ILASS Americas 19th Annual Conference on Liquid Atomization and Spray Systems, Toronto, Canada, May 2006

Axisymmetric impact of a drop onto a solid spherical target

S. Bakshi! †, I. V. Roisman , C. TropeaChair of Fluid Mechanics and Aerodynamics

Darmstadt University of Technology, Petersenstr. 30, 64287, Darmstadt, Germany

AbstractAn experimental and theoretical study of the impact of a droplet onto a spherical target has been performed.The droplet size in this study is around 2.4-3.2 mm and the target size is varied between 1.7-10 mm.Experiments are conducted to study the e!ect of droplet Reynolds number and target-to-droplet size ratioon the dynamics of the film flow on the surface of the target. Spatial and temporal variation of film thicknesson the target surface is measured. Three distinct temporal phases of the film dynamics are clearly visible fromthe experimental results, namely the free fall phase, the inertia dominated phase and viscosity dominatedphase. It is observed that for a given target-to-drop size ratio, the non-dimensional temporal variationof film thickness collapse onto a single curve in the first and second phases. But, the transition to theviscosity dominated regime occurs earlier for the low Reynolds number cases with higher residual thickness.A theoretical model is proposed to explain the film flow on the surface of the target. The theory predictsthe evolution of the film thickness and the film profile quite accurately inspite of the fact that no adjustableparameters are used in the model.

!Corresponding Author†On leave from Indian Institute of Technology-Madras,

Chennai-600036, India

Introduction

Droplet impaction onto a solid surface is awidely investigated area of research in fluid dynam-ics considering its extensive application in industrialprocesses and devices. Spray cooling, spray paint-ing, fuel-air mixing in internal combustion engines,ink-jet printing, spray forming, fire suppression viasprinkler systems etc are few examples where thedroplet impaction process is considered important[1, 2]. An extensive investigation in this area haveresulted in state-of-the-art experimental techniquesusing high speed photography and advanced compu-tational methodologies. Rein [3] and Yarin [4] havepresented a comprehensive review on this subject.A systematic study has been carried out by vari-ous researchers [3, 5, 6, 7] to characterize the post-impingement behaviour of the droplets and spraysin terms of the spreading process, crown formation,wetting e!ect, breakup and disintegration. A largemajority of these studies have focused primarily onthe impact behaviour of droplets on large plane sur-faces (both dry and wetted). Recently, the im-paction of droplets onto small targets has also beeninvestigated. A small target is defined as one, forwhich the size is comparable to the size of the im-pacting drops. Hung and Yao [8] have conductedexperiments on the impaction of water droplets of110, 350 and 680 µm in diameter on cylindrical wiresof various diameter between 0.1 to 1.5 mm at animpaction velocity between 2 to 7 m/s. They ob-served two distinct modes of the impaction outcome,namely finely disintegrated drops at high dropletvelocity and large dripping drops at low velocity.Hardalupas et al. [9] have reported experiments onliquid drop (160-230 µm diameter) impacting on thesurface of small solid sphere (0.8-1.3 mm diameter)at an impaction velocity of 6 to 13 m/s. They ob-served a retraction of liquid crown at low droplet im-pact velocity and disintegration from cusps locatedon the crown rim at a high impact velocity. Theyalso noticed that the increase in sphere curvature re-sulted in promoting the onset of splashing. Rozhkovet al. [10, 11] have presented results of drops ofdiameter ranging between 2.8-4.0 mm onto a stain-less steel disk of 3.9 mm diameter at an impact ve-locity of 3.5 m/s [10]. Thus, though the impactingsurface is flat, its size is comparable to the dropletsize. This condition resembles the impingement ofan ink drop from an ink-jet printer onto the paperas the length scale of the droplet is comparable tothe roughness size of the paper. The experimentsof Rozhkov et al. [10] showed that the collision ofthe droplet onto a small target induces liquid ejec-tion in the shape of a central cap surrounded by a

