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Analysis of bubble coalescence dynamics and postrupture oscillation of capillary-held bubbles in water Yesenia Saavedra Moreno, Ghislain Bournival, and Seher Ata Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03197 • Publication Date (Web): 21 Nov 2017 Downloaded from http://pubs.acs.org on November 26, 2017
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Analysis of bubble coalescence dynamics and post-
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rupture oscillation of capillary-held bubbles in water
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Yesenia Saavedra Moreno, Ghislain Bournival, and Seher Ata*
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School of Mining Engineering, University of New South Wales, Sydney, New South Wales,
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2052, Australia
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Keywords: bubble coalescence; neck expansion; post-rupture oscillation; VOF method
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Corresponding author
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*
E-mail:
[email protected] 9
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ABSTRACT
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This paper presents a three-dimensional computational study of coalescence dynamics of two
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capillary-held air bubbles in water using the volume of fluid (VOF) method. The interface
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motion of the newly formed bubble indicated that smaller initial separation distances resulted in
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a slightly faster expansion of the neck. The velocity vectors showed that the inward motion of
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the air at the contact point of the two bubbles favors the generation of small eddies at the top and
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bottom part of the neck at the early stage of coalescence and bulges at the edges of the bubble at
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a later stage. The release of free-surface energy drove the bubble contraction and expansion and
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consequently the oscillation motion, which created differences in pressure across the bubble
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interface. The computationally simulated bubbles followed the oscillatory motion observed
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experimentally, but with lower damping constant and lower angular frequency.
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1. INTRODUCTION
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The coalescence of air bubbles is an important phenomenon in a number of industrial
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processes that use bubbling in a liquid flow such as in froth flotation, wastewater treatment and
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paper recycling. Bubble coalescence is initiated by the thinning of the liquid separating the two
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bubbles followed by a rupture of the liquid film 1. Once coalescence occurs, a neck is formed
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joining the two bubbles. As the neck opens, the surface area and the curvature of the two initial
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bubbles decrease. The newly formed bubble expands horizontally and contracts vertically and
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vice versa until a stable spherical shape is achieved. The release of free-surface energy due to
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this coalescence process is imparted to the surrounding liquid as kinetic energy 2.
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The effects of inorganic electrolytes and surfactants in inhibiting bubble coalescence have
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been widely studied experimentally and theoretically. Many dissolved salts have a critical
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concentration beyond which the likelihood of bubble coalescence is significantly reduced
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Lessard and Zieminski 3 introduced a transition concentration for each electrolyte studied where
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the probability of coalescence was 50%, using 100% as the coalescence of a bubble pair in pure
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water. They found that electrolytes with valence combinations of 3-1 and 2-2 were the most
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effective electrolytes with transition concentrations of 0.032 M and 0.035 M, respectively. These
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findings were corroborated by Craig et al. 4, who observed that highly charged salts were more
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effective as bubble coalescence inhibitors at lower transition concentrations. More recently,
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Yaminsky et al. 7 reported that the speed at which the bubbles were brought together played an
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important role in bubble coalescence, finding that instantaneous coalescence of bubbles occurred
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at greater approach speed (150 µm/s). However, they found that high concentration of NaCl
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electrolyte increased film stability by immobilizing the interface and therefore inhibiting
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coalescence. For non-ionic surfactants, it is believed that the adsorption at the liquid-air interface
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is mainly driven by the hydrophobic forces
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surface properties of the interface, causing the Gibbs-Marangoni phenomena, which is
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commonly used to explain the effect of surface tension on bubble coalescence. However, surface
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forces have also been recognized as an important contributor in preventing film rupture in some
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non-ionic surfactant systems
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increasing surfactant concentration and coalescence time until a maximum coalescence time was
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reached. However, all surfactants investigated had a maximum coalescence time until no change
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with increasing surfactant concentration was observed.
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8,9
. The amount of surfactant adsorbed affects the
. Bournival et al.
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observed a direct relationship between
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High-speed cameras and laser technologies are major optical technique developments which
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have been used as tools to improve understanding of bubble coalescence dynamics. Thoroddsen
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et al.
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approximately 2.4 mm in average diameter using a high speed imaging technique with a capture
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rate of 1×106 frames/second (fps). The frame sequences taken allowed the authors to analyze and
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quantify the motion of the neck and bubble surface shape. More recently, Bournival et al. 11 used
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a high speed camera to study the coalescence dynamics of bubbles in non-ionic surfactant
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solutions. The authors generated two bubbles attached to the tips of a pair of fine capillaries to
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examine the relationship between coalescence time, defined as the time interval from the
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moment at which the bubbles are in contact to the rupture of the liquid film separating the two
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investigated the motion of the neck during the merging of two air bubbles of
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bubbles, and surfactant concentration. Coalescence and oscillation of the entire bubble were
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captured and analyzed by tracking the newly formed bubble’s change in projected surface area
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over time. A direct relationship between increasing surfactant concentration and coalescence
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time was observed and the post-rupture dampening of the oscillation of the newly formed bubble
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increased with surfactant concentration. The study suggested that surface elasticity partly
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dampened the oscillation of the newly formed bubble while other factors like surface viscosity
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may contribute to the dissipation of energy. The experimental work provided an analysis of the
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surface area of the oscillated bubble in two dimensions, quantifying the expansion and
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contraction of the coalesced bubble as a projected area. However, there are still questions on how
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fluid dynamics properties, such as flow velocity and dynamic pressure, control the bubble
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coalescence and post-rupture oscillation. The objective of the paper is to explain bubble
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coalescence dynamics by characterizing the flow velocity field vectors and local dynamic
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pressure inside the bubble and in the surrounding liquid across the bubble interface using
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computer simulation.
