Buoyancy-Driven Motion of Bubbles in the Presence of Soluble

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Thermodynamics, Transport, and Fluid Mechanics

Buoyancy-driven motion of bubbles in the presence of soluble surfactants in a Newtonian fluid Weihua Li, and Nivedita Gupta Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04788 • Publication Date (Web): 22 Jan 2019 Downloaded from http://pubs.acs.org on January 26, 2019

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Industrial & Engineering Chemistry Research

Buoyancy-driven motion of bubbles in the presence of soluble surfactants in a Newtonian fluid Weihua Li and Nivedita R. Gupta∗ Department of Chemical Engineering University of New Hampshire Durham, NH 03824

Abstract We present our experimental results for the effect of bulk-soluble surfactants on the buoyancy-driven motion of an air bubble rising in square and rectangular channels filled with an immiscible liquid. Glycerol and glycerol-water solutions were chosen as the bulk fluid and Triton X-100 and Aerosol-OT were used as surfactants. Bulk surfactant concentrations below as well as above the critical micelle concentration were considered. Even at very low bulk surfactant concentrations where the equilibrium surface tension did not change, bubbles rising through the surfactant solutions showed reduced mobility, especially for small bubbles. At larger surfactant concentrations, bubbles of all sizes showed lower terminal velocities. At a large enough bulk surfactant concentration, two peaks were seen in the velocity-volume curve. Increasing the bulk surfactant concentrations above the critical micelle concentration increased the mobility of the bubbles but still showed two peaks in the velocity-volume curve.

Keywords: Buoyancy-driven, Bubble motion, Surfactants, Marangoni stress, Critical Micelle Concentration

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1. Introduction

The buoyancy-driven motion of a long gas bubble through a vertical cylindrical tube in a Newtonian fluid has been studied theoretically, experimentally, and numerically for several decades1-12.

Several

experimental and numerical studies have considered bubble sizes that are comparable to the tube size in the low and intermediate Reynolds number regimes13-16. Studies of motion of bubbles in non-circular channels have gained momentum due to their relevance in the field of droplet microfluidics and as better model descriptions for porous media than circular capillaries17. Kolb and Cerro18-20 studied the pressure-driven motion of long bubbles in a capillary of square cross section and found that transition from a nonaxisymmetric to an axisymmetric bubble occurred at capillary number, Ca > 0.1. Liao and Zhao21 presented a theoretical model that predicted the drift velocity of a Taylor bubble in vertical mini triangular and square channels filled with a stagnant liquid. They found that the drift velocities in a triangular channel are substantially higher than those in a square channel with the same hydraulic diameter. Clanet et al.22 experimentally studied the buoyancy-driven motion of long bubbles in vertical tubes of different crosssections (rectangles, regular polygons, toroidal tubes) filled with Newtonian fluids and developed relationships between the rise velocity and the cross-section of the tube in the viscous and inertial limits. Taha and Cui23 used a volume-of-fluid method to study the rise of long bubbles in square channels and showed that bubbles develop a negative curvature at the rear stagnation point at large capillary numbers. Li et al.24 experimentally studied the buoyancy-driven motion of bubbles in square channels over a range of Reynolds numbers with bubble sizes comparable to the channel size. They found that bubbles rose faster and the maximum bubble width was higher in a square channel compared with those in a circular channel with the same hydraulic diameter. When the Weber number was large, bubbles showed a negative curvature at the rear and showed a maximum in the velocity-volume curve. Most of their experimental results were numerically confirmed by Amaya-Bower and Lee25 using a lattice-Boltzmann method. They further showed that the deformation of the bubbles depended on not only the bubble size and Weber number but also the Reynolds number.


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Surfactants are present in most multiphase fluid system as impurities or additives. For buoyancy-driven flows, surfactants adsorbed at the air-water interface advect along the interface and accumulate at the rear stagnation point26. The non-uniform surfactant distributions result in Marangoni stresses that affect the bubble shape and mobility. A few studies have considered the effect of surfactants on the buoyancy-driven motion of bubbles at low Reynolds number27,28 as well as at finite Reynolds number29-32. Almatroushi and Borhan27 conducted experiments on drops and bubbles rising in a vertical tube to show that the presence of surfactants retarded the motion of small bubbles due to Marangoni stresses but the retardation was not as strong for larger bubbles as they deformed more readily. Tasoglu et al.29 numerically studied the effect of soluble surfactants on the mobility of small bubbles with different deformations rising in a tube. The presence of surfactants retarded the mobility of the bubbles though the retardation was more significant for the less deformed spherical bubble compared to the ellipsoidal and dimpled ellipsoidal cap bubbles. Kurimoto et al.31 experimentally and numerically studied the terminal velocity of clean and contaminated drops rising in a vertical tube. They found that the changes in deformation and mobility of Taylor drops in the presence of surfactants became more significant when the drop viscosity was increased or the Bond number was reduced. Cui and Gupta32 numerically studied the effect of surfactants on buoyancy-driven motion of drops in a tube at intermediate Reynolds numbers. They also confirmed that Marangoni stresses slowed down drop velocity, especially for drop sizes that are comparable to the tube diameter.

At surfactant bulk concentrations above the critical micelle concentration (CMC), surfactant molecules self assemble to form aggregates or micelles in solution. When the bulk surfactant concentration is increased beyond CMC, the added surfactant increases the concentration of micelles while the monomer concentration remains the same33. If the equilibrium between the monomer concentration and the micelle concentration is disturbed, aggregates either break down or reassemble to return to the critical micellar concentration. Even above the CMC concentration, surfactants adsorb at the interface in monomer form. For a rising bubble, surfactants that accumulate at the rear get desorbed from the interface to the sublayer



and diffuse away to the bulk. In the bulk, the monomer - micelle equilibrium is disturbed and micelles form to restore the monomer concentration.