liquid lamella. Two free surfaces are formed as aresult of transformation of the droplet into a freeliquid lamella. Hence, the e!ect of surface tensionforce is almost doubled. Moreover, the role playedby viscous forces are substantially reduced as theliquid lamella is a free shear flow with no contactwith the solid surface. In fact, Rozhkov et al. [10]state this as a uniqueness of their study, where thee!ect of capillarity and inertia forces in the drop im-pact process can be studied in isolation, artificiallyremoving the e!ect of viscosity. In a later study bythe same investigators [11], the e!ect of high mole-cular weight polymeric additives on the impactionprocess was investigated. With polymer additives,the secondary jet, after being issued from the rim,transformed into liquid fingers consisting of a thin-ning filament with an attached drop. Depending onthe polymer concentration, the attached droplet ei-ther escaped or is pulled back to the parent drop.With this background study, it is clear that the im-paction process of droplet onto small targets, likespheres and particles, is quite di!erent from the im-pact of a droplet onto a large substrate. However,the numerical and experimental tools developed overthe years for investigating the droplet impact ontoa large substrate can be extensively used for this re-search. In fact, these studies will have enormous util-ity in the context of new areas in drop/spray impact,namely tablet coating and encapsulation processes.The main interest in the present study lies in thecoating of particles like detergents and pharmaceu-tical products using small droplets of coating materi-als. Additionally, it became apparent in the courseof this investigation that unlike drop impact ontoflat surfaces, the spherical geometry of the target inthis study o!ers the possibility to view and measurethe temporal and spatial variation of film thicknessafter the drop impact. In contrast, with drop impactonto a flat surface, the lamellae are surrounded bya thick rim making it di"cult to measure the filmthickness. This provides an excellent opportunity tovalidate and improve existing theories [12, 13] de-scribing the fluid flow in the lamella after the dropimpaction process. This has been the focus of thepresent study.

Experimental setup and method

The experimental set-up used is a typical one asused widely for investigating drop impacts onto flatsurfaces. It mainly constitutes of two high-speedCMOS cameras (Vosskuhler HCC 1000 (Strobe))with 8 bit grey-scale resolution and frame grabbingrates of 922 fps (1024 ! 512 pixels) or 1825 fps (1024! 256 pixels). One camera, i.e., the camera-1 is fit-

Figure 1. Schematic of the set-up used in the dropimpact experiments.

ted with a long distance (LD) microscope to measurethe film thickness accurately with high spatial res-olution. The other camera (camera-2) without themicroscope provides a larger field of view; hence isuseful to record the process as a whole. The imagesfrom the camera-2 are generally used to measure thevelocity and diameter of the impinging droplet andalso to measure the film thickness when the film isoutside the field of view of the camera-1. Back-lighting from an LED stroboscope with an illumi-nation of 10 µs is triggered by the camera. Di!usersin front of both the stroboscopes help to obtain auniform lighting. The droplet is generated using ahypodermic syringe with 0.4 mm diameter needlewhich yields around 2.6 mm diameter distilled waterdroplets (2.4 mm droplet for glycerine-water solu-tion, 50/50 %Volume). The droplet passes througha light barrier before impinging on the spherical tar-get. The light barrier triggers both the cameras af-ter a delay as set in the delay generator. The im-ages captured in the camera are eventually trans-ferred to the computer for further post-processing.The image processing is done by using the imageprocessing tool box as available with Matlab (ver-sion:7.0.0.19920 (R14)) of The MathWorks. Thetarget was mounted on a two-way traverse to fa-cilitate accurate alignment of the droplet centrelinewith that of the target. Proper alignment was onlypossible after several trials and constantly observ-ing the images captured by both the cameras. Thetarget surface was cleaned and dried before everyexperiment.

A sequence of images as captured by the camera-2 for a droplet of aqueous glycerine solution ontoa target of 3.2 mm diameter is shown in Fig. 2.The figure clearly shows the formation of the liquidlamella surrounded by the rim on the surface of thetarget. It is also quite apparent from these figures

Figure 2. Impact of a droplet (Glycerine + Wa-ter, 50/50 V%, 2.4 mm diameter) onto a sphericaltarget (3.2 mm diameter). The impact velocity is1.8 m/s. From left to right of the figure, snap-shotswith increasing time are shown.

1)

3)

2)

−

−

−

=

=

=

Figure 3. Temporal and spatial variation of theflowing lamella on a spherical target can be trackedby subtracting two images.

that subtracting the post-impingement images fromthe image of the target alone will yield the imageof the liquid film on the surface of the target. Thesame is illustrated in Fig. 3. The rows 1, 2 and 3shows the image subtraction done at three di!erentpoints of time. This exemplifies the manner in whichthe spatial and temporal distribution of the film onthe target surface is obtained from the images. Toobtain an enhanced resolution in terms of time, thesame experiments were repeated with a di!erent de-lay setting. The images during the time of interestwere observed to be su"ciently repeatable to adoptthis strategy. The spatial resolution for the camera-1was set between 5-9 µ m/pixel. The higher resolu-tion was possible for the smaller targets. But forrelatively large targets the resolution was reduced(to around 9 µ m/pixel) to insure su"cient field ofview. The spatial resolution of the camera-2 wasfixed at around 26 µ m/pixel.