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Several numerical investigations have used computational fluid dynamics (CFD) as a tool for 13
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studying multiphase flows
. Most of these studies use multiphase methods such as volume of
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fluid (VOF) method 14-22, front tracking method 23,24 and level set method 25-27. The VOF method,
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first introduced by Hirt and Nichols
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volume of each fluid phase in a specific computational cell. Using this method, Tomiyama et al.
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14
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rising in a laminar flow. The computational results were in reasonable agreement with the
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experimental data suggesting that VOF may predict the qualitative and quantitative behavior of
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bubble motion under different conditions. Nevertheless, further development of the numerical
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, tracks the liquid-air interface by defining a fractional
investigated the effect of fluid properties and surface tension on the motion of a single bubble
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model to consider a larger computational domain and a three-dimensional model is required for a
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precise quantitative prediction of the bubble rising motion.
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The study of coalescence dynamics of a bubble pair rising freely in a liquid phase using the 15,16,18
18
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VOF method has been well documented
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dimensional model of two co-axial free bubbles rising in a liquid phase and used the VOF
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method to track the motion, shape, and velocity of the two bubbles during the interaction and
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coalescence processes. Although the two-dimensional numerical study did not fully represent the
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experimental conditions, the findings corroborated the experimental data available in the
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literature such as Chen et al. 15. The study of Hasan and Zakaria 18 showed that the VOF method
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may be considered an accurate tool for modeling bubble coalescence time and the subsequent
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oscillation behavior. Some authors have carried out qualitative analysis of the interaction of a
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pair of bubbles rising side by side
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investigate the effect of approach and rise velocity on bubble coalescence, considering liquid
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film drainage before the film rupture.
17,19,29,30
. Hasan and Zakaria
developed a two-
. Such computational studies were intended only to
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These studies show that no research has numerically investigated the oscillation following the
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merging of two captive bubbles. The present study provides insight into bubble coalescence
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dynamics by mapping the flow velocity vectors inside the bubble and in the surrounding liquid,
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and local dynamic pressure across the bubble interface. This objective was achieved by modeling
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two air bubbles fixed to adjacent capillaries in three dimensions using the VOF method. The
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developed model was validated by comparing the overall oscillatory motion of the bubbles with
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experimental data produced in a well-controlled system. The analysis provided further
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understanding of post-rupture bubble oscillation and the fluid dynamics properties regulating the
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coalescence process.
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2. METHODOLOGY
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2.1 Computational Solution Method
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The VOF method developed by Hirt and Nichols
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was used to simulate the coalescence of
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two air bubbles fixed to adjacent capillaries in water. The computational model was implemented
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in the commercial code ANSYS-Fluent v 17.2.
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2.1.1 Governing equations
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In the VOF method, both phases were assumed to be insoluble and all computational cells in
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the domain were either occupied by the water or air phase or a combination of both 28. The water
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phase was defined as the primary phase and the air phase as the secondary phase. A VOF
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function F was defined to track the fractional volume of a particular fluid in any specific cell of
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the computational domain at time t. If the cell was completely empty of air, F = 0; if the cell was
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completely full of air phase, F = 1; and if the cell contained the interface between the water and
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air phase, 0 < F < 1. Assuming that the mass of the fluid was preserved, the VOF method tracked
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the water-air interface by solving a continuity equation for the secondary phase (air) given by: ∇·V=0
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1
where V is the velocity vector of the air phase in the entire domain. A momentum equation was solved for both water and air phases by the following equation: ∂ρV V ∇ρV V∙V V - ∇p ρg g ∇∙μ ∇V V∇V VT F Fσv ∂t
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where ρ and t represent the density and time; p, g, µ, Fσv, are pressure, gravity force, viscosity
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and surface tension force per unit volume, respectively; and T is the matrix transpose operator.
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The fluid properties, ρ and µ, were determined by the properties of each phase as:
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ρ= Fair ρair +(1-Fair )ρwater
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µ= Fair µair +(1-Fair )µwater
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The surface tension force per unit volume (Fσv) was calculated by adopting the continuous surface force model developed by Brackbill et al. 31 and assuming a constant surface tension: Fσv = σkn
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5
In eq 5, σ is the surface tension, k is the surface curvature and n is the normal vector to the interface. The surface curvature of the interface, k, was calculated by: k=
1 n ∇ |n|-∇n |n| |n|
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The normal vector, n, was evaluated by the following equation, which took into account the
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three-phase contact perimeter formed between the bubble interface (i.e. water and air) and the
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capillary tube, known as the wall adhesion effect: n= nwall cos θeq +twall sin θeq
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where nwall and twall are the unit vectors normal and tangent to the capillary tube respectively, and θeq is the equilibrium contact angle between the water-air interface and the capillary tube.