Stebe and coworkers26,34 showed that the interface can be

remobilized in the presence of surfactants if the desorption rate as well as the bulk diffusion of surfactants is fast. Then, the surface concentration of surfactant remains uniform and the interfacial mobility is restored. Kurimoto et al.31 considered silicone oil drops rising in glycerol-water solutions in the presence of surfactants well above the CMC concentrations and compared the experimental shapes with the numerical calculations. They found that the terminal velocities of the fully contaminated Taylor drops were higher than the clean drops and the difference increased with viscosity ratio and decreased with the Bond number.

In this study, we experimentally investigate buoyancy-driven motion of bubbles through vertical channels with circular, square, and rectangular cross sectional areas in the presence of surfactants at concentrations below and above the critical micelle concentration. The bubble sizes are typically comparable to the channel size and therefore, wall effects are significant.

2. Experimental

2.1 Experimental Setup The schematic of experimental setup, shown in Figure 1, consists of an acrylic channel positioned vertically in front of a monochrome CMOS digital video camera (PixeLink PL-A741 with a maximum speed of 27 frames per second at a resolution of 1280×1024). A fiberglass light source provides uniform backillumination to capture the images of the bubbles. A personal computer is used to control the camera and capture a video of the rising bubble. Vision Assistant software from National Instruments is used to obtain and analyze individual frames from the recorded movie. Acrylic plastic channels (TAP Plastics) with various cross sectional geometries were used to study the dynamics of bubbles over a large range of parameters and are shown in Table 1. The circular tubes were enclosed inside a 15 mm square channel filled with the corresponding suspending fluid to remove optical distortion. The length of each channel


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was at least sixty times the hydraulic diameter. The bottom end of each channel was sealed with an acrylic end fitted with an 11 mm diameter rubber septum (RESTEK) to serve as the injection port. 2.2 Materials and Chemicals The channels were filled with glycerol (Aldrich) or glycerol-water (GW) solutions in ultrapure water (Millipore Direct-Q 3 UV with resistivity of 18.2 MΩ•cm). Air was used as the drop fluid. We used a water soluble nonionic surfactant, Triton X-100 (Alfa Aesar, > 99%) and a water soluble ionic surfactant, Aerosol-OT (EM Science, > 99%) without further purification. All solutions were prepared on per weight basis.

Table 1: Channel geometries considered for studying dynamics of bubbles rising in a channel filled with an immiscible fluid. Number

Geometry

1

Circular

2

Square

3

Square

4

Rectangular

Dimensions

Hydraulic Diameter, DH

Diameter, D = 10 mm Length = 813 mm Width, h = 10 mm Length = 927 mm Width, h = 15 mm Length = 950 mm Width, w = 16 mm Depth, t = 3 mm Length = 1232 mm

10 mm 10 mm 15 mm 5.1 mm

2.3 Operating Procedure The suspending fluids were mixed for 24 hours in a covered beaker to prevent evaporation and were placed in a vacuum degasser to remove any air bubbles from the fluid before use. Before each experiment, the channel was pretreated by filling them with the experimental suspending fluid at a surfactant concentration well above the critical micellar concentration (CMC) for 24 hours. The channel was cleaned with warm tap water six times and Millipore-Q water six times with 3 hours in between each wash. The channel was then dried using the air supply in the lab. This ensured that surfactants did not adsorb/desorb to the channel wall while the experiment was conducted. The surface tension of the experimental solution was also tested before and after the experiment to ensure the pretreatment was effective.



Figure 1: Schematic of the experimental setup used to study the buoyancy-driven motion of bubbles in channels.

There was no external temperature control for this system and all of the experimental data presented in this paper was collected between 21 and 22 °C.

The experimental temperature was determined by a

thermocouple placed near the outlet regions of the channel. To conduct the experiment, the dried channel was filled slowly with the suspending fluid to minimize bubble regeneration, clamped to a stand, and leveled. The channel was placed in front of the camera and the system was allowed to equilibrate thermally before the experiments were conducted. For each run of experiments, the desired volume of the drop fluid was injected at the bottom of the channel using a micrometer syringe along the symmetry axis of the channel. The motion of the bubble through the channel was recorded using the camera. The camera was placed on a leveled tripod high enough from the injection point to capture the steady shape and velocity of the bubbles. The captured images were analyzed to determine the steady shape and speed of bubbles.


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2.4 Measurements and Analysis The schematic of the front and cross-sectional view of a bubble rising in a confined channel is shown in Figure 2. The density and viscosity of the bulk phase is designated as  and  while that of the bubble phase is represented by  b and b , respectively. The bubble moves with a steady velocity, U b along the axis of the channel. The equilibrium surface tension between the bulk phase and the drop phase is represented by  eq . The size of the bubble is characterized by the radius of a spherical bubble,

a   3Vb 4  of the same volume, Vb . In the cross-sectional view the bubble may be axisymmetric as 1/3

seen in Figures 2(a),(b) and (d) or non-axisymmetric as seen in Figures 2 (c) and (e). The surface tension as a function of time for a given bulk surfactant concentration was measured using a pendant drop method on an optical tensiometer (Attension Theta) till the surface tension reached an equilibrium value. The equilibrium surface tension values are reported in Table 2 and Figure 3. The bubble velocity, U b is chosen as the characteristic velocity scale to non-dimensionalize the system variables. The characteristic length scale, Rc , is chosen as half of the hydraulic diameter, DH such that Rc  R for a cylindrical tube of radius

R , Rc  h 2 for a square channel of side, h and Rc  wt  w  t  for a rectangular channel with width, w and depth, t. The dimensionless bubble size, viscosity ratio, and density ratio are defined as,   a Rc ,

  b  , and   b  , respectively. The dynamic parameters that affect the bubble shape and mobility in the channels are the Reynolds number, Re  U b Rc  , the capillary number, Ca  U b  eq , and the Bond number, Bo     b  gRc2  eq .



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Circular channel

Ud (a) σ

D

g Square channel

L 2a ρd

μd

(c) h

B μ

(b)

h ρ

Rectangular channel

D t h (d)

(e)

Figure 2. Schematic of the front view and the cross-sectional view of a bubble rising through a channel. This bubble may be axisymmetric as in (a), (c) and (e) or non-axisymmetric as in (b) and (d) in the crosssectional view.

