Dynamics of bubble formation in highly viscous liquids in a flow-focusing device

Accepted Manuscript Short Communication Dynamics of bubble formation in highly viscous liquids in a flow-focusing device Chong Zhang, Taotao Fu, Chunying Zhu, Shaokun Jiang, Youguang Ma, Huai Z. Li PII: DOI: Reference:

S0009-2509(17)30416-5 http://dx.doi.org/10.1016/j.ces.2017.06.026 CES 13670

To appear in:

Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

23 November 2016 14 June 2017 16 June 2017

Please cite this article as: C. Zhang, T. Fu, C. Zhu, S. Jiang, Y. Ma, H.Z. Li, Dynamics of bubble formation in highly viscous liquids in a flow-focusing device, Chemical Engineering Science (2017), doi: http://dx.doi.org/10.1016/ j.ces.2017.06.026

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Dynamics of bubble formation in highly viscous liquids in a flow-focusing device

Chong Zhanga, Taotao Fua *, Chunying Zhua, Shaokun Jiangb, Youguang Maa*, Huai Z. Lic a

State Key Laboratory of Chemical Engineering, Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China

b

The 718th Research Institute of China Shipbuilding Industry Corporation, No.17 Zhanlan Road, Handan City, Hebei Province 056027, China

c

Laboratory of Reactions and Process Engineering, University of Lorraine, CNRS, 1, rue Grandville, BP 20451, 54001 Nancy Cedex, France * Corresponding authors: [email protected] (T. Fu); [email protected] (Y. Ma)

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Abstract This article reports the dynamics and mechanism for N2 bubble formation in highly viscous glycerol-water mixtures in a flow-focusing device by using a high-speed digital camera. The evolution of the volume for the gaseous thread during bubble formation is highlighted. A square microchannel with 400 μm×400 μm is used. The bubble formation process can be divided into an expansion stage and a breakup stage. The volume of the gaseous thread increases linearly with time in the two different stages, however, the growth rate in the breakup stage is always greater than that in the expansion stage. The growth rate for the evolution of the gaseous thread is controlled by the capillary number and the gas-liquid flow rate ratio, with varied exponents for the two stages. The turning point of the two stages occurs at the moment when the width of gaseous neck reaches its maximum, and the dimensionless time at the turning point is related to the gas-liquid flow rate ratio. Finally, the volume of generated bubbles in highly viscous liquids in the flow-focusing device is scaled with the capillary number and the gas-liquid flow rate ratio.

Key words microchannels; viscous liquids; gas-liquid two-phase flow; bubble; dynamics

1. Introduction Microchemical engineering and technology has attracted increasing attention because of its high rates for the mass and heat transfer, the numbering-up facility, safety and flexible controllability (Elvira et al., 2013). The dynamics of multiphase flows in confined spaces is one of the key problems

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in the microchemical technology, such as the formation of bubbles and droplets in microchannels. Microbubbles have numerous applications in medicine (Lindner, 2004; Rodríguez-Rodríguez et al., 2015), food engineering (Zuniga and Aguilera, 2008), and materials synthesis (K. S. Suslick and Price., 1999). Therefore, many contributions have been devoted to the formation and manipulation of bubbles in microchannels (Cao et al., 2006; Castro-Hernández. et al., 2011; Hoeve et al., 2011; Marín et al., 2009; Xu et al., 2006; Chen et al., 2013).

Bubble formation in microchannel was found to be mainly controlled by two mechanisms (Garstecki et al., 2006; Thorsen et al., 2001): the shearing mechanism in unconfined flow and the squeezing mechanism in completely confined flow. The shearing mechanism indicates that the bubble breaks up in the equilibrium of the surface tension and viscous force, and the liquid viscosity plays an important role in bubble formation. Thorsen et al. (2001) proposed that the radius r of the generated droplet in oil in the T-shaped microchannel is inversely proportional to the capillary number Ca: r  Ca-1, with capillary number Ca representing the ratio of the viscous force over the surface tension force, and Ca=μlu/σ. μl, u and σ represent the viscosity and velocity of the continuous phase and the surface tension between the two phases, respectively. The squeezing mechanism implies that the bubble formation is controlled by the accumulated pressure in the blocked liquid phase around the growing gaseous thread, and the bubble size is determined by the gas-liquid flow rate ratio and the geometry of the channel. In this case, the bubble size is not dependent of the liquid viscosity. Garstecki et al. (2006) proposed that the bubble size was proportional to the gas-liquid flow rate ratio in a T-shaped microchannel in completely confined flow at low capillary numbers (Ca≤10-2):

