cong. on Optic Particle Sizing, 1990.

Bernardin, J. D. and Mudawar, I., The Leidenfrost Point: Ex-perimental study and assessment of existing models,ASMEJ. Heat Transfer, vol. 121, pp. 894–903, 1999.

Cooper-White, J. J., Crooks, R. C., and Boger, D. V., A dropimpact study of worm-like viscoelastic surfactant solutions,Colloids Surf., A, vol. 210, pp. 105–123, 2002.

Cossali, G. E., Coghe, A., and Marengo, M., The impact of asingle drop on a wetted solid surface,Exp. Fluids, vol. 22,pp. 463–472, 1997.

Cossali, G. E., Brunello, G., Coghe, A., and Marengo, M., Im-pact of a single drop on a liquid film: Experimental analysisand comparison with empirical models,Italian Congress ofThermofluid Dynamics UIT,Ferrara, Italy, 30 June 1999.

Cossali, G. E., Marengo, M., Coghe, A., and Zhdanov, S., Therole of time in single drop splash on thin film,Exp. Fluids,vol. 36, pp. 888–900, 2004.

Fujimatsu, T., Fujita, H., Hirota, M., and Okada, O., Interfacialdeformation between an impacting water drop and a silicone-oil surface,J. Colloid Interface Sci., vol. 264, pp. 212–220,2003.

Fujimoto, H., Ogino, T., 1, Takuda, H., and Hatta, N., Collisionof a droplet with a hemispherical static droplet on a solid,Int.J. Multiphase Flow, vol. 27, pp. 1227–1245, 2001.

Ghafouri-Azar, R., Mostaghimi, J., Chandra, S., and Charm-chi, M., A stochastic model to simulate formation of a ther-mal spray coating,J. Therm. Spray Technol., vol. 12, pp. 53–69, 2003.

Gu, Y., Drop size dependence of contact angles of oil drops on asolid surface in water,Colloids Surf., A, 181:215–224, 2001.

Hatta, N., Fujimoto, H., and Takuda, H., Deformation processof a water droplet impinging on a solid surface,Trans. ASMEJ. Fluids Eng., 117:394–401, 1995.

Hatta, N., Fujimoto, H., Kinoshita, K., and Takuda, H., Experi-mental study of deformation mechanism of a water dropletimpinging on hot metallic surfaces above the Leidenfrosttemperature,Trans. ASME J. Fluids Eng., vol. 119, pp. 692–699, 1997.

Hobbs, P. V. and Kezweeny, A. J., Splashing of a water drop,Science, vol. 155, pp. 1112–1114, 1967.

Kang, B., Zhao, Z., and Poulikakos, D., Solidification of liquid



metal droplets impacting sequentially on a solid surface,Int.J. Heat Mass Transfer, 116:436–45, 1994.

Kang, B., Waldvogel, J., and Poulikakos, D., Remelting phe-nomena in the process of splat solidification,J. Mater. Sci.,vol. 30, pp. 4912–4925, 1995.

Levin, Z. and Hobbs, P. V., Splashing of water drops on solidand wetted surfaces: Hydrodynamics and charge separation,Philos. Trans. R. Soc. London, Ser. A, vol. 269, p. 555, 1971.

Macklin, W. C. and Metaxas, G. J., Splashing of drops on liquidlayer,J. Appl. Phys., vol. 47, no. (9), p. 3963, 1976.

Mao, T., Kuhn, D. C. S., and Tran, H.,AIChE J., vol. 43, p.2169, 1997.

Moreira, A. L. N, Moita, A. S., Cossali, G. E., Marengo, M., andSantini, M., Secondary atomization of water and isooctanedrops impinging on tilted heated surfaces,Exp. Fluids, vol.43, pp. 297–313, 2007.

Muller, A., Schumann, F., Dullenkopf, K., and Bauer, H. J.,Analysis of droplet wall interaction with graded surfaceroughness,Proc. 22nd ILASS Europe, 2007.

Naber, J. D. and Farrell, P. V., Hydrodynamics of droplet im-pingement on a heated surface, SAE Technical paper no.930919, 1993.

Oguz, H. N. and Prosperetti, A., Bubble entrainment by the im-pact of drops on liquid surfaces,J. Fluid Mech., vol. 219, p.143, 1990.

Okawa, T., Shiraishi, T., and Mori, T., Production of secondarydrops during the single water drop impact onto a plane watersurface,Exp. Fluids, vol. 41, p. 965, 2006.