The physical properties of the various two-phase systems used in the experiments are shown in Table 2. The viscosities of all the suspending fluids were measured using the Brookfield cone and plate viscometer (LVDV-III+ with Rheocalc software). The density,  , of the 75 wt%, 90 wt% glycerol-water, and pure glycerol suspending fluids was 1192 kg/m3 , 1223 kg/m3 , and 1265 kg/m3

respectively and the

viscosity,  , of the 75 wt%, 90 wt% glycerol-water, and pure glycerol suspending fluids was 33 mPa.s ,

204 mPa.s , and 968 mPa.s respectively. Air density and viscosity are assumed to be b  1.2 kg/m3 and b  0.02 mPa.s . Figure 3 shows the equilibrium surface tension,  eq as a function of the bulk surfactant concentration, C for Triton X-100 (TX-100) in 75 wt%, 90 wt% glycerol-water, and pure


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glycerol solutions and Aerosol-OT (AOT) in 75 wt% glycerol-water solution. The CMC values of TX-100 and Aerosol-OT in glycerol-water solutions occurred at ~ 450 ppm and ~ 3000 ppm, respectively. The various two-phase system experimental runs and the corresponding ranges of dimensionless numbers are listed in Table 2. In Table 2, the Weber number, We, is also presented for long bubbles rising in each of the two-phase systems. The Weber number is a measure of the inertial force as compared to the interfacial force and is defined as We  UT2 Rc /  eq , where U T is the steady velocity of a long bubble rising in the channel.

Figure 3: Surface tension of Aerosol-OT (AOT) and Triton-X100 (TX-100) as a function of varying bulk concentrations in glycerol and glycerol-water (GW) solutions.

3. Results and Discussion The experimental results for the motion of bubbles rising in Newtonian fluids in the absence and presence of surfactants are presented. For all the experiments conducted, data are reported only for the bubbles that were rising at the centerline of the channels. In all experiments, we calculate the bubble volume assuming that the shapes of the bubbles are axisymmetric as they rise through the channels. For each experimental



run, we conducted a steady state test on at least five bubbles ranging from the smallest to the largest to ensure that the deformations and velocities obtained were indeed steady. The dimensionless terminal velocity is defined as U  U d U c where U c     d  gRc2 3 is the characteristic buoyancy rise velocity. The edges of the images presented in this paper do not correspond to the channel wall. As shown in Figure 2, the bubble shape is characterized by the maximum axial bubble length, L, the maximum radial length, B. The deformation parameter is defined as   ( L  B) / ( L  B) . For prolate bubbles elongated in the flow direction,   0 while   0 corresponds to oblate bubbles elongated perpendicular to the flow direction. A spherical bubble with L  B results in   0 .

We will first consider bubble dynamics in surfactant solutions at concentrations below the CMC which indicates that all surfactant molecules were present in monomer form. Figure 4 shows the dimensionless terminal velocity as a function of dimensionless bubble size for bubbles rising in 75 wt% glycerol water solution with different concentrations of AOT in a 10 mm × 10 mm square channel. For comparison, the corresponding velocity-volume curve in the absence of surfactants is also shown on the same graph. As seen in Figure 4, the terminal velocity of small bubbles for all cases increases linearly with increasing bubble volume because of the increased buoyancy force. As the bubble size becomes comparable to the channel size, the drag force due to the confining walls increases resulting in a decrease in the terminal velocity. Beyond a critical bubble volume, the bubble velocity reaches a constant plateau value, UT, where the bubble velocity is independent of the bubble volume. In surfactant-contaminated systems, surfactants adsorb at the front end of the bubble, are transported along the interface due to surface convection and diffusion, and desorb at the rear end of the bubble. Adsorbed surfactants tend to accumulate near the stagnation point at the trailing pole as there is converging flow at this end. At steady state, a surface tension gradient is established along the interface due to the nonuniform distribution of surfactants. The surface tension is higher at the leading end and lower at the trailing end of the bubble. The interface pulls toward the high tension region at the front end of bubble, exerting a Marangoni stress along the interface and


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retarding the surface velocity of the bubble. This in turn lowers the terminal velocity of the rising bubble. The terminal velocity of small bubbles even in a 1 ppm surfactant solutions is lower than similar sized bubbles in surfactant-free solutions due to Marangoni effect. The terminal velocities for long bubbles does not seem to be affected by the presence of small amounts of surfactants. As the bulk concentration of AOT in the 75 wt% glycerol-water solution is increased to 10 ppm, the terminal velocity of small as well as large bubbles reduces. The dimensionless terminal velocity for small bubbles in 100 ppm AOT solution is lower than the corresponding bubbles in 0, 1, and 10 ppm AOT solutions as expected due to the increased Marangoni effect. The terminal velocity increases with bubble volume initially, reaches a peak at  ~ 0.5 and then starts decreasing again till  ~ 0.57. Beyond this bubble size, the bubble velocity increases again, reaches a second peak at  ~ 0.75 and then reduces again to reach the terminal velocity for long bubbles. The long bubble velocity in the 100 ppm AOT solution is less than the long bubble velocities in other AOT solutions. As the AOT concentration is increased further to 1000 ppm, the retardation in the terminal velocity of small as well as long bubbles reduces causing the bubble velocities to approach that of surfactant-free bubbles but the two peaks in the velocity-volume curve are still observed. To the best of our knowledge, the existence of two peaks in the velocity-volume curve has not been reported to date. Two peaks in the velocity-volume curve were observed for higher surfactant concentrations of Triton-X100 in 75 wt% GW solutions in the 10 x 10 mm and 15 x 15 mm square channels as well as 10 mm cylindrical channels (see Supporting Information Figures S1, S2, and S3).



Table 2: The two-phase systems and the range of dimensionless parameters for the experimental runs conducted. Drop fluid is air with density ratio for all the systems,   b   0.001 .