Experimental results

The experimental results will be mainly pre-sented in the form of the distribution of the fluidmass on the target surface after the impingementprocess at di!erent points of time. For the temporalvariation it will be more convenient and still mean-ingful to look at the variation of the film thickness

at the North Pole of the target sphere. Figure 4shows the images for di!erent target sizes at simi-lar dynamic conditions. Unlike the small targets thelarger ones do not entirely get coated with the im-pinging droplets, rather the film only spreads over apart of the surface.

The film profiles at di!erent points of time afterthe droplet impact for a Reynolds number of around4806 (Weber number=131) and Reynolds number of2560 (Weber number=35) are shown in Fig. 5 andFig. 6 respectively. The di!erent line types in thefigures indicates the di!erent values of delay settingsas used in the experiments. These figures show thehigh degree of repeatability in the experiments asall the curves with di!erent delay settings appearsto arrive from a single experiment. The horizontalaxis of this figure is the angle ! as shown in the top-right corner of the figure. The case with low Webernumber and Reynolds number, i.e., Fig. 6 shows thepresence of many capillary waves on the surface ofthe film, which is not seen for the other case. It isalso clear that towards the final stages of spreading,the film distribution becomes almost uniform on thesurface, particularly for the high Reynolds numbercase. The temporal variation of the film thickness

A)

B)

C)

Figure 4. Sequence of images (from left to right) forimpact of a water droplet (2.6 mm diameter) onto aspherical target (di!erent diameter for di!erent row)is shown. The target diameter for row-A is 3.2 mm,for row-B is 6.12 mm and for row-C is 10.2 mm. Theimpact velocity is 2 m/s.

at the North Pole of the spherical target is shownin Fig. 7 in a log-log axes. Hereby the film thick-ness is non-dimensionalized with the impacting dropdiameter and the time is non-dimensionalized usingthe ratio of impacting drop diameter and the impactvelocity. The drop diameter is 2.6 mm and the tar-get size is 3.2 mm. Three distinct temporal phasesof the film dynamics can be easily recognized fromthis figure. The first phase, i.e., the phase-1 exhibits

−150 −100 −50 0 50 100 1500

500

1000

1500

θ (Degree)

Film

thic

knes

s in

mic

ron

θ

Figure 5. Spatial variation of the flowing lamellaon the spherical target(3.2 mm diameter) at di!er-ent points of time after the droplet impact. Thedroplet size is 2.6 mm and droplet velocity is 2 m/s.The dashed and solid lines indicate measurementscorresponding to two di!erent delay settings.

−100 −50 0 50 1000

200

400

600

800

1000

1200

1400

1600

1800

θ

Film

thic

knes

s in

mic

ron

θ

Figure 6. Spatial variation of the flowing lamellaon the spherical target(3.2 mm diameter) at di!er-ent points of time after the droplet impact. Thedroplet size is 2.6 mm and droplet velocity is 1 m/s.Each linetype (solid, dash, dot, dash-dot) indicatemeasurements corresponding to four di!erent delaysettings.

variation corresponding to the equation h! = 1 " t!

(h! and t! are non-dimensional film thickness andtime respectively). During the second phase (phase-2), the temporal variation is given by the equationh! = 0.15

(t!)2 . Finally, during the third phase the filmthickness almost reaches a residual value. So, the ini-tial phase is the free fall period during which the freesurface of the deforming drop is negligibly influencedby the presence of the target. In the second phase in-ertia dominates the viscous forces; hence the natureof variation resembles the inviscid theory of Yarin etal. [12]. The precise comparisons with this inviscidtheory will be shown in the next section. The thirdphase is governed by the balance of the viscous andgravity forces. The dynamics is very slow during thisphase and a slow thinning of the lamella takes placedue to the gravitational draining. Similar experi-ments are conducted for di!erent values of dropletReynolds number for the same set of the droplet andtarget. The Reynolds number is varied by changingthe impact velocity and the working liquid. Di!erentvelocities are achieved with di!erent falling heightsand the di!erent liquid used is an aqueous solutionof glycerine. A spread of Reynolds number is ob-tained in this manner. The results for the tempo-ral variation of film thickness for di!erent Reynoldsnumber is shown in Fig. 8. Interestingly, the non-

