Experimental Thermal and Fluid Science 55 (2014) 150–157

Contents lists available at ScienceDirect

Experimental Thermal and Fluid Science

journal homepage: www.elsevier .com/locate /et fs

Experimental investigation of a drop impacting on wetted spheres

http://dx.doi.org/10.1016/j.expthermflusci.2014.03.0080894-1777/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +86 0411 84708464.E-mail addresses: gtliang@126.com (G. Liang), zzbshen@dlut.edu.cn (S. Shen).

Gangtao Liang, Yali Guo, Xingsen Mu, Shengqiang Shen ⇑Key Lab. of Ocean Energy Utilization and Conservation of Ministry of Education, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 December 2013Received in revised form 15 March 2014Accepted 16 March 2014Available online 24 March 2014

Keywords:Drop impactWetted sphereSpreadingSplashing

Numerous experiments were performed to investigate a heptane drop impact dynamics on wettedspheres using a high speed camera. Outcomes after impact include spreading at a low impact Webernumber and splashing at a high value. Limits between the two outcomes can be greatly affected bythe sphere-drop curvature ratio ranging in 0.090–0.448. Additionally, the spreading process on wettedspherical surfaces is discussed in detail. The spreading factor defined as the ratio between the spreadingarea and the drop surface area can be increased by increasing the curvature ratio or by reducing liquidviscosity, while the effect of the increment in the Weber number is minor. It is found that the spreadingfactor follows a linear law with dimensionless time, which is confirmed by the butanol drop spreading aswell. Finally, concerning different curvature ratios and fluids, many coefficients with respect to the linearlaw are obtained to predict the spreading scale by regressing the experimental data.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The drop impact phenomenon is common in industry, such asdrops impact on surfaces of heat transfer tubes in horizontal-tubefalling film evaporators [1], spraying cooling [2], ink jet printing[3], plasma spraying technique [4] as well as various fire safety sit-uations [5], etc. Rein [6] summarized the main studies focused onthe liquid drop impact phenomenon and made comprehensive re-views about this subject. Some research in the public literatureprecisely inspires the present investigation.

For the drop impact on dry solid surfaces, Fukai et al. [7] pre-sented a theoretical study on the spreading process accountingfor the surface tension, and obtained the recoiling occurrenceand mass accumulation around the spreading film periphery. Theirresults also show that the dependence of the maximum spreadingradius on time is non-monotonic. Later, Fukai et al. [8] proposedanother theoretical model, considered the presence of inertia, vis-cosity, gravitation, surface tension and wetting effects. Their theo-retical model predicts well the deformation of the impacting drop,not only in the spreading phase, but also during recoiling and oscil-lation. Rioboo et al. [9,10] conducted many experiments and pro-vided qualitative and quantitative analysis. The results show thatoutcomes after impact include splashing, rebound, partial reboundand deposition. The time evolution of the spreading factor is di-vided into four distinct phases: the kinematic phase, the spreadingphase, the relaxation phase and the wetting/equilibrium phase. Xu

et al. [11] investigated the influence of the surrounding gas pres-sure on splashing limits and found a striking phenomenon: splash-ing can be suppressed by decreasing the surrounding gas pressure.Afterwards, Xu et al. [12,13] reported the interplay between sub-strate roughness and the surrounding gas pressure. They associ-ated two distinct types of splashing with each parameter:prompt splashing is due to surface roughness, while corona splash-ing is resulted from instabilities produced by the surrounding gas.Bi et al. [14] experimentally found that liquid viscosity plays adecisive role in the spreading process, and surface tension has aleading influence on the recoiling process. Both the two propertiesjointly determine the oscillation characteristics. For a single dropimpact on hot surfaces, Negeed et al. [15] discussed thermal prop-erties of the hot surface and drop characteristics on the drop evap-oration. Later, Negeed et al. [16,17] presented solid–liquid contacttime and the maximum drop spreading diameter, concerning ef-fects of the surface roughness amplitude, the oxide layer thickness,the We, and surface superheat.

