Breakup dynamics of gas-liquid interface during Taylor bubble formation in a microchannel flow-focusing device

Journal Pre-proofs Breakup dynamics of gas-liquid interface during Taylor bubble formation in a microchannel flow-focusing device Xingchen Li, Yiyong Huang, Xiaoqian Chen, Bengt Sunden, Zan Wu PII: DOI: Reference:

S0894-1777(19)31294-4 https://doi.org/10.1016/j.expthermflusci.2020.110043 ETF 110043

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Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

8 August 2019 1 January 2020 8 January 2020

Please cite this article as: X. Li, Y. Huang, X. Chen, B. Sunden, Z. Wu, Breakup dynamics of gas-liquid interface during Taylor bubble formation in a microchannel flow-focusing device, Experimental Thermal and Fluid Science (2020), doi: https://doi.org/10.1016/j.expthermflusci.2020.110043

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Breakup dynamics of gas-liquid interface during Taylor bubble formation in a microchannel flow-focusing device Xingchen Li1,2, Yiyong Huang3, Xiaoqian Chen3*, Bengt Sunden2*, Zan Wu2 1College

of Aeronautics and Astronautics, National University of Defense Technology, Changsha, China

2Department

of Energy Sciences, Lund University, Box 118, Lund, Sweden

3Defense

Technology Academy of Military Sciences PLA China Beijing, China

Abstract This work aims to investigate the breakup dynamics of the gas-liquid interface during bubble formation in a microchannel flow-focusing device. An interface tracking method is developed to capture the profiles of the gaseous thread evolution. The results show that the pinch-off period can be further divided into a liquid squeezing stage and a free pinch-off stage in both the radial and axial directions. The time domain criterion between these two stages in a low viscous liquid, with Ohnesorge numbers 𝑂ℎ ≪ 1, is proved to be shorter than the capillary time. The effects of surface tension, viscosity and gas inertial force exerting on the interface during the free pinch-off stage are proved similar to those in a quiescent liquid pool. The power law of the minimum diameter at the gaseous thread to the pinch-off remaining time in the present experiments agrees with previous studies in both ranges (1/3 to 1/2) and tendency.

To whom all corresponding should be addressed. [email protected] & [email protected] *

Keywords Microfluidics, Multiphase flow, Nonlinear dynamics, Interface, Confinement, Pinch-off

1. Introduction With the significant progress on microfabrication and microelectronic techniques, the microfluidics involving lab-on-a-chip have been applied in various branches, such as chemical engineering, bio-chemical engineering, pharmaceutical technology, medicine science, microelectromechanical system (MEMS), etc. [1–6]. Gas-liquid two-phase flow is of vital importance in these microfluidic devices concerning mixing, sorting, performance evaluation and reaction which all benefit from the enlarged unit contact surface area and the relatively stable transportation. Thus, the evolution of a gas-liquid interface and its dynamics during bubble breakup and formation in microchannels have attracted much attention [7,8], especially in a flow-focusing device, which is one of the most popular bubble generating devices. In flow-focusing devices, the pinch-off period, also named as the nonlinear breakup stage, indicates the time period from the moment when the gas-liquid interface detaches from the wall to the moment when the gaseous thread breaks up into a bubble or slug. [9,10]. By analogy with the controlled bubble pinch-off in a quiescent liquid pool, the interface transition has been theoretically investigated and predicted under two assumptions, (i) the shape of the bubble is locally thin, slender and symmetric around the gaseous thread neck, and (ii) the velocity field is irrotational [11–15]. The minimum radial radius of the neck region 𝑅0 is generally acknowledged as 𝑅0 = 𝐴𝜏𝛼.

τ is the remaining time to pinch-off. The power exponent 𝛼 was observed as 0.56~0.58 by [16] in a low-viscous liquid (water) and 0.5~1 by [14] in a viscous liquid (silicone oils and glycerol solutions). After a Rayleigh-like pair of equations of the interface profiles in the pinch-off period deduced by Gordillo (2008), a smooth transition of the power exponent 𝛼 from an inviscid liquid to a viscous liquid in a quiescent pool is described naturally. Compared to the bubble pinch-off theory in static liquid pools without confinement, effects of the wall confinement and the microfluidic inlet junction on bubble pinch-off are scarcely investigated in the literature[17]. With the additional liquid filling effect and the wall confinement effect in the microchannel flowfocusing device, the power exponent 𝛼 was observed within to be 1/3 to 1/2 in various experiments but could not reach a uniform mathematical explanation [18– 20]. For a better understanding of the Taylor bubble pinch-off in microchannel flowfocusing devices, the pinch-off period was divided into a liquid squeezing stage and a free pinch-off stage. The time criterion was established as the capillary time tcap to pinch-off [9,19]. This criterion was compared by [10] upon the 𝛼 mutation in the radial radius in a viscous liquid. The results stated that 𝛼 was 1/3 in the liquid squeezing stage and 1/2 during the free pinch-off period. However, with the axial direction deflection in flow-focusing devices, not only the radial motion of the neck point of the gaseous thread but also the axial motion and evolution should be tracked and analyzed to establish a uniform description. The neck region evolution of the gas-liquid interface and the breakup dynamics during the pinch-off period in a flowfocusing device for both inviscid and viscous liquids are still unclear so far. Therefore, investigations on the interface evolution and the effect of different forces are quite essential.