10−1 100 101 102 10310−3

10−2

10−1

100

101

Non−dimensional time ( t* )

Non−d

imen

sion

al fi

lm th

ickn

ess

( h* )

Phase−1

Phase−2

Phase−3

h*=1−t*

h*=0.15/(t*)2

Figure 7. Temporal variation of film thickness atthe North Pole of the target sphere (3.2 mm diam-eter). The value of the droplet (2.6 mm diameter)Reynolds number is 4806 and Weber number is 131.

dimensional temporal variation of film thickness fordi!erent values of Reynolds number collapse onto asingle curve in the first and second phases. But, thetransition to the viscosity dominated regime occursearlier for the low Reynolds numbers case. Also, theresidual thickness in the viscosity dominated phase

10−2 10−1 100 101 102 10310−3

10−2

10−1

100

101

Non−d

imen

sion

al fi

lm th

ickn

ess

( h* )

Non−dimensional time ( t* )

Re=525, We=90Re=635, We=132Re=2560, We=35Re=4806, We=131Re=6108, We=205

h*=0.15/(t*)2

h*=1−t*

Figure 8. Temporal variation of film thickness atthe North Pole of the target sphere (3.2 mm diame-ter) for di!erent values of droplet Reynolds and We-ber number.

reduces with increasing Reynolds number. At thelowest value of Weber number, i.e., 35 and with aReynolds number of 2560, as indicated by trianglesin this figure an instability could be observed in thefilm in the residual stage. This is due to the pres-ence of the capillary waves as was also observed inthe film profiles in Fig. 6. The variation of themean film thickness in the residual stage with theReynolds number is shown in Fig. 9. The nature ofthis curve is of the form, h!

residual = 0.39!Re"0.37.Next, the e!ect of target-to-droplet size ratio onthe dynamics of film flow is investigated. Figure 10shows the temporal variation of the film thicknessat the North Pole of the target sphere for di!er-ent values of target-to-droplet size ratios (R!) withthe same value of droplet Reynolds and Weber num-ber. It is clear from this figure that the rate of film-thinning becomes slower with increasing R!. Theresidual thickness also increases with increasing R!,the change becoming less towards the higher valuesof R!. The receding phase, marked by the thickeningof the film, is also clearly visible for the higher valuesof R!. The line corresponding to h! = 1 " t!, i.e.,the free fall of the droplet (as shown by the solid linein the figure) is the fastest temporal variation. Thiscan be imagined as a case with zero target size. Asthe target size is gradually increased from zero value,the temporal variation slows down gradually, beingthe slowest at the largest R! value. In terms of theresidual thickness, increasing R! has the same e!ectas that of decreasing the Reynolds number. Figure11 shows the variation of film thickness (h!) and thespreading (!!) of the lamella in the same plot for

0 1000 2000 3000 4000 5000 6000 70000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Reynolds number

Non−d

imen

sion

al fi

lm th

ickn

ess

( h* )

h*residual=0.39 Re−0.37

Figure 9. Variation of the residual film thicknesson the surface of the target sphere with the dropletReynolds number.

two di!erent values of R!. The non-dimensionalizedspreading is defined as, !! = 2#R#!

Do. Where R is

the radius of the target, Do is the droplet diame-ter and ! is the spreading as shown in Fig. 6. Thenature of the spreading curve resembles that for theimpingement of a droplet onto a flat surface, i.e., thespreading is proportional to t

1

2 . For lower values ofR! the rim leaves the target surface faster makingit di"cult to measure the temporal spreading be-haviour. The several features as visible in the filmthickness variation (like inertia and viscosity dom-inated phases) is also apparent from the spreadingcharacteristics. In general, these two curves seem tobe quite representative of one another.

Theory for the flowing lamella on the targetsurface

A. Inviscid Solution

The governing equations for flow of the lamellaafter the drop impact was given by Yarin et. al.[12] and later generalized by Roisman et. al. [13].Here the same equations are used in the sphericalco-ordinate system and the equations are as below.

"u!

!

"t!+

2u!

!

R!

"u!

!

"!= 0, (1)

"h!

"t!+

2u!

!

R!

"h!

"!+

2h!

R!

"u!

!

"!+

2h!u!

! cos !