In some industrial equipments, the impact target surfaces arenot always planar. For example, in horizontal-tube falling filmevaporators, liquid drops impact on heat transfer tubes. The targetsurfaces are curved instead of planar. Concerning a drop impact oncurved dry surfaces, Pasandideh-Fard et al. [18] simulated a 2 mmwater drop impact on tubes with the diameter in 0.5–6.35 mm andlow velocity of 1 m/s. They found that drops landing on the largesttube cling to the solid surface, but for smaller tubes, there are notenough surface areas for the liquid to remain attached, and dropsfall off after impact, disintegrating into several smaller drops. Hungand Yao [19] studied experimentally water drops with a diameter

Nomenclature

A area, mm2

d diameter, mmh film thickness, mmk coefficientOh Ohnesorge number, l/(qrddrop)1/2

Ra surface roughness, lmRe Reynolds number, qvddrop/lt time, msv impact velocity, m/sWe Weber number, qv2ddrop/r

Greek symbolsd dimensionless film thickness, h/ddrop

l liquid viscosity, Pa sq liquid density, kg/m3

r surface tension, N/ms dimensionless time, vt/ddrop

U spreading factor, As/Adrop

- sphere-drop curvature ratio, ddrop/dsphere

Subscriptsc criticaldrop liquid droph horizontals spreadingsphere solid spherev vertical

G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157 151

of 110–680 lm impacting on isothermal cylindrical wires. Their re-sults show that outcomes after impact include disintegration anddripping. Smaller drops are disintegrated if the incoming dropshave high velocity or the wire diameter is small. Larger drippingdrops are formed when the velocity is low or the wire diameteris large. Shen et al. [20] studied influences of several dimensionlessparameters on the drop deformation after impact on a two-dimen-sional round surface using lattice Boltzmann implementation ofthe pseudo-potential model. Four typical deformation processescan be found in their research: moving, spreading, nucleatingand falling. In Chow and Attinger [21], the drop diameter was80 lm and the target sphere diameter was in the range of 0.06–10 mm. Their visualization experiments show that the sphere cur-vature has no significant influences on the maximum spreadingfactor for a substrate-drop curvature ratio below 0.3. Hardalupaset al. [22] reported experiments on liquid drops with the diameter160–230 lm impacting on small solid spheres with the diameter0.8–1.3 mm at impact velocity 6–13 m/s. They observed a retrac-tion of the liquid crown at low drop impact velocity and disintegra-tion from cusps located on the crown rim at high impact velocity.They also pointed out that the increase in the sphere curvaturepromotes the splashing onset. Bakshi et al. [23] reported experi-mental investigations of drops with the diameter 2.4–2.6 mmimpacting onto a spherical target of 3.2 mm in diameter. Spatialand temporal variations of the film thickness on the target surfacewere measured. Three distinct temporal phases of the film dynam-ics are clearly visible from their experimental results: initial dropdeformation, inertia dominating and viscosity dominating.

However, the research above mentioned is limited to a drop im-pact on dry surfaces. For the impact on wetted surfaces, i.e., solidsurfaces covered by thin liquid films, a lot of work was also com-pleted. Cossali et al. [24] and Motzkus et al. [25] defined the impacttarget as a thin liquid film when the dimensionless film thickness dvaries in the range of 0–1. Rioboo et al. [26] found experimentallythree outcomes including deposition, crown formation withoutsplashing and splashing by varying impact velocity (0.44–3.14 m/s), the drop diameter (1.42–3.81 mm) and the dimensionless filmthickness (0.004–0.189). Particularly for a dimensionless filmthickness less than 0.02, crowns without splashing could almostno longer be observed. Okawa et al. [27] and Shi et al. [28] gainedthe same results by experiments and three-dimensional simula-tions. Cossali et al. [24] and Vander Wal et al. [29] associatedsplashing with the production of satellite drops separating fromthe crown liquid sheet after the impact, which were named as sec-ondary drops. Cossali et al. [24] firstly distinguished two kinds ofsplashing: prompt splashing and delayed splashing. The prompt

splashing is associated with ejected drops from the crown edgewhen it is still advancing, while the delayed splashing occurs nearor after the crown maximum expansion and is associated with thecrown wall breakup. Motzkus et al. [30] also demonstrated out-comes of coalescence, prompt splashing and delayed splashing intheir work.

From the above reviews on a single drop impact phenomenon,we note that most impact targets are dry planar or curved surfaces,and studies focused on the impact on wetted surfaces are limitedto planar wetted surfaces. Inspired by the above research, it isfound that there are few studies especially focused on the impactdynamics for a single drop impinging on wetted curved surfaces.In the previous study [31], the outcomes after a single drop impacton wetted cylinders were presented. Later, the rebound andspreading processes were discussed in detail in [32]. However,the drop impact on cylinders is a three-dimensional problem. Fora drop impact on wetted spheres with a two-dimensional geome-try, the impact behavior is still unclear. Thus, in the present re-search, outcomes after a single heptane drop impact on wettedspheres are presented through experimental observations using ahigh speed camera. In addition, the spreading factor is analyzed,with respect to influences of the We and the sphere-drop curvatureratio -.