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Figure 4: Dimensionless velocity as a function of dimensionless bubble size for Aerosol-OT in 75 wt% GW solution in 10 x 10 mm square channel. Figure 5 compares the shapes of bubbles and shows the corresponding deformation parameters,  as a function of bubble sizes,  for the bubbles rising in 0, 10, and 100 ppm AOT in 75 wt% GW solutions in 10 x 10 mm square channels, respectively. Please note that the volumes of the bubbles for the three surfactant concentration solutions are not identical and variation of up to ± 0.02 in  exists. The shape of the front of the bubble is nearly identical for all the bubbles in different AOT concentration solutions. The rear of the bubbles is more flat for bubbles in the 0 ppm solution as compared to the 100 ppm solutions. It is interesting that the more prolate bubbles at the higher surfactant concentrations rise at lower speeds as compared to less prolate bubbles at the lower concentrations. This is due to the presence of surfactants that accumulate at the rear of the bubble. Increased surfactant concentration at the rear of the bubble reduces the surface tension locally. A normal stress jump across the interface requires that as the surface tension decreases, the curvature must increase locally to maintain the same pressure drop across the interface. Thus,



the bubbles moving through higher concentration surfactant solutions have a more curved interface at the rear compared to bubbles travelling in solutions in the absence of surfactants resulting in more prolate shapes. However, the accumulation of surfactants at the rear of the interface also renders the interface immobile in this region reducing the circulation within the bubble and the terminal velocity of the bubble. For small bubble sizes ( < 0.9), the bubbles become more prolate as the surfactant concentration increases. The deformation parameters for larger bubbles ( > 0.9) at different concentrations are similar.

The effect of the bulk fluid viscosity on the mobility and shape of the bubbles is shown in Figures 6 and 7 respectively. As expected, the higher viscosity of the bulk fluid reduces the terminal velocity of the bubbles rising in it due to the increased drag. The bubbles are more prolate in the higher viscosity bulk fluid as seen by the higher value of deformation parameter for bubbles rising in the 90 wt% glycerol-water solution seen in Figure 7. In the presence of surfactants, a further retardation of the bubbles is seen, though the effect is more significant for bubbles that are smaller and comparable to the channel size than long bubbles. Furthermore, the existence of two peaks in the velocity-volume curve is not seen for bubbles rising in the higher viscosity bulk fluid. Similar trends were seen for bubbles rising in pure glycerol bulk solution (see Supporting Information Figure S4).


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(a) 0 ppm

κ = 0.26

κ = 0.62

κ = 0.78

κ = 0.90

κ = 1.02

κ = 0.63

κ = 0.76

κ = 0.91

κ = 1.04

κ = 0.62

κ = 0.78

κ = 0.93

κ = 1.02

10 ppm

κ = 0.24

100 ppm

κ = 0.26

(b)

Figure 5: Comparison of (a) shape and (b) deformation parameter as a function of dimensionless bubble size for varying concentrations of AOT in 75 wt% GW solution.



Figure 6: Dimensional velocity as a function of bubble volume for TX-100 in 75 wt% and 90 wt% glycerol-water solutions in 10 x 10 mm square channel.

Figure 7: Deformation parameter as a function of dimensionless bubble size for TX-100 in 75 wt% and 90 wt% glycerol-water solutions in 10 x 10 mm square channel.


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Figures 8-11 show the effect of a non-uniform film surrounding the bubble on the velocity and shape of the rising bubbles in the presence of surfactants. In Figure 8, we compare the dimensionless terminal velocity as a function of dimensionless bubble size obtained for steady bubbles rising in a square and a circular channel with a hydraulic diameter, DH = 10 mm in the absence of any surfactants and in the presence of 10 ppm TX-100. It should be noted that the cross-sectional area of a square channel with 10 mm side length is larger than the cross-sectional area of a circular tube with 10mm diameter. The trend of the velocityvolume curve for the bubbles rising in a square channel is similar to that of the velocity-volume curve for a circular channel either with or without surfactants. However, the terminal velocity for all bubbles moving in a square channel are larger than those in a circular channel due to a leakage flux at the corners of the channel, which allows the bubbles to expand more radially outward. The terminal velocities of a long bubble rising in a circular channel are approximately 23% and 21% lower than the velocities of a long bubble rising in a square channel with the same hydraulic radius in the absence and presence of surfactants, respectively.

Figure 8: Comparison of dimensionless bubble velocity as a function of dimensionless bubble size for a square versus a circular channel with DH  10 mm for TX-100 in 75 wt% GW solution.



A comparison of the shapes as a function of the bubble size for bubbles rising in 75 wt% glycerol water solution in a circular and square channel with DH = 10 mm is shown in Figure 9. For small bubble sizes, the bubbles are nearly spherical and unaffected by the shape of the confining walls. For larger bubbles, however, the bubbles in a circular channel are more prolate than bubbles in the square channel in the absence and presence of surfactants. Because of the extra cross-sectional area near the corners of a square channel, the air bubble expands more radially outward in a square channel. These results are consistent with the experimental observations of Li et al.24 for bubbles rising in square channels. The presence of 10 ppm surfactant does not seem to affect the bubble shape noticeably.

Figure 9: Comparison of bubble shapes as a function of dimensionless bubble size for varying concentrations of TX-100 in 75 wt% GW solution in channels with DH  10 mm.


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Figure 10: Dimensionless velocity as a function of dimensionless bubble size for TX-100 in 75 wt% GW solution in 16 x 3 mm rectangular channel.

Figure 10 compares the velocity profile of the rising bubbles in 16 mm x 3 mm rectangular channels with various concentrations of surfactants as a function of bubble sizes. In the presence of very small amounts of surfactants (10 ppm), the terminal velocity of small bubbles (< 0.9) increases linearly with increasing bubble volume. Further increasing bubble sizes (0.9 < < 1.2) results in a steeper rise in the bubble velocity. As the bubble size becomes comparable to the channel size, the drag force due to the confining walls increases resulting in a decrease in the terminal velocity. Beyond a critical bubble volume, the bubble velocity reaches a constant plateau value, where the bubble velocity is independent of the bubble volume. The terminal velocities of all bubble sizes decrease with the increase of surfactant concentration because of Marangoni effect though the effect is more significant for smaller bubbles. Two peaks are also seen in the velocity-volume curve for a high enough surfactant concentration (100 ppm).