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2.1.2 Geometry and mesh design
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A rectangular structure of dimensions 18×30×8 mm with two cylindrical capillaries of 20 mm
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long was created in ANSYS-Design modeler. The geometry used was sufficiently large with the
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ratio of the average diameter of the two initial bubbles (Db,ave) to the domain diameter being less
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than about 0.25 to eliminate any wall effects on the bubble coalescence dynamics 32. A schematic
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representation of the geometry used in ANSYS-Fluent is presented in Figure 1. The two vertical
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capillaries were chosen to have an outer diameter of 1.07 mm and inner diameter of 0.69 mm to
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replicate the experimental conditions of Bournival et al.
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offered a well-controlled experimental system, where the neck expansion and the bulging of the
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extremities of bubbles were not as constrained by the capillary tubes as other configurations.
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This capillary arrangement also allowed the (almost free) surface motion of the newly formed
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bubble, minimizing the effect of the capillary tubes on the horizontal expansion of the coalesced
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bubbles. Practically, the vertical capillaries effectively allowed the coating of the bubbles with
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hydrophobic particles. Thus, that configuration offered practical advantages in a stirred system
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and for the packing of particles on the surface of the bubbles 33-36.
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, which for experimental purposes
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The domain was discretized in ANSYS-Meshing using a multizone mesh method to generate
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hexahedral meshes. A local mesh refinement was implemented by increasing the number of cells
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per average diameter of the bubbles to improve the resolution of the interface curvature around
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the bubbles and reduce the computational error of the interface tracking. The mesh refinement
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was created in ANSYS-Fluent by marking a hexahedron with dimensions of 2.5Db,ave × 2.5Db,ave
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× 2.5Db,ave, as depicted in Figure 2a. A case with 23 cells per 2 mm was compared against two
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cases with mesh refinement of 46 and 92 cells per 2 mm. The mesh independence solution was
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evaluated by extracting the mean velocity profiles at the cross section plane. A small difference
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of 4.1% was observed in the mean velocity for the finest mesh (92 cells per 2 mm) with respect
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to the case with no mesh refinement (23 cells per 2 mm), indicating a mesh independent solution.
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Consequently, for computational efficiency and numerical accuracy, a mesh density of 92 cells
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per 2 mm was selected for all subsequent analysis in this study. A high level of grid quality
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throughout the domain was achieved where the aspect ratio of the cells was below 5.2 and the
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skewness was 0.51 37.
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Isometric view
Lateral view
Top view, capillary inner diameter (ID) and outer diameter (OD)
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Figure 1. Schematic representation of rectangular structure with two cylindrical capillary tubes
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for the 3D computational simulations.
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Time-step sizes below 1.5×10-6 s were selected to satisfy a courant number (CN= |V|∆t/∆X)
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below 0.6. This dimensionless number quantifies the velocity magnitude at a specific cell and the
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ratio of the time-step size to the element size. To compare the computational simulations with the
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experimental results, a simulation time of 0.025 s was chosen.
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a
b
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Figure 2. (a) Cross section view of the region prior to mesh refinement and after the first and
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second mesh refinement. (b) Boundary conditions used in the computational domain.
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2.2 Computational Parameters in ANSYS-Fluent
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Two air bubbles, each with 2 mm diameter, were patched into the domain and placed at the
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tips of the capillary tubes. The other ends of the capillary tubes were closed to the atmosphere.
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The immersion depth of the capillary tubes in water was 20 mm and the air phase was patched
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into the capillary tubes. A contact angle (θeq) of 160° between the capillary tube and the water-
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air interface was defined. It was measured inside the air phase and calculated from the
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experimental images by using an image processing technique (ImageJ). The rectangular structure
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was open to the atmosphere by a pressure outlet set at the top of the rectangular structure as
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shown in Figure 2b. Both water and air phases were assumed to be Newtonian fluids and
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incompressible flow, which, by definition, assumed the speed of the flow was significantly lower
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than the speed of sound
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that these assumptions are valid for simulating the motion of coalesced bubbles
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flow was considered laminar and unsteady. The physical properties for the water and air phases
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specified in the computational simulations are shown in Table 1.
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Table 1. Physical properties of water and air phases used in the computational simulations.
38
. Previous computational studies on bubble coalescence have shown
Physical property
Water phase
Fluid density (ρ), kg m-3
998.2
Dynamic viscosity (µ), mPa s
1.002 39
Surface tension (σ), mN m-1
72.8 11
Temperature (T), °C
20 11
39
18,27,29,30
. The
Air phase 1.204 39 1.805×10-2 39
20 11
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2.2.1 Solution methods
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A pressure-base solver and a time dependent solution were chosen as a solver approach for the
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model. The pressure implicit with splitting operator (PISO) was applied as the pressure velocity
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coupling in eq 2. A geo-reconstruction scheme was selected for tracking the water-air interface
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as it is the most precise reconstruction scheme available in ANSYS-Fluent
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based on a piecewise linear approximation of the water-air interface at a specific cell by a plane
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in three dimensions 41. All the parameters used in ANSYS-Fluent are shown in Table 2.
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Table 2. Parameters used in ANSYS-Fluent.
Parameter
Setting
Solver
Pressure-base
40
. This scheme is
Transient Solution method
Pressure velocity coupling: PISO
Discretization
Momentum: second order upwind Volume fraction: geo-reconstruction
197 198
2.2.2 VOF computational cases
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The liquid film thickness between two bubbles at the rupture could not be measured in the
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experimental case. Thus, two computational cases with initial separation distances (h) between
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the two bubbles of 0.05 mm and 0.02 mm were generated. These separation distances were in the
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range of 0.1 mm, which is the film thickness of the initial liquid drainage estimated by
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Kirkpatrick and Lockett 42, and 10-5 mm which is the critical film thickness in water 43. It was not
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possible to simulate smaller separation distances, as the cell size was 0.02 mm.