In this paper, an experimental facility is built up and an interface tracking process is established to capture and coordinate gaseous threads profiles during the pinch-off period. The results are displayed and analyzed considering the surface tension, gas inertial force and viscosity effect. By scaling the motion of the neck region in both axial and radial directions, it is evident that the viscosity effect should be included to describe the transitional time criterion between the liquid squeezing stage and the free pinch-off stage. A dynamic analysis of the pinch-off period has been implemented. The viscous shear force direction is changed between the two pinch-off stages, which is caused by the axial direction motion. Subsequently, the axial direction motion of the neck region during the pinch-off period, which has not been quantified in previous studies, is analyzed covering various surface tensions, gas inertial forces and viscosity effects. The results indicate that the time criterion between the liquid squeezing stage and the free pinch-off stage can be correlated with the Ohnesorge number, especially for low viscosity liquids. Finally, some conclusions are obtained to describe the rules from inviscid liquids to viscous ones.

2. Experimental procedures This section presents the experimental setup and how to acquire effective information by image processing of different experimental data, as well as the rationality of the image processing method.

2.1. Experimental setup The rig of the experiment facility used in the present study is shown in Fig. 1a. It mainly consists of two syringe pumps (New Era, NE-4000) equipped with 50 mL syringes, a Nitrogen loaded high pressure gas cylinder, a gas mass flow meter/controller (Bronkhorst, EL-Flow prestige FG-200CV), and a high-speed digital camera (VITcam CTC) for flow visualization. The square cross-section

microchannel with a width W of 600 μm (the accuracy is within ±10 μm) was fabricated by Little Things Factory GmbH in a 140 mm × 50 mm borosilicate glass plate. The edge at the junctions is rounded and the radius is within 100 μm, which is decided by the processing technique of the factory. Taking advantage of the material property and the thermal bonding sealing technique, the microchannel has excellent chemical resistance, a relatively low surface roughness and a high transmittance of light. The specially designed connectors were employed to convert 1.5 mm O.D. flexible tubes screwed on the microchannel’s inlets and outlet. The connectors are fastened to the microchannel by PDMS (polydimethylsiloxane). Particularly, the length of the flexible tubes at the inlets and the outlet was set as 100 mm, guaranteeing a uniform status of the pressure drop. The gas phase (dispersed phase) was fed to the main channel under control of a gas meter/controller and the liquid phase (continuous phase) was fed to two side channels by individual syringe pumps. As shown in Fig. 1b, the dispersed phase and the continuous phase met at the junction region of the flow-focusing microchannel and formed Taylor bubbles as described by [21]. The flow direction and flow rates of both phases were shown in Fig. 1b as well. The liquid flow rate was Ql and the gas flow rate was Qg. All experiments were conducted at Ql = 40 mL/h while Qg varied from 40 mL/h to 160 mL/h. The observed scenes were magnified by a manual lens (Laowa, FF65mm F2.8) and recorded by a high-speed digital camera at 6 kfps (kilo frame per second). At least a 300 s time elapse was applied between two different flow conditions to ensure the stabilization of the flow pattern. A differential pressure sensor (Yokogawa, EJA110E) was connected to the inlet of the gas (Nitrogen) and the outlet tube for monitoring the pressure drop in the microchannel. Data were captured when the pressure drop only fluctuated within 0.01 bar.

(a)

(b) Fig. 1 (a) Schematic diagram of the experimental apparatus for bubble formation in a flow-focusing device; (b) the image for bubble formation and the coordination setup of the picture

Six immiscible liquid phases were considered, with Nitrogen as the gas phase were tested in order to study the effect of physical properties on the neck region evolution during the pinch-off period in this flow-focusing device. Nitrogen was used as the dispersed phase, while de-ionized water and five aqueous solutions including 50% glycerol (MP Biomedicals, LLC), 80% glycerol, 90% glycerol, 0.1% SDS (Sodium dodecyl sulfate) and 0.5%SDS solutions were employed as the continuous phase. Properties of the liquids are listed in Table 1. The capillary time is the characteristic length W divided by the capillary velocity 𝑣cap = 𝜎/(𝜌𝑊). The local capillary time is 𝑡cap = 𝑊/ 𝜎/(𝜌𝑊) = 𝜌𝑊3/𝜎). These parameters are also shown in Table 1. Table 1 Physical properties and capillary parameters of various continuous phases Liquid phase

surface tension, σ (mN/m)

Viscosity, µ (mPa s)

Density, ρ (kg/m3)

Capillary velocity, 𝑣cap (m/s)

Capillary time, tcap (ms)

Water

72

0.92

1000

0.35

1.73

0.1% SDS/water

39

0.92

1000

0.25

2.35

0.5% SDS/water

33

0.92

1000

0.23

2.56

90% glycerol

64.2

163

1190

0.30

2.00

80% glycerol

64.9

54.4

1090

0.32

1.90

50% glycerol

66.5

5.7

1050

0.32

1.85

The viscosities of the SDS solutions and the water/glycerol mixtures were measured by a DV-III Brookfield viscometer at room temperature. The measurement uncertainty is within ±5%. The interfacial tension of the SDS solutions and the water/glycerol mixtures was measured by a drop volume tension meter (Lauda TVT 2). The uncertainty of the measurement is within ±5% [22]. The data were also compared with others [23]. As pointed out in previous works [14,20,24], the variation range of the surface tension and the liquid density is much narrower than that of the liquid viscosity. In order to compare the gas-liquid interface evolution at

different properties, two liquid groups were prepared, including an SDS/water group with almost the same viscosity but different surface tension values and a glycerol/water group with similar surface tension values while the viscosity varies within a wide range. By defining the mean velocity of the gas-liquid two-phase flow as U = (Qg +Ql)/W2, the capillary number Ca (Ca = Uμl/ σ) and the Reynolds number Re (Re = ρUW/μl) at Ql = 40 mL/h, Qg = 40, 80, 120 and 160 mL/h are listed in Table 2. W is the width of the square microchannel. Table 2 Reynolds number and capillary number of various continuous phases Liquid phase

