NUMERICAL SIMULATION OF DROPLET COALESCENCE AND BUBBLE ENTRAPMENT

NUMERICAL SIMULATION OF DROPLET COALESCENCE AND BUBBLE ENTRAPMENT J. Esmaeelpanah *, A. Dalili, S. Chandra, J. Mostaghimi Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada H. Fan Manufacturing Systems Research Lab General Motors R&D Warren, Michigan, USA ABSTRACT When liquid droplets are sprayed onto solid surfaces during spray painting and coating, defects may be created due to air bubbles trapped in the liquid film. Bubble entrapment during single droplet impact onto a solid surface or in a liquid pool has been studied extensively, but in spray applications most droplets land on each other and bubbles are formed at the interface between droplets. In this paper bubble entrapment during droplet coalescence was modeled numerically using a three-dimensional parallel code, based on a two-step projection method with a volume-of-fluid technique used to model free surface flow. Simulations and experiments were performed with a high viscosity liquid (87%wt glycerin in water solution) with the same viscosity as industrial automotive paint, and the effect of gravity and contact angle on bubble entrapment studied. Predictions from the numerical model of droplet spread diameters agreed well with experimental measurements (within 8.8%). Simulation showed that during impact of a droplet on a solid surface an air film is trapped at the solid-liquid interface during the initial stages of impact. The air film contracts, forms a torus, and finally collapses into a bubble at the droplet center. When a second droplet hits the first sessile droplet on the substrate, bubbles are entrapped at the liquid-liquid interface that move within the coalescing droplets, break up and merge with each other. Calculated pressure contours showed that bubble motion was driven by pressure gradients in the liquid. Gravity had a negligible effect on bubble entrapment or movement so that the orientation of the spray (onto an upward or downward facing surface) made little difference. Increasing liquid-solid contact angle was found to enhance bubble entrapment. * Corresponding author: javade@mie.utoronto.ca

INTRODUCTION Although droplet impact has been the subject of numerous studies, dark spots remain which need to be discovered. Bubble entrapment is a phenomenon that happens frequently during the impact and spreading of a droplet. In coating technologies such as thermal spray coating and paint spray, bubble entrapment creates defects in the coating and should be prevented while in some other applications it should be enhanced such as air entrapment in the oceans during rain droplet impact. The literature has proposed various justifications and solutions to the air entrainment for cases of droplet impact onto a solid surface and a liquid pool. Morton et al. [1] used a validated numerical approach to study flow regimes resulting from the impact of a 2.9 mm water drop on a deep water pool at velocities in the range of 0.8 2.5 m/s. The results were used to predict conditions for vortex ring formation, bubble entrapment and growth of vertical Rayleigh jets. The researchers concluded that bubble entrapment was the result of a capillary wave that propagated along the walls of the cylindrical crater formed during droplet impact. The cylindrical shape of the crater would be due to the Weber and Froude number combinations of the impact. Vortex ring formation depended on the time at which vorticity of the appropriate sign was generated and its subsequent transport. Eggers [2] studied air entrainment through free-surface cusps. The author predicted a critical curvature and capillary number based on the relative viscosity of the air to the fluid and flow characteristics above which the stationary cusp is lost and a sheet emanates from the cusp s tip through which air is entrained. Eggers confirmed his analytical theory by quantitative comparison with numerical simulations of flow equations and qualitative comparison with experimental results. Elmore et al. [3] took high speed images of airwater interface dynamics of drop impact that lead to reproducing bubbles and discussed the various phenomena seen during the process. The paper discussed the differences between the computational and experimental methods. Specifically, the experiments showed