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(a)

0 ppm

κ = 1.10

κ = 1.41

κ = 1.76

κ = 2.20

κ = 2.58

κ = 1.41

κ = 1.76

κ = 2.20

κ = 2.59

κ = 1.38

κ = 1.73

κ = 2.20

κ = 2.59

10 ppm

κ = 1.10

100 ppm

κ = 1.10

(b)

Figure 11: Comparison of (a) shape and (b) deformation parameter as a function of dimensionless bubble size for varying concentrations of TX-100 in 75 wt% GW solution in 16 mm x 3 mm rectangular channel.


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Figure 11 shows the images of bubble shapes and the deformation parameter as a function of bubble size in 16 mm × 3 mm rectangular channels for 0 ppm, 10, and 100 ppm TX-100 solutions. In a rectangular channel, the bubble takes on a pancake-like shape as shown in the schematic in Figure 2(e). As the bubble rises through the channel, fluid flows along the sides of the bubble as well as in the thin fluid film separating the bubble from the walls. The bubble shape is not axisymmetric and the distribution of surfactant at the rear of the bubble depends on the magnitude of the flow along the sides and the thin film. As a result, the shapes of bubbles as bubble size increases shows trends not seen in bubbles rising through square and circular channels where the bubble shapes are axisymmetric. In the absence of surfactants, the bubbles are nearly spherical and as the bubble size increases ( > 1.5) the bubbles become prolate. As small amount of surfactant is added, the bubbles become prolate upto  ~ 1.0 beyond which the bubbles become more spherical in shape till  ~ 1.5. For larger bubbles ( > 1.5), the bubbles again become prolate in shape. However, even in the presence of very small amount of surfactant, bubble shapes are more prolate in the surfactant solution than in the clean solution. The effect of increased curvature in the presence of surfactants is enhanced at low surfactant concentrations due to the stronger flow in the thin film region across the depth of the channel. As the surfactant concentration is increased to 100 ppm TX-100, the bubble shapes become oblate for bubbles with  < 1.1 where the bubble shape suddenly becomes very prolate and then continues to remain prolate for long bubbles similar to the surfactant-free case. It is not clear why the bubble shape changes suddenly. It should be noted that in this study, the surfactants were used without any further purification. It is known that even trace amounts of impurities present in surfactants that strongly adsorb at the gas-liquid interface can significantly affect the interfacial behavior35-37. The sudden changes in bubble shapes and deformations seen in this study may be attributed to the presence of trace amounts of impurities in the surfactants.

When surfactant concentration is raised to values above the CMC, the bubble interface remobilizes such that the terminal velocity of long bubbles increases to equal the terminal velocity of clean bubbles. This is seen clearly in Figure 12 where the terminal velocity is plotted as a function of bubble size for bubbles ACS Paragon Plus Environment


rising in 75 wt% glycerol-water solution with concentration of TX-100 above CMC (1000ppm) in a 10 mm × 10 mm square channel. For comparison, the velocity-volume curves in the absence of surfactant (0 ppm) as well as surfactant concentration below CMC (100 ppm) are also plotted on the same graph. For small bubbles, however, complete remobilization is not seen. The bubbles rising in the 1000 ppm TX-100 solution are more oblate than the bubbles rising in a 0 ppm solution at small bubble sizes as seen in Figure 13. This is opposite to all our observations in surfactant solutions below CMC where bubbles became more prolate with the addition of surfactants. A look at the shape of the bubbles rising in the 1000 ppm solution (see Figure 13) shows a distinct bell-like shape at the rear of the bubble, specifically for bubbles with  < 0.8 rising in the 1000 ppm TX-100 solution. The shape and deformation of long bubbles remains largely unaffected by the presence of surfactants above CMC. The velocity-volume curve for bubbles rising in 1000 ppm TX-100 solution also shows two peaks similar to the 100 ppm solution. Remobilization of the interface for long bubbles and the existence of two peaks in the velocity-volume curve is also seen for surfactant runs above CMC in 75 wt% glycerol-water solution in 15 mm square channels (see Figure S5 in Supporting Information).


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Figure 12: Comparison of the dimensionless velocity as a function of dimensionless bubble size for air bubbles rising in a 75 wt% glycerol-water solution in the absence (0 ppm) and presence of TX-100 below (100 ppm) and above (1000 ppm) CMC in a 10 x 10 mm square channel.

Figure 13: Comparison of shapes as a function of dimensionless bubble size for air bubbles rising in a glycerol-water solution in the absence and presence of TX-100 above CMC in a 10 x 10 mm square channel.



Figure 14: Comparison of the dimensionless velocity as a function of dimensionless bubble size for air bubbles rising in a glycerol-water solution in the absence and presence of TX-100 above CMC in a 16 x 3 mm rectangular channel.

For bubbles rising in a TX-100 solutions in a 16 mm x 3 mm rectangular channel remobilization of long bubbles is seen for bubbles rising in a 1000 ppm TX-100 solution (above CMC). This is seen clearly in Figure 14 where the dimensionless velocity is plotted as a function of bubble size for bubbles rising in 0, 500, and 1000 ppm TX-100 solutions. However, for bubbles rising in a 500 ppm TX-100 solution (just above the CMC), the bubble terminal velocity is retarded for the entire range of bubble sizes studied. As the surfactant concentration increases above the CMC value, large bubbles seem to completely remobilize while small bubbles do not (see Supporting Information Figures S6 and S7). A comparison of shapes as a function of bubble size for bubbles rising in 0 ppm, 500 ppm, and 1000 ppm TX-100 solution is seen in Figure 15. The bubbles rising in 500 ppm TX-100 solution are more prolate than bubbles in the absence of surfactants. As the surfactant concentration increases to 1000 ppm, the bubbles again become flat at the rear end. Very small as well as very large bubbles in a 1000 ppm surfactant solution are more oblate than bubbles rising in a clean solution. The thin film that arises between the pancake-like shape of the bubble


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and the channel wall is likely to affect greatly the surfactant distribution along the bubble interface, which in turn affects its steady shape and velocity. We expect the bubble behavior in a rectangular channel to differ significantly from bubble dynamics in a square or circular channel. It is unclear, however, what leads to this drastic change in shape and velocity of the bubble at surfactant concentrations close to the CMC value in rectangular channels.

