Drop formation from a wettable nozzle

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

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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

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Article history: Available online 27 August 2011 Keywords: Capillary rise Drop formation Dripping Wetting

a b s t r a c t The process of drop formation from a nozzle can be seen in many natural systems and engineering applications. However, previous research focuses on the pinch-off mechanism of drops from a non-wettable nozzle. Here we investigate the formation of a liquid droplet from a wettable nozzle. In the experiments, drops forming from a wettable nozzle initially climb the outer walls of the nozzle due to surface tension. Then, when the weight of the drops gradually increases, they eventually fall due to gravity. By changing the parameters such as the nozzle size and fluid flow rate, we have observed different behaviors of the droplets. Such oscillatory behavior is characterized by an equation that consists of capillary force, viscous drag, and gravity. Two asymptotic solutions in the initial and later stages of drop formation are obtained and show good agreement with the experimental observations. Ó 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-jet printing, spray cooling, emulsion formation, 3D micro-printing, and more. In most applications, uniform size distribution and 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 and chemical properties of the nozzle [1]. In particular, the motion of a slowly emitted liquid strongly depends on the nozzle-exit surface 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 called jetting mode because the drops separate themselves at the tip of a long fluid column far away from the nozzle [1,9]. The other mode requires a lower flow speed so that drops can detach themselves directly from the orifice, often called dripping mode [10–13]. Drop formation in jetting mode has been extensively studied from the perspective of capillary stability of jet breakup [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 contact line between the interface and the nozzle [10,17]. The drop dynamics produced by a wettable surface has not been studied extensively. This study requires analyzing the interaction between the liquid and the outer walls of the nozzle. This is where capillary rise comes into play. The natural phenomena of capillary rise has been studied for almost a century now [18,19]. It is a common mechanism used 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 or a porous material. However, when a drop is forming outside a wettable nozzle, the capillary force is observed to be able to pull a drop upwards before gravity plays a role. ⇑ Corresponding author. Address: 228 Norris Hall, Virginia Tech, Blacksburg, VA 24061, United States. Tel.: +1 540 231 5146; fax: +1 540 231 4574. E-mail address: [email protected] (S. Jung). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.08.023

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Previous studies focus on either the formation of a drop from a non-wettable orifice [6,11,20] or capillary rise inside a wettable tube [19,21,22]. In this article, we investigated the dynamics of drop formation from a wettable nozzle. In contrast to drops forming from a non-wettable nozzle, drops initially rise due to the capillary force when emerging from a wettable nozzle. The drops then fall as soon as the weight is sufficiently large enough. In Section 2, the experimental setup and procedure 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. A schematic of the experimental apparatus is shown in Fig. 1. By changing the flow rate and nozzle diameter, we have tested the dynamics of drops forming from wettable nozzles. In order to vary the flow rates, the silicone fluid with a viscosity of 10 cSt was pumped by a syringe pump (new era system; NE-1000) in the low velocities range (from 50 to 250 mL/hr with 50 mL/hr intervals) in order to avoid the jetting mode. The syringes used were 19, 20 and 21 gauges (corresponding inner diameters 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 highcontrasted movies, we were able to analyze the motion of the droplets using digital imaging processing in MATLAB. Then, the motion 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 and white image in order to identify the boundaries of the drop (not including the nozzle). After the boundaries were found, the code 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 trapezoidal integration method. Our experiments have been carried out with low flow rates through a wettable orifice, as shown in Fig. 2a–c. Fig. 2d exhibits drop forming from a non-wettable nozzle for comparison purposes. In the initial stages of drop formation, as shown in 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 changed on drop formation at a set flow rate, whereas Fig. 2c demonstrates the effect of flow rate on drop formation with a set nozzle size. In Fig. 2d, a nozzle is coated with a hydrophobic material and water is pushed through the nozzle. No capillary rise along the hydrophobic nozzle is observed. In comparison with the drops from a wettable nozzle, Fig. 2d demonstrates the 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 flow rate increases, the effect of capillary rise is less prevalent. The weight of the drop dominates the surface tension and viscous drag at high flow rates; the drop has less time to climb up the outer surface of the nozzle. Eventually, as we continue to increase the flow rate, the fluid does not make contact with the outer surface of the nozzle and goes into jetting mode. The motions 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 is characterized by the pressure at given point at altitude z according to the Young–Laplace equation. The fluid pressure is larger than the ambient atmospheric pressure due to surface tension, thereby forming a spherical shape. Previous studies

Fig. 1. Schematic of the experimental setup.

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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.

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

dr ¼ cos h; ds

dz ¼ sin h; ds

  dh qg sin h z ¼ 2j0  ; ds r c

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

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(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.