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2.3 Experimental data
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The experimental work of Bournival et al. 11 was used for validating the computational results
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of the present study. These authors studied the coalescence dynamics of capillary bubbles in non-
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ionic surfactant solutions by evaluating the oscillation of the resultant bubble. They generated
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two bubbles attached to the tips of a pair of fine, vertical capillaries. The two air bubbles were
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brought together in a controlled environment to allow coalescence using an electronic linear
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actuator (T-LA28A, Zaber Technologies Inc.). A resolution time of 3.33 ms, which was the time
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taken for the two bubbles to coalesce, was set for all experiments to evaluate cleanliness of the
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system. The bubbles were left ageing for 90 s in Milli-Q water (resistivity of 18.2 MΩ cm)
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before being brought in contact. The system was deemed cleaned if the bubbles coalesced within
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one frame interval (i.e. between two frames). The details of the cleaning procedure may be found
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in Bournival et al.
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was maintained during all experiments. Five independent trials were run for replicating the same
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experimental conditions. The authors used a high speed camera (Phantom 5, Vision Research
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Inc., USA) to study the coalescence dynamics of the bubbles with a capture rate of 6024 frames
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per second (fps). The analysis of the oscillation was performed as described in section 3.1.2.
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3. RESULTS AND DISCUSSION
11
. All the experiments were conducted at a temperature of 20±2 °C, which
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3.1 Analysis of bubble coalescence dynamics
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Two capillary-held bubbles were simulated using the VOF method in ANSYS-Fluent to 11
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evaluate the bubble coalescence dynamics, similar to Bournival et al.
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computational study were firstly used to evaluate the effect of the initial separation distance
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between the two bubbles on the neck expansion by calculating and comparing the Weber (We),
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Reynold (Re) and Ohnesorge (Oh) numbers for both computational cases and the experimental
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trial. The post-rupture oscillation behavior of coalesced bubbles was assessed by measuring the
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projected surface area of the coalesced bubbles and comparing it to the experimental case.
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3.1.1 Neck expansion during coalescence
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The neck expansion during coalescence can be characterized by the Weber (We), Reynold (Re)
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and Ohnesorge (Oh) numbers. The We number quantifies the relative magnitude of the inertial
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force with the surface tension, as follows: Inertial force ρwater Vr 2 λ We Surface tension force σ
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The Weber number takes into account the velocity of the neck radius (Vr), the density of the
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water phase (ρwater), the horizontal width of the two coalesced bubbles (λ), which is determined
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by the neck radius (r), and the surface tension (σ).
237 238
The Reynold number (Re) describes the ratio of inertial force to viscous force and it is given by the following equation: Re
Inertial force ρwater V+ λ Viscous force μ,-./+
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where µwater is the dynamic viscosity of the water phase. The importance of the surface tension,
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viscous and inertial forces on the motion of the neck expansion can be measured by Ohnesorge
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number (Oh), defined as: Oh
μ,-./+ √We Re 3ρwater σλ
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Frames of the VOF computational cases and experimental trial were produced and then
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analyzed using ImageJ. The neck expansion of the two coalesced bubbles is illustrated in Figure
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3a. The neck grows rapidly in both vertical directions until a maximum neck radius is achieved,
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at 2.0 ms. Figure 3b illustrates the contour of the bubbles prior to coalescence compared with the
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bubble shape at t = 0.8 ms. The initial contact point between the two bubbles was established as
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the origin of the Cartesian coordinates system. The neck radius (r) was measured from the initial
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contact in a downward vertical direction to eliminate the neck growth restriction due to the
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capillaries, as shown in Figure 3b. The initial shape of the two bubbles at t = 0 ms was used to
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measure the horizontal width of the coalesced bubbles (λ). a
b
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Figure 3. (a) the neck expansion contours as a function of time and (b) comparison of the shape
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of the two bubbles (black continuous line) prior to coalescence with the bubble shape at t = 0.8
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ms (blue dashed line), for the experimental trial conducted by Bournival et al. 11.
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The motion of the neck radius (r) for two liquid drops in an inviscid system has been described
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as a power-law dependence on time, r ∝ tA, where A is the power exponent
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bubble coalescence have also reported that the growth of the neck radius follows a power-law
257
dependence on time during the early stage of coalescence 2,12,47.
44-46
. Studies on
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In this study, the neck radius (r) was normalized by the average radius of the two initial
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bubbles (Rb,ave=(Rb1+Rb2)/2). The time (t) was normalized by the capillary-inertial time
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(τ=3ρwater Rb,ave3 / σ) which compared the surface tension force per unit area (σ/Rb,ave) with the
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inertial force per unit area 47. A prefactor, C, was calculated through the best fit of the data to the
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power-law scaling: t A = C Rb,ave τ r
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The growth of the neck for the experimental and simulated data was fitted to eq 11 and is
264
plotted in Figure 4. All three cases followed the power-law scaling with exponents in a range
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between 0.40 and 0.55 for r/Rb,ave < 0.85. The observed initial neck radius in all three cases
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deviated slightly, leading to different power-law exponents. However, these exponents were
267
relatively close to 0.5, which is the theoretical power exponent suggested by Eggers et al.
