Commun Nonlinear Sci Numer Simulat 17 (2012) 2045–2051

Contents lists available at SciVerse ScienceDirect

Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Drop formation from a wettable nozzle

Brian Chang, Gary Nave, Sunghwan Jung ⇑Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, United States

a r t i c l e i n f o

Article history:Available online 27 August 2011

Keywords:Capillary riseDrop formationDrippingWetting

1007-5704/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.cnsns.2011.08.023

⇑ Corresponding author. Address: 228 Norris HallE-mail address: sunnyjsh@vt.edu (S. Jung).

a b s t r a c t

The process of drop formation from a nozzle can be seen in many natural systems and engi-neering applications. However, previous research focuses on the pinch-off mechanism ofdrops from a non-wettable nozzle. Here we investigate the formation of a liquid droplet froma wettable nozzle. In the experiments, drops forming from a wettable nozzle initially climbthe outer walls of the nozzle due to surface tension. Then, when the weight of the drops grad-ually increases, they eventually fall due to gravity. By changing the parameters such as thenozzle size and fluid flow rate, we have observed different behaviors of the droplets. Suchoscillatory behavior is characterized by an equation that consists of capillary force, viscousdrag, and gravity. Two asymptotic solutions in the initial and later stages of drop formationare obtained and show good agreement with the experimental observations.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The mechanism of drops forming from a nozzle has been employed in many industrial applications [1–4]; i.e. ink-jetprinting, spray cooling, emulsion formation, 3D micro-printing, and more. In most applications, uniform size distributionand fast formation rate of droplets are required for yielding predictable high-quality products and reducing operation time[5,6]. The key parameters for designing such mechanisms are the ejecting speed of a fluid as well as the physical shape andchemical properties of the nozzle [1]. In particular, the motion of a slowly emitted liquid strongly depends on the nozzle-exitsurface property i.e. a tea-pot effect [7,8].

In the process of drop formation, there are two primary modes in which a drop can be produced. The first mode is calledjetting mode because the drops separate themselves at the tip of a long fluid column far away from the nozzle [1,9]. Theother mode requires a lower flow speed so that drops can detach themselves directly from the orifice, often called drippingmode [10–13]. Drop formation in jetting mode has been extensively studied from the perspective of capillary stability of jetbreakup [1]. The transition from dripping to jetting mode has also been observed [14–16].

In dripping mode, the size and shape of the drops become highly dependent on the nozzle-exit condition. However,previous studies have been made using a non-wettable orifice which provides a simple nozzle-exit condition on the contactline between the interface and the nozzle [10,17]. The drop dynamics produced by a wettable surface has not been studiedextensively. This study requires analyzing the interaction between the liquid and the outer walls of the nozzle. This is wherecapillary rise comes into play.

The natural phenomena of capillary rise has been studied for almost a century now [18,19]. It is a common mechanismused by plants to distribute water throughout the body. Such capillary forces are also utilized by industrial systems; i.e.,dying colors, driving ink in pens, and chromatography. Mostly, capillary rise is observed in confined spaces like a tube ora porous material. However, when a drop is forming outside a wettable nozzle, the capillary force is observed to be ableto pull a drop upwards before gravity plays a role.

. All rights reserved.

, Virginia Tech, Blacksburg, VA 24061, United States. Tel.: +1 540 231 5146; fax: +1 540 231 4574.

2046 B. Chang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2045–2051

Previous studies focus on either the formation of a drop from a non-wettable orifice [6,11,20] or capillary rise inside awettable tube [19,21,22]. In this article, we investigated the dynamics of drop formation from a wettable nozzle. In contrastto drops forming from a non-wettable nozzle, drops initially rise due to the capillary force when emerging from a wettablenozzle. The drops then fall as soon as the weight is sufficiently large enough. In Section 2, the experimental setup and pro-cedure is described. Then the theoretical model and its comparison with the experimental results are discussed in Section 3.In Section 4, we discuss the conclusions of our findings and the future direction of this research.