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L/wc=1+αφ, where L, wc and φ represent the bubble length, channel width and the gas-liquid flow rate ratio, respectively; and α is an adjustable parameter related to the geometry of the channel. In addition, some studies showed that the bubble formation was controlled by the joint shearing mechanism and the squeezing mechanism for partly confined breakup of gaseous thread during bubble formation (Christopher et al., 2008; De Menech et al., 2008; Fu et al., 2010; Xu et al., 2008). Castro-Hernández. et al. (2011) reported a new regime for bubble formation in low viscous fluids in a flow-focusing microfluidic device, and scaled the bubble size with the flow rate ratio and the viscosity ratio of gas and liquid phases: db/wc=2.75(μg/μl)1/12φ5/12, in which, db and μg represent bubble diameter and viscosity of gas phase, respectively.

Although a lot of efforts have been devoted to the bubble formation mechanism and the prediction of bubble size in microfluidic devices, most of these studies have focused on bubble formation in low viscous fluids (Elvira et al., 2013). Highly viscous fluids are frequently encountered in polymer engineering and food engineering (

- m

., 2009; Thoroddsen et al., 2007), but there

are few reports for bubble formation in highly viscous fluids in microdevices (Lu et al., 2014). Furthermore, the literature mainly considers the final volume of the generated bubbles, however, the mechanism and detailed information for bubble formation process have not been fully explored yet. This work investigates the dynamics for bubble formation in highly viscous liquids in a flow-focusing device. The evolution of the volume of the gaseous thread during bubble formation is clarified, and the effect of the operating conditions such as the gas-liquid flow rate ratio, and the liquid viscosity on the dynamics of bubble formation is highlighted.

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2. Experimental procedures Experimental facilities include a microfluidic flow-focusing device, a fluid control system and an image acquisition system, as shown in Figure 1(a). Gas is fed from a N2 cylinder and the gas flow rate is controlled by a micrometering value (KOFLOC, Japan). The pressure Pg at the exit of the N 2 cylinder is maintained at a constant value of 0.7 MPa. Thus, the actual volumetric gas flow rate is estimated by the ratio of the volume of generated bubbles to the bubble formation period as the actual flow rate varied with the change of the resistance in the downstream channel, according to the Hagen-Poiseuille relationship (Garstecki et al., 2005). Liquid is pumped into the horizontal microchannel by a syringe pump (PHD 2000, Harvard Apparatus, USA) through polyethylene rubber tube. The square microchannel with a cross-section of 400 μm × 400 μm

f br c

d

a

polymethyl methacrylate (PMMA) plate (45 mm×27.5 mm×2 mm) by milling. The dispersed gas phase is introduced from the main channel with a volumetric flow rate of Qg, and the continuous phase liquid is fed from the two lateral channels with a volumetric flow rate of Ql/2, as shown in Figure 1(b). The process of the bubble formation is magnified by a microscope (ECLIPSE Ti-U, Nikon, Japan) and recorded by a high-speed digital camera (MotionProY5, IDT, USA). The recording rate of the camera is 4000 fps (frames per second). Halogen lamp is placed at the other side of the microfluidic device to provide sufficient light for the image acquisition. The images for bubble formation are captured when the flow reaches stable after a new flow condition is set. All the experiments are conducted at room temperature and atmospheric pressure.

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Highly viscous glycerol and 99%, 93%, 90%, 87%, 80%, 75% glycerol-water solutions are used as the continuous phase. The density of liquids ρl is measured by using a pycnometer. The surface tension of the liquid in air σ is measured by the pendant drop method via a tensiometer (OCAH200， Data Physics instruments GmbH, Germany). The viscosity of the liquid μl is measured by Ubbelohde viscometer (iVisc, LAUDA, Germany). The physical properties of the continuous phase are shown in Table 1. The range of the gas-liquid flow rate ratio φ is from 0.03 to 0.56. The range of the capillary number Ca is from 0.02 to 1.35. The range of the Reynolds number Re (Re=wcρlu/μl) is from 0.02 to 3.12.