Ca Qg =40 mL/h

Re Qg=40 mL/h

Ca Qg =80 mL/h

Re Qg =80 mL/h

Ca Qg=120 mL/h

Re Qg=120 mL/h

Ca Qg=160 mL/h

Re Qg =160 mL/h

Water 0.1% SDS/water 0.5% SDS/water Glycerol 90% Glycerol 80% Glycerol 50%

0.0008

40.23

0.0012

60.36

0.0016

80.48

0.0020

100.60

0.0015

40.23

0.0022

60.36

0.0029

80.48

0.0036

100.60

0.0017

40.23

0.0026

60.36

0.0034

80.48

0.0043

100.60

0.1567

0.27

0.2350

0.4054

0.3133

0.5405

0.3916

0.6757

0.0517

0.74

0.0776

1.113

0.1034

1.484

0.1293

1.854

0.0053

6.82

0.0079

10.23

0.0106

13.64

0.0132

17.05

2.2. Data processing method The photographic sequences captured by the high-speed digital camera were first processed by an image analysis software ImageJ’s plugin named Kappa and then by a home-made Matlab program for the curvature analysis. The processing steps are organized as follows: 1) uniform and crop the sequences as standard images along the main channel in ImageJ, 2) establish the ROI (region of interest) to extract the contour of the bubbles neck region by the key frame tracking method in the ImageJ’s plugin Kappa, 3) use the gray scale intensity thresholding and interface identical algorithm to recognize the interface in each key frame, 4) between each frame and pixel, the time and space scale linear interpolation is employed to achieve

subpixel and accurate frequencies for smoothing the fitting curve, then output the data, 5) a derivable curve function, R= f(Z), is fitted and obtained to describe the contour development of the neck region during the pinch-off period by an in-house Matlab program. The coordinate system was set alongside in the corner of the junction, as shown in Fig. 1b. The processing detail of the 50% Glycerol solution at Ql = Qg = 40 mL/h is shown in Fig. 2.

Fig. 2 The image tracking method of the gas-liquid interface for the pinchoff during bubble formation at the inlet junction. The green points are the control points. The area surrounded by blue lines is ROI. The pink region is an identification of the interface. The vertical blue line was the original Z position of

the narrowest neck region when the pinch-off started. τ is the remaining time to p

i

n

c

h

-

o

f

f

,

m

s

.

The green points in the magnified view are the tracking control points on the curve and modified on each key frame based on the frame tracking method. The three blue lines around the tracking control points were sketched ROI for gray scale intensity thresholding, after that the pink points were identified as the component of the bubble neck region contour, which numerically fits with the curve. The interface identical algorithm established the interface profiles within three pixels which were defined by the pink region in ROI. The gradient of pixel intensity was employed in this identical algorithm. Five gray values were used to differentiate the boundary. The error of this method is ± 7.5 μm. The beginning of the pinch-off period in this work is the same as that in previous references [19,23], when the interface of the gaseous thread starts to detach from the main channel wall. The pinch-off period ends at the neck breakup, so-called τ = 0 (τ is the remaining time to pinch-off, ms). The vertical blue line was the original Z position of the narrowest neck point when the pinch-off started, which indicated that the neck point not only moved in the radial direction but also in the axial direction. The fitting contours in each frame at 6 kfps were batch obtained by the Matlab program which also collected the coordinates as well as the curvature of the narrowest neck for subsequent analysis. Another manual tracking method for recognizing the neck point was also implemented for comparison. The result of the 50% Glycerol solution at Ql = Qg = 40 mL/h under manual tracking is shown in Fig. 3.

Fig. 3 The image manual tracking method of the gas-liquid (50% Glycerol) interface for the pinch-off during bubble formation at the junction. The blue and green points were manually picked tracking points. The manual tracking method caused the inevitable error in points picking and gray scale judgement, which led to feasible trend demonstrations but inaccurate quantitative analyses. Thus, for the following analysis of the neck region movement, only the programmed method by Matlab and ImageJ was applied while the manual tracking results are listed in Appendix as a contrast.

3. Experimental results and discussions This section mainly illustrates the photographic sequences and the processed results of the pinch-off period under designed gas-liquid two-phase flow in the flowfocusing microchannel. The bubbles pinch-off phenomenon on the neck region, the interface contour transformation and the scaling law are analyzed in this part as well for providing further understanding.

3.1. Dynamics of bubble pinch-off The dynamics of Taylor bubbles pinch-off in the 600 μm flow-focusing device with different designed liquid phases at Ql = Qg = 40 mL/h are shown in Fig. 4. The

error of the breakup point judgement is within 0.16ms, which is the time gap between two frames. However, one of the previous studies [10] also claimed the whole pinchoff period as a nonlinear collapse breakup compared to the device confinement as a first-stage linear collapse. From these image sequences, the gaseous thread becomes thinner and more slender with the decrease of remaining time to pinch-off. The different time gap of τ shown in Fig. 4 is determined by the scale of the pinch-off time of different continuous phases. The gas-liquid interface morphology affected by surface tension is illustrated in Fig. 4(a)(b)(c). Compared to water, the SDS aqueous solutions have lower surface tension values while the viscosity is barely changed. Comparing the tendency in the pictures, the neck regions become less slender at the last moment of the breakup with decreasing surface tension. At the axisymmetric gaseous neck region, from [9], the capillary pressure p is: 𝑝 = 𝜎(