Prosperetti, A. and Oguz, H. N., The impact of drops on liquidsurfaces and the underwater noise of rain,Annu. Rev. FluidMech., vol. 25, p. 577, 1993.

Rahimi, S. and Weihs, D., Experimental investigation ofmagneto-rheological droplet on a smooth surface,J. Magn.Magn. Mater., vol. 321, pp. 3178–3182, 2009.

Rein, M., Phenomena of liquid drop impact on solid and liquidsurfaces,Fluid Dyn. Res., vol. 12, p. 61, 1993.

Richter, B., Dullenkopf, K., and Wittig, S., Wall impact of singledroplets under conditions of DISI-engines,9th Int. Conf. onLiquid Atomization and Spray systems, Sorrento, Italy, 2003.

Richter, B., Dullenkopf, K., and Bauer, H. J., Investigation ofsecondary droplet characteristics produced during wall im-pact,12th Int. Symp. on Applications of Laser Techniques toFluid Mechanics, Lisbon, Portugal, 2004.

Richter, B., Dullenkopf, K., and Bauer, H. J., Investigation ofsecondary droplet characteristics produced by an isooctanedrop chain impact onto an heated piston surface,Exp. Fluids,vol. 39, pp. 351–363, 2005.

Rioboo, R., Marengo, M., and Tropea, C., Outcomes from a

drop impact on solid surfaces,Atomization Sprays, vol. 11, p.155, 2001.

Rioboo, R., Bauthier, C., Conti, J., Voue, M., and Coninck, J.,Experimental investigation of splash and crown formationduring single drop impact on wetted surfaces,Exp. Fluids,vol. 35, pp. 648–652, 2003.

Rodriguez, F. and Mesler, R., Some drops don’t splash,J. Col-loid Interface Sci., vol. 106, pp. 347–352, 1985.

Roisman, I.V., Horvat, K., and Tropea, C., Spray impact: Rimtransverse instability initiating fingering and splash and de-scription of a secondary spray,Phys. Fluids, vol. 18, 2006.

Shin, J. and McMahon, T. A., The tuning of a splash,Phys.Fluids A, vol. 2, pp. 1312–1317, 1990.

Stanton, D. W. and Rutland, C. J., Multi-dimensional modelingof thin liquid films and spray-wall interactions resulting fromimpinging sprays,Int. J. Heat Mass Transfer, vol. 41, p. 3037,1998.

Vander Wal, R. L., Berger, G. M., and Mozes, S. D., Dropletssplashing upon films of the same fluid of various depths,Exp.Fluids, vol. 40, pp. 33–52, 2006.

Wachters, L. H. J. and Westerling, N. A. J., The heat transferfrom a hot wall to impinging water drops in the spheroidalstate,Chem. Eng. Sci., vol. 21, pp. 1047–1056, 1966.

Wang, A. B. and Chen, C.C., Splashing impact of a single droponto very thin liquid films,Phys. Fluids, vol. 12, pp. 2155,2000.

Wang, M., Hung, L., Lin, F., and Lin, S., Dynamic behavioursof droplet impact and spreading: A universal study of dimen-sionless wetting diameter and droplet height,Exp. Therm.Fluid Sci., vol. 33, pp. 1112–1118, 2009.

Werner, S. R., Jones, J. R., Paterson, A. H., Archer, R. H., andPearce, D. L., Droplet impact and spreading: Droplet forma-tion effects,Chem. Eng. Sci., vol. 62, pp. 2336–2345, 2007.

Worthington, A. M., A Study of Splashes, Longmans,Green/London, 1908.

Yarin, A. L. and Weiss, D. A., Impact of drops on solid surfaces:Self-similar capillary waves, and splashing as a new type ofkinematic discontinuity,J. Fluid Mech., vol. 283, pp. 141–173, 1995.

Yarin, A., Drop impact dynamics: Splashing, spreading, reced-ing, bouncing,Annu. Rev. Fluid Mech., vol. 38, pp. 159–192,2006.

Yu, Z., Ge, Y., and Fan, L. S., Multi-scale simulation of obliquecollisions of a droplet on a surface in the Leidefrost regime,Chem. Eng. Sci., vol. 62, pp. 3462–3472, 2007.

Zhang, X. and Basaran, O. A., Dynamic surface tension effectsin impact of a drop with a solid surface,J. Colloid InterfaceSci., vol. 187, pp. 166–178, 1997.


EXPERIMENTAL INVESTIGATION OF A SINGLE DROPLET IMPACT ONTO A SESSILE DROP

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