The remaining question is how to determine the vertical position of the drop, which is critical to describe the shape as well as the 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 a drop emerging from a nozzle, we express a governing equation as:

dðMVÞ V ¼ cð2pR0 Þ cosðhc Þ  2plL    Mg dT ln a

ð1Þ

R0

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, and L the length of the wetted nozzle. The fluid mass in a drop linearly increases with time by pumping a fluid at a constant rate _ _ is the mass flux rate and T0 is the initial time at the beginning of drop formation. The viscous drag at M ¼ MðT  T 0 Þ, where M

Fig. 5. 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– Laplace equations.

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is evaluated by using the boundary conditions of vertical velocity (v) equals to zero along the outer nozzle (r = R0) and equals ¼  lV a . After to the drop velocity at the drop’s interface (v = V at r = a). Then, the shear stress on the nozzle yields: ldv dr r¼R0

R0 lnR

0

integrating the stress over the surface around the nozzle, the viscous drag force becomes 2plL lnVa . Typically, the speed of R0

the drop is so low that the viscous drag is insignificant compared to other terms. _ v ¼ V=ðM= _ qR2 Þ, and l = L/R, the governBy introducing non-dimensionalized parameters as m ¼ M=ðqR3 Þ; t ¼ T=ðqR3 =MÞ; ing equation becomes:

dðmv Þ 2p cosðhc Þ 2p 1   vl  2 m ¼  dt We Fr Re ln a

ð2Þ

R0

Here, the Weber number is the ratio of inertial forces to the surface tension, defined as We = qV2R/c, where R is the charac_ qR2 Þ; q the fluid density, and c the surface tension. The mass of the drop teristic size, V the characteristic velocity ðV ¼ M= (m(t) = qqt) is determined by the flow rate, q. The Reynolds number is the ratio of inertia to the viscous forces and is defined as 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 acceleration due to gravity. Compared to the Froude and Weber number (104  105), the Reynolds number (102) is large enough 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 was observed that as the emerging fluid wets the outer wall of the nozzle, both the contact line and the drop are accelerated upwards 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:


d z dz 2p cosðhc Þ dt ¼ ¼ const: q dt We

ð3Þ

Assuming that z  ta, we find that a = 1 at the beginning of the stage and the vertical position of the drop increases as z(t)  We1/2t, which has a different scaling compared to Washburn’s result describing capillary rise inside a tube (z(t) = We1/2t1/2). This behavior of a = 1 indicates that a drop moves upward at constant velocity and zero acceleration due to the capillary force. The comparison between the Weber number and upward velocity (dz/dt) at the initial stage is shown in Fig. 6. This comparison was typically made using the average velocity between the first 0.02 s of each case. The dotted line in the figure is the rearrangement of z(t)  We1/2t 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 matches well with this predicted behavior. The R2 value for our distribution of data is 0.789. As the Weber number increases, equivalently 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 weight of the drop dominates the motion. The drop then slides downward due to gravity. After the moment of the highest vertical position (t = tm), the fluid mass still increases as m(t) = m(tm) + (t  tm) but its growth rate is smaller compared to the fluid mass m(tm) at the low pumping rate, and z(t) starts from z(tm). The above equation by assuming the constant mass m predicts the asymptotic solution of the vertical position as:

zðtÞ ¼ zm  Fr 2 t 2 2

ð4Þ 2

2

By taking dz/d(t ), this equation becomes jdz/d(t )j  Fr . This relation corresponds to the dotted line in Fig. 7. The experiment showed that this prediction was accurate for the downward motion of the drop. By taking the absolute value of the average velocity within an interval of 0.02 s in the later stage (t = tm), the experimental data points are obtained and follow 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, as shown in Fig. 7. These asymptotic analyses predict the capillary-driven upward motion and the gravity-driven downward motion. The detailed 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 single droplet from a wettable orifice, the drop initially undergoes an upward movement due to surface tension and then, once the droplet is sufficiently large enough, falls downward due to gravity. The third and final force acting on the droplet is viscous drag along the boundary between the droplet and the surface; however it was neglected in our calculations because of its insignificance compared to the capillary force and the gravitational force in the regime of our experiments. Asymptoti1 cally, the drop starts out with an upward rise based on z  t=We2 , 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 =Fr 2 . Experimental observations were made by varying diameter and volumetric flow rate in experiments, and are in good agreement with the asymptotic solutions found in the theoretical model. The determination of such a model for this behavior is also relevant in applications with flows under the

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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  We1/2 predicted from our calculation. The plotted points are the averaged velocity from the early stage of the droplet motion. R2 = 0.789.

−2.5

−3

log(Fr)

−3.5

−4

−4.5

−5

−5.5 −2

50 mL/hr 100 mL/hr 150 mL/hr 200 mL/hr 0.406 mm 0.324 mm 0.292 mm −1

0

1

2

3

2

log(|dz/d(t )|) 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  Fr2 predicted from our calculation. The points on the graph are taken from the average velocity near t = tm  R2 = 0.748.

electric field, such as microfluidics and ink-jet printer technology. In this context, we plan to combine the proposed experiments 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 wild world of science with his thoughtful (on-going) insights into seeking a theory of everything.

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