2. Experimental apparatus and procedures

The experimental apparatus is similar with that in [31], andshown in Fig. 1. The main components include a syringe, a hypo-dermic needle connected with the syringe by a latex tube to gener-ate drops, a high speed camera, a wetted sphere, a xenon lampused to provide illumination for photography, a light diffuser,and a data acquisition computer.

A single drop can be formed by forcing the liquid in the syringeat a certain pressure through the stainless steel hypodermic nee-dle. The needle is flat tipped, with an inner diameter of 0.50 mm.The drop is formed at the tip of the needle and detaches whenthe gravity exceeds the surface tension force. The impact behavioris recorded by a Phantom V12.1 high speed camera with capacityof 106 frames per second, equipped with a 100 mm, f-2.8 Tokinamacro lens. The camera is aligned horizontally. In order to obtainphotographs with sufficient image resolution, the shooting speedis set as 10,000 frames per second, with 1024 � 512 pixels in eachimage. The back light method is employed in the experiments toexpose the impact images and the cold light source is providedby a xenon lamp XD-300 with a power of 350 W. A light diffuseris used between the sphere and the xenon lamp to make the light

0.0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

h (μ

m)

ϖ

butanol heptane

Fig. 2. The pre-existing film thickness.

Table 1Liquid properties and experimental conditions.

Liquid r (N/m) l (Pa s) q (kg/m3) ddrop (mm) h (lm) v (m/s)

Heptane 0.0201 0.000409 684 1.79 12–33 0.19–2.14Butanol 0.0201 0.00295 810 1.82 36–61 0.34–1.46

Fig. 1. The schematic diagram of the experimental apparatus.

152 G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157

be distributed uniformly on the wetted spherical surface. Becausethe whole impact process occurs in a very short time, the triggermode is selected as post trigger. Namely, after the drop impacton the wetted sphere, the trigger is launched, then the signal isdelivered to the data acquisition computer and the impact processis recorded.

Before each experiment, several drops are cleared away to en-sure that the liquid remains free of any air bubbles. Seven spheresare adopted with the diameter ranging from 4 mm to 20 mm. Thespheres are polished by the Cw 1500 silicon carbide electro coatedabrasive paper to assure that the average roughness of the spheresurfaces Ra is less than 0.05 lm. Heptane is selected as the exper-imental fluid. The drop diameter can be acquired by pixel analyz-ing and calibration is performed by using a reference substance.The software MATLAB 7.1 is used to fulfill the pixel analyzing pro-cess. The drop is similar to an ellipse and the diameter is measuredin both the horizontal and vertical directions. In Rioboo et al. [10],Stow and Hadfield [33], the equivalent diameter of the ellipse isdefined as (dh

2dv)1/3. The equivalent diameter of the heptane dropis 1.79 mm. The uncertainty is 1 pixel, and the error is 0.025 mm,which corresponds to a 1.40% relative error of the real diameter.Hence, the sphere-drop curvature ratio is ranging from 0.090 to0.448. The distance between the needle tip and the sphere is ad-justed to vary drop impact velocity. The impact velocity v is de-rived by tracking the location of the drop centroid in two imageswith 0.5 ms time spacing before impact, which ranges from0.19 m/s to 2.14 m/s with an accuracy of ±0.05 m/s.

Before the experiment, a thin liquid film is spread on the spheresurface uniformly by using a high-quality painting brush. Both thedrop and the film are heptane in the experiments. When the liquidfilm becomes stable and has a relatively uniform thickness, theexperiment can be started. The thickness of the liquid film is mea-sured directly by comparing an image of the film surface to a ref-erence image of the dry sphere. Before each impact test, the pre-existing film thickness is measured by a SLR digital camera (CanonEOS 5D Mark II) with a larger resolution than the high speed one.The film thickness is in the range 12–33 lm with an accuracy±4 lm, depending on the sphere diameter. Fig. 2 shows the pre-existing film thickness at the impact point with different curvatureratios. It can be seen that the larger curvature ratio results in thesmaller film thickness, caused by the larger curvature of thesphere. Cossali et al. [24], Stow and Stainer [34] pointed out that,when the surface roughness is comparable with the pre-existingfilm thickness, the roughness influence becomes remarkable. How-ever, the roughness 0.05 lm is much lower than the smallest filmthickness 12 lm. Besides, all the spheres are polished with thesame type of the abrasive paper, so the roughness influence isnot considered in this research. In order to further verify the pro-posed linear rule of the spreading factor, the other fluid of butanolis selected, the pre-existing film thickness of which is alsopresented in Fig. 2. Some experiments are done with smallimpact velocity. Table 1 summarizes the liquid properties and

experimental conditions adopted in the present study, whilst Ta-ble 2 presents ranges of dimensionless parameters.

