Submerged injection of gas into a thin liquid sheet

Accepted Manuscript

Submerged injection of gas into a thin liquid sheet Mingbo Li , Liang Hu , Wenyu Chen , Haibo Xie , Xin Fu PII: DOI: Reference:

S0301-9322(18)30291-X https://doi.org/10.1016/j.ijmultiphaseflow.2018.09.009 IJMF 2887

To appear in:

International Journal of Multiphase Flow

Received date: Revised date: Accepted date:

16 April 2018 11 September 2018 21 September 2018

Please cite this article as: Mingbo Li , Liang Hu , Wenyu Chen , Haibo Xie , Xin Fu , Submerged injection of gas into a thin liquid sheet, International Journal of Multiphase Flow (2018), doi: https://doi.org/10.1016/j.ijmultiphaseflow.2018.09.009

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Highlights •

Our study provides the first quantitative and qualitative overview of the events following submerged injection of gas into a thin quiescent liquid sheet with a thickness of a few millimeters.

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For the experimentally available range, three different flow regimes are identified and,

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particularly, we build a phase diagram outlining these regimes in terms of the Weber number and Bond number. •

We investigated the complete evolution of periodic bubbling-bursting behavior systematically. The bubbling characteristics, which show great differences with

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traditional bubbling in deep liquid, are depicted and analyzed.

Using dimensional arguments, we proposed a scaling law for the rupture radius of bubbles which brings out the effects of gravity and inertia.

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Focusing on the subsequent liquid jet dynamics, we unravel experimentally the intricate

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roles of capillary wave velocities, cavity morphology, liquid thickness and gas momentum in bubble collapse.

Our results and conclusions herein provide a new method for the control of the

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•

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bursting-bubble jet and an important supplement to the physical problem of submerged

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gas injection.

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Submerged injection of gas into a thin liquid sheet Mingbo Li, Liang Hu*, Wenyu Chen, Haibo Xie, Xin Fu The State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University, Hangzhou 310027, China *

Corresponding author.

E-mail address: [email protected] (L. Hu)

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ABSTRACT

Submerged injection of gas into a thin liquid sheet is frequently observed in industrial processes. Most studies on submerged gas injection, however, focus on the phenomena happened in deep liquid pool. Few studies have been devoted to a thin liquid layer with a thickness of a few millimeters. Here,

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we study submerged injection of gas into a thin liquid sheet, where we quantify the behaviors of bubbling/collapse and the resulting liquid jet with a high-speed video system. For the experimentally available range of liquid depths and gas velocities, three different flow regimes are identified and we build a phase diagram outlining these regimes in terms of the Weber number and Bond number.

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Particularly, the complete evolution of periodic bubbling-bursting behavior is investigated systematically. The bubbling characteristics, which show great differences with traditional bubbling

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in deep liquid, are depicted and analyzed. Using dimensional arguments, we propose a scaling law for the rupture radius of bubbles which brings out the effects of gravity and inertia. Focusing on the

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subsequent liquid jet dynamics, we unravel experimentally the intricate roles of capillary wave

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velocities, cavity morphology, liquid thickness and gas momentum in bubble collapse. With increase of the liquid sheet thickness, jets first become fat and small and then ends up thinner, detaching more

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and smaller droplets due to Rayleigh-Plateau instability. Our study firstly provides the quantitative overview of the events following submerged gas injection into thin quiescent liquid and provide guides for the control of the bursting-bubble jet. Keywords: Submerged injection, Thin liquid sheet, Bubble dynamics, Capillary wave, Bursting-bubble jet

1. Introduction 2

ACCEPTED MANUSCRIPT Submerged injection of gas into a quiescent liquid pool is a very important physical problem in the dynamics of two-phase flow and has been well investigated in industrial applications, including heat exchange, biochemical operations, metallurgy, waste water treatment and chemical processing among others (Schügerl et al., 1978; Kantarci et al., 2005; Gulawani et al, 2007). Generally, previous studies have confirmed the presence of two regimes that characterize the development of the gas flow after leaving the nozzle/orifice (Mori et al., 1982; McNallan and King,

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1982; Chen and Richter, 1997). At low gas injection velocities, the bubbling regime is observed, characterized by the sequential formation of bubbles that break near the nozzle/orifice and rise in the direction dictated by gravitational or density effects. With increased gas velocity, a transition from bubbling to jetting occurs when the bubbles in the plume coalesce, thus a continuous cone jet is

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produced. The bubble formation in bubbling regime occurs in many modes viz. single bubbling, chain bubbling (with coalescence), and finally the chaotic bubbling. This process is largely governed by the balance of six different forces of which three are aiding (buoyancy, gas momentum, and pressure) and the remaining three (surface tension, viscous drag, and inertia) are retarding forces

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(Sunder and Tomar, 2013). The extent to which these forces control the bubbling characteristics depends on the nozzle/orifice configuration, the gas velocity, gas-liquid system properties, and

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finally magnitude of gravitational force acting on the system (Badam et al., 2007; Fan et al., 2008; Stanovsky et al., 2011; Chen et al., 2013; Hu et al., 2016). Typically, the bubble formation can be

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naturally divided into two different stages: expansion stage and collapse stage (Bolaños-Jiménez et al., 2009). During the expansion stage, the bubble grows quasi-statically until buoyancy forces

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overtake surface tension forces. Afterward, the bubble equilibrium shape becomes unstable with a

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neck formed, and the collapse stage takes place, during which the neck shrinks until the main bubble detaches from the nozzle/orifice in a finite time due to the self-accelerating nature of the so-called pinch-off phenomenon. Since the size of bubble after its detachment and its wake decide the rise velocity, trajectory and shape deformation in the liquid, it even influences the dynamic processes of coalescence and break-up, and the overall turbulence of the gas-liquid two-phase system (Zhang and Shoji, 2001; Zhu et al., 2014; Lee and Park, 2017). There is a considerable amount of work about bubble formation and its modelling, covered by several reviews (Kumar and Kuloor, 1970; 3

ACCEPTED MANUSCRIPT Jamialahmadi et al., 2001; Kulkarni and Joshi, 2005; Yang et al., 2007). Under most circumstances, the liquid height is large enough and several-times the bubble size that the bubbles can grow fully at the nozzle/orifice and detach before breaking the surface. While bubbling is a topic widely studied, surprisingly few papers deal with it formed in a shallow pool of water. It is well-known that the modes of bubbling in a given system are a strong function of the gas velocity and liquid depth. The liquid height affects the size of the produced

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bubbles and the character of the bubble