that the cavity depth stagnated before bubble entrapment; a phenomenon that is not seen in the simulations. The authors indicate that surface-tension smoothing or neglect of vorticity maybe the cause for this discrepancy. Thoroddsen et al. [4] studied bubble entrapment during droplet impact onto a liquid surface. They proposed a model for the initial bubble thickness and radius based on the bottom curvature of droplet and the Reynolds number. They concluded that most of the bubble entrapment occurs due to the secondary droplets as their impact velocity is lower than the main droplet. Deng et al. [5] investigated bubble entrapment for droplet impact onto a deep liquid pool of the same liquid as the droplet. They found that viscosity has a weakening effect on capillary waves around the crater which is responsible for the bubble pinching. They concluded that bubble size decreases exponentially with increasing Capillary number (increasing the liquid viscosity and decreasing the surface tension). Chandra and Avedisian [6] photographed the impact of a n-heptane droplet onto a stainless steel surface at room temperature and observed the presence of a single bubble at the point of impact. Mehdi-Nejad et al. [7] numerically simulated the droplet impact for water, n-heptane and molten nickel droplets to investigate the effect of viscosity, velocity and contact angle on bubble entrapment. They performed a mathematical analysis to find the bubble pressure and concluded that the bubble size decreased with decreasing gas viscosity and droplet impact speed. In addition, in the case of n-heptane, they observed bubble breakup (due to its lower surface tension) and bubble detachment from the solid surface (due to its lower contact angle). Van Dam et al. [8] experimentally studied micro-sized water droplet impact and observed bubble entrapment in almost all cases. They concluded that bubble size increases with increasing Weber number and impact velocity and proposed a model for the bubble volume based on impact speed. Thoroddsen et al. [9] observed the formation of the air disk under a droplet impacting onto a surface for a range of Weber and Reynolds numbers. They found that the contraction speed of the bubble is independent of the wettability of the liquid. The micro-bubble formation on the initial ring location was seen during some experiments. In addition, the capillary wave propagation from the edge of the air disk leaves a small droplet in the middle of the bubble. Thoroddsen et al. [10] showed the primary mechanism for the bubble entrapment at the tip of lamella. They demonstrated that the bubble size increases with distance from the impact point due to the thickening of the air cushion. Lee et al. [11] studied bubble entrainment during water droplet impact onto a solid surface and proposed a critical Ohnesorge number (0.026±0.003) below which daughter droplets were observed. They also suggested a threshold contact angle (40±5) below which detachment occurs, but the threshold may decrease with increasing Ohnesorge number. In most industrial applications such as spraying process, only a few percentage of droplets hit solid surface and most of droplets land over each other. The novelty of this paper lies in the fact that bubble entrapment at a droplet-droplet interface has been studied contrary to previous researches that focused on bubble entrapment in droplets impacting a solid surface or a deep liquid pool. This paper reports the effect of contact angle and gravity on bubble entrapment. It was shown that gravity has no effect while an increase in

contact angle led to an increase in bubble entrapment. Bubble migration within the droplet was also studied. EXPERIMENTAL SETUP The experimental droplet deposition system for 87 wt% glycerin in water solutions, which has a viscosity of 124 centistokes, consists of two major components: droplet generator and translation motion system. A schematic of the experimental system is given below in Fig. 1. Figure 2. Final diameter of the first droplet before the impact of a second droplet, Ds, droplet center-to center distance, L, stage speed, u The extent of overlap between droplets deposited on the surface was described by the overlap ratio [13]: (1) Figure 1. Schematic of the 87 wt% glycerin experimental system The droplet generation