Splashing of impacting drops

0.73 MB·

by  T. C. de Goede

Splashing of impacting drops
T. C. de Goede,1,∗ K.G. de Bruin,1,2 and D. Bonn1,†
1Van der Waals-Zeeman Institute, Institute of Physics,
University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
2Netherlands Forensic Institute, Laan van Ypenburg 6, 2497 GB The Hague, Netherlands
(Dated: January 11, 2017)
We investigate the impact velocity beyond which the ejection of smaller droplets from the main
droplet (splashing) occurs for droplets impacting a smooth surface. We examine its dependence on
the surface wetting properties and droplet surface tension. We show that the splashing velocity is
independent of the wetting properties of the surface, but increases roughly linearly with increasing
surfacetensionoftheliquid. Apreexistingsplashingmodelisconsideredthatpredictsthesplashing
7
velocitybyincorporatingtheairviscosity. Themodelisconsistentwithourdata,butthecalculation
1
iscomplex. Toaddressthisissue,weproposeasimplification,assumingatmosphericconditionsand
0
low viscosities, and show that the simplification gives an equally good prediction of the splashing
2
velocity from a simple analytical formula.

n
a
J When impacting a dry, smooth surface, a droplet are closely linked through Young’s Law [23]. Therefore,

0 either spreads over the surface for low impact velocities the dependence of the splashing velocity on the surface
1 or disintegrates into smaller droplets for high impact tension should also be taken into account.
velocities. This so-called splashing phenomenon has
]
n been the subject of numerous studies for the last In this paper, we systematically investigate the effect
y several decades and is of relevance for a wide range of of the surface tension of the liquid and the wetting
d practicalapplicationslikecropspraying[1,2],raindrops properties of the surface on the splashing velocity of
- impacting on porous stones [3, 4] and forensic research the droplet. Using high-speed camera footage, we
u
l [5–7]. Thesplashingvelocityvsp isdefinedasthecritical measure the splashing velocity of a set of ethanol-water
f value of the impact velocity of the droplet beyond which mixtures impacting on different surfaces. We compare
.
s splashing occurs. Many studies have tried to find an our results with the splashing model of Riboux and
c
i empirical relation between splashing velocity and fluid Gordillo [20]. Although the model is consistent with
s
parameters [8–13] and between splashing velocity and the experimental data, the calculation is complex
y
h surface properties [14–17]. In addition, Xu et al. [18] and depends on several parameters that have to be
p showed that the atmospheric conditions have a signifi- inferred from the experimental conditions and have to
[ cant influence on droplet splashing [13, 17, 19], implying be calculated separately. To address this shortcoming,

thatthe airviscosityis alsoanimportant parameter. To we propose a simplification valid for low Ohnesorge
1
include the air viscosity, Riboux and Gordillo recently numbers and atmospheric conditions, which is the
v
4 proposed a theoretical model for impact and splashing situation that pertains to most practical applications
0 on smooth surfaces [20]. They postulated that splashing andshowthatitpredictsthesplashingvelocityverywell.
5 occurs due to the break up of a small liquid film that
2 lifts off the surface just after impact due to the lift force In order to measure the splashing velocity v , droplet
sp
0
generatedbythesurroundingair. Inthemodel,thefluid impactswererecordedusingahigh-speedcamera(Phan-
.
1 and substrate properties govern the splashing, as well tom Miro M310). The droplets were generated from a
0 as the air viscosity; in [20] quantitative agreement was blunt tipped needle (needle diameter 0.4 mm) using a
7 found between the model and experiments on a single syringe pump, where the needle was suspended above
1
substrate. the substrate at a certain height. By systematically
:
v increasingtheheightoftheneedle,andcheckingwhether
i Numerous studies have investigated splashing [8–19]; the droplet merely spreads over the surface (Fig. 1a) or
X
however, the influence of the wetting properties of the splashes (Fig. 1b) for each height, we determined the
r
a surface has not been considered in detail. It has recently initial droplet diameter D0 and impact velocity v at the
beenshownthatthewettingpropertiesofthesurfaceaf- onset of splashing for each liquid. The fluid parameters
fect droplet spreading at low impact velocities [21, 22]. of each liquid are given in Table I. Three surfaces were
In the model of Riboux and Gordillo, the splashing ve- investigated: two hydrophilic surfaces (stainless steel
locity also depends on the surface properties [20]. The and borosilicate glass) and one hydrophobic surface
surface tension and the wetting properties of the surface (parafilm).