R! sin != 0. (2)

All the quantities in the above equations are non-dimensionalized (Velocity with drop impact velocityand length with drop size). From a physical per-spective, equation (1) signifies that the net force onthe fluid element in the flowing lamella is zero. The

10−2 10−1 100 101 10210−2

10−1

100

Non−d

imen

sion

al fi

lm th

ickn

ess

( h* )

Non−dimensional time ( t* )

R* = 0.68

R* = 1.23

R* = 1.98

R* = 3.9

h*=1−t*

Figure 10. Temporal variation of film thickness atthe North Pole of the target sphere for di!erent tar-get size. The value of the droplet (2.6 mm diameter)Reynolds number is 4806 and Weber number is 132.

10−1 100 101 10210−2

10−1

100

101

Non−dimensional time ( t* )

h* a

nd θ

*

R* = 1.98R* = 3.9

2.5 (t*)0.5

1.3 (t*)0.5

h*

θ*

Figure 11. Temporal variation of film thickness(h!) and film spreading (!!) for two di!erent targetsizes are shown in the same plot. The value of thedroplet (2.6 mm diameter) Reynolds number is 4806and Weber number is 132.

next equation, i.e., equation (2) is the mass conserva-tion equation and signifies that the convective massfluxes balances the rate of change of volume due tothe thinning (or thickening) of the lamella. Below isa list of assumptions for which the above equationsare valid.

• The above equations are only valid in the wettedpart, where "h!

"!1

R!<< 1.

• The flow is inviscid.

• The radial component of velocity is negligibleas compared to the tangential component.

• The inertia forces are much larger than the cap-illary forces (Weber number is high). Thus, thepressure gradient due to surface tension is neg-ligible.

• The film thickness is much smaller as comparedto the radius of the target-sphere so the pressuregradient in the radial direction on the spheresurface can be neglected.

The first equation has an analytical solution withthe conditions as below.

u!

!(0, t!) = 0. u!

!(!, 0) = sin !.

One partial analytical solution is given as,

u!

! =R!!

2(t! + R!!2 sin ! )

(3)

This can be directly substituted into equation (2)to solve for h!. It can be further observed fromequation (3) that as ! tends to zero, u!

! = R!!2(t!+ R!

2).

So, using this value of u!

! in equation (2), we canget an exact solution for equation (2) as well. Thissolution for h! which is valid in the region near theNorth Pole of the target sphere is given as,

h!(!, t!) =A ! !(n"1)

sin !(t! + R!

2 )n(4)

Where A and n are arbitrary constants. Now, forh! to have a non-zero, real value at the North Poleof the target sphere, n should be equal to 2. So, thesolution for the film thickness at the North Pole ofthe sphere is given as,

h! =A

(t! + R!

2 )2. (5)

B. Initial Conditions

It should be noted here that the t! in all theabove expressions is the time starting from the initi-ation of the film flow regime on the target sphere sur-face. Now, a time t!i is introduced which is the timefrom the first contact of the droplet with the targetsurface. So, the di!erence between t!i and t! is #k,which is the time after the droplet impact for whichit travels with the impacting velocity. It is better toexpress h! in terms of t!i as this is the time whichwe get directly from the measurements. In termsof t!i the temporal variation of non-dimensional filmthickness at the North Pole of the target sphere isgiven as,

h! =A

(t!i + (R!

2 " #k))2, for t!i # #k. (6)

Now, it is also known from the experimental resultsthat the drop travels with the impacting velocityfrom 0 $ t!i $ #k. Hence, at t!i = #k,

h!(#k) = 1 " #k cos !

anddh!

dt!= " cos !

At the North Pole of the sphere these equations takesthe form h! = 1"#k and dh!

dt! = "1. So, using this we

can find the value of A and #k and they are A = R!3

16

and #k = (1" R!

4 ). With these, the final form of theequation (6) is given as

h! =R!3

16

(t!i + 3R!

4 " 1)2, for t!i # #k. (7)

But, one deficiency in the expression for #k is that itbecomes negative for R! greater than 4. This fact in-dicates that the proposed model for the initial phaseof drop impact is valid only for cases where the dropand target sizes are comparable. In fact, the descrip-tion of geometry and flow during this initial stage ofdrop deformation is a rather complicated problemand only a very rough estimation of the initial con-ditions is possible. It was shown before [14] that fordrop impact onto a flat surface the initial drop defor-mation is governed by inertia, viscous and capillaryforces. However, for smaller values of R!, inertialforces dominate the other forces substantially.