system consists of a stainless steel tank filled with liquid and connected to a centrifugal pump which passes the liquid through stainless steel and plastic tubing towards a solenoid valve. A partially open needle valve maintains a closed loop for flow recirculation to the tank and ensures no back flow into the pump. A pressure regulator was used to control the pressure upstream of the needle. The solenoid valve was normally closed and could be opened for a pre-determined period of time with a timer circuit that was triggered by a computer. When the valve was opened, fluid passed through a 17 gage needle which had a 1.47 mm outer diameter. The translation motion system consists of an x-y motion stage with 200 mm x 200 mm (8 in. x 8 in.) travel. Computer software developed by Fang [12] was used to control the system. The solenoid valve was opened for 13 ms to allow liquid at 30 kpa pressure to pass through a 17 gage needle (with 1.47 mm outer diameter) and detach from the tip as a droplet. The average diameter and average impact velocity of fourteen droplets created were calculated to be 3.43 mm and 0.844 m/s respectively. The final diameter of the first droplet before the impact of the second droplet, Ds, was measured to be 5.98 mm. The droplet center-to-center distance, L, was varied in our study. If the centers of the droplets coincide, then there is complete overlap and λ = 1. If there is no overlap then λ < 0. By varying the speed of the motion stage, the center-to-center distance, L, was controlled. Droplets were deposited at a frequency of 1 Hz. NUMERICAL SCHEME Real-time multiphase flows were modeled using the continuity and Navier-Stokes equations for incompressible flow as follows: ( ) (2) ( ) (3) where the continuum-surface-force (CSF) scheme [14] was used to model surface tension as a body force ( ). Therefore, equation (2-3) can be solved in the omnidomain for all phases. Eq. (2-3) were discretized for a 3D Cartesian staggered grid and the two-step projection method was employed to solve the coupled equations (2-3). The pressure Poisson equation (PPE) was solved using incomplete-cholesky-conjugatedgradient (ICCG) solver. Since the convective and diffuse terms were discretized explicitly, the time step restrictions were applied to the program. In addition, in order to track the interface the volume-of-fluid (VOF) technique was employed, ( ) (4) where is a scalar variable. The piecewise linear interface construction (PLIC) method has been used to construct the surface slope of interfacial cells. The density and viscosity of the fluids were averaged as follows: ( ) (5) ( ( ) ) (6)

For high-viscosity liquids, harmonic viscosity averaging was used as seen in equation (6). Following the method of Bussmann et al. [15] the liquid-solid contact angle value was set to either the advancing, receding or equilibrium value depending on the direction of motion of the liquid-solid contact line. Measurements of liquid-solid contact angle were made from video images and the advancing, equilibrium and receding contact angles were determined to be 145, 50, and 10 respectively. Fig. 3 illustrates the grid independency for the droplet impact for mesh size of the 26 (120x120x120), 40 (180x180x180) and 53 (240x240x240) CPR (cell per radius), respectively. The error corresponding to 26 and 53 CPR is 3.2 % and therefore the simulations were performed with 30 CPR. Fig. 4 shows the mesh size in droplet coalescence. As shown in the figure, the mesh size is capable of capturing the bubbles. As mentioned, once a droplet hits a solid surface a bubble entraps at the center of each droplet. Various explanations have been proposed for this phenomenon. Mehdi-Nejad [7] proved that the pressure rise in the air gap at the interface leads to bubble entrainment. VALIDATION The numerical results were compared with those of the experiment for various cases. Fig. 5 shows the droplet coalescence and bubble entrapment for the case of 87wt% glycerine droplet impact onto a steel substrate with diameter 3.4 mm, velocity 0.84 m/s and droplet center-to-center distance of 4.82 mm. The numerical model is capable of predicting the same bubbles observed in the corresponding experiments. In addition, an error of 8.8% is seen in the droplet spreading distance which illustrates the accuracy of the