3. Results and discussions

3.1. Outcomes after impact

The number embedded in the following images is the evolutiontime with the unit ms. The time sequence 0.0 ms is set as the dropexactly contacts with the wetted surfaces.

For the small impact We corresponding to - = 0.149, the dropspreading process can be observed in Fig. 3. When the drop touchesthe wetted spherical surface, liquid in the drop that firstly contact-ing with the wetted surface begins to flow around on the surface(0.4 ms). As the spreading continues, the periphery of the spread-ing film extends outward continuously (1.6 ms). Namely, the sur-face area of the wetted sphere covered by the spreading filmbecomes larger. Then the periphery becomes more and moreinconspicuous and a stable film is formed on the wetted sphereat last (11.0 ms). It is noted that the rebound phenomenon at thelow impact We still does not occur for the heptane drop throughmany repeatable experiments.

For the same curvature ratio, by increasing the impact We to acertain high value, the splashing phenomenon can be seen in Fig. 4.An extremely thin jet is generated immediately after impact in theregion where the drop connects with the wetted spherical surface

Table 2Ranges of dimensionless parameters.

Liquid We Re Oh -

Heptane 2–279 569–6406 0.0026 0.090–0.448Butanol 8–156 170–730 0.0171 0.091–0.455

Fig. 5. Spreading of the heptane drop with - = 0.448 and We = 67.

Fig. 6. Splashing of the heptane drop with - = 0.448 and We = 278.

00.0 0.1 0.2 0.3 0.4 0.5

90

180

270

360

450

We c

ϖ

Oh = 0.0026 Present experiments Oh = 0.0175 Wang and Chen(2000) Oh = 0.0026 Asadi and Passandideh-Fard (2009) Oh = 0.0018 Asadi and Passandideh-Fard (2009) Oh = 0.0028 Motzkus et al. (2011)

Fig. 7. Splashing limits of the heptane drop on wetted spheres.

G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157 153

(0.1 ms). Then the jet develops into an unobvious liquid sheet at0.3 ms. However, this liquid sheet only sustains an extremely shortduration and quickly cracks into many secondary drops (0.5 ms).By 0.9 ms, the sheet has collapsed completely. Thus, the splashingin Fig. 4 pertains to the typical prompt splashing defined in Cossaliet al. [24], caused by low viscosity of the heptane drop.

For the higher curvature ratio - = 0.448, the drop spreadingprocess is presented in Fig. 5. Compared with Fig. 3, experimentalobservations in Fig. 5 suggest that the spreading qualitative behav-ior varies less with different curvature ratios. However, for thedrop splashing in Fig. 6 with - = 0.448, it appears a few differ-ences. In Fig. 6, the secondary drops are much smaller than thatin Fig. 4. Moreover, the tiny secondary drops move horizontallyafter impact in Fig. 6. While in Fig. 5, the secondary drops scattermore irregularly.

When splashing occurs, the critical impact velocity can be ac-quired by fine adjustment of the impact height. Fig. 7 shows curvesof the critical Weber number Wec for heptane at different curva-ture ratios. It shows that the Wec decreases with the decrementin the curvature ratio when the curvature ratio is larger than0.224, while for - < 0.224 the Wec varies little and almost keepsa constant 124. For the large curvature ratio, the sphere diameteris small, so there is drop downward slippage after impact alongthe wetted surface due to gravity. One part of the impact energyhas not been applied to overcome the adverse effect of the flowresistance and surface tension force. Hence, more impact energyis required to attain the splashing occurrence, i.e., the Wec is highfor the large curvature ratio. With the decrement in the curvatureratio, such downward slippage is alleviated, as a consequence theWec is reduced. However, when the curvature ratio is less than0.224, the wetted surface approximates to a planar surface andthe curve effect fades away gradually. Hence, the Wec does notchange with the curvature ratio. To make this interpretation bemore conclusive, several splashing limits for a single drop impacton a horizontal liquid film (- = 0) in the literatures are selectedto make a brief comparison with the present results in Fig. 7. Refs.[35–37] suggest that the splashing limits grow with the incrementin the Oh or the viscosity. In Fig. 7 it is noted that for Oh = 0.0175 inWang and Chen [35] and Oh = 0.0028 in Motzkus et al. [30], theirthresholds are larger than Oh = 0.0026 in the present experimentswith - < 0.224, while the critical value for Oh = 0.0018 in Asadi andPassandideh-Fard [37] is smaller than the present value. However,for the same Oh = 0.0026 in [37], their result agrees extremely wellwith the present one. Thus, when the curvature ratio is larger than

Fig. 3. Spreading of the heptane dro

Fig. 4. Splashing of the heptane dro

0.224, the curve effect should be taken into account with respect tothe splashing process.