1 1 ― ) 𝑅0 𝑅c

where, 𝑅0 is the minimum radial radius of the neck and 𝑅c is the minimum axial radius of the neck. As the inertial force and the viscosity are barely changed, the decrease in surface tension leads to 𝑅c rising at the neck region to guarantee that the capillary pressure can keep the balance of the three interaction forces at the breakup moment. The length of the neck region and the whole time period of pinch-off also decline with decreasing surface tension. The vertical blue lines in Fig. 4(a)(b)(c) illuminate the original position of the Z direction of the narrowest neck region, which shows that the pinch-off point movement is affected by the variation of surface tension. The quantitative analysis will be given in the following paragraph.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4. Dynamics of Taylor bubble pinch-off in the flow-focusing device of different liquids. (a) water; (b) 0.1% SDS solution; (c) 0.5% SDS solution; (d) 50% glycerol solution; (e) 80% glycerol solution; (f) 90% glycerol solution. The blue lines indicate the original neck thinning position. The unit for τ is ms. The effect of liquid viscosity on the gas-liquid interface morphology is illustrated in Fig. 4(d)(e)(f). Previous studies [14,25,26] focused on the continuous phase viscosity variation, because its viscosity can be tailored within a wide range. The glycerol solutions applied in the present experiments have a viscosity range from 5.7 to 163 mPa s with negligible changes in density and surface tension. In these sequences, the Taylor bubble size shrinks with the increase of glycerol concentration. The thickness of the liquid film alongside the wall of the main channel

increases as well. Compared to water and SDS solutions, glycerol solutions have smaller bubble sizes and thicker liquid films, as evident in the obtained images. The gaseous threads tend to be slender and the pinch-off time tends to be shorter with increasing viscosity. It has been reported that the gas-liquid interface is more hyperbolic in the low-viscosity liquid and is more parabolic in the high-viscosity liquid [20]. This tendency is quite similar to the phenomenon in bubble generation experiments without wall constraints [25]. The blue lines also indicate that the pinchoff point movement in the Z direction is affected by the viscosity of the liquid as well.

3.2. Interface evolution during the pinch-off period The interfaces of these gas-liquid flows in photographic sequences during the pinch-off period were captured by the method developed in chapter 2.2, as shown in Fig. 5. Considering the symmetry of interface evolution along the axial direction in the 2D image [20], a continuous sequence of one curve can indicate the whole period development.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5. The one-side profiles evolution of the gas-liquid interface during bubble pinch-off at the junction for different liquids. The horizontal curves are the interface of one-side profiles in each frame. They vary from the beginning of the pinch-off to the end, basically from top to bottom alongside the vertical red lines. The vertical red lines are the connection of the neck points in each frame during the pinch-off period. The curvatures of the neck point at thread breakup are shown at

the bottom of the profiles series. The frame rate is 6000 Hz. (a) water; (b) 50% glycerol solution; (c) 80% glycerol solution; (d) 90% glycerol solution; (e) 0.1% SDS solution; (f) 0.5% SDS solution. In the interface spatial evolution, the asymmetry of the curve along the radial direction at the minimum neck is quite obvious, especially for low-viscosity liquids like water and SDS solutions. This resembles the observation of bubble pinch-off in an unconfined tank [11]. This asymmetry is mainly dominated by the interaction of the inertial force and the buoyancy force at the neck region. However, considering that the buoyancy might be negligible in microchannels, the wall confinement effect and the inertial force positively contribute to the curve asymmetry. In addition, as shown in Fig. 5(a)(b)(c)(d), the viscous shear increases the elongation effect exerted on the gas-liquid interface during the gaseous shrinking process, which indirectly results in less asymmetric profiles not only at the minimum neck but also the entire gaseous threads in the cross junction as well. The slendering characteristic is also indicated by the curvature of the finest point at the gaseous thread neck region. The minimum curvature of the 90% glycerol solution at the breakup moment is computed as 0.00287 μm-1, which is 37.2% of the average curvature of water and the other two glycerol solutions. The minimum curvatures of the tested gas-liquid flows at the pinch-off moment have the same order of magnitude, which also confirms that the slendering assumption could be valid for a wide range of liquid viscosities from water to glycerol solutions and for various surface tension values, e.g., SDS solutions. The self-similarity of the profiles mentioned in previous studies [14,16] during the pinch-off period is also evident in the present study, which indicates that the profiles can be fitted by two main parameters. In addition, the parabola assumption of the profiles mentioned by [12,13] in a quiescent liquid pool, 𝑅(𝑡,𝑧) = 𝑅0(𝑡) + 𝑟𝑐(𝑡)𝑧2, in which 𝑅0(𝑡) is the dimensional minimum radial radius of the