2. Experimental methods

The experimental setup was designed to visually analyze the behavior of droplets emerging from a wettable orifice. Aschematic of the experimental apparatus is shown in Fig. 1. By changing the flow rate and nozzle diameter, we have testedthe dynamics of drops forming from wettable nozzles. In order to vary the flow rates, the silicone fluid with a viscosity of10 cSt was pumped by a syringe pump (new era system; NE-1000) in the low velocities range (from 50 to 250 mL/hr with50 mL/hr intervals) in order to avoid the jetting mode. The syringes used were 19, 20 and 21 gauges (corresponding innerdiameters are Ri = 0.406, 0.324, and 0.292 mm).

The motion of the droplets was recorded with a high speed camera (MotionXtra N3) at 500 frames per second. With high-contrasted movies, we were able to analyze the motion of the droplets using digital imaging processing in MATLAB. Then, themotion and shape of each drop was analyzed frame by frame and the centroidal position was calculated.

Details on the image analysis using the MATLAB code are as follows. Each individual frame was converted into a black andwhite image in order to identify the boundaries of the drop (not including the nozzle). After the boundaries were found, thecode averaged the radius at each point along the vertical axis. This assumes an axisymmetric shape about the vertical axis,but does not assume a spherical shape. This information allowed the program to calculate the centroid using the trapezoidalintegration method.

Our experiments have been carried out with low flow rates through a wettable orifice, as shown in Fig. 2a–c. Fig. 2dexhibits drop forming from a non-wettable nozzle for comparison purposes. In the initial stages of drop formation, as shownin Fig. 2a, a small amount of fluid climbs the sides of the tube against gravity by wetting the outer surface of the nozzle.Gradually, the drop increases in weight and ultimately falls due to gravity. Fig. 2b shows the effect of the nozzle size is chan-ged on drop formation at a set flow rate, whereas Fig. 2c demonstrates the effect of flow rate on drop formation with a setnozzle size. In Fig. 2d, a nozzle is coated with a hydrophobic material and water is pushed through the nozzle. No capillaryrise along the hydrophobic nozzle is observed. In comparison with the drops from a wettable nozzle, Fig. 2d demonstratesthe absence of capillary rise in a non-wettable nozzle.

Fig. 3 shows the behaviors of the centroid and the contact line as a function of time with the wettable nozzles. As the flowrate increases, the effect of capillary rise is less prevalent. The weight of the drop dominates the surface tension and viscousdrag at high flow rates; the drop has less time to climb up the outer surface of the nozzle. Eventually, as we continue to in-crease the flow rate, the fluid does not make contact with the outer surface of the nozzle and goes into jetting mode. Themotions of the drop’s centroid and contact line are observed with different nozzle sizes at a fixed flux in Fig. 4. Initially,the drop climbs up in a similar fashion regardless of the nozzle size, but the later dropping behaviors are different. Overall,the period and maximum drop height decrease with decreasing nozzle size.

3. Theoretical model

First, we account for the shape of drop moving along the nozzle. In a quasi-static equilibrium state, its interfacial shape ischaracterized by the pressure at given point at altitude z according to the Young–Laplace equation. The fluid pressure is lar-ger than the ambient atmospheric pressure due to surface tension, thereby forming a spherical shape. Previous studies

Fig. 1. Schematic of the experimental setup.

Fig. 2. Drop formed from different nozzles with (a) Ri = 0.292 mm at 50 mL/hr, (b) Ri = 0.406 mm at 50 mL/hr, (c) Ri = 0.292 mm at 250 mL/hr, (d)Ri = 0.292 mm at 250 mL/hr with non-wettable nozzle.