The cross section of the microchannel is square, but the channel is not fully filled with the gas, and the cross section of bubble is supposed to be circular. The light shines on the microdevice vertically, so it is bright without refraction at the center of the bubble, and at the rest of the bubble it is dark because the light is radiated to the outside of the microchannel through refraction and total reflection (Mac Giolla Eain et al., 2013). When the light does not pass through the bubble, it will not refract because the light is perpendicular to the channel and the liquid, so the liquid part is bright as shown in Figure 2(a). This means that the gas-liquid interface is not influenced by optical distortion and the captured one is the actual interface for the bubble. The contour of the gas-liquid interface for the bubble is extracted by comparing the captured picture containing bubbles with that only containing liquid in the microchannel. And the gas-liquid interface is recognized with a gray scale of 0.5 by a Matlab program. Then the interior of the bubble is filled with black as shown in Figure 2(b). Thus, the error is caused by the interface recognition, and the absolute error is ±2 pixels (±2.4μm). When

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the radius of the bubble is the smallest, the maximum error is calculated as 3.6%, which can be ignored in our experiments.

3. Results and discussion A typical process for N2 bubble formation in glycerol in the flow-focusing device is shown in Figure 3. The bubble formation process can be divided into the expansion stage and the breakup stage. (  ) The process of 0~22 ms is named as the expansion stage. After the detachment of the last generated bubble, the tip of the gaseous thread retracts due to the surface tension from 0 to 2 ms. The thread represents the slender gaseous part from the entrance to the bubble tip as shown in Figure 2(a). The gaseous thread is deformed by the viscous force of the liquid to form a cone-shaped tip at 2ms. Then the gaseous thread expands due to the continuous supply of N2 from 2 ms to 22 ms. Due to the obstruction by the liquid in the radial direction, the gaseous thread expands mainly in the axial direction to reach its maximum width near the cross-junction of the flow-focusing device, and then a neck of the gaseous thread is formed at 22 ms. (  ) The process of 22~35.5 ms is named as the breakup stage. The gaseous neck thins gradually under the action of the liquid. Meanwhile, the volume of the gaseous thread continues to increase owing to the continuous supply of the dispersed phase. At last, the gaseous thread breaks up to form a new bubble with a sharp tail.

Several expressions for the prediction of bubble volume were reported in literature (Castro-Hernández. et al., 2011; De Menech et al., 2008; Dietrich et al., 2008; Fu et al., 2010; Ganan-Calvo and Gordillo, 2001; Garstecki et al., 2006; Thorsen et al., 2001), but the related

7

mechanism and the detailed information for the evolution of gaseous thread during bubble formation remain unclear. Thus, we obtain the evolution of the volume of gaseous thread during bubble formation to explore the bubble formation mechanism as shown in Figure 4. The gaseous thread during bubble formation is supposed to be axisymmetric and its volume V can be therefore calculated as:

 w2 ( l ) V  dl 4

(1)

where l represents the distance from the entrance of the junction at the left side to a certain position of the gaseous thread, and w(l) is the width of the gaseous tip for a certain l as illustrated in Fig. 2b. It should be noted that the actual volume of the gaseous thread during bubble formation is obtained by deducting the initial one just after the pinch-off of the formed bubble from the estimated value by Eq. (1). It can be seen from Figure 4 that the evolution of the volume for the gaseous thread during the bubble formation can be divided into two linear stages: the expansion stage with a smaller slope and the breakup stage with a higher slope. In the expansion stage (corresponding to 0~22 ms in Figure 3), with the growing of the volume of the gaseous thread, the width of the gaseous thread increases. In this stage, the growing gaseous thread is obstructed by the flowing liquid on both sides from the lateral microchannels, so the liquid phase acts as the resistance to the growing of the gaseous thread. In the breakup stage, after the gaseous neck reaching its maximum width (corresponding to 22~33.5 ms in Figure 3), the volume of the gaseous thread continues to increase and the gaseous neck thins gradually. In this stage, the flowing liquid drives the gaseous neck to collapse in the radial direction and pushes the gaseous thread to propagate in the axial direction, thus the liquid phase acts as the driving role. It is therefore understood that the bubble formation process 8

can be divided into two stages and the slope of breakup stage is higher than that in the expansion stage, owing to the change of the role of the continuous phase in the bubble formation.