neck (𝑅0(𝑡) = 𝑅0(𝑡)/𝑊) and 𝑟𝑐(𝑡) is 1/2 of the dimensional neck region curvature (𝑟𝑐(𝑡) = 𝑟𝑐(𝑡)𝑊), is consistent with profiles of the last few frames before breakup in Fig. 5. This conformity of the bubble pinch-off between the confinement channel and the unconstrained bulk indicates that the evolution of the interface at the last moment of the pinch-off is quite independent of the wall effect and mainly driven by the balance of viscosity, inertial force and surface tension. This time period is divided from the whole pinch-off period by [19] and [10], which is named as the free pinch-off stage based on the time criterion of tcap. The stage before the free pinch-off stage is named as the squeezing collapse stage where the liquid squeezing effect plays the key role. They also fitted one of the profile parameters 𝑅0(𝑡) evolution into two stages with the remaining pinch-off time τ, as a power law, 𝑅0(𝑡)~𝐴𝜏𝛼, with different rules at 𝛼 = 1/3 for the free pinch-off stage and 𝛼 = 1/2 for the liquid squeezing collapse stage. However, these criteria mainly focus on the development of the radial radius 𝑅0 and the timescale of the interface evolution during the pinchoff period. According to Table 1, the capillary time of different gas-liquid systems in the same flow-focusing device varies slightly with a small range from 1.73 ms to 2.56 ms. Thus, a similar time criterion can not state the interface evolutionary process mutation over a big range of liquid property variation comprehensively. The line chart of the minimum neck point shown in Fig. 5 reveals that the axial direction movement should not be ignored when dividing the pinch-off stage, especially for inviscid liquids and the low-viscosity liquids. As for the power rules over the radial radius, in previous studies [13,25] 𝛼 varies with τ continuously during the free pinchoff stage. This 𝛼 variation can also be fitted by several exponential functions during different periods. Figure 6 shows the neck point at the capillary time before the breakup, which is marked by the orange line. The obvious mutation of the interface evolution always slightly lags than the capillary time, which infers that the free

pinch-off stage may be shorter than τ, especially for the low-viscosity liquids. This indicates that the time criterion is affected by surface tension, density, the characteristic length, as well as the liquid viscosity. The classical theory of capillary breakup explains that the shrinking rate of the gaseous thread is correlated with the 𝐶𝑎

Ohnesorge number, 𝑂ℎ2 = 𝑅𝑒 =

𝜇2 . 𝜌𝜎𝑙

Therefore, the time criterion should include the

Ohnesorge number of the liquid phase, which requires further theoretical and experimental investigations. The interface profiles shown in Fig. 5 also indicate the time domain evolution of the gaseous threads during the pinch-off period. The increase of the gaseous thread shrinking speed and tendency movement of the SDS solutions is slower than other solutions at the same flow rate obviously, which might be caused by an insufficient capillary driving force due to surface tension reduction. This tendency infers that surface tension has a positive role to improve the radial velocity gradient of the neck region, 𝑅0(𝑡), during the pinch-off period. Besides, the gaseous thread expansion at the junction area during the pinch-off period appears in water, 0.1% SDS solution and 50% glycerol. This phenomenon may lead to a reduction of the gas flow volume through the thread and an increase of the liquid jetting speed into the neck portion and thus improve the pinch-off efficiency. This process was also recorded by [18] in one case, but further study of this phenomenon is still needed.

3.3. Effect of different forces on the neck point evolution during the pinch-off period

The effect of surface tension, viscosity and the gas inertial force on the neck point evolution during the pinch-off period will be discussed in this section based on the image tracking data processing in section 2.2.

(a)

(b) Fig. 6 The motion of the neck point in the coordination setup in 2.2. (a) water, 0.1% SDS solution, and 0.5% SDS solution; (b) 50% glycerol solution, 80% glycerol solution, and 90% glycerol solution. Ql = Qg = 40 mL/h. The orange marks show the position of the neck point at the capillary time just before pinch-off.

The evolution of the neck region in water and SDS solutions is shown in Fig. 6(a). The properties of water, 0.1% SDS and 0.5% SDS solutions are listed in Table 1 with the surface tension varying from 72 to 33 mN/m. First, the final breakup point moves towards the junction inlet with decreasing surface tension, in which the end position of the 0.1% SDS solution is almost at the end of the junction and that of the

0.5% SDS solution is located at the right middle of the junction. Second, opposite to water and the 0.1% SDS solution, the neck region motion of the 0.5% SDS solution deflects towards the upstream of the axial direction at the liquid squeezing stage, which may be due to the partial detachment at the end corner of the junction caused by contact angle changing. However, the gaseous thread thinning process in the free pinch-off stage of the 0.5% SDS solution is similar to the others in the way that the inward radial motion of the minimum radius at the gas-liquid interface deflects towards the downstream along the axial direction [16]. In summary, the above analysis infers that a decrease in surface tension tends to switch the flow pattern from dripping in the channel to dripping in the junction. In a previous work [22], with increasing capillary number of the continuous phase in liquid-liquid systems, the dripping flow changed to jetting flow within the same liquid system, but in some situations, even an increase in capillary number increase in liquid-liquid two-phase flow systems may lead a transition from jetting flow into dripping. In Fig.6(a), the neck point motion of the gaseous thread shows a great difference from the 0.1% SDS solution to the 0.5% SDS solution with a 15% decrease in static surface tension. However, in this case, the static surface tensions listed in Table 1 are not the only parameters that indicate the interface tension. The dynamic interfacial tension shows an important influence on bubble formation and breakup at microscale. The transformation law of dynamic interfacial tensions varies with the surfactant concentration when it is below or higher than the critical micelle concentration (CMC). The pinch-off phenomenon changes with the dynamic interfacial tension. Wang et al. [27] tested the dynamic interfacial tensions over different surfactant concentrations. They figured out that when the concentration was higher than the CMC, the dynamic interfacial tension would drop a lot, around 50%. However, the dynamic interfacial tension remained almost the same when the