numerical scheme. Figure 3. The mesh independency study for 26, 40 and 53 CPR (D=3.4 mm, V= 1.1 m/s) Figure 4. The bubble dynamics of droplet coalescence and mesh size RESULTS AND DISCUSSION Using a high speed camera (FASTCAM SA5, Photron, San Diego, CA, USA), the impact of fourteen 87 wt% glycerin droplets onto a steel substrate with roughness of 0.22 μm was videotaped. The videos were taken at 4000 frames per second, 1024x1024 pixel resolution and 1/7000 shutter speed. Figure 5. The comparison the numerical results with those of the experiments (D=3.4 mm, V= 0.84 m/s, Offset= 4.82 mm (λ = 0.19)) Fig. 6-9 shows the effect of gravity, and contact angle on bubble entrapment during droplet coalescence. The first droplet is in diameter with a spreading diameter of. The second droplet is half of the first droplet in volume with

diameter of and spreading diameter of. The impact speed for both droplets is and the droplet center-to-center distance is i.e. the center of the second droplet is located at edge of the first one. However, as shown in the figure, the two droplets interact with each other. g GRAVITY EFFECT Fig. 6-7 illustrates the surface evolution and pressure contour of droplet coalescence for two cases of positive and negative gravitational acceleration. As shown in the figure, once the first droplet hits the solid surface, an air film is created in both cases. In case, a torus forms after 6ms and finally a semi-spherical bubble is created at 7ms. The same behaviour can be observed in case but the process is faster than case by ~0.25ms. The main reason could be due to the lower pressure in case which accelerates the air film contraction. The pressure contours illustrate the behaviour accordingly. Once the droplet hits the surface a high pressure region is created adjacent to the wall. The pressure is higher at the central region of impact (i.e. the air film) and at the edges of spreading area. The viscous effect and the surface tension force dampen the pressure. However, the pressure in the bubble region remains high which agrees with Mehdi-Nejad et al. [2003] results that showed high pressure at interface led to bubble entrapment. Once the second droplet impacts the first sessile droplet, bubbles entrap at the interface. In case the number of the bubbles is greater than case. The bubbles in both cases oscillate in the liquid. The high pressure of the impact region drives the bubbles towards the wall where the pressure is low. As soon as the distance between the bubble and the wall reduces, the pressure bounces the bubble off the wall. If the gravitational acceleration is downward, the buoyancy force supports the bubbles to move upward (i.e. towards the free surface). Therefore, the first bubble escapes the liquid in 31 ms. In case, the bubbles continuously oscillate and finally escape in 38 ms. Comparison between case and shows that more bubbles entrain in than case but they escape faster. However, the simulation illustrates that the gravity has minor effect on the bubble entrapment. The reason is that the strong oscillations dominate the buoyancy force. Figure 6. The bubble dynamics of droplet coalescence for positive and negative gravitational acceleration; g=-9.81m/s2; g=+9.81m/s2; (1st droplet: D=3.4mm; V=1.1m/s and 2nd droplet:

liquid. Comparatively, the contact angle is higher in the case which makes the surface energy dominant. Therefore, the bubble initially bounces upward for a short period of time before returning back to the wall and finally settling. g Figure 7. The pressure contour of droplet coalescence for positive and negative gravitational acceleration; g=-9.81m/s2; g=+9.81m/s2; (1 st droplet: D=3.4mm; V=1.1m/s and 2 nd droplet: EFFECT OF CONTACT ANGLE Figure 8-9 illustrate the surface evolution and pressure contour of bubble entrainment and droplet coalescence for two different contact angles ( θeq=45 ; θeq=90 ). Since the contact angle only changes the condition adjacent to the wall, bubbles far from the wall remain unaffected. However, contact angle has a minor effect on the flow field which causes a small change in bubble behaviour. As shown in the figure, once the second droplet hits the sessile droplet, the same number of bubbles is formed for case and. The bubbles push towards the low pressure regions. The bubble moves towards the wall at 24 ms in both cases and contacts the wall for a short period of time. The interaction between the surface tension and the pressure (between the bubble and the wall) finally pushes the bubbles upward. The surface tension tends to settle the bubble on the surface while the pressure drives it away from the surface. Due to lower contact angle in case, the pressure force dominates the surface energy and causes the bubble to escape the Figure 8. The bubble dynamics of droplet coalescence for the effect of contact angle; θeq=45 ; θeq=90 ; (1 st droplet: D=3.4mm; V=1.1m/s and 2 nd droplet:

contact angle increases the bubble entrapment. The reason could be due to higher surface energy in the case of higher contact angle. Figure 9. The pressure contour of droplet coalescence for the effect of contact angle; θeq=45 ; θeq=90 ; (1st droplet: D=3.4mm; V=1.1m/s and 2nd droplet: CONCLUSIONS A three dimensional numerical simulation has been performed to investigate the bubble entrapment during the droplet coalescence. The model verified and validated against the experiments. Numerical simulation of droplet coalescence for a high viscous liquid was performed and the bubble entrapment and bubble dynamics were studied. The results showed that gravitational acceleration (keeping the rest of properties constant) will affect the bubble dynamics but it does not influence the bubble entrapment. Base on Bo~0.006 and Fr~100 for the bubbles, the gravitational force is smaller than the surface tension force (next to wall) and convective terms (within the liquid). Therefore, the gravitational force is less influential on the bubble dynamics and settlement with respect to other forces. However, the numerical results showed that increasing REFERENCES [1] D. Morton, M. Rudman, and L. Jong-Leng, "An investigation of the flow regimes resulting from splashing drops," Physics of Fluids, vol. 12, pp. 747-763, 2000. [2] J. Eggers, "Air Entrainment through Free-Surface Cusps," Physical Review Letters, vol. 86, pp. 4290-4293, 2001. [3] P. A. Elmore, G. L. Chahine, and H. N. Oguz, "Cavity and flow measurements of reproducible bubble entrainment following drop impacts," Experiments in Fluids, vol. 31, pp. 664-673, 2001/12/01 2001. [4] S. T. Thoroddsen, T. G. Etoh, And K. Takehara, "Air entrapment under an impacting drop," Journal of Fluid Mechanics, vol. 478, pp. 125-134, 2003. [5] Q. DENG, A. V. ANILKUMAR, and T. G. WANG, "The role of viscosity and surface tension in bubble entrapment during drop impact onto a deep liquid pool," Journal of Fluid Mechanics, vol. 578, pp. 119-138, 2007. [6] S. Chandra and C. T. Avedisian, "On the Collision of a Droplet with a Solid Surface," Proceedings: Mathematical and Physical Sciences, vol. 432, pp. 13-41, 1991. [7] V. Mehdi-Nejad, J. Mostaghimi, and S. Chandra, "Air bubble entrapment under an impacting droplet," Physics of Fluids, vol. 15, pp. 173-183, 2003. [8] D. B. van Dam and C. Le Clerc, "Experimental study of the impact of an ink-jet printed droplet on a solid substrate," Physics of Fluids, vol. 16, pp. 3403-3414, 2004. [9] S. T. THORODDSEN, T. G. ETOH, K. TAKEHARA, N. OOTSUKA, and Y. HATSUKI, "The air bubble entrapped under a drop impacting on a solid surface," Journal of Fluid Mechanics, vol. 545, pp. 203-212, 2005. [10] S. T. Thoroddsen, K. Takehara, and T. G. Etoh, "Bubble entrapment through topological change," Physics of Fluids, vol. 22, pp. 051701-4, 2010. [11] J. S. Lee, B. M. Weon, J. H. Je, and K. Fezzaa, "How Does an Air Film Evolve into a Bubble During Drop Impact?," Physical Review Letters, vol. 109, p. 204501, 2012. [12] M. Fang, S. Chandra, and C. B. Park, "Building three-dimensional objects by deposition of molten metal droplets," Rapid Prototyping Journal, vol. 14, pp. 44-52, 2008. [13] R. Li, N. Ashgriz, S. Chandra, J. R. Andrews, and J. Williams, "Drawback During Deposition of Overlapping Molten Wax Droplets," Journal of Manufacturing Science and Engineering, vol. 130, pp. 041011-10, 2008. [14] J. U. Brackbill, D. B. Kothe, and C. Zemach, "A continuum method for modeling surface tension," Journal of Computational Physics, vol. 100, pp. 335-354, 1992. [15] Bussmann, M., Mostaghimi, J., Chandra, And S., On A Three-Dimensional Volume Tracking Model Of Droplet Impact vol. 11. Melville, NY, ETATS-UNIS: American Institute of Physics, 1999.