3.2. Spreading factor

The foremost parameter in drop spreading is the spreadingmagnitude. A broad spreading can be desired to enhance the trans-port phenomena between the drop and the substrate in someapplication aspects such as ink jet printing [38]. When the drop im-pinges on the horizontal surface, the spreading factor, which is de-fined as the ratio between the spreading film diameter and thedrop diameter, is used to measure the spreading magnitude [14],whereas for the inclined surface, back and front spreading factorsare introduced [39]. However, when the drop spreads on thecurved surface, it is inappropriate to use these parameters to mea-sure the spreading because the contact surface is not planar any-more: it is a spherical cap. Thus, in this research, the spreading

p with - = 0.149 and We = 47.

p with - = 0.149 and We = 191.

Fig. 8. The spreading model after impact.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

ϖ = 0.448 ϖ = 0.224ϖ = 0.149ϖ = 0.090

h s/ t (

m/s

)

t (ms)

Fig. 10. The specific value between hs and t with We = 42–67.

154 G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157

area instead of the spreading diameter is used here. Fig. 8 is thespreading model after impact, where the point O is the impactpoint, the points A and B determine the raised spreading rim andthe point C is the foot point from O to the segment AB. The spread-ing height hs is the length of AB, so the spreading area As can be ex-pressed as

As ¼ phsdsphere; ð1Þ

and the spreading factor U is defined as

U ¼ As

Adrop: ð2Þ

The spreading factor in the following figures is only plotted if itcan be clearly measured.

Fig. 9 shows the spreading factor at different conditions. FromFig. 9(a)–(c) it is found that the We has minor effects on the spread-ing factor in the present curvature ratio range, though some dis-crepancies emerge in Fig. 9(a) with - = 0.448, caused by therelatively shorter lifetime of the spreading film. However,Fig. 9(d) shows that the curvature ratio effect is remarkable. Whens is less than 0.7, the curvature ratio does not influence U, while fors > 0.7, with the decreasing of the curvature ratio, the spreadingfactor is increased. It is considered that at the incipient stage, the

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Φ

τ

We = 67We = 116We = 209

ϖ = 0.448

(a)

0

2

4

6

8

10

ϖ = 0.112

Φ

τ

We = 7We = 38We = 77

0.0 0.3 0.6 0.9 1.2 1.5

0 1 2 3 4 5

(c)

Fig. 9. The heptane spreading factor, (a)–(c) the We effect with different

impact drop just spreads in a small region with the area about12.5 mm2 and this curved region can be deemed as a flat surface,so the spreading factors are almost the same with different curva-ture ratios. Nevertheless, after the initial stage, the effect of thecurvature ratio becomes prominent. As described in introduction,the wetted surface is the solid surface covered by a very thin film.If it is assumed that the impact point corresponds to the 0� in-cluded angle, when this pre-existing film becomes relatively stable,the film thickness increases with angles for the left or right hemi-sphere due to gravity. Taking - = 0.224 for example, at the angles45�, 90� and 135�, the film thickness is about 26 lm, 30 lm and41 lm, respectively. The smaller curvature ratio means that thereis enough space for the drop to spread on the upper position ofthe wetted sphere. While for the large curvature ratio, the

0

2

4

6

8

10

ϖ = 0.224

Φ

τ

We = 8We = 42We = 87

(b)

0 1 2 3 4 5

0 1 2 3 4 50

3

6

9

12

heptane

Φ

τ

ϖ = 0.448 ϖ = 0.224ϖ = 0.149ϖ = 0.090

(d)

curvature ratios and (d) the curvature ratio effect with We = 42–67.

0 2 4 6 80

3

6

9

12

15Φ

τ

heptane

Fig. 11. Experimental data of the spreading factor for heptane.

0.0 0.1 0.2 0.3 0.4 0.50.0

0.6

1.2

1.8

2.4

3.0

k

ϖ

Fig. 12. The k value for heptane.

0 1 2 3 4 50

2

4

6

8

butanol

Φ

τ

Fig. 14. Experimental data of the spreading factor for butanol.

0.0 0.1 0.2 0.3 0.4 0.50.0

0.6

1.2

1.8

2.4

3.0

k

ϖ

Fig. 15. The k value for butanol.