surfactant concentration was less than the CMC. For SDS solution cases, the CMC is 8.0-8.3×10-3 mol/L and the molecular weight is 288.4 g/mol. The concentration of the 0.1% SDS solution is 3.5×10-3 mol/L and for the 0.5% SDS solution it is 17.3 ×10-3 mol/L. As the concentration of the 0.5% SDS solution is higher than the CMC, the dynamic interface tension decreases during the pinch-off period, which leads to the very different pinch-off processes in this case. The evolution of the neck region in water and glycerol solutions is shown in Fig. 6(b) regarding the effect of viscosity. The properties of water and different glycerol solutions are listed in Table 1 with the viscosity varying from 0.92 to 163 mPa s. First, the movements of the neck region of nitrogen-glycerol solution interfaces are located within the junction, which indicates that the dripping flow prevails in the junction during this range of viscous shear variation. Second, with increasing glycerol concentration in the solution, the motion and orientation of the neck region gradually evolve from downstream to upstream along the axial direction in the liquid squeezing stage. Especially during the transition, the neck region of 80% glycerol solution barely shows the fixed axial direction tendency and shrinks almost alongside the radial direction. However, the gaseous threads maintain a consistency of motion trends during the free pinch-off period; meanwhile, the downstream deflection is also suppressed by at least 30% in comparison to water and SDS solutions as the continuous phase. In addition, the mutation of the axial direction motion in the squeezing stage for SDS solutions does not survive in viscous liquids, even for the 50% glycerol solution.

Fig. 7 A schematic illustration for the dynamics of bubble pinch-off. The directions of the viscous shear force shown in the two pinch-off stages are different because of the axial deflection in the liquid squeezing stage.

After the spatial domain analysis of the neck region during the pinch-off period, the forces exerted on the thread are further described in Fig. 7. The forces include the force due to the flow of the liquid phase (Fl), the force due to the flow of the gas phase (Fg), the interfacial tension force of the neck (Fs_in and Fs_out) and the viscous shear force (Fμ). These forces are in an equilibrium process during the evolvement of the interface. In the pinch-off period, the equilibrium state is gradually destroyed because of the change of the curvature in the neck and then interfacial tension force dominates over the others to break up the thread. However, the shear force, which acts as the resistance force during the breakup, behaves differently between the liquid squeezing stage and the free pinch-off stage. In the liquid squeezing stage, the axial deflection of the squeezing liquid gives rise to the resistance to the interfacial tension force, which slows down the pinch-off process.

However, in the free pinch-off stage, the perpendicular angle between the shear force and the interfacial tension force leads to barely no resistance on the thread breakup. Therefore, the axial deflection of the neck point is an important basis for distinguishing the two stages. In the time domain, the whole period of bubble formation and the time period of the pinch-off stage with increasing gas flow rates and fixed liquid flow rates are compared in Fig. 8. The whole period life cycle of the Taylor bubble decreases with increasing gas flow rate; meanwhile, the pinch-off time slightly declines in every tested gas-liquid system. This comparison infers that the gas inertial force promotes the generation efficiency of Taylor bubbles in flow-focusing devices while the pinch-off period is independent of this effect. Considering the relatively small value of the gas-liquid density ratio (Λ =

𝜌𝑔 𝜌l

= 0.00105~0.00125), the gas inertial force

is not competitive with the interfacial force during the pinch-off stage. The gas inertial force mainly affects the flow pattern transition and slug length variation. This phenomenon in profiles transition was also realized during the pinch-off period in a quiescent liquid pool without confining boundaries by [13], which may indicate that in the same flow pattern, such as slug flow, the gas flow rate barely influences the pinch-off period in both the time and spatial domains. As for the viscosity effect on the bubble formation cycle and the pinch-off period shown in Fig. 8(a), both life time and pinch-off time decrease with increasing viscosity from 5.7 mPa s to 54.4 mPa s, in which the life time declines to 70.5% and pinch-off time decreases to 76.0%. In contrast, both the life time and the pinch-off time vary in a negligible range between 80% glycerol and 90% glycerol solutions when the viscosity rises from 54.4 mPa s to 163 mPa s. It seems that there exists a transitional viscosity regarding the bubble formation cycle, while the time of the pinch-off period does not change. This might be due to two reasons. First, the

junction area expansion of the 50% glycerol solution leads to the delay of bubble formation and pinch-off for longer interface traveling distances and larger volumes in the radial direction. Second, the obvious mutation of the bubble generation between 50% and 80% glycerol solutions shown in Fig. 4 initiates pinch-off in higher viscous liquids, which is quite different from the less viscous ones. For less viscous liquids the gaseous threads are thinner and the Taylor bubbles are more slender.

40 50% Gly whole 50% Gly pinch-off 80% Gly whole 80% Gly pinch-off 90% Gly whole 90% Gly pinch-off

35

Time (ms)

30 25 20 15 10 5 0

40

60

80

100

120

Gas Flow Rate (mL/h) (a)

140

160

40 water whole water pinch-off 0.1% SDS whole 0.1% SDS pinch-off 0.5% SDS whole 0.5% SDS pinch-off

35

Time (ms)

30 25 20 15 10 5 0

40

60

80

100

120

140

160

Gas Flow Rate (mL/h) (b) Fig. 8. Evolution of bubble lifetime and pinch-off time in different liquids under a series of gas flow rates. (a) 50% glycerol solution, 80% glycerol solution, and 90% glycerol solution; (b) water, 0.1% SDS solution, and 0.5% SDS solution. The effect of surface tension on the time domain of Taylor bubbles formation and pinch-off with increasing gas flow rates is shown in Fig. 8(b). The surface tension shows the same tendency with both life cycle and pinch-off period duration, decreasing at the same pace, without mutation motion as the glycerol cases. The average decline percentage from water to 0.1% SDS solution (surface tension varies from 72 mN/m to 39 mN/m) is 17.8% for the life cycle and 39.4% for the pinch-off period. The average decline percentage from 0.1% SDS to 0.5% SDS (surface