The measured splashing velocities are plotted as
function of surface tension in Fig. 1c. The graph shows
∗ [email protected] that the splashing velocity increases roughly linearly
† [email protected] with the liquid surface tension. For pure water droplets,
2
TABLE I: Ethanol mass fraction, density, surface tension and viscosity values of the water, ethanol and
ethanol-water mixtures used in this study. Source: [24, 25]
wt(%) Density (cid:0)kg(cid:1) Surface tension (cid:0)mN(cid:1) Viscosity (mPa·s)
m3 m
0 997.0 71.99 0.89
5 989.0 56.41 1.228
10 981.9 48.14 1.501
15 975.3 42.72 1.822
20 968.7 37.97 2.142
40 935.3 30.16 2.846
60 891.1 26.23 2.547
80 843.6 23.82 1.881
100 789.3 21.82 1.203
no splashing was observed within the velocity range is caused by the negative pressure difference above the
investigatedhere(0.1<v <4.7m/s). Nosignificantdif- liquidsheetduetoBernoulliprinciple, whilethelubrica-
ference between the three evaluated surfaces is observed, tion force is generated by the air moving underneath the
implying that the splashing velocity is independent of liquid sheet, creating a positive pressure difference that
the wetting properties of the substrate. pushes the lamella upward. K and K are constants.
u l
WhileK (cid:39)0.3wasdeterminedin[20]bynumericalcal-
u
Todescribesplashing,theairviscosityneedstobecon- culations,K canbecalculatedusingthesheetthickness,
l
sidered. To do so, in [20] the ejection time t (the mo- mean free path of the molecules λ in the surrounding air
e
ment a thin liquid sheet appears from the droplet after and the wedge angle α, which is the angle between the
impact) is calculated numerically using the momentum lifted sheet and the surface:
balance equation:
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
6 λ λ
√ K =− ln 19.2 −ln 1+19.2 (3)
3Re−1te−21 +Re−2Oh−2 =1.21te32 (1) l tan2α Ht Ht
2 According to [20], the wedge angle is equal to 60◦ and
where Re = ρRv and Oh = √µ are the Reynolds should be dependent on the wetting properties of the
µ ρRσ surface [20].
and Ohnesorge numbers, respectively; ρ the density, µ
the viscosity, σ the surface tension, R the radius and v
Having determined these parameters, a dimensionless
the impact velocity of the droplet. Using the ejection
number defined as the splashing ratio β is calculated,
time, the velocity V and thickness H of the thin liquid
t t whichindicatesthemagnitudeoftheaerodynamicforces
sheet can be calculated:
needed toovercomethe surfacetension inorder tobreak
up the liquid sheet into smaller droplets:
1√ −1
Vt = 2 3v te 2 (2a)
(cid:18)K µ V +K ρ V2H (cid:19)12
β = l g t u g t t (4)
√ 2σ
12 3
Ht = π Rte2 (2b) Comparing Equations (1) and (4) to both their own
experiments and previous work [13, 18, 26], Riboux and
Using the sheet velocity and thickness, the aerody- Gordillo determined find that the value of the splashing
namic lifting force (see Fig. 2), consisting of the suc- ratio should be around 0.14.
tion (∼ K µ V ) and lubrication force (∼ K ρ V2H ),
l g t u g t t
can be determined. Here, µ and ρ are the viscosity To calculate the splashing velocity from the splashing
g g
and density of the air, respectively. The suction force model of Riboux and Gordillo, Equations (2a) and (2b)
3
(a)v<vsp
(b)v≥vsp (c)
FIG. 1: (a) Droplet impact for v <v where no splashing occurs. (b) Droplet impact for v ≥v where splashing
sp sp
occurs. (c) Measured critical velocity as function of the surface tension for a stainless steel (blue circles) and
borosilicate glass (yellow squares). The green line depicts the splashing velocity as given by the full splashing model
[Eq. (5)] while the red dashed line shows the splashing velocity as calculated from the simplified model [Eq. (12)].