B. Chang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2045–2051 2047

showed that numerical results with the Young–Laplace equation well predicted experimental drop shapes. For axisymmetricshapes like the drop attached to the nozzle, this set of differential equations is used to be solved:

drds¼ cos h;

dzds¼ sin h;

dhds¼ 2j0 �

qgc

� �z� sin h

r;

where s is the arc length, q the fluid density, c the surface tension, j0 the curvature at the bottom, h the tangential angle, andr & z are radial and vertical positions of the interface, respectively. The equations are numerically solved with initial condi-tions as r (0) = 0, z(0) = zmin, h (0) = 0 and compared with experimental observations in Fig. 5. Since the contact angle variesbetween advancing and receding angles while a drop moves up and down, it is not used for a boundary condition. The timeevolution of the drop shape shows that the interface is well described by the Young–Laplace equation with the known ver-tical position. The shape of the interface is simply described by the Young–Laplace equation at given boundary conditions.

(a) (b)

Fig. 3. Motions of drop’s (a) centroid and (b) contact line with different flow rates with Ri = 0.292 mm.

(a) (b)

Fig. 4. Motions of drop’s (a) centroid and (b) contact line with different nozzle sizes with 100 mL/hr flow rate.

2048 B. Chang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2045–2051

The remaining question is how to determine the vertical position of the drop, which is critical to describe the shape as well asthe motion of the drop.

Surface tension, viscous drag and gravity are the main forces controlling the motion of the drop. For the above system of adrop emerging from a nozzle, we express a governing equation as:

Fig. 5.Laplace

dðMVÞdT

¼ cð2pR0Þ cosðhcÞ � 2plLV

ln aR0

� ��Mg ð1Þ

where M is the fluid mass, V the vertical velocity, hc the contact angle, a the drop radius, R0 the outer radius of the nozzle, andL the length of the wetted nozzle. The fluid mass in a drop linearly increases with time by pumping a fluid at a constant rateat M ¼ _MðT � T0Þ, where _M is the mass flux rate and T0 is the initial time at the beginning of drop formation. The viscous drag

Simulation of drop shape with Ri = 0.292 mm at 50 mL/hr. The red outline is the predicted shape of the drop by numerically solving the Young–equations.

B. Chang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2045–2051 2049

is evaluated by using the boundary conditions of vertical velocity (v) equals to zero along the outer nozzle (r = R0) and equalsto the drop velocity at the drop’s interface (v = V at r = a). Then, the shear stress on the nozzle yields: ldv

dr

��r¼R0¼ � lV

R0 ln aR0

. After

integrating the stress over the surface around the nozzle, the viscous drag force becomes �2plL Vln a

R0

. Typically, the speed of

the drop is so low that the viscous drag is insignificant compared to other terms.By introducing non-dimensionalized parameters as m ¼ M=ðqR3Þ; t ¼ T=ðqR3= _MÞ; v ¼ V=ð _M=qR2Þ, and l = L/R, the govern-

ing equation becomes:

dðmvÞdt

¼ 2p cosðhcÞWe

� 2pRe ln a

R0

� �vl� 1Fr2 m ð2Þ

Here, the Weber number is the ratio of inertial forces to the surface tension, defined as We = qV2R/c, where R is the charac-teristic size, V the characteristic velocity ðV ¼ _M=qR2Þ;q the fluid density, and c the surface tension. The mass of the drop(m(t) = qqt) is determined by the flow rate, q. The Reynolds number is the ratio of inertia to the viscous forces and is definedas Re = qVR/l. The Froude number is a ratio of inertial to gravitational forces and is defined as Fr = V/(gR)1/2, g being the accel-eration due to gravity. Compared to the Froude and Weber number (10�4 � 10�5), the Reynolds number (10�2) is large en-ough so that the viscosity term can be neglected in the regime of current experiments.