tension varies from 39 mN/m to 33 mN/m) solutions are 28.5% for the life cycle and 26.0% for the pinch-off period, which infers that surface tension has a vital role during the pinch-off period but is less important for the bubble formation efficiency. However, the decrease in surface tension promotes the generation of Taylor bubbles and the pinch-off efficiency. By comparing (a) and (b) in Fig. 8, it is obvious that the surface tension variation within a small range influences the pinch-off and bubble formation in flow-focusing devices in time domain more than the viscosity variation within a wide range. These phenomena did not show up in the previous study. In Table 3, the pinch-off frequency of the slug bubbles is listed at Ql = Qg = 40 mL/h. These results are calculated from Fig. 8. At the same flow rate for the glycerol solutions, as the viscosity increases, the pinch-off frequency rises from 36.97 Hz to 49.48 Hz. However, the pinch-off frequency in 80% and 90% glycerol solution is almost the same. For water and SDS solutions, the frequency rises from 28.11 Hz to 49.90 Hz as the static surface tension of the liquid decreases from 72 mN/m to 33 mN/m. In addition, the dynamic interfacial tension, which is mentioned in section 3.3, may also affect the pinch-off frequency. About a 15% decrease in surface tension between 0.1% SDS and 0.5% SDS solutions leads to a 34% increase in the pinch-off frequency. The 0.5% SDS solution has a lower dynamic interface tension than its static surface tension.

Table 3 The pinch-off frequency of the slug bubbles in different liquids at Ql = Qg = 40 mL/h 50%

80%

90%

glycerol

glycerol

glycerol

Water

0.1%

0.5%

SDS

SDS

Pinch-off frequency

36.97

49.90

49.48

28.11

36.97

49.90

(Hz)

3.4. Scaling law for the neck region during the pinch-off period The evolution of the minimum radial radius 𝑅0 of the neck region with the remaining time for a series of glycerol concentration is illustrated in Fig. 9. The development rules for the radial radius, 𝑅0 = 𝐴𝜏𝛼, have been established and discussed over decades. Both the 𝛼 value and its increasing tendency with the remaining time approaching zero have been investigated theoretically and experimentally in previous studies [10,11,13]. Consistent with pinch-off power rules, 𝑅0 values during the whole pinch-off period of different glycerol contents are fitted in the Matlab curve fitting tool with a power function model that employs a trustregion algorithm. The power fitting results, coefficients of determination (R-square) and root mean square errors (RMSE) are listed in Table 4. The experimental power fitting results of glycerol-ethanol [10] are also shown in Table 4 for comparison. The similar range of 𝛼 indicates consistent results. The logarithmic coordinate system, as shown in Fig. 9, illuminates the mutation of 𝛼 at the stage split point between the free pinch-off stage and the liquid squeezing stage. The orange points of 80% and 90% glycerol solutions are located around the capillary time τcap. However, there is an undeniable deflection between the stage split point and the τcap range for the 50% glycerol solution. This result is in agreement with the analysis of Fig. 6, that the time criterion should involve the viscosity effect, or the deviation

from stage division could be enlarged in the low viscosity liquid-gas flow focusing. Other than this, either the power index value or its tendency shows a good agreement with previous studies [10,19,28].

Fig. 9. The evolution of the minimum radius of the gaseous neck 𝑅0 with remaining time until the pinch-off (τ) in logarithmic coordinate system for the 50% glycerol solution, the 80% glycerol solution and the 90% glycerol solution. The orange points are the mutation points of different liquids. Ql = Qg = 40 mL/h

Table 4 The fitting results and errors of gaseous threads’ minimum radial radius during the pinch-off period in a series of glycerol solutions and the comparison with a previous work 50% glycerol

80% glycerol

90% glycerol

glycerol-ethanol [10]

𝛼

0.2854

0.2736

0.3719

1/3, followed by 1/2

R-square

0.9925

0.9774

0.9993

--

RMSE

2.858

4.927

2.3

--

To further validate the mutation between the two periods and the motion of the neck point, the axial direction Z(τ) motion of the neck point is displayed in the semi-logarithmic coordinate system in Fig. 10. The remarkable stage split points between the free pinch-off stage and the liquid squeezing stage are determined by a sharp turn in the scatter plot. A similar inference can be presented in the way that high-viscous liquids transit into the free pinch-off stage at around τcap while the lowviscous liquid transits into the free pinch-off stage in a shorter time than the capillary time. In addition, the instability of thread shrinking during the free pinch-off period is also indicated in this figure, which is shown as the fluctuation motion in the axial direction for each solution.

Fig. 10. The evolution of the axial position of the gaseous neck 𝑍0 with remaining time until the pinch-off (τ) in semi-logarithmic coordinate system for the 50% glycerol solution, the 80% glycerol solution and the 90% glycerol solution. The orange points are the mutation points of different liquids. Ql = Qg = 40 mL/h If we consider the time domain criterion between the liquid squeezing stage 𝐶𝑎

and the free pinch-off stage by using the Ohnesorge number, 𝑂ℎ2 = 𝑅𝑒 =

𝜇2 , 𝜌𝜎𝑙

and

the capillary time tcap, the viscosity effect can be compared in these three glycerol solutions. In the dimensionless study of ‘self-thinning’ process by [29], it was indicated that capillary-thinning and break-up processes are mainly governed by three main characteristic time scales: a viscous time scale 𝑡𝑣𝑖𝑠𝑐 = 𝜇𝑙/𝜎, the