are substituted into Eq. (4), for which a quadratic equa- the wedge angle should be 60◦. Therefore, to investigate
tion for the splashing velocity can be found: the time dynamics of the wedge angle during splashing,
we measure the wedge angle: in our experiments, we
v = −√23Klµgt−e 12 +(cid:113)34(Klµg)2t−e1+ 6√π3KuρgD0te21σβ2 laent iemthpaancotlvdelroocpitleytscoimmppaarcatbalesttoaitnhleessspslateshelinsgurvfealcoeciatyt
sp 3√3K ρ D t12 (v = vsp). Recording the impact with the high-speed
2π u g 0 e camera gives the time evolution of the liquid sheet
(5)
expansion, which is depicted in Figures 2a to 2d. We
where vsp is the splashing velocity. observe that after the moment of the sheet ejection
(Fig. 2a), the liquid sheet starts to radially expand
Tocomparethesplashingmodelwithourexperiments, outward (Fig. 2b). After a certain time, liquid fingers
wefirstcalculatetheejectiontimeforeachethanol-water start to form at the edge of the sheet (Fig. 2c), after
mixture by substituting Eq. (5) into Eq. (1). From the which satellite droplets detach from the sheet (Fig. 2d):
obtained ejection time, the splashing velocity can be splashing occurs.
determined using Eq. (5). The best fit of the splashing
model on our data (green line, Fig. 1) was determined
by minimising the sum of square residuals using the From each recorded frame we extracted the instanta-
splashing ratio as a fit parameter. The obtained best neous wedge angle, which is shown in Fig. 2e. In this
fit value is equal to 0.127 ± 0.004, which is close to graph, the wedge angle seems to increase linearly with
the value 0.14 found by Riboux and Gordillo. While time up until the moment of finger formation (Fig. 2c).
the splashing model has been verified for borosilicate Assuming that the wedge angle keeps increasing linearly
glass substrates before [13, 26], our data for splashing until droplet detachment, we can fit a simple linear
on stainless steel and parafilm indicate that splashing function (α=At) to the data of Fig. 2e and extrapolate
should be independent of the wetting properties of the the wedge angle to the moment the droplets detach
impacted surfaces. from the liquid sheet. Using this method, we obtain an
average wedge angle of α = 59◦ ±8◦ at the moment of
The wetting properties of the surface should, accord- droplet detachment (Fig. 2d), which is almost identical
ing to [20], determine the wedge angle α. Riboux and to the value of the wedge angle postulated by Riboux
Gordillo however gave no argument why the value of and Gordillo. These results therefore show that the
4
Then, taking a constant wedge angle of ≈60◦, we can
simplify the splashing model. Most practical situations
deal with fluids with a low Ohnesorge number (Oh(cid:28)1)
and take place at atmospheric conditions. If we evaluate
Eq.(1)forlowOhnesorgenumbers, thefirsttermonthe
(a)t=0.137ms (b)t=0.342ms
left side in the equation dominates, allowing us to write
the ejection time as a function of the Weber number We
(cid:16) (cid:17)
We=Oh2·Re2 = ρRv2 as:
σ
t ≈We−2/3 (6)
e
(c)t=0.548ms (d)t=0.685ms
Consequently, the lamella thickness and spreading ve-
locity are given by:
√ √
Vt = 23vspt−e 12 = 23vspWe1/3 (7a)
√ √
Ht =R π12te32 ≈ π3D0We−1 (7b)
where R= 1D .
2 0
A further simplification can be made for the constant
K [Eq.(3)]. IfH iscalculatedwiththemeasuredsplash-
l t
ing velocity and initial diameter of the drops, an average
valueontheorderof10−5 misobtained. Sinceλ∼10−8
m , the ratio λ is in the order of 10−3. As λ is small,
Ht Ht
the second logarithmic term can be approximated with
(e) the first term of the Taylor approximation:
FIG. 2: (a)-(d) High-speed camera footage of the lifted
liquid sheet after impact (with t=0s the moment of (cid:18) λ (cid:19) λ
droplet impact). (e) Measured wedge angle α as ln 1+19.2 ≈19.2 (8)
H H
t t
function of the time as deduced from three independent
splashing experiments on stainless steel. The black (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:12)
Since (cid:12)ln 19.2 λ (cid:12) (cid:29) (cid:12)19.2 λ (cid:12) for λ ∼ 10−3, the
dotted line is the best linear fit of the wedge angle (cid:12) Ht (cid:12) (cid:12) Ht(cid:12) Ht
measurements. The vertical red lines depict the abovetermcanbeneglected,givingasimplifiedequation
different stages of the liquid sheet spreading as given in for Kl:
Figures (a)-(d): the emergence of the liquid sheet from
the droplet (a), radial expansion (b), formation of liquid (cid:18) (cid:19)
−6 λ 9.9
fingers at the front of the liquid (c) and droplet K ≈ log 19.2 ≈ (9)