0 ppm

κ = 1.07

κ = 1.41

κ = 1.76

κ = 2.20

κ = 2.58

κ = 1.36

κ = 1.77

κ = 2.13

κ = 2.57

κ = 1.73

κ = 2.19

κ = 2.61

500 ppm

κ = 1.05

1000 ppm

κ = 1.02

κ = 1.43

Figure 15: Comparison of shapes as a function of dimensionless bubble size for air bubbles rising in a glycerol-water solution in the absence and presence of TX-100 above CMC in a 16 x 3 mm rectangular channel.

4. Summary In this paper, we studied the effect of changing bulk surfactant concentration on the shape and mobility of buoyancy-driven bubbles at finite Reynolds numbers. Channels with circular, square, and rectangular cross-section were considered with surfactant concentrations below and above the critical micelle concentration. In the presence of very small quantities of surfactants, although the equilibrium surface tension did not change, the terminal velocity for small bubbles was lower than that of the surfactant-free system due to the non-equilibrium effects. Long bubbles seemed to be unaffected by the presence of small



amounts of surfactants. When the surfactant concentration was increased further, the terminal velocity of small as well as long bubbles was retarded as compared to the clean bubbles. Two peaks in the velocityvolume curve were observed at higher surfactant concentrations still below the CMC concentration. The terminal velocities for bubbles moving in square channel were larger than those in circular channel even though the bubbles were more prolate in circular tubes. The deformation for bubbles moving in a rectangular channel in the presence of surfactants differs from bubbles rising through square and circular channels where the bubble shapes are axisymmetric.

Some of the unusual behavior observed for

intermediate-sized bubbles may be attributed to the presence of trace amounts of impurities in the surfactants. When the sufactant concentration was above CMC, the interface for long bubbles appeared to be remobilized while complete remobilization was not seen for small bubbles.

Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: XXX. Additional figures of dimensionless bubble velocity as a function of bubble size for bubbles rising in a variety of surfactant solutions (PDF).

Author Information Corresponding Author *E-mail: [email protected] ORCID Nivedita R. Gupta: 0000-0002-6742-8445 Notes The authors declare no competing financial interest


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Acknowledgements This research was supported by PRF grant 47612-AC9.

Reference

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(2) Goldsmith, H. L.; Mason, S. G. The movement of single large bubbles in closed vertical tubes. J Fluid Mech. 1962, 14, 42–58. (3) Reinelt, D. A. The rate at which a long bubble rises in a vertical tube. J Fluid Mech 1962, 175, 557–565. (4) Nigmatulin, T. R.; Bonetto, F. J. Shape of Taylor bubbles in vertical tubes. Int. Comm. in Heat Mass Transfer 1997, 24, 1177-1185. (5) Bugg, J. D.; Mack, K.; Rezkallah, K. S. A numerical model of Taylor bubbles rising through stagnant liquids in vertical tubes. Int. J. Multiphase Flow 1998, 24, 271-281. (6) Polonsky, S.; Shemer, L.; Barnea, D. The relation between the Taylor bubble motion and the velocity field ahead of it. Int. J. Multiphase Flow 1999, 15, 957-975. (7) Bi, Q. C.; Zhao, T. S. Taylor bubbles in miniaturized circular and noncircular channels. Int. J. Multiphase Flow 2001, 27, 561-570. (8) Bugg, J. D.; Saad, G. A. The velocity field around a Taylor bubble rising in a stagnant viscous fluid: numerical and experimental results. Int. J. Multiphase Flow 2002, 28, 791-803. (9) Viana, F.; Pardo, R.; Yanez, R.; Trallero, J. L.; Joseph, D.D. Universal correlation for the rise velocity of long gas bubbles in round pipes. J. Fluid Mech 2003, 494, 379-398. (10) Funada, T.; Joseph, D. D.; Maehara, T.; Yamashita, S. Ellipsoidal model of the rise of a Taylor bubble in a round tube. Int. J. Multiphase Flow 2005, 31, 473-491. (11) Taha, T.; Cui, Z., CFD modelling of slug flow in vertical tubes. Chem. Eng. Sci. 2006, 61, 676– 687. (12) Gutiérrez, E.; Balcázar, N.; Bartrons, E.; Rigola, J. Numerical study of Taylor bubbles rising in a stagnant liquid using a level-set/moving-mesh method. Chem. Eng. Sci. 2017, 164, 158–177. (13) White, E. T.; Beardmore, R. H. The velocity of rise of single cylindrical air bubbles through liquids contained in vertical tubes. Chem. Eng. Sci. 1962, 17, 351–361. (14) Borhan, A.; Pallinti, J. Buoyancy-driven motion of viscous drops through cylindrical capillaries at small Reynolds numbers. Ind. Eng. Chem. Res. 1995, 34, 2750-2761.