polymeric time scale 𝑡𝑝𝑜𝑙𝑦 = λ and the capillary time 𝑡𝑐𝑎𝑝 = 𝜌𝑊3/𝜎. The 𝑡𝑣𝑖𝑠𝑐

Ohnesorge number can also be presented as 𝑂ℎ = 𝑡 = 𝜇/ 𝜌𝜎𝑙, which means the 𝑐𝑎𝑝 ratio of the viscosity time domain to the capillary time domain. The Ohnesorge numbers of 90%, 80% and 50% glycerol solutions are 0.761, 0.264 and 0.028, respectively. For the 50% glycerol solution, the Ohnesorge number 0.028 is one order of magnitude less than the other two, which infers that the viscosity effect is far less than the capillary driven force. This shows that for the low viscosity fluid, 𝑂ℎ ≪ 1, the viscosity acts as an unmatched resistance to the capillary driven force, leading to a shorter free pinch-off period, which is called the free surface selfthinning. However, in a viscous Newtonian fluid, 𝑂ℎ~1, the viscosity resistance force may reach a balance with the driving surface tension, which shows that the mutation point is at around tcap.

4. Conclusions To conclude, an interface tracking process has been established, which involves gray scale intensity thresholding and a key frame tracking method, in order to capture and analyze the interface profiles of the gaseous thread transition during the Taylor bubbles pinch-off period in a microchannel flow-focusing device. The evolution of the gas-liquid interface and dynamics of the pinch-off period were analyzed in six designed gas-liquid systems with both surface tension and viscosity variation. The profiles spatial analysis, especially the evolution of the slenderness between SDS and glycerol solutions validates that surface tension dominates the final breakup while viscosity promotes the slenderness, which is also confirmed by comparing the curvatures at the finest neck region. The self-similarity of the profiles

at the free pinch-off stage is consistent with the parabolic characteristic of free surface bubbles breakup in a quiescent liquid pool. In the free pinch-off stage, the axial direction motion only fluctuates at a negligible range caused by fluid oscillation. In contrast, the big axial direction deflection during the liquid squeezing stage is under influence of both surface tension and viscosity, in which the surface tension mainly acts on the flow pattern variation and the viscosity acts more on the axial direction and displacement. Through the comparison between the whole bubble formation life cycle and the pinch-off time period with increasing gas flow rates for different liquids, the gas inertial force is quite small during the pinch-off period. The time domain criterion between the free pinch-off stage and the liquid squeezing stage is validated concerning both the capillary time tcap and the Ohnesorge number of the liquid phase by scale law of both the minimum neck radial radius and the axial mutation. For a low viscosity fluid in the flow-focusing device, 𝑂ℎ ≪ 1, the mutation is shorter than that of a high viscosity fluid, 𝑂ℎ~1, which is reported by the capillary time, caused by the reduction of viscous shear resistance to the driving force, i.e., surface tension. These results will assist with the further understanding of the dynamics in bubble breakup and transportation in micro-scale flow-focusing devices. For the Taylor bubbles in the experiments, the ratio of the bubble size (or length) to the channel width ranges from 2.0 to 20.0.

Nomenclature W

the width of microchannel, μm

g

gravitational constant, m s-2

𝑅0

the minimum radial radius of the gas thread, μm

𝑅c

the minimum axial radius of the gas thread, μm

Ql

liquid volumetric flow rate, mL/h

Qg

gas volumetric flow rate, mL/h

𝑣cap

the local capillary velocity, m s-1

τ

the remaining time to pinch-off, ms

tcap

the local capillary time, ms

𝑡𝑣𝑖𝑠𝑐

the viscous time, ms

𝑡𝑝𝑜𝑙𝑦

the polymeric time scale, ms

U

the mean velocity of the gas-liquid two-phase flow, m s-1

p

the capillary pressure, Pa

l

the characteristic length, m

R

radial direction, μm

Z

axial direction, μm

Λ

the gas-liquid density ratio

𝜌𝑔

gas phase density, kg m-3

𝜌l

liquid phase density, kg m-3

Fl

force due to flow of the liquid phase

Fg

force due to the flow of the gas phase

Fs_in

interfacial tension force of the neck in inlet

Fs_out

interfacial tension force of the neck in outlet

Fμ

viscous shear force

Greek letters σ

surface tension, mN m-1

μ

viscosity, mPa s

ρ

density, kg m-3

Dimensionless groups Ca

capillary number

Re

Reynolds number

Oh

Ohnesorge number

Acknowledgments The financial supports for this project are from Swedish Research Council, the Crafoord foundation, the ÅForsk Research Foundation and the National Science Foundation for Young Scientists of China (Grant No. 11702320). X.C. Li appreciates the scholarship from China Scholarship Council. The authors would also like to thank Qi Liu at Xi’an Satellite Control Center for his help during the course of this work.

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Appendix

(a)

(b)

(c)

(d)

(e)

(f) Fig. A1 The image manual tracking method of the gas–liquid interface for the pinch-off during bubble formation at the junction. (a) water; (b) 0.1% SDS solution; (c) 0.5% SDS solution; (d) 50% glycerol solution; (e) 80% glycerol solution; (f) 90% glycerol solution. The blue and green points were manually picked tracking points.

Highlights (1) Breakup dynamics for bubble formation in a flow-focusing device is studied. (2) An interface tracking and processing method is developed to coordinate the profiles of the gaseous thread evolution during the pinch-off period. (3) Interface profile evolves at different gas-liquid systems in the axial and radial directions. (4) The effects of surface tension, viscosity and gas inertial force on the interface in the pinchoff period are discussed. (5) The free pinch-off period of low viscous liquid is proved to be shorter than the capillary time and concerned with Ohnesorge numbers.