G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157 155

spreading rim arrives at the lower position quickly, where the filmis thicker and the spreading film will encounter more resistance, asa consequence the spreading expands slowly. Therefore, thespreading factor becomes larger with the decrement in the curva-ture ratio.

It is also noted that the spreading factor almost increases line-arly with dimensionless time in Fig. 9. To verify this linear law,the definition of the spreading factor in Eq. (2) is derived:

U ¼ As

Adrop¼ phsdsphere

pd2drop

¼ hsdsphere

d2drop

: ð3Þ

0.0

0.6

1.2

1.8

2.4

3.0

ϖ = 0.455

Φ

τ

We = 37We = 80We = 130

0.0 0.4 0.8 1.2 1.6 2.0

(a)Fig. 13. The butanol spreading factor, (a) the We effect with -

Substituting the impact velocity and time yields

U ¼ hsdsphere

d2drop

¼ hsdsphere

vtddrop� vt

ddrop: ð4Þ

Then combining the dimensionless time definition yields

U ¼ dsphere

vddrop� hs

t� s: ð5Þ

For the given curvature ratio and the impact We, the first termfrom the equation right side in Eq. (5) is a constant. If the secondterm (hs/t) can keep a constant, the linear law of the spreading fac-

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

1.5

3.0

4.5

6.0

butanol

Φ

τ

ϖ = 0.455ϖ = 0.228ϖ = 0.152ϖ = 0.091

(b) = 0.455 and (b) the curvature ratio effect with We = 37–57.

156 G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157

tor with time can be affirmed. Fig. 10 presents the specific valuebetween hs and t with different curvature ratios. It suggests thatthe second term (hs/t) varies less with time and the value growswith the increasing of the curvature ratio. Hence, a linear spreadingmodel for a drop impact on wetted spheres can be proposed in Eq.(6):

U ¼ ks; ð6Þ

where k can be acquired by regressing the experimental data.Fig. 11 provides the spreading factor data in the present exper-

imental range shown in Tables 1 and 2. Due to influences of thecurvature ratio, the data are scattered. Hence, it is not proper tofit all the data by using one correlation. Based on this situation,more experiments are supplemented concerning different curva-ture ratios, in order to regress the k value for each curvature ratioby utilizing the linear model in Eq. (6). Fig. 12 shows the k value forheptane at different curvature ratios. It is found that the k valuefluctuates a little, but the overall trend is still increasing with thedecrement in the curvature ratio. This result coincides well withthat in Fig. 9(d).

Many experiments are also performed by adopting butanol asthe experimental fluid due to the same surface tension with hep-tane, so that the linear rule of the spreading factor can be furtherconfirmed. The experimental conditions are presented in Tables 1and 2. These experiments are focused on the spreading process,so the impact velocity is low and the splashing process is not con-sidered in this research. Fig. 13 shows the spreading factor forbutanol at different conditions. The effects of the We and the cur-vature ratio are the same with that in Fig. 9: the spreading factorcan be increased by decreasing the curvature ratio, while the Weeffect is minor. It is still found that the butanol spreading factor fol-lows the linear rule with dimensionless time as well. Figs. 14 and15 present the experimental data of the spreading factor and thek value for butanol. It can be seen that the k value in Fig. 15 is moreinerratic than that in Fig. 12, while the k value still increases withthe decrement in the curvature ratio. Thus, Figs. 14 and 15 wellconfirm the linear model proposed in Eq. (6) and this linear modelis recommendable for the spreading process after a single drop im-pact on wetted spheres.

Comparing Figs. 12 and 15, the k value for heptane is larger thanthat for butanol. Table 1 indicates that butanol viscosity is muchhigher than heptane viscosity. Higher viscosity means that moreimpact energy will be decreased by viscous dissipation when thedrop spreads on the wetted surface, and the spreading extent is re-duced a lot. Hence, the heptane spreading factor is larger than thebutanol value.