(15) Bozzi, L.; Feng, J.; Scott, T.; Pearlstein, A. Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number. J. Fluid Mech. 1997, 336, 1–32. (16) Feng, J. Q. Buoyancy-driven motion of a gas bubble through viscous liquid in a round tube. J. Fluid Mech. 2008, 609, 377-410. (17) Ajaev, V. S.; Homsy, G. M. Modeling Shapes and Dynamics of Confined Bubbles. Ann. Rev. Fluid Mech. 2006, 38, 277–307. (18) Kolb, W. B.; Cerro, R. L. Coating the inside of a capillary of square cross-section. Chem. Eng. Sci. 1991, 46 (9), 2181-2195. (19) Kolb, W. B.; Cerro, R. L. The motion of long bubbles in tubes of square cross section. Am. Inst. Physics 1993, 5 (7), 1549-1557. (20) Kolb, W. B.; Cerro, R. L. Film flow in the space between a circular bubble and a square tube. J. Colloid Interface Sci. 1993, 159(2), 302-311. (21) Liao, Q.; Zhao, T. S. Modeling of Taylor bubble rising in a vertical mini noncircular channel filled with a stagnant liquid. Int. J. Multiphase Flow 2003, 29, 411-434. (22) Clanet, C.; Hėraud, P.; Searby, G. On the motion of bubbles in vertical tubes of arbitrary crosssections: some complements to the Dumitrescu–Taylor problem. J. Fluid Mech. 2004, 519, 359-376. (23) Taha, T.; Cui, Z. CFD modelling of slug flow inside square capillaries. Chem. Eng. Sci. 2006, 61(2), 665-675. (24) Li, J.; Bulusu, V.; Gupta, N. R. Buoyancy-driven motion of bubbles in square channels. Chem. Eng. Sci. 2008, 63, 3766-3774. (25) Amaya-Bower, L.; Lee, T. 2011. Numerical simulation of single bubble rising in vertical and inclined square channel using lattice Boltzmann method. Chem. Eng. Sci. 2011, 66, 935-952. (26) Chen, J.; Stebe, K. J. 1996. Marangoni retardation of the terminal velocity of a settling droplet: the role of surfactant physico-chemistry. J. Colloid Interface Sci. 1996, 178, 144-155. (27) Almatroushi, E.; Borhan, A. Surfactant effect on the buoyancy-driven motion of bubbles and drops in a tube. Ann. NY Acad. Sci. 2004, 1027, 330-341. (28) Daripa, P.; Pasa, G. The effect of surfactant on long bubbles rising in vertical capillary tubes. J. Stat. Mech.: Theory Exp. 2011, 2, L02003. (29) Tasoglu, S.; Demirci, U.; Muradoglu, M. The effect of soluble surfactant on the transient motion of a buoyancy-driven bubble. Phys. Fluids 2008, 20, 040805. (30) Hayashi, K.; Tomiyama, A. Effects of surfactant on terminal velocity of a Taylor bubble in a vertical pipe. Int. J. Multiph. Flow 2012, 39, 78–87. (31) Kurimoto, R.; Hayashi, K.; Tomiyama, A. Terminal velocities of clean and fully-contaminated drops in vertical pipes. Int. J. Multiph. Flow 2013, 49, 8–23.


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(32) Cui, Y.; Gupta, N. R. Numerical study of surfactant effects on the buoyancy-driven motion of a drop in a tube. Chem. Eng. Sci. 2016, 144, 48-57. (33) Nagarajan, R.; Ruckenstein, E. Theory of surfactant self-assembly: a predictive molecular thermodynamic approach. Langmuir 1991, 7, 2934 – 2968. (34) Stebe., K. J., Maldarelli, C., Remobilizing surfactant retarded fluid partial interfaces II. Controlling the surface mobility at interfaces of solutions containing surface active components. J. Colloid Interface Sci. 1994, 163, 177-189. (35) Mysels, K. J., Florence, A. T., The effect of impurities on dynamic surface tension – basis for a valid surface purity criterion. J. Colloid Interface Sci. 1973, 43, 577-582. (36) Kralchevsky, P. A., Danov, K. D., Kolev, V. L., Broze, G., Mehreteab, A., Effect of nonionic admixtures on the adsorption of ionic surfactants at fluid interfaces. 1. sodium dodecyl sulfate and dodecanol. Langmuir 2003, 19, 5004-5018. (37) Basarova, P., Suchanova, H., Souskova, K., Vachová, T., Bubble adhesion on hydrophobic surfaces in solutions of pure and technical grade ionic surfactants. Colloids and Surfaces A: Physicochem. Eng. Aspects 2017, 522, 485-493.



Abstract Graphics


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Figure 1: Schematic of the experimental setup used to study the buoyancy-driven motion of bubbles in channels. 136x93mm (300 x 300 DPI)



Figure 2: Schematic of the front view and the cross-sectional view of a bubble rising through a channel. This bubble may be axisymmetric as in (a), (c) and (e) or non-axisymmetric as in (b) and (d) in the crosssectional view. 144x121mm (600 x 600 DPI)


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Figure 3: Surface tension of Aerosol-OT (AOT) and Triton-X100 (TX-100) as a function of varying bulk concentrations in glycerol and glycerol-water (GW) solutions. 124x96mm (300 x 300 DPI)



Figure 4: Dimensionless velocity as a function of dimensionless bubble size for Aerosol-OT in 75 wt% GW solution in 10 x 10 mm square channel. 124x96mm (300 x 300 DPI)


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Figure 5: Comparison of (a) shape and (b) deformation parameter as a function of dimensionless bubble size for varying concentrations of AOT in 75 wt% GW solution. 111x195mm (300 x 300 DPI)



Figure 6: Dimensional velocity as a function of bubble volume for TX-100 in 75 wt% and 90 wt% glycerolwater solutions in 10 x 10 mm square channel. 124x98mm (300 x 300 DPI)


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Figure 7: Deformation parameter as a function of dimensionless bubble size for TX-100 in 75 wt% and 90 wt% glycerol-water solutions in 10 x 10 mm square channel. 124x96mm (300 x 300 DPI)



Figure 8: Comparison of dimensionless bubble velocity as a function of dimensionless bubble size for a square versus a circular channel with for TX-100 in 75 wt% GW solution. 124x96mm (300 x 300 DPI)


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Figure 9: Comparison of bubble shapes as a function of dimensionless bubble size for varying concentrations of TX-100 in 75 wt% GW solution in channels with DH = 10 mm. 81x123mm (300 x 300 DPI)



Figure 10: Dimensionless velocity as a function of dimensionless bubble size for TX-100 in 75 wt% GW solution in 16 x 3 mm rectangular channel. 124x96mm (300 x 300 DPI)