268
bubbles in water. Duchemin et al. 46 pointed out that the growth of the neck radius resulting from
269
the coalescence of two identical drops in an inviscid flow had a power-law exponent of 0.5. The
270
experimental and computational results were consistent with the experiments of Thoroddsen et
271
al. 12 who also showed an overall agreement with a power-law exponent of 0.5 for the expansion
272
of the neck radius of two bubbles in ethyl alcohol. The prefactors C in the present study were
273
slightly smaller, i.e., 1.10 – 1.25, than the prefactor observed by Thoroddsen et al. 12 who found a
274
value of 1.39 ±0.03. Therefore, the results presented in Figure 4 not only represented well the
275
motion of the neck radius as a power-law dependence but were also in good agreement with the
276
other theoretical and experimental findings.
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for
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1.0 0.8
r/Rb,ave
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0.6 0.4 0.2 0.0 0.0
0.1
277
0.2
t/τ
0.3
0.4
0.5
278
Figure 4. Neck radius (r) normalized by average radius of the two initial bubbles (Rb,ave) as a
279
function of t/τ for (solid red triangle) experimental trial with A=0.55 and C=1.10, (solid green
280
square) VOF computational case with h = 0.02 mm, A=0.40 and C=1.23, and (solid blue circle)
281
VOF computational case with h = 0.05 mm, A= 0.46 and C=1.25. The solid lines show the best
282
fit to eq 11. The initial radii of the bubbles for the experimental trial were Rb1 = 0.971 mm and
283
Rb2 = 0.982 mm. Both computational cases had two bubbles of equal radii of 1 mm.
284
Figure 5 illustrates the horizontal width of the two coalesced bubbles as a function of the
285
normalized neck radius for all three cases. Although the initial horizontal width observed in the
286
computational case with an initial separation distance of 0.05 mm was smaller than that for the
287
computational case with a separation distance of 0.02 mm, the two cases had nearly the same
288
horizontal width-neck radius curves. Based on this result, the computational case with h = 0.05
289
mm was selected for further analysis.
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1.2 1.0 0.8
λ (mm)
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0.6 0.4 0.2 0.0 0.0
290
0.2
0.4
0.6
0.8
1.0
r/Rb,ave
291
Figure 5. The horizontal width of the two coalesced bubbles (λ) as function of r/Rb,ave for (solid
292
red triangle) experimental trial, (solid green square) VOF computational case with h = 0.02 mm
293
and (solid blue circle) VOF computational case with h = 0.05 mm.
294
Snapshots of the experimental trial and contours of air volume fraction for the VOF case with
295
h = 0.05 mm are presented in Figure 6. Three representative r/ Rb,ave values were selected for
296
comparison. As the neck radius grew, the horizontal width between the coalesced bubbles
297
increased and therefore more surrounding liquid was pushed outwards. This motion created a
298
surface wave across the newly formed bubble surface. The VOF case captured the bubble surface
299
deformation very well as illustrated in Figure 6. However, the experiments had wider horizontal
300
widths than the computational case as shown graphically in Figure 5. The deviation could be
301
expected due to the piecewise linear scheme used for calculating the interface, which may have
302
compromised the reconstruction of the sharp edges during the neck expansion 48.
303 304
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a
b r/Rb,ave = 0.39
r/Rb,ave = 0.65
r/Rb,ave = 0.76
305
Figure 6. Comparison of the neck radius for (a) experimental trial conducted by Bournival et al.
306
11
307
the bubbles before coalescence while the green arrows indicate the horizontal width between the
308
two coalesced bubbles. The capillary tubes are 1.07 mm in diameter and act as a scale.
and (b) computational case with h = 0.05 mm. The dashed red lines represent the contour of
309
The neck growth velocity during the early stages of bubble coalescence was calculated from
310
the derivative of the neck expansion equation, Vr = dr/dt, and used to calculate the Weber and
311
Reynold numbers (eqs 8 and 9). The time at which the rupture of thin liquid film occurred was
312
chosen as time zero. For both computational cases and the experimental trial, the Weber number
313
was compared with the Reynold number (Figure 7). It is important to note that the experiments
314
had a time resolution of 0.2 ms from the rupture of the thin liquid film. For the experimental
315
data, the motion of the neck in the early stages of expansion was not fully counter-balanced by
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316
the inertial force, although the value was close to unity as stressed by the Reynold number
317
(Re≤0.24). This finding is corroborated by the results of Stover et al. 2 who found that the motion
318
of the resultant bubbles is the result of the reduction in total surface area, which causes a
319
decrease in the surface energy of the system. As such the motion of the interface is driven by the
320
surface tension, which determines the amount of surface energy to be released. The excess
321
energy is passed to the surrounding liquid where it is opposed by the viscosity and the inertia of
322
the liquid.
323
On the other hand, the Weber number for the computational cases differed slightly from the
324
experimental results, showing the inertial force as superior in magnitude to the driving force (i.e.
325
surface tension) followed by surface tension for longer times although all results are close to
326
unity. Since the motion is driven by the surface tension, the small discrepancy could be attributed
327
to the poor capability of VOF to simulate the initial complex phenomenon of the liquid film
328
rupture. Previous numerical studies have pointed out the limitation of VOF to accurately
329
calculate the local curvature near the interface, which is then used for calculating the surface
330
tension force per unit volume 49,50. Moreover, water, in the absence of any surfactant, could have
331
small variations in surface tension
332
tension in the computational cases could also lead to small deviation in the results specifically at
333
the early stage of the neck expansion.