The dependence of the evolution of the gaseous thread during bubble formation on the operating conditions such as the liquid viscosity and the gas-liquid flow rate ratio is shown in Figure 5. It can be also found that the bubble formation process can be divided into two linear stages and the slope of the second stage is higher than the first one under various operating conditions. For a given liquid viscosity, the growth rate for the volume of the gaseous thread during bubble formation in the two stages increases and the dimensionless time for the turning point for the two stages decreases with the increase of the gas-liquid flow rate ratio. For a given gas-liquid flow rate ratio, the liquid viscosity also affects the evolution of the gaseous thread during bubble formation. For example: when φ=0.31, the growth rate for the volume of the gaseous thread decreases with the increase of the liquid viscosity as shown in Figure 5 (c), (f) and (g). For bubble formation in highly viscous liquids at high capillary numbers, the gaseous thread does not obstruct the entire cross-section of the microchannel so that the viscous liquid can still flow pass through the film between the gaseous thread and the channel wall. In addition, the curvature of the gas-liquid interface varies with time. Therefore, the viscous force and surface tension interact, and the shearing mechanism is involved for the bubble formation process (Lu et al., 2013; van Hoeve et al., 2011). Furthermore, due to the obstruction of the liquid phase by the gaseous thread at the junction, the pressure in the liquid phase around the gaseous thread is accumulated to drive the breakup of the gaseous thread, signifying that the squeezing mechanism is also involved for the partly confined breakup of gaseous thread during

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bubble formation. In this partly confined breakup case, the segregation of the gaseous thread and the wall is induced by the thick liquid film between the gas-liquid interface and the channel wall due to the highly viscous liquids and high capillary numbers. Therefore, the evolution of the volume of the gaseous thread during bubble formation in highly viscous liquids in a flow-focusing device is controlled by the joint squeezing - shearing mechanism, which is also reported for bubble and droplet formation in microdevices under partly confined breakup situation at intermediate capillary numbers and low liquid viscosities (Christopher et al., 2008; De Menech et al., 2008; Fu and Ma, 2015; Fu et al., 2010; Garstecki et al., 2006; Thorsen et al., 2001).

The dependence of the growth rate for the volume of the gaseous thread with time during bubble formation on the capillary number and the gas-liquid flow rate ratio is shown in Figure 6, and their relationship is obtained by using the least squares fitting method for the expansion stage and the breakup stage respectively. For the expansion stage: k1  φ0.8Ca-0.04

(2)

R2=0.91, R2 represents the square of linear regression correlation coefficient. For the breakup stage: k2  φ0.52Ca-0.06

(3)

R2=0.86. The slopes of two stages are positively correlated with the gas-liquid flow rate ratio and negatively correlated with the capillary number, and the exponent for the gas-liquid flow rate ratio varies much more largely than the exponent of the capillary number for the two stages. This phenomenon can be explained by the fact that the liquid flow plays different roles in the bubble

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formation: the resistance to the gaseous thread in expansion stage; while the driving force towards the gaseous thread in breakup stage.

The dimensionless time Tc (Tc=t/T, here t represents time and T represents the period of the bubble formation.) for the turning point of the expansion and breakup stages decreases with the increase of the gas-liquid flow rate ratio for a given liquid viscosity. When the gaseous neck reaches the maximum width wmax, the bubble formation turns from the expansion stage to the breakup stage, and the pressures around the gaseous neck should satisfy the Laplace disjunction: Pg-Pl=2σ/wmax

(4)

where Pg and Pl represent the pressure in the gas and liquid phases, respectively. The dependence of the maximum width wmax of the gaseous neck on φ in glycerol is shown in Figure 7, and wmax increases only slightly with φ. When the gas-liquid flow rate ratio increases, (Qg-Ql) increases, so (Pg-Pl) increases. This leads to a shorter expansion stage as wmax shoud be smaller as shown in Eq. (4). Thus the percentage of the expansion stage in the whole period for bubble formation decreases. The expression for the dimensionless Tc at the turning point can be obtained by the least square fitting method as shown in Figure 8: Tc  φ-0.34Ca0.002  φ-0.34

(5)

with R2=0.89. The influence of the capillary number on Tc is negligible compared to the gas-liquid flow rate ratio. This may be due to the fact that the dimensionless Tc is the ratio of the time to the period for bubble formation, and the period is also affected by the capillary number (Lu et al., 2014).