The beginning of the process takes place at a low Weber number and the surface tension plays a dominant role. It wasobserved that as the emerging fluid wets the outer wall of the nozzle, both the contact line and the drop are accelerated up-wards by the capillary force. The drop is approximated as an annular shape with volume approximately proportional to zR2,where z is the vertical distance traveled by the drop. Then, the above equation in the low Weber limit becomes:

qd z dz

dt

� �dt

¼ 2p cosðhcÞWe

¼ const: ð3Þ

Assuming that z � ta, we find that a = 1 at the beginning of the stage and the vertical position of the drop increases asz(t) �We�1/2t, which has a different scaling compared to Washburn’s result describing capillary rise inside a tube(z(t) = We�1/2t1/2). This behavior of a = 1 indicates that a drop moves upward at constant velocity and zero accelerationdue to the capillary force. The comparison between the Weber number and upward velocity (dz/dt) at the initial stage isshown in Fig. 6. This comparison was typically made using the average velocity between the first 0.02 s of each case. Thedotted line in the figure is the rearrangement of z(t) �We�1/2t as log (We) � �2log(dz/dt). Flow rates between 50–200 mL/hr were used in the Weber number analysis because capillary rise is more prevalent. Overall, the experimental data matcheswell with this predicted behavior. The R2 value for our distribution of data is 0.789. As the Weber number increases, equiv-alently as diameter increases or flow rate decreases, the upward velocity of the drop decreases along the predicted slope.

Later, the Froude number becomes large and the surface tension and viscous drag terms become negligible as the weightof the drop dominates the motion. The drop then slides downward due to gravity. After the moment of the highest verticalposition (t = tm), the fluid mass still increases as m(t) = m(tm) + (t � tm) but its growth rate is smaller compared to the fluidmass m(tm) at the low pumping rate, and z(t) starts from z(tm). The above equation by assuming the constant mass m predictsthe asymptotic solution of the vertical position as:

zðtÞ ¼ zm � Fr�2t2 ð4Þ

By taking dz/d(t2), this equation becomes jdz/d(t2)j � Fr�2. This relation corresponds to the dotted line in Fig. 7. The exper-iment showed that this prediction was accurate for the downward motion of the drop. By taking the absolute value ofthe average velocity within an interval of 0.02 s in the later stage (t = tm), the experimental data points are obtained and fol-low the predicted slope of �1/2 nicely, supporting our prediction with an R2 value of 0.748. As the Froude number increases(as diameter decreases or flow rate increases), the downward velocity of the drop decreases along the predicted slope, asshown in Fig. 7.

These asymptotic analyses predict the capillary-driven upward motion and the gravity-driven downward motion. The de-tailed dynamics can be obtained by solving the governing equation numerically.

4. Conclusion

In this article, we investigated the process of drop formation from a wettable nozzle. In the process of forming a singledroplet from a wettable orifice, the drop initially undergoes an upward movement due to surface tension and then, oncethe droplet is sufficiently large enough, falls downward due to gravity. The third and final force acting on the droplet is vis-cous drag along the boundary between the droplet and the surface; however it was neglected in our calculations because ofits insignificance compared to the capillary force and the gravitational force in the regime of our experiments. Asymptoti-cally, the drop starts out with an upward rise based on z � t=We

12, as it is more accurately approximated by a cylinder than

a sphere, and finishes with a free fall and follows the behavior as z � �t2=Fr2. Experimental observations were made by vary-ing diameter and volumetric flow rate in experiments, and are in good agreement with the asymptotic solutions found in thetheoretical model. The determination of such a model for this behavior is also relevant in applications with flows under the

Fig. 6. Weber number vs. drop velocity (dz/dt) with different radii and flow rates. The dotted line is from the asymptotic behavior dz/dt �We�1/2 predictedfrom our calculation. The plotted points are the averaged velocity from the early stage of the droplet motion. R2 = 0.789.