Now, it is also important to identify the pointbeyond which the viscosity becomes important in theflow of the lamella. For this a simple scaling analysishas been used. It is clear that at the point beyondwhich viscosity becomes important the viscous termand the inertia term in the momentum conservationequation are of the same order of magnitude. It can

be easily shown that this happens at a time givenby, t! = Re! h2, Where Re is the droplet Reynoldsnumber. In terms of t!i this equation becomes,

t!i +R!

4" 1 = Re ! h2. (8)

Using equation (8) and equation (7) the values of(h!,t!i ) from which the transition to the viscos-ity dominated regime takes place can be obtained.These two equations gives a fifth-order polynomialwhich can be solved numerically and it always pos-sess only one real root.

C. General Solution of the Film Profile in theInviscid Regime

Next, an alternative simple Lagrangian ap-proach will be presented for arriving at the sameequations as above and it can be noticed that amore general prescription of the spatial film thick-ness variation could be obtained using this approach.Here, a solution for the velocity as obtained beforewill be used. The velocity field as obtained before isgiven as, u!

! = R!!2(t!+ R!

2). Now in a Lagrangian frame

this will signify a velocity distribution as, u! = KS,where K is a constant (reciprocal of a time scale)and S is the arc length on the spherical surface overwhich the fluid element flows. So, the velocity of thetwo edges of a fluid element is given as u!o

= KSo

and u!o+!!o= K(So + #So). After a time t, the

fluid element #So changes to a length #S and thenby using mass conservation it can be easily shownthat,

h!(!, t!) =h!

osin!

!1+K!t!

"

(1 + K!t!) sin !. (9)

where h!

o is the initial film thickness and K! is thenon-dimensional value of K and it is given as 1

R!

2

.

With this, the above expression for film thicknessbecomes

h!(!, t!) =

h!

oR!

2 sin

#

R!

!

2

R!

2+t!

$

(R!

2 + t!) sin !. (10)

Now, as ! tends to zero, i.e., close to the North Poleof the sphere, the expression for the film thicknessbecomes,

h!(t!) =h!

oR!2

4

(R!

2 + t!)2. (11)

Using the fact that dh!

dt! = "1 at t! = 0., we get that

h!

o = R!

4 . Substituting this value of h!

o, the same

equation as equation (7) is obtained for the tempo-ral variation of film thickness. Assuming that h!

o isuniformly distributed over !, the final expression forthe spatial distribution of film thickness at any pointof time is given as below.

h!(!, t!) =

R!2

8 sin

#

R!

!

2

R!

2+t!

$

(R!

2 + t!) sin !. (12)

D. Viscous flow in the film near the NorthPole of the sphere

Now, we consider the viscous film flow in theneighborhood of the North Pole, ! = 0, of the sphereat large times. The inviscid flow approximation isvalid only at times when the wall boundary layer ismuch thinner than the film thickness. At a partic-ular time instant, t = tBL, the thickness of the ex-panding boundary layer is comparable with the filmthickness. At times, t > tBL, the flow in the filmis assumed to be developed and thus significantlyinfluenced by the viscous terms.

D.1 Evolution equation for the film thickness

The azimuthal component of the velocity, u!,and the gradient of the film thickness, "h/"!, vanishat the axis ! = 0 due to the symmetry condition.Therefore, at small angles, i.e., ! % 0, the velocityfield can be linearized and assumed in the form:

ur = U0G(r, t), u! = U0!F (r, t), u# = 0(13)

The linearized continuity equation at ! % 0

"

"r(r2!ur) +

"

"!(r!u!) = 0 (14)

yields

2G + 2F + r"G

"r= 0 (15)

Next, the pressure distribution in the film canbe estimated using the velocity field (13) in the rcomponent of the linearized Navier-Stokes equation:

1

$

"p

"r= !2 F 2

r+ f(r, t) (16)

where f(r, t) is some function of r and t only. Sincethe expression for the pressure will be then used inthe momentum equation in the azimuthal direction,only the terms containing ! are important in thepresent model.

The normal stress vanishes at the free surface ofthe film, r = 1 + h, therefore the pressure field canbe expressed in the form:

p = $ U20 !2

r%

R+h

F 2

rdr + f $(r, t) (17)

where f $(r, t) is another function of r and t.Consequently, consider the azimuthal, ! compo-

nent of the momentum equation at small !. Thislinearized equation obtained with the help of the ve-locity field (13) and the expression for the pressureprofile (17) can be written in the dimensionless form

L(r, t) = F (F + G) + r"F

"t+ rG

"F

"r+ 2

r%

1+h

F 2

rdr

"1

Re

&

2"F

"r+ r

"2F

"r2

'

"r

Fr= 0 (18)

where Re is the Reynolds number and Fr is theFroude number.