51
, which means that the assumption of a constant surface
334
Figure 8 shows that the Ohnesorge number was between 15 and 3. It indicates that the neck
335
motion driven by surface tension was initially restrained by a greater magnitude of the fluid
336
viscosity than the inertial force, followed by an exponential decay of viscous force to inertial
337
force ratio. Therefore, it appears that the simulation of the coalescence dynamics by VOF is
338
relatively accurate since the Weber numbers are relatively close to unity and the Ohnesorge
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21
numbers are consistent. However, the early stage of coalescence (first 1 ms) represented by the
340
initial expansion of the neck may lack in computational accuracy.
We
339
1.6
1.6
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 0.0
0.5
1.0
1.5
2.0
Re
2.5
t (ms)
341 342
Figure 7. Comparison of (closed markers) Weber number and (open markers) Reynold number
343
as function of time for (red triangle) experimental trial, (green square) VOF computational case
344
with h = 0.02 mm and (blue circle) VOF computational case with h = 0.05 mm.
16.0 14.0 12.0 10.0 Oh
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8.0 Viscous force
6.0 4.0 2.0 0.0 0.0 345
0.5
1.0
1.5 t (ms)
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346
Figure 8. Ohnesorge number as function of time for (solid red triangle) experimental trial, (solid
347
green square) VOF computational case with h = 0.02 mm and (solid blue circle) VOF
348
computational case with h = 0.05 mm.
349 350
3.1.2 Oscillation behavior of coalesced bubbles
351
After obtaining the VOF results, the oscillating behavior was quantified following the 52
352
approach which has its basis in the equation proposed by Schulze
353
approximated the surface oscillation of a coalescing bubble using the following equation: A' A0 ' × e-δt sinω0 tφ B
. Bournival et al.
11
12
354
where A' and A0' denote the normalized relative projected area and initial amplitude,
355
respectively, δ is the damping constant in ms-1, φ is phase shift, t is time in ms, ω0 is the angular
356
frequency in ms-1 and B is an integration constant.
357
It is important to state that the normalized relative projected area A' in eq 12 (which was
358
calculated by the instantaneous projected area minus the average projected area towards an
359
infinite time divided by the average projected area towards an infinite time) is characterized by
360
five parameters, with three highly significant parameters. The initial amplitude describes the
361
magnitude of the deviation of the initial projected area of the bubble from the final equilibrium
362
state. The damping constant, which depends on the fluid and interfacial properties, shows if the
363
oscillation is slowly (low damping constant) or rapidly dampening (high damping constant). The
364
angular frequency gives an indication of how fast the oscillation completes one period and is
365
important for regulating the violent nature of the process at extreme points (peaks and troughs)
366
of the oscillation.
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367
The phase shift describes only the starting position of the bubble’s oscillation. It is not a very
368
critical parameter in this study and therefore not meaningful in comparing the simulations and
369
the experimental data. The integration constant is generally very close to zero and therefore not
370
relevant in the present analysis.
371
The calculation assumed equilibrium surface tension at the start of the simulation, which did
372
not change after coalescence. As such, no surface tension gradient caused by a change in surface
373
area was generated, limiting any Gibbs elasticity effect although relaxation may in fact occur 51.
374
High-frame-rate animations of the VOF numerical cases were produced and then analyzed
375
using ImageJ to track the changes in the projected surface area over time. The five parameters in
376
eq 12 were fitted using the solver command in Microsoft Excel and minimizing the sum of the
377
square error. An oscillation period of 1.5 was selected for fitting the simulation data to eq 12. A
378
previous study determined that 1.5 – 2 periods are sufficient to detect small deviations for the
379
damping constant and angular frequency 53.
380
The projected area curves for both computational case and experimental trials are illustrated in
381
Figure 9. In the VOF case, the average projected surface area towards an infinite time was
382
calculated by assuming that the total volume of the bubbles was preserved during the post-
383
rupture oscillation. From these two curves, it can be seen that the computational case followed
384
the oscillatory motion of the bubble after coalescence. The VOF results showed a higher initial
385
amplitude peak of the bubble oscillation compared to the experimental trial and indicated that the
386
coalesced bubbles in the VOF case were initially further from the final equilibrium state at which
387
a stable shape was reached. In the experiments, the damping surface oscillation was rapid
388
suggesting that the coalesced bubbles expanded and contracted quickly. The VOF computational
389
case showed a lower resistance of the bubble surface on the neck expansion after the film
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390
rupture, leading to a lower damping constant (error of 13.3 %) and slower angular frequency
391
(error of 18.6 %) as shown in Table 3. 0.4
Normalized relative projected area
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0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 0
5
10
15
20
25
t (ms)
392 393
Figure 9. Comparison of modeled data (eq 12) for (solid red triangle) experimental data and
394
(solid blue circle) VOF computational case with h = 0.05 mm.
395
Table 3. Calculated parameters from eq 12 for experimental and VOF case with an initial
396
separation distance of 0.05 mm.