l tan2(α) H tan2(α)
detachment (d). t
Afinalassumption,thatwasalsosuggestedbyRiboux
andGordillo,isthatthelubricationforcedominatesover
wedge angle given in [20] should be the wedge angle the suction force under atmospheric conditions [20], im-
at the moment of droplet detachment, since the angle plying that Eq. (4) can be simplified to:
various continuously in time. The wedge angle at the
(cid:18) (cid:19)1
moment of droplet detachment does not depend on β ≈ KlµgVt 2 (10)
the surface or the fluid, it is simply given by the air 2σ
viscosity, which explains why the wetting properties
are unimportant: the droplet appears to land on an Finally, by substituting the simplified terms of V
t
air cushion. It was shown recently that the maximum [Eq. (7a)] and K [Eq. (9)], β can be rewritten as a func-
l
radius of impacting drops in this high velocity impact tion of the splashing velocity v , initial diameter D ,
sp 0
regime does not depend on the substrate properties density ρ, surface tension σ, wedge angle α and the vis-
(wettability, roughness) either [22], for the same reason. cosity of the air µ :
g
5
velocity. Furthermore, the predicted values of the full
splashing model and the simplification are similar, with
√
β ≈2.07 µg (ρD )1/6 vs5p/6 (11) the largest relative error of around 17.7% [27]. There-
tanα 0 σ2/3 fore, this result shows that the splashing velocity can
bepredictedeasiercomparedtothefullsplashingmodel.
Thus, it is possible to significantly simplify Riboux
and Gordillo’s splashing model for low Ohnesorge To summarise, we systematically investigated the in-
number fluids and assuming atmospheric conditions, fluence of the surface tension of the droplet and wetting
where the splashing ratio is only dependent on the properties of the surface on the splashing velocity of
fluid parameters and the viscosity of the air, as the droplets impacting a smooth surface. We showed that
wedgeangleseemstobeidenticalforallsmoothsurfaces. the wetting properties do not influence the splashing
velocity. Second, we compared experimental with a
In order to compare the simplification with the exper- preexisting splashing model and showed that the model
imental data, Eq. (11) can be rewritten into a simple can be applied to both hydrophilic as hydrophobic
analytical expression for the splashing velocity: surfaces. By measuring the wedge angle, we confirmed
that the wedge angle used in the splashing model is
(cid:32) (cid:33)6 equal60◦ atthemomentofdropletdetachmentfromthe
v = tanασ23β 5 (12) liquidsheet. Finally, weproposedasimplificationonthe
sp √ 1
2.07 µg(ρD0)6 splashing model based on low Ohnesorge numbers and
atmospheric conditions and obtained a simple equation
Then, the splashing velocity only depends on the for the splashing velocity, which both agree very well
density, surface tension and initial diameter of the with the experimental data.
droplet, the viscosity of the air, the wedge angle and the
splashing ratio. The simplification (red dashed line) is
plotted together with the experimental data and the full
splashing model in Fig. 1c. Again, the splashing ratio ACKNOWLEDGEMENTS
wasusedasfittingparameter, wherethebestfitvalueof
β isequalto0.127±0.003,identicaltothesplashingratio We would like to thank Detlef Lohse, who pointed us
of the full splashing model. The simplification also gives towards the work of Guillaume Riboux and Jos´e Manuel
an equally good prediction of the measured splashing Gordillo.
[1] W.Wirth,S.Storp,andW.Jacobsen. Mechanismscon- drop with an unyielding dry surface. Proceedings of the
trollingleafretentionofagriculturalspraysolutions.Pes- Royal Society of London. A. Mathematical and Physical
tic. Sci., 33(4):411–420, 1991. Sciences, 373(1755):419–441, 1981.
[2] V.Bergeron,D.Bonn,J.Y.Martin,andL.Vovelle.Con- [9] AL Yarin and DA Weiss. Impact of drops on solid sur-
trolling droplet deposition with polymer additives. Na- faces: self-similarcapillarywaves,andsplashingasanew
ture, 405:772–775, 2000. typeofkinematicdiscontinuity.JournalofFluidMechan-
[3] NoushineShahidzadeh-Bonn,SalimaRafaı,DanielBonn, ics, 283:141–173, 1995.
and Gerard Wegdam. Salt crystallization during evap- [10] C. H. R. Mundo, M. Sommerfeld, and C. Tropea.
oration: impact of interfacial properties. Langmuir, Droplet-wall collisions: experimental studies of the de-
24(16):8599–8605, 2008. formation and breakup process. International journal of
[4] JB Lee, D Derome, and J Carmeliet. Drop impact on multiphase flow, 21(2):151–173, 1995.