The equation (18) is non-dimensionalized usingthe sphere radius R as a length scale, the impactvelocity U0 as a velocity scale, and the expressionR/U0 as a time scale. All the expressions below aregiven in this dimensionless form.

The system of integro-di!erential equations (15)and (18) can be solved numerically accounting forthe boundary conditions

F = 0, G = 0 at r = 1 (19)

"F

"r"

F

r= 0,

dh

dt= G at r = 1 + h (20)

if the initial velocity distribution in the film isknown.

In our case, however, the initial velocity field innot defined precisely. In the present study the veloc-ity field in the film satisfying the momentum equa-tion (18) in ! direction cannot be satisfied pointwise.In the present study the momentum equation is sat-isfied only in integral sense:

1+h%

1

Ldr = 0 (21)

where L is defined in (18).In order to roughly estimate the temporal evolu-

tion of the film thickness in the regime correspondingto the fully developed flow, the equations (15) and(18) are satisfied in the integral form, assuming aparabolic profile for the function F .

Moreover, in the present work the film thicknessh is assumed to be much smaller than the sphere ra-dius R. This condition is always satisfied in our ex-periments associated with high Reynolds numbers.

Neglecting the terms of order h/R and neglect-ing the gravity, the evolution equation for the filmthickness takes the following form:

h "9h2

5h+

3h

Reh2= 0 (22)

which must be solved subject to the matching con-ditions with the inviscid solution hI(t):

h = hI(tBL) & hBL, h = dhI/ dt & "VBL at t = tBL

(23)The solution of the di!erential equation (22) is

h%

h0

dx

1514Re x "

!

xhBL

"9/5 !

1514Re hBL

+ VBL

"

= t(h)"tBL.

(24)It can be also shown that the solution (24) of

the gravity-free equation (22) yields the finite valueof the residual film thickness hr at t % ':

hr =h9/14

BL

(1/hBL + 14Re VBL/15)5/14(25)

This means that at high Reynolds number theresidual film thickness at constant droplet-to-targetratio behaves as hr ( Re5/14 ( Re0.36. This scalingagrees well with the experimental results shown inFig. 9.

D.2 Boundary layer

At very high Reynolds numbers the thickness ofthe boundary layer associated with drop impact ontoa dry wall can be roughly estimated by the solutionof the Stokes’ first (or Rayleigh’s) problem [12]. Inthe dimensionless form this thickness is evaluated bythe following relation

hBL(t) = 6(

t/Re (26)

The time tBL is the root of the equation

hBL(tBL) = hI(tBL) (27)

which can be easily solved numerically.

E. Comparison of the Theory and the Mea-surements

From the inviscid theoretical analysis, threemain outputs are obtained. First, the temporal vari-ation of the non-dimensional film thickness at theNorth Pole of the target sphere, which is given byequation (7). We also get an estimation of time forwhich this inviscid temporal law is valid and thisis obtained from the solution of equation (7) andequation (8). Finally, the spatial variation of thefilm thickness at a particular point of time is alsoobtained from the equation (12). From the viscousanalysis there are two important outcomes. The firstone is the scaling law of the residual thickness withthe Reynolds number, which was discussed in sectionD.1 and it was found to be in agreement with the

measurements. The second outcome is the temporalevolution of the film thickness at the North Pole ofthe target for both the inviscid and viscous regime.All these predictions will be compared with the ex-perimental observations and the limitations of thisanalysis and its range of validity will be clarified.

Figure 12 shows the comparison of the measuredtemporal variation of film thickness with the modelfor di!erent values of R!. The thin lines in thefigure represents the output from the inviscid the-ory. Big circles on these lines indicate the pointbeyond which viscosity becomes important. Thethicker lines in Fig. 12 represents the output fromthe viscous model. The inviscid model gives a goodcomparison for values of R! close to one. However,the model substantially overpredicts the film thick-ness for higher values of R!. For lower values ofR! the film thickness is little underpredicted. Theviscous theory seems to capture the overall natureof the variation of film thickness. But, quantita-tive comparison is only good for smaller values ofR!. There is overprediction for higher values of R!.Overall, it can be summarized that this theory isquite useful for deriving scaling laws. And for val-ues of R! close to one the theory compares extremelywell with the experiments. The predictions of thepoint beyond which viscosity becomes important inthe flow (shown by open circles in the figure) is alsogood. It is important to notice that all the expres-sions compared here are fully analytical and withoutany constants whatsoever. Finally, the prediction ofthe spatial distribution of the film on the surface ofthe target (as given by equation (12)) will be com-pared with the measurements. It could be observedfrom Fig. 13 that for a value of R! = 1.23, thepredicted spatial distribution matches the measure-ments for three di!erent points of time quite well.However, it is clear that the rim formation is notcaptured by this equation as the necessary physicsfor this is not incorporated in the model. It couldbe stated again here that the model itself has notused any fitting data or information from the ex-periments but is developed independently from themeasurements. But, the comparison of the tempo-ral and spatial variation of the film thickness for thevalues of R! close to one is quite good.