Parameter
Experimentsa
VOF case
Initial amplitude, A0'
0.259
0.346
Damping constant, δ (ms-1)
0.060
0.052
Angular frequency, ω0 (ms-1)
0.586
0.477
Phase shift, φ
0.938
0.798
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Integration constant, B 397
a
1.5×10-3
5.8×10-3
Obtained from averaging 5 trials in Bournival et al. 11
398 399
3.2 Flow dynamic analysis
400
Flow dynamic analysis provides an insight into the coalescence dynamic experienced by two
401
bubbles during the first cycle of post-rupture oscillation. Analysis of velocity vectors and
402
dynamic pressure gradients are presented below.
403
3.2.1 Velocity field analysis
404
Simulation snapshots of velocity vectors predicted at a vertical plane along the central axis of
405
the capillary tubes are illustrated in Figure 10a. The motion of the interface was captured in the
406
change of the projected surface area over time. Nine points in time during the initial stage of
407
bubble coalescence were chosen. As can be seen in Figure 10a, the velocity vectors indicate the
408
direction and the velocity magnitude of the airflow inside the coalesced bubbles and the
409
surrounding liquid. After the rupture of the liquid film separating the two bubbles, the air phase
410
flowed towards the bubbles’ initial contact point and then it moved in the vertical opposite
411
direction, as shown during the first 1.55 ms
412
producing the neck expansion. Small eddies with velocity magnitudes between 0.25 m/s and 0.75
413
m/s were observed at the top and bottom part of the neck as the curvature of the bubble surface
414
decreased. The neck expansion generated a surface wave that propagated horizontally in the
415
opposite direction, creating bulges at the edges of the bubble, as illustrated at 3.16 ms. The air
416
phase in the bulging area moved at much higher velocity, in the range of 1.75 m/s to 2.5 m/s,
417
than the rest of the bubble surface. The bubble contracted vertically as the surface wave recoiled
418
and the air phase flowed perpendicular towards the top of the neck creating eddies with small
54
. This airflow pushed the surrounding liquid,
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419
velocity magnitude at the bottom of the neck and leaving a stagnant void in the center of the
420
bubbles. It can be seen from 4.76 ms to 5.57 ms that the air phase near the bubble interface–
421
capillary tube flowed towards the top of the neck as the bubble continued to vertically contract. a
b
t = 0.14 ms
t = 0.74 ms
t = 1.55 ms
t = 2.35 ms
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t = 2.96 ms
t = 3.16 ms
t = 3.96 ms
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t = 4.76 ms
t = 5.57 ms
422
Figure 10. (a) Velocity vectors colored by velocity magnitude and (b) contour of dynamic
423
pressure gradients for VOF computational case with h = 0.05 mm. A logarithmic scale was
424
selected for the color map of the dynamic pressure contours. The capillary tubes are 1.07 mm
425
outer diameter and 0.69 mm inner diameter as indicated by the black contour lines. The
426
capillaries act as a scale.
427
3.2.2 Dynamic pressure gradient analysis
428
In addition to the velocity field analysis, the dynamic pressure gradients of fluid were also
429
assessed in ANSYS-Fluent to quantify the pressure associated with the difference of velocities at
430
the bubble interface. Dynamic pressure (pd=0.5×ρ|V|2), which represents the difference between
431
the total and static pressure in the system, is proportional to the density and square of the velocity
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432
vector magnitude of the fluid. Snapshots of dynamic pressure contours along a cross section
433
plane during the first 5.57 ms are shown in Figure 10b. The dynamic pressure scale was set
434
between 50 and 800 Pa for better comparison, and the white color in the snapshots represents
435
values in the range of 0 to 50 Pa. It is evident from the snapshot contours that the neck expansion
436
created dynamic pressure gradients at the interface where the motion was constrained by the
437
surrounding liquid. There were contours with higher dynamic pressure gradients in the bubble
438
interface, where the air phase flowed with high velocity, creating eddies as shown in Figure 10a.
439
Consequently, vertical bubble contraction was rapidly driven by the high-pressure gradients
440
around the bulged area that pushed the air phase inward, as shown at 3.16 ms. The high dynamic
441
pressure could be useful in understanding the detachment of particles when two bubbles coated
442
with hydrophobic particles coalesce. Bournival et al. 35 studied the coalescence dynamics of two
443
bubbles coated with silica particles in water.
444
Some snapshots of the experimental study were compared against the VOF computational case
445
to obtain an insight into the velocity fields and dynamic pressure gradients. The comparison was
446
based on the oscillation time after coalescence. The selected time of the surface contours for the
447
two uncoated bubbles presented in Figure 11 differed slightly from the coated bubbles. Apart
448
from the numerical error observed in the VOF results, one reason is the particle layer around the
449
loaded region, which may restrict the bubble surface deformation
450
detachment of a large number of particles from the bubble surface as the bulging area contracted.
451
At 3.36 ms, the bulged area moved at much higher velocity magnitude, in the range of 1.75 m/s
452
to 2.5 m/s, than the rest of the bubble surface. Therefore, a dynamic pressure greater than 800 Pa
453
is observed at the bulging area. The vertical contraction also created a difference in dynamic
454
pressure at the interface, which may explain the results by Bournival et al.