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The dependence of the final volume of the generated bubble Vf on the gas-liquid flow rate ratio and the capillary number can be further correlated by the least square fitting method: Vf/wc3  φ0.73Ca-0.07

（6）

with R2=0.87, as shown in Figure 9. The bubble volume increases with the increase of the gas-liquid flow rate ratio, which is consistent with the squeezing mechanism (Garstecki et al., 2006). However, the exponent for the gas-liquid flow rate ratio is 0.73, which is different from the prediction by the squeezing mechanism for the completely confined breakup for bubble formation in microdevices. Garstecki et al.(2006) obtained α=1 for bubble formation in low viscous liquids in T-junctions at low capillary numbers under completely confined situation. Fu et al. (2010) got the prediction for the bubble length in low viscous liquids (μl ＜11 mPa·s) in T-junctions at low capillary numbers: L/wc=0.32φ+0.64. Dietrich et al. (2008) and Ganan-Calvo and Gordillo (2001) found that the exponent of the gas-liquid flow rate ratio for bubble formation in low viscous liquids in flow-focusing devices is 0.25 and 0.37, respectively. The divergence on the exponent of the gas-liquid flow rate ratio stems from the confinement degree of the channel wall to the gas-liquid interface during bubble formation, which is due to the liquid viscosity, the capillary number and the geometry of the junctions (Garstecki et al., 2006; Dietrich et al., 2008). The exponent for the capillary number is -0.07, which is different from -1 obtained by Thorsen et al. (2001) for the shearing mechanism for droplet formation in a T-junction. De Menech et al. (2008) proposed a relationship between the radius of droplet and the capillary number as r  Ca-0.25 in T-shaped microchannel. These differences could probably be attributed to the fact that the accumulated pressure in the obstructed liquid also plays a role in bubble formation in the partly confined breakup

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of the gaseous thread during bubble formation, and thus the varied exponent reflects the degree of the confinement of the channel wall to the gas-liquid interface. Therefore, the results show that for the partly confined breakup of the gaseous thread for bubble formation in highly viscous liquids at high capillary numbers in a flow-focusing device is controlled by the joint squeezing - shearing mechanism, which is consistent with the conclusion of Castro-Hernández. et al. (2011). In both studies, the absolute value of the exponent of the viscosity term is smaller than 0.1. The result in the present study is also similar to the expression in Lu et al.(2014): Vf/wc3=1.12φ0.52Ca-0.29. The divergence of the exponent stems from the difference in the range of the liquid viscosity, the gas-liquid flow rate ratio and the capillary number.

It is noteworthy to point out that the gas flow rate is estimated by the ratio of the volume of the generated bubble to the bubble formation period, the same as the method employed in Castro-Hernández. et al. (2011), and the actual gas flow rate Qg follows the Hagen-Poiseuille relation as Qg  Pg/μl (Garstecki et al., 2005). So the liquid viscosity also influences the gas flow rate at a fixed inlet pressure for the gas phase. Thus, the effect of liquid viscosity on the bubble formation is already reflected in Qg, and the absolute value of the exponent on Ca is much smaller than that of the gas-liquid flow rate ratio in Eqs. (2)-(3) and (5)-(6). This result is also similar to the conclusion of Castro-Hernández. et al. (2011). Thus, to some extent, the bubble formation is also affected by the liquid viscosity via a hydrodynamic feedback mechanism that characterizes the feedback of the fluid dynamics in the downstream channel towards the bubble formation at the upstream junction (Fu and Ma, 2015; Sullivan and Stone, 2008).