natural porous stones. Journal of colloid and interface [11] M Bussmann, S Chandra, and J Mostaghimi. Modeling
science, 469:147–156, 2016. thesplashofadropletimpactingasolidsurface. Physics
[5] C.KnockandM.Davison. Predictingthepositionofthe of Fluids (1994-present), 12(12):3121–3132, 2000.
source of blood stains for angled impacts. J. Forensic [12] Randy L Vander Wal, Gordon M Berger, and Steven D
Sci., 52(5):1044–1049, 2007. Mozes. Thesplash/non-splashboundaryuponadrysur-
[6] Craig D Adam. Experimental and theoretical studies of faceandthinfluidfilm. Experiments in fluids,40(1):53–
thespreadingofbloodstainsonpaintedsurfaces.Forensic 59, 2006.
science international, 229(1):66–74, 2013. [13] Cacey S. Stevens. Scaling of the splash threshold
[7] NickLaan,KarlaG.deBruin,DeniseSlenter,JulieWil- for low-viscosity fluids. EPL (Europhysics Letters),
helm,MarkJermy,andDanielBonn. Bloodstainpattern 106(2):24001, 2014.
analysis: implementation of a fluid dynamic model for [14] CD Stow and RD Stainer. The physical products of a
position determination of victims. Scientific reports, 5, splashing water drop. Meteorological Society of Japan
2015. Journal, 55:518–532, 1977.
[8] C. D. Stow and M. G. Hadfield. An experimental inves- [15] ZiNiuWu.Mod´elisationetcalculimplicitemultidomaine
tigationoffluidflowresultingfromtheimpactofawater d’´ecoulements diphasiques gaz-gouttelettes. PhD thesis,
6
Universit´e Pierre et Marie Curie, Paris, France, 1992. [22] J. B. Lee, N. Laan, K. G. de Bruin, G. Skantzaris,
[16] Kai Range and Fran¸cois Feuillebois. Influence of surface N.Shahidzadeh,D.Derome,J.Carmeliet,andD.Bonn.
roughnessonliquiddropimpact. Journal of Colloid and Universalrescalingofdropimpactonsmoothandrough
Interface Science, 203(1):16–30, 1998. surfaces. Journal of Fluid Mechanics, 786:R4, 2015.
[17] Andrzej Latka, Ariana Strandburg-Peshkin, Michelle M [23] DanielBonn,JensEggers,JosephIndekeu,JacquesMeu-
Driscoll,CaceySStevens,andSidneyRNagel. Creation nier,andEtienneRolley.Wettingandspreading.Reviews
of prompt and thin-sheet splashing by varying surface of modern physics, 81(2):739, 2009.
roughnessorincreasingairpressure. Physical review let- [24] William M Haynes. CRC handbook of chemistry and
ters, 109(5):054501, 2012. physics. CRC press, 2014.
[18] Lei Xu, Wendy W Zhang, and Sidney R Nagel. Drop [25] Gonzalo Vazquez, Estrella Alvarez, and Jose M Navaza.
splashing on a dry smooth surface. Physical review let- Surfacetensionofalcohol+waterfrom20to50.degree.
ters, 94(18):184505, 2005. c. Journal of chemical and engineering data, 40(3):611–
[19] Cacey S Stevens, Andrzej Latka, and Sidney R Nagel. 614, 1995.
Comparison of splashing in high-and low-viscosity liq- [26] Julia´nPalacios,JulioHern´andez,PabloGo´mez,Claudio
uids. Physical Review E, 89(6):063006, 2014. Zanzi,andJoaqu´ınLo´pez. Experimentalstudyofsplash-
[20] Guillaume Riboux and Jos´e Manuel Gordillo. Exper- ing patterns and the splashing/deposition threshold in
iments of drops impacting a smooth solid surface: A drop impacts onto dry smooth solid surfaces. Experi-
model of the critical impact speed for drop splashing. mental Thermal and Fluid Science, 44:571–582, 2013.
Physical review letters, 113(2):024507, 2014. [27] Here, the relative error is defined as vsimpl−vfull ·100%,
[21] NickLaan,KarlaGdeBruin,DenisBartolo,Christophe vfull
where v and v are the splashing velocity pre-
simpl full
Josserand, and Daniel Bonn. Maximum diameter of
dictedbythesimplificationandfullsplashingmodel,re-
impacting liquid droplets. Physical Review Applied,
spectively.
2(4):044018, 2014.