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. Figure 11 illustrates the
36
, who pointed out
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455
that the difference in pressure force between the air and the surrounding liquid exerts a force on
456
the particles, leading to particle detachment. Therefore, it could be argued that the pressure
457
gradients generated during the post-rupture oscillation of the coalesced bubble affect the stability
458
of particles attached at the interface. However, other factors such as particle size, density and
459
hydrophobicity may also help quantify the detachment of particles. t = 2.67 ms
t = 2.96 ms
t = 3.0 ms
t = 3.36 ms
Dynamic pressure > 800 Pa
t = 3.17 ms
t = 3.56 ms
t = 4.0 ms
t = 4.16 ms
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460
Figure 11. Velocity vectors and contours of dynamic pressure gradients for coalescence
461
dynamics of two uncoated bubbles, using the VOF computational case with h = 0.05 mm,
462
compared with an experimental study of bubble coalescence dynamics of two bubbles coated
463
with hydrophobized silica particles carried out by Bournival et al. 35. The capillary tubes may be
464
used to scale the images where capillary tubes of 1.04 mm in diameter were used in the
465
experiments and 0.69 mm inner diameter as indicated by the black contour lines on the
466
simulation.
467
4. CONCLUSIONS
468
The flow field arising from the merging of two equal-size bubbles was numerically
469
investigated for two separation distances using a finite volume methodology and was validated
470
against experimental data. The use of the VOF method allowed the detailed investigation of the
471
flow field and pressure gradients in the interface regions of the coalescing bubble. The results
472
indicated that there was a small deviation of the neck expansion motion showing a faster neck
473
growth at the smaller separation distance.
474
The dynamics of the coalesced bubbles were studied by evaluating the velocity fields and the
475
dynamic pressure gradients. The velocity vector showed that as the neck grew the surrounding
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476
liquid was pushed outwards. The chaotic motion created a surface wave across the newly formed
477
bubble surface, which propagated in the opposite direction towards the edges of the bubble. The
478
neck growth created high dynamic pressure gradients at the top and bottom of the bubble
479
interface, where the air phase flowed with higher velocity than the surrounding liquid, which
480
created eddies. Overall, it was found that the computational case followed the oscillatory motion
481
very well, as observed in experiments, but with lower damping constant and lower angular
482
frequency. Further studies of the velocity fields and pressure gradients during the merging of two
483
bubbles with different types of surfactants are needed. These studies should be complemented by
484
investigation of the changes in surface energy during post-rupture oscillation to further elucidate
485
the oscillation mechanism. The distribution of surfactants along the interface during bubble
486
coalescence should also be studied numerically. The variation in the distribution of surfactant
487
may be resolved by using a non-constant surface tension force, which would take into account
488
surface elasticity. A similar approach may be needed to reconcile the initial growth of the neck,
489
which computer simulations considered to be inertia driven as opposed to surface tension driven.
490
Overall, the VOF method successfully tracked the interface motion of the water-air phases in the
491
current system.
492
SUPPORTING INFORMATION
493
Supplementary video of the VOF simulation with an initial separation distance of 0.05 mm can
494
be found in the online version.
495
ACKNOWLEDGEMENT
496
The authors wish to thank the following people: Mr Collin Turner of McGill University,
497
Canada for his contribution to the settings of the preliminary VOF computational case; Dr
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498
Yuqing Feng of CSIRO Mineral Resources (Australia) for helpful discussion on the model
499
settings and validation; and Dr Francisco Trujillo of the School of Chemical Engineering,
500
University of New South Wales (Australia) for facilitating access to a computer for running
501
preliminary models and vital discussion on the model settings. Mrs Yesenia Saavedra Moreno
502
would also like to thank the Australian government and University of New South Wales
503
(Australia) for financial support through the Australian Government Research Training Program
504
(RTP) scholarship.
505
ABBREVIATIONS 3D
three-dimensional
A
power exponent
A0'
initial amplitude
A'
normalized projected area
B
integration constant
C
prefactor
CFD
computational fluid dynamics
CN
courant number
Db,ave
average diameter of the two initial bubbles, mm
F
function of volume fraction
Fair
volume fraction of air phase
Fσv
surface tension force per unit volume, N m-3
g
gravity, m s-2
h
initial separation distance between the two bubbles, mm
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n
normal vector to the interface
nwall
vector normal to the capillary tube
Oh
Ohnesorge number
p
pressure, Pa
pd
dynamic pressure, Pa
PISO
pressure implicit with splitting operator
r
neck radius, mm
Rb,ave
average radius of the two bubbles, mm
Rb1
radius of bubble 1, mm
Rb2
radius of bubble 2, mm
Re
Reynold number
t
time, ms
twall
vector tangent to the capillary tube
T
matrix transpose operator
T
temperature, °C
V
velocity vector
|V|
velocity vector magnitude, m s-1
Vr
velocity of the neck radius, m s-1
VOF
volume of fluid
We
Weber number
Greek letters ∆t
time-step size, s
∆X
element size, m
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δ
damping constant, ms-1
θeq
equilibrium contact angle, °
k
surface curvature of the interface
λ
horizontal width of the two coalesced bubbles, mm
µ
dynamic viscosity, mPa s
µair
dynamic viscosity of air phase, mPa s
µwater
dynamic viscosity of water phase, mPa s
ρ
density, kg m-3
ρair
density of air phase, kg m-3
ρwater
density of water phase, kg m-3
σ
surface tension, mN m-1
τ
capillary-inertial time, ms
φ
phase shift
ω0
angular frequency, ms-1
∇
gradient operator
506 507 508 509 510 511 512 513 514 515 516
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