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4. Conclusions The linear expansion stage and breakup stage are highlighted for the evolution of the volume of gaseous thread during bubble formation in highly viscous liquids in a flow-focusing device at low Reynolds numbers and high capillary numbers. The growth rate for the volume of the gaseous thread during bubble formation differs for the two stages, and the slope for the evolution of the volume of the gaseous thread in the breakup stage is always greater than that in the expansion stage, due to the different role of the liquid phase in the evolution of the gaseous thread for the two stages. In the expansion stage, the liquid phase around the gaseous thread constraints its augmentation; while in the breakup stage, the liquid phase around the gaseous thread drives its thinning by the squeezing force in the radial direction and the shearing force in the axial direction. The slope in the two stages can be scaled with the gas-liquid flow rate ratio and the capillary number as k1  φ0.8Ca-0.04 for the expansion stage, and k2  φ0.52Ca-0.06 for the breakup stage. The dimensionless time at the turning point of the two stages is controlled by the gas-liquid flow rate ratio as Tc  φ-0.34. The final volume for the generated bubble in highly viscous liquids in a flow-focusing device can be scaled with the gas-liquid flow rate ratio and the capillary number as Vf/wc3  φ0.73Ca-0.07, signifying that the bubble formation in highly viscous liquids in a flow-focusing device is controlled by the joint squeezing shearing mechanism, which is owing to the partly confinement of the channel to the gas-liquid interface that is induced by the viscous property of the liquids and high capillary numbers. Furthermore, the feedback effect of the fluid dynamics in the downstream channel on the bubble formation in the upstream junction is explored to take consideration of the resistance change due to

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the variation of the liquid viscosity. Nevertheless, this mechanism needs to be explored systematically to take into consideration of the characteristics of the segmented flow in highly viscous liquids in downstream channels. This study paves the way for further experimental and theoretical studies on bubble formation dynamics in complex fluids within microfluidics to gain insight into potential applications in chemical engineering, materials synthesis, polymer engineering and lab-on-a-chip.

Acknowledgment The financial supports for this project from the National Natural Science Foundation of China (No. 21576186, 91634105, 91434204, 21276175), the Research Fund for the Doctoral Program of Higher Education (20110032120010), the Tianjin Natural Science Foundation (17JCQNJC05300), the aid of Opening Project of State Key Laboratory of Chemical Engineering (No. SKL-ChE-13T04, SKL-ChE-16B03) and the Programs of Introducing Talents of Discipline to Universities (Grant No.B06006) are gratefully acknowledged.

Nomenclature db.

diameter of bubble (μm)

k1.

growth rate in the expansion stage

k2.

growth rate in the breakup stage

L.

g h f bubb (μm)

l.

g h fg

u

p (μm)

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Pg.

pressure of gas (Pa)

Pl.

pressure of liquid (Pa)

Qg.

volumetric flow rate of gas phase (mL·h-1)

Ql.

volumetric flow rate of liquid phase (mL·h-1)

r.

r du

t

time (ms)

T.

formation period of bubbles (ms)

Tc .

dimensionless time at the turning point, Tc=t/T

u.

superficial velocity of liquid (m·s-1)

V.

volume of gaseous thread during bubb f rm

Vf.

f

wc.

w d h f h ch

wmax.

m x mum w d h f h g

w(l).

wd h fg

α.

prefactor of gas-liquid flow rate ratio

σ.

surface tension (mN·m-1)

μg.

viscosity of gas (mPa·s)

μl.

viscosity of liquid (mPa·s)

φ.

gas-liquid flow rate ratio (φ=Qg/Ql)

ρg.

density of gas (kg·m-3)

ρl.

density of liquid (kg·m-3)

f bubb

v um

r dr p

f h g

u

(μm)

r

d bubb (μm3)

(μm) u

ck (μm)

p (μm)

16

(μm3)

Ca.

capillary number (Ca=μlu/σ)

Re.

Reynolds number (Re=wcρlu/μl)

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v

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Caption of Figures Fig. 1. (a) Experimental procedure; (b) Diagram of the flow-focusing device used for the generation of bubbles. The dispersed gas phase is introduced from the main channel with a volumetric flow rate of Qg, and the continuous liquid phase is fed from the lateral channels with a volumetric flow rate of Ql/2.

Fig. 2. (a) Experimental picture for bubble formation; (b) Post-processing picture for a filled black bubble. l and w(l) represent the length and width of the gaseous tip at a certain axial position, respectively.

Fig. 3. Bubble formation process in a viscous liquid in the flow-focusing device. Dispersed phase: N2. Continuous phase: glycerol. μl = 1000 mPa·s, Ql = 50 mL·h-1, Qg = 4.5 mL·h-1, φ = Qg/Ql = 0.09, T=35.5 ms. T is the formation period of bubbles. The recording rate of the high-speed camera is 4000 fps. The time is designated on the right-hand of the images with the unit of ms. The time zero is selected as the moment for the bubble that just breaks up.