−2 −1 0 1 2 3−5.5

−5

−4.5

−4

−3.5

−3

−2.5

50 mL/hr100 mL/hr150 mL/hr200 mL/hr0.406 mm0.324 mm0.292 mm

log(

Fr)

log(|dz/d(t2)|)

Fig. 7. Froude number vs. drop motion (dz/dt2) with different radii and flow rates. The dotted line is from the asymptotic behavior jdz/d(t2)j � Fr�2 predictedfrom our calculation. The points on the graph are taken from the average velocity near t = tm � R2 = 0.748.

2050 B. Chang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2045–2051

electric field, such as microfluidics and ink-jet printer technology. In this context, we plan to combine the proposed exper-iments with electro-wetting techniques in the future, in order to vary the surface tension while the drop is at the orifice.

Acknowledgement

The author, S. Jung, celebrates the 60th birthday of Prof. Philip Morrison, a great advisor for guiding S. Jung into the wildworld of science with his thoughtful (on-going) insights into seeking a theory of everything.

B. Chang et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2045–2051 2051

References

[1] Bogy D. Drop formation in a circular liquid jet. Ann Rev Fluid Mech 1979;11(1):207–28.[2] Beale J, Reitz R. Modeling spray atomization with the Kelvin–Helmholtz/Rayleigh–Taylor hybrid model. Atom sprays 1999;9(6):623–50.[3] Thorsen T et al. Dynamic pattern formation in a vesicle-generating microfluidic device. Phys Rev Lett 2001;86(18):4163–6.[4] Hon K, Li L, Hutchings I. Direct writing technology–advances and developments. CIRP Annal Manufact Tech 2008;57(2):601–20.[5] Ganan-Calvo AM. Generation of steady liquid microthreads and micron-sized monodisperse sprays in gas streams. Phys Rev Lett 1998;80(2):285–8.[6] Basaran OA. Small scale free surface flows with breakup: drop formation and emerging applications. AIChE J 2002;48(9):1842–8.[7] Reiner M. The teapot effect: a problem. Phys Today 1956;9:16.[8] Duez C et al. Wetting controls separation of inertial flows from solid surfaces. Phys Rev Lett 2010;104:084503.[9] Rayleigh L. On the instability of jets. Proc Lond Math Soc 1878;1(1):4.

[10] Tate T. On the magnitude of a drop of liquid formed under different circumstances. Phil Mag Ser 4 1864;27(181):176–80.[11] Burton JC, Waldrep R, Taborek P. Scaling and instabilities in bubble pinch-off. Phys Rev Lett 2005;94(18):184502.[12] Gekle S et al. Approach to universality in axisymmetric bubble pinch-off. Phys. Rev. E 2009;80(3):036305.[13] Gordillo JM, Gekle S. Generation and breakup of Worthington jets after cavity collapse. Part 2. Tip breakup of stretched jets. J Fluid Mech

2010;663:331–46.[14] Clanet C, Lasheras J. Transition from dripping to jetting. J Fluid Mech 1999;383:307–26.[15] Ganan-Calvo AM, Rieso-Chueca P. Jetting-dripping transition of a liquid jet in a lower viscosity co-flowing immiscible liquid: the minimum flow rate in

flow focusing. J Fluid Mech 2006;553(-1). 75–75.[16] Utada A et al. Absolute instability of a liquid jet in a coflowing stream. Phys Rev lett 2008;100(1):14502.[17] Wilson S. The slow dripping of a viscous fluid. J Fluid Mech 1988;190:561–70.[18] Lucas R. Rate of capillary ascension of liquids. Kolloid Z 1918;23:15–22.[19] Washburn E. The dynamics of capillary flow. Phys Rev 1921;17(3):273–83.[20] Wilkes ED, Phillips SD, Basaran OA. Computational and experimental analysis of dynamics of drop formation. Phys Fluids 1999;11(12):22.[21] Zhmud BV, Tiberg F, Hallstensson K. Dynamics of capillary rise. J Colloid Interf Sci 2000;228(2):263–9.[22] Stange M, Dreyer ME, Rath HJ. Capillary driven flow in circular cylindrical tubes. Phys Fluids 2003;15(9):15.