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Figure 11: Comparison of (a) shape and (b) deformation parameter as a function of dimensionless bubble size for varying concentrations of TX-100 in 75 wt% GW solution in 16 mm x 3 mm rectangular channel. 110x200mm (300 x 300 DPI)



Figure 12: Comparison of the dimensionless velocity as a function of dimensionless bubble size for air bubbles rising in a 75 wt% glycerol-water solution in the absence (0 ppm) and presence of TX-100 below (100 ppm) and above (1000 ppm) CMC in a 10 x 10 mm square channel. 124x96mm (300 x 300 DPI)


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Figure 13: Comparison of shapes as a function of dimensionless bubble size for air bubbles rising in a glycerol-water solution in the absence and presence of TX-100 above CMC in a 10 x 10 mm square channel. 86x63mm (300 x 300 DPI)



Figure 14: Comparison of the dimensionless velocity as a function of dimensionless bubble size for air bubbles rising in a glycerol-water solution in the absence and presence of TX-100 above CMC in a 16 x 3 mm rectangular channel. 124x96mm (300 x 300 DPI)


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Figure 15: Comparison of shapes as a function of dimensionless bubble size for air bubbles rising in a glycerol-water solution in the absence and presence of TX-100 above CMC in a 16 x 3 mm rectangular channel. 114x133mm (600 x 600 DPI)



Number

Geometry

1

Circular

2

Square

3

Square

4

Rectangular

Dimensions Diameter, D = 10 mm Length = 813 mm Width, h = 10 mm Length = 927 mm Width, h = 15 mm Length = 950 mm Width, w = 16 mm Depth, t = 3 mm Length = 1232 mm


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Hydraulic Diameter, DH 10 mm 10 mm 15 mm 5.1 mm

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Channel size

Drop fluid

Surfactant

σeq (mN/m)

λ

Bo

Re

Ca

We T

10 mm circular 10 mm circular 10 mm circular 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 10 × 10 mm square 15 × 15 mm square 15 × 15 mm square 15 × 15 mm square 15 × 15 mm square 15 × 15 mm square 16 × 3 mm rectangular 16 × 3 mm rectangular 16 × 3 mm rectangular 16 × 3 mm rectangular 16 × 3 mm rectangular 16 × 3 mm rectangular 16 × 3 mm rectangular 16 × 3 mm rectangular

75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 90 wt% glycerol-water 90 wt% glycerol-water 90 wt% glycerol-water 90 wt% glycerol-water Glycerol Glycerol Glycerol Glycerol 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water 75 wt% glycerol-water

No surfactant 10ppm TX 100ppm TX No surfactant 1ppm AOT 10ppm AOT 100ppm AOT 1000ppm AOT 1ppm TX 10ppm TX 100ppm TX 1000ppm TX No surfactant 10 ppm TX 100 ppm TX 1000 ppm TX No surfactant 10 ppm TX 100 ppm TX 1000 ppm TX No surfactant 1ppm TX 10ppm TX 100ppm TX 1000ppm TX No surfactant 1ppm TX 10ppm TX 100ppm TX 250ppm TX 500ppm TX 750ppm TX 1000ppm TX

65.2 56 40.3 65.2 64 60 50.9 39.2 58 56 40.3 31.4 62.6 61.7 47.7 32.2 63.8 59.8 44.4 32 65.2 58 56 40.3 31.4 65.2 58 56 40.3 40 30 31 31.4

0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0001 0.0001 0.0001 0.0001 0.00002 0.00002 0.00002 0.00002 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006

0.04 0.05 0.07 4.43 4.53 4.84 5.71 7.45 5.01 5.23 7.3 9.39 4.79 4.86 6.28 9.32 4.86 5.18 6.98 9.69 9.98 11.26 11.76 16.41 21.12 1.13 1.28 1.3 1.71 1.75 2.5 2.42 2.39

8.6 - 16.3 5.9 - 13.4 4.2 - 11.0 5.9 - 9.2 2.5 - 17.2 5.6 - 15.4 1.9 - 14.0 2.5 - 16.2 4.8 - 17.6 4.8 - 15.1 3.6 - 15.0 7.4 - 15.5 0.08 - 1.35 0.27 - 1.24 0.35 - 1.32 0.20 - 1.08 0.03 - 0.06 0.00 - 0.02 0.03 - 0.05 0.03 - 0.04 20.2 - 44.7 10.0 - 43.8 8.5 - 40.2 13.2 - 37.5 15.6 - 37.2 2.8 - 6.7 2.2 - 6.3 1.5 - 6.3 3.3 - 6.5 0.3 - 5.6 1.6 - 4.2 1.7 - 5.9 2.0 - 6.0

0.02 - 0.05 0.02 - 0.04 0.02 - 0.05 0.02 - 0.05 0.01 - 0.05 0.02 - 0.05 0.01 - 0.05 0.01 - 0.08 0.02 - 0.06 0.02 - 0.05 0.02 - 0.07 0.004 - 0.08 0.01 - 0.15 0.03 - 0.14 0.05 - 0.19 0.04 - 0.23 0.07 - 0.14 0.06 - 0.13 0.09 - 0.18 0.12 - 0.21 0.04 - 0.08 0.02 - 0.09 0.02 - 0.09 0.04 - 0.11 0.06 - 0.14 0.02 - 0.06 0.01 - 0.04 0.01 - 0.04 0.03 - 0.05 0.02 - 0.05 0.02 - 0.05 0.02 - 0.07 0.02 - 0.07

0.29 0.36 0.42 0.49 0.5 0.51 0.53 0.88 0.55 0.57 0.77 1.06 0.13 0.12 0.17 0.21 0.005 0.005 0.007 0.009 2.38 2.6 2.65 3.72 4.98 0.19 0.2 0.21 0.29 0.27 0.2 0.31 0.33



Graphical Abstract 98x57mm (300 x 300 DPI)


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Buoyancy-Driven Motion of Bubbles in the Presence of Soluble

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