Fig. 4. Evolution of the dimensionless volume V/wc3 for the gaseous thread during bubble formation in the flow-focusing device with dimensionless time t/T. Gas phase: N2. Liquid phase: glycerol. μl = 1000 mPa·s, Ql = 50 mL·h-1, Qg = 4.5 mL·h-1, φ = 0.09, T = 35.5 ms. The rate of the high-speed camera is 4000 fps. The inset shows the variation of the volume for the gaseous thread V with time t. wc is the width of the microchannel.

22

Fig. 5. Evolution of the dimensionless volume V/wc3 with dimensionless time t/T for the gaseous thread during bubble formation in the flow-focusing device for various gas-liquid flow rate ratios and liquid viscosities. k1 is the growth rate in the expansion stage that is illustrated by the hollow points. k2 is the growth rate in the breakup stage that is represented by the solid points. The red star shows the turning point from the expansion stage to the breakup stage. Tc is the dimensionless time at the turning point.

Fig. 6. The dependence of the growth rates for the evolution of the volume of gaseous thread during bubble formation on the capillary number and the gas-liquid flow rate ratio: (a) for the expansion stage, the solid line presents the relationship k1= 2.39φ0.8Ca-0.04; (b) for the breakup stage, the solid line corresponds to k2= 2.54φ0.52Ca-0.06.

Fig. 7. The dependence of the maximum width wmax for the gaseous neck during bubble formation on the gas-liquid flow rate ratio φ. Liquid phase: pure glycerol.

Fig. 8. The relationship between the dimensionless time at the turning point and the gas-liquid flow rate ratio. The solid line represents Tc=0.3φ-0.34.

23

Fig. 9. The volume of bubbles generated in highly viscous liquids in the flow-focusing device as a function of the gas-liquid flow rate ratio and the capillary number. The solid line denotes Vf/wc3= 2.56φ0.73Ca-0.07.

Caption of Tables Table 1. Physical properties of glycerol-water mixtures used in the experiment.

24

Fig. 1.

25

Fig. 2.

26

Fig. 3.

27

0.8 0.07

Experimental data V=0.0009t+0.0004 V=0.0017t-0.0165

3

0.6

V (mm )

0.06 0.05 0.04 22ms

V/wc

3

0.03

0.4

0.02

0

5

10

15

20

25

30

35

t (ms)

0.2

0.62

Experimental data 3 V/wc =0.51t/T+0.01 3

V/wc =0.91t/T-0.24

0.0 0.0

0.2

0.4

0.6 t/T

Fig. 4.