4. Conclusions

In this work, a series of experiments are performed focused onthe less concerned investigation of a single liquid drop impact onthe wetted spheres using a high speed camera. After the heptanedrop impacting on the wetted sphere, spreading and splashingphenomena can be observed. The sphere-drop curvature ratiocan greatly influence the splashing thresholds: with the incrementin the curvature ratio larger than 0.224, the critical We can be in-creased due to the drop downward slippage, while the criticalWe almost keeps a constant as the curvature ratio smaller than0.224. The spreading process is quantitatively discussed in detailand the spreading factor defined as the ratio between the spread-ing area and the drop surface area is introduced to measure thespreading amplitude. Experimental results indicate that thespreading factor can be increased by decreasing the curvature ra-tio, while the We influences are minor. The interesting linear modelof the spreading factor with dimensionless time is proposed, which

is also verified by the derivation in the definition of the spreadingfactor. The k value corresponding to different curvature ratios isobtained by regressing the experimental data. To further confirmthe linear rule, more experiments are conducted by adopting buta-nol as the experimental liquid. The trends are similar with thatwhen the liquid is heptane, while the k value is smaller causedby its higher viscosity. Hence, the linear model proposed in this re-search is recommendable.

Acknowledgement

Support of the Key Project of the National Natural Science Foun-dation of China (No. 51336001) is gratefully acknowledged.

References

[1] S. Shen, G. Liang, Y. Guo, R. Liu, X. Mu, Heat transfer performance and bundle-depth effect in horizontal-tube falling film evaporators, Desalin. Water Treat.51 (4–6) (2013) 830–836.

[2] M. Pasandideh-Fard, S.D. Aziz, S. Chandra, J. Mostaghimi, Cooling effectivenessof a water drop impinging on a hot surface, Int. J. Heat Fluid Fl. 22 (2) (2001)201–210.

[3] M.R. Davidson, Spreading of an inviscid drop impacting on a liquid film, Chem.Eng. Sci. 57 (17) (2002) 3639–3647.

[4] G. Liang, Y. Guo, S. Shen, Y. Yang, Crown behavior and bubble entrainmentduring a drop impact on a liquid film, Theor. Comp. Fluid Dyn. 28 (2) (2014)159–170.

[5] S. Chandra, M. Di Marzo, Y.M. Qiao, P. Tartarini, Effect of liquid–solid contactangle on droplet evaporation, Fire Safety J. 27 (2) (1996) 141–158.

[6] M. Rein, Phenomena of liquid drop impact on solid and liquid surfaces, FluidDyn. Res. 12 (2) (1993) 61–93.

[7] J. Fukai, Z. Zhao, D. Poulikakos, C.M. Megaridis, O. Miyatake, Modeling of thedeformation of a liquid droplet impinging upon a flat surface, Phys. Fluids 5(11) (1993) 2588–2599.

[8] J. Fukai, Y. Shiiba, T. Yamamoto, O. Miyatake, D. Poulikakos, C.M. Megaridis, Z.Zhao, Wetting effects on the spreading of a liquid droplet colliding with a flatsurface: experiment and modeling, Phys. Fluids 7 (2) (1995) 236–247.

[9] R. Rioboo, C. Tropea, M. Marengo, Outcomes from a drop impact on solidsurfaces, Atomization Spray 11 (2) (2001) 155–165.

[10] R. Rioboo, M. Marengo, C. Tropea, Time evolution of liquid drop impact ontosolid, dry surfaces, Exp. Fluids 33 (1) (2002) 112–124.

[11] L. Xu, W. Zhang, S.R. Nagel, Drop splashing on a dry smooth surface, Phys. Rev.Lett. 94 (18) (2005) 184505.

[12] L. Xu, L. Barcos, S.R. Nagel, Splashing of liquids: interplay of surface roughnesswith surrounding gas, Phys. Rev. E 76 (6) (2007) 066311.

[13] L. Xu, Liquid drop splashing on smooth, rough, and textured surfaces, Phys.Rev. E 75 (5) (2007) 056316.

[14] F. Bi, Y. Guo, S. Shen, J. Chen, Y. Li, Experimental study of spread characteristicsof droplet impact on solid surface, Acta Phys. Sin-Ch Ed 61 (18) (2012) 184702.

[15] E.R. Negeed, N. Ishihara, K. Tagashira, S. Hidaka, M. Kohno, Y. Takata,Experimental study on the effect of surface conditions on evaporation ofsprayed liquid droplet, Int. J. Therm. Sci. 49 (12) (2010) 2250–2271.

[16] E.R. Negeed, S. Hidaka, M. Kohno, Y. Takata, Effect of the surface roughness andoxidation layer on the dynamic behavior of micrometric single water dropletsimpacting onto heated surfaces, Int. J. Therm. Sci. 70 (2013) 65–82.

[17] E.R. Negeed, S. Hidaka, M. Kohno, Y. Takata, High speed camera investigation ofthe impingement of single water droplets on oxidized high temperaturesurfaces, Int. J. Therm. Sci. 63 (2013) 1–14.