Conclusions

In this work, the impact of droplet with spher-ical targets of di!erent diameters are studied. Thefilm thickness variation on the surface of the sphereshows that the dynamics comprise three distinctphases namely the free fall phase, the inertia domi-nated phase and the viscosity dominated phase. The

10−2 10−1 100 101 10210−3

10−2

10−1

100

t*

h*

R* = 0.68R* = 1.23R* = 1.98

R*=0.68R*=1.23

R*=1.98

Figure 12. Comparison of the theoretical temporalvariation of film thickness with the measurementsfor di!erent values of R!. Big circles show the pointbeyond which viscosity becomes important in theflow.

−150 −100 −50 0 50 100 1500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

θ

Non−d

imen

sion

al fi

lm th

ickn

ess

( h* ) R*=1.23

Re = 4806We = 132

t*=1.33

t*=2.21t*=3.11

Figure 13. Comparison of the theoretical spatialvariation of film thickness with the measurementsfor di!erent values of t!.

first two phases collapse onto each other for di!erentvalues of Reynolds number for a given droplet-targetcombination. But, the transition to the viscos-ity dominated phase occurs earlier at low Reynoldsnumber resulting in larger residual film thickness.When the target size is increased with respect to thedroplet size the film thinning process is slower witha larger value of residual thickness. These trends arewell explained by the theoretical analysis.

Acknowledgments

One of the authors (S.B.) would like to acknowl-edge the financial support from Alexander von Hum-boldt Foundation through the Research Fellowshipprogram. The authors also wish to thank Dr. Ver-duyn of Degussa for discussions and help during thecourse of this work.

Nomenclatureh film thicknessv velocityR target radiusD drop diametert timeRe Reynolds numberWe Weber numberFr Froude numberS arc length on the target surface! angular position on the target#k time span from the initial contact

between the droplet and the targetand the commencement of thefilm flow regime

Subscriptso time=0Superscripts) non-dimesionalized quantity (length

with drop diameter and time with theratio of the drop diameter and theimpacting velocity)

References

[1] C. Crowe, M. Sommerfeld, and Y. Tsuji. Multi-phase Flows with Droplets and Particles. CRCPress, Boca Raton,, 1998.

[2] A. H. Lefebvre. Atomization and Sprays. Hemi-sphere, New York, 1989.

[3] M. Rein. Fluid Dynam. Res., 12:61–93, 1993.

[4] A. L. Yarin. Ann. Rev. Fluid Mech., 38:159–192, 2006.

[5] S. Chandra and C. T. Avedisian. Proc. R. Soc.Lond. A, 432:13–41, 1991.

[6] C. Mundo, M. Sommerfeld, and C. Tropea. Int.J. Multiphase Flow, 21(2):151–173, 1995.

[7] R. Rioboo, M. Marengo, and C. Tropea. Exper-iments in Fluids, 33:112–124, 2002.

[8] L. S. Hung and S. C. Yao. Int. J. MultiphaseFlow, 25:1545–1559, 1999.

[9] Y. Hardalupas, A. M. K. P. Taylor, and J. H.Wilkins. Int. J. Heat and Fluid Flow, 20:477–485, 1999.

[10] A. Rozhkov, B. Prunet-Foch, and M. Vignes-Adler. Physics of Fluids, 14(10):3485–3501,2002.

[11] A. Rozhkov, B. Prunet-Foch, and M. Vignes-Adler. Physics of Fluids, 15(7):2006–2019,2003.

[12] A. L. Yarin and D. A. Weiss. J. Fluid Mech.,283:141–173, 1995.

[13] I. V. Roisman and C. Tropea. J. Fluid Mech.,472:373–397, 2002.

[14] I. V. Roisman, R. Rioboo, and C. Tropea. Proc.R. Soc. Lond. A, 458:1411–1430, 2002.