28

0.8

1.0

（ a）1.5 1.2

1.13 1.02 0.92 0.71 0.51

k2

（ b） 1.2

Tc T/ms

1.69 1.50 1.37 1.17 0.91

0.4 0.43 0.46 0.55 0.62

40 42 41 38 35.5

k1

0.25 0.23 0.16 0.14

0.9

k2

0.94 0.82 0.58 0.40

Tc T/ms

1.50 1.47 0.94 0.76

0.49 0.54 0.62 0.68

44 39 29 28

V/wc

3

V/wc

3

0.9

k1

0.56 0.35 0.26 0.15 0.09

0.6

0.3

0.3

（ c） 1.2

3

0.9

0.2

k1

0.31 0.23 0.15 0.09 0.07

1.00 0.62 0.44 0.29 0.26

0.4

k2

t/T

0.6

0.8

0.53 0.59 0.70 0.78 0.83

（ d）

34 22 18 15 11.5

V/wc

V/wc

0.4

t/T

0.6

0.8

0.36 0.18 0.11 0.07

1.19 0.55 0.35 0.28

k2 2.10 0.97 0.79 0.60

（ f）

Tc T/ms 0.4 0.53 0.72 0.77

3

0.2

0.4

t/T

0.6

0.8

0.0 0.0

1.0

2.0 k1

0.31 0.22 0.16 0.09

1.40 0.85 0.72 0.47

k2 2.24 1.77 1.15 0.93

0.8

1.0

0.8

1.0

31.5 22.5 15 12 8.5

37.5 27.5 18 13

0.8 0.4

0.2

0.4

t/T

0.4

t/T

1.24 0.83 0.62 0.35

k2 1.97 1.20 0.97 0.60

Tc T/ms 0.39 0.48 0.56 0.68

34 21 16.5 11

0.2

0.4

t/T

0.6

(a) 100%gly l=1000mPa·s

Tc T/ms 0.43 0.52 0.64 0.77

3

V/wc

0.6

Tc T/ms 0.45 0.53 0.72 0.81 0.85

0.9

0.3

0.0 0.0

k2 1.64 1.07 0.80 0.66 0.50

0.2

0.31 0.22 0.16 0.08

1.2

0.3

1.2

1.04 0.68 0.38 0.24 0.16

k1

0.6

1.6

1.0

1.8

0.6

（ g）

0.8

0.6

1.5

44 19 14 10

0.9

0.0 0.0

0.39 0.25 0.12 0.06 0.03

0.0 0.0

1.0

V/wc

3

V/wc

0.2

k1

1.2

0.6

0.3

1.8 1.5

k1

0.9

0.3

（ e）

0.4

1.5 1.2

0.6

0.0 0.0

0.2

t/T

Tc T/ms

1.35 0.96 0.81 0.75 0.72

0.0 0.0

1.0

3

0.0 0.0

0.6

0.6

0.8

1.0

Fig. 5.

29

(b) 99%gly

l=630mPa·s

(c) 93%gly

l=216mPa·s

(d) 90%gly

l=150mPa·s

(e) 87%gly

l=94mPa·s

(f) 80%gly

l=83mPa·s

(g) 75%gly

l=26mPa·s

（ a）

（ b）

1.5 l=1000mPa·s

1.2

2.1

l=630mPa·s l=216mPa·s

1.8

l=150mPa·s

0.9

l=94mPa·s

1.5

l=26mPa·s

k2

k1

l=83mPa·s

0.6

l=1000mPa·s l=630mPa·s l=216mPa·s l=150mPa·s l=94mPa·s l=83mPa·s

1.2

l=26mPa·s

2.39

0.9 0.3

2.54

1

1

0.6 0.0 0.0

0.1

0.2

0.3

0.4

 Ca

-0.04

0.8

0.5

0.6

0.7

0.1

0.2

0.3

0.4

0.5

 Ca 0.52

Fig. 6.

30

-0.06

0.6

0.7

0.8

0.25

wmax (mm)

0.20

0.15

0.10

0.05

0.00

0.1

0.2

0.3

Fig. 7.

31



0.4

0.5

0.6

0.8

l=1000mPa·s l=630mPa·s l=216mPa·s

0.7

l=150mPa·s

Tc

l=94mPa·s

0.6

l=83mPa·s l=26mPa·s

0.5

0.3 1

0.4 1.2

1.6

-0.34 Fig. 8.

32

2.0

2.4

1.8

l=1000mPa·s l=630mPa·s

1.5

l=216mPa·s l=150mPa·s

1.2

l=83mPa·s

3

Vf /wc

l=94mPa·s

0.9

l=26mPa·s

2.56 0.6

1

0.3 0.0

0.1

0.2

0.3

0.4

 Ca 0.73

Fig. 9.

33

0.5 -0.07

0.6

0.7

Table 1. Solution (wt%)

ρl (kg·m-3)

μl (mPa·s)

σ (mN·m-1)

glycerol

1210

1000

64.5

99%gly

1210

630

62.8

93%gly

1200

216

64.8

90%gly

1190

150

64.9

87%gly

1150

94

65

80%gly

1090

83

65.7

75%gly

1080

26

66.3

34

1.13 1.02 0.92 0.71 0.51

k2 1.69 1.50 1.37 1.17 0.91

1.8

Tc T/ms 0.4 0.43 0.46 0.55 0.62

l=1000mPa·s l=630mPa·s

40 42 41 38 35.5

1.5

l=216mPa·s l=150mPa·s

1.2

V/wc

3

0.9

k1

0.56 0.35 0.26 0.15 0.09

0.6

l=94mPa·s l=83mPa·s

3

1.2

Vf /wc

1.5

0.9

l=26mPa·s

2.56 0.6

0.3

1

0.3

0.0 0.0

0.2

0.4

0.6

t/T

0.8

1.0

0.0

0.1

0.2

0.3

0.4

0.73Ca-0.07

0.5

0.6

0.7

Highlights: 1. Bubble formation in highly viscous liquids is studied 2. Liquid viscosity and gas-liquid flow rate ratio affect the bubble formation dynamics 3. The linear expansion stage and breakup stage for bubble formation are highlighted 4. A scaling law is proposed to predict the size of bubbles

35