[18] M. Pasandideh-Fard, M. Bussmann, S. Chandra, J. Mostaghimi, Simulatingdroplet impact on a substrate of arbitrary shape, Atomization Spray 11 (4)(2001) 397–414.

[19] L.S. Hung, S.C. Yao, Experimental investigation of the impaction of waterdroplets on cylindrical objects, Int. J. Multiphase Flow 25 (8) (1999) 1545–1559.

[20] S. Shen, F. Bi, Y. Guo, Simulation of droplets impact on curved surfaces withlattice Boltzmann method, Int. J. Heat Mass Transfer 55 (23–24) (2012) 6938–6943.

[21] C.K. Chow, D. Attinger, Visualization and measurements of microdropletimpact dynamics on a curved substrate, in: 4th ASME_JSME Joint FluidsEngineering Conference, ASME, Hawaii, USA, 2003, pp. 405–413.

[22] Y. Hardalupas, A.M.K.P. Taylor, J.H. Wilkins, Experimental investigation of sub-millimetre droplet impingement on to spherical surfaces, Int. J. Heat Fluid Fl.20 (5) (1999) 477–485.

[23] S. Bakshi, I. Roisman, C. Tropea, Investigations on the impact of a drop onto asmall spherical target, Phys. Fluids 19 (3) (2007) 032102.

[24] G.E. Cossali, A. Coghe, M. Marengo, The impact of a single drop on a wettedsolid surface, Exp. Fluids 22 (6) (1997) 463–472.

[25] C. Motzkus, F. Gensdarmes, E. Géhin, Parameter study of microdropletformation by impact of millimetre-size droplets onto a liquid film, J. AerosolSci. 40 (8) (2009) 680–692.

G. Liang et al. / Experimental Thermal and Fluid Science 55 (2014) 150–157 157

[26] R. Rioboo, C. Bauthier, J. Conti, M. Voué, J. De Coninck, Experimentalinvestigation of splash and crown formation during single drop impact onwetted surfaces, Exp. Fluids 35 (6) (2003) 648–652.

[27] T. Okawa, T. Shiraishi, T. Mori, Production of secondary drops during the singlewater drop impact onto a plane water surface, Exp. Fluids 41 (6) (2006) 965–974.

[28] Z. Shi, Y. Yan, F. Yang, Y. Qian, G. Hu, A lattice Boltzmann method forsimulation of a three-dimensional drop impact on a liquid film, J. Hydrodyn. B20 (3) (2008) 267–272.

[29] R. Vander Wal, G. Berger, S. Mozes, Droplets splashing upon films of the samefluid of various depths, Exp. Fluids 40 (1) (2006) 33–52.

[30] C. Motzkus, F. Gensdarmes, E. Géhin, Study of the coalescence/splash thresholdof droplet impact on liquid films and its relevance in assessing airborneparticle release, J. Colloid Interface Sci. 362 (2) (2011) 540–552.

[31] G. Liang, Y. Guo, Y. Yang, S. Guo, S. Shen, Special phenomena from a singleliquid drop impact on wetted cylindrical surfaces, Exp. Therm. Fluid Sci. 51(2013) 18–27.

[32] G. Liang, Y. Yang, Y. Guo, N. Zhen, S. Shen, Rebound and spreading during adrop impact on wetted cylinders, Exp. Therm. Fluid Sci. 52 (2014) 97–103.

[33] C.D. Stow, M.G. Hadfield, An experimental investigation of fluid flow resultingfrom the impact of a water drop with an unyielding dry surface, Proc. R. Soc. A-Math. Phys. 373 (1755) (1981) 419–441.

[34] C.D. Stow, R.D. Stainer, The physical products of a splashing water drop, J. Met.Soc. Japan 55 (1977) 518–532.

[35] A.B. Wang, C.C. Chen, Splashing impact of a single drop onto very thin liquidfilms, Phys. Fluids 12 (9) (2000) 2155–2158.

[36] K.L. Pan, C.Y. Hung, Droplet impact upon a wet surface with varied fluid andsurface properties, J. Colloid Interface Sci. 352 (1) (2010) 186–193.

[37] S. Asadi, M. Passandideh-Fard, A computational study of droplet impingementonto a thin liquid film, Arab. J. Sci. Eng. 34 (2B) (2009) 505–517.

[38] A. Karl, A. Frohn, Experimental investigation of interaction processes betweendroplets and hot walls, Phys. Fluids 12 (4) (2000) 785–796.

[39] Š. Šikalo, C. Tropea, E.N. Ganic, Impact of droplets onto inclined surfaces, J.Colloid Interface Sci. 286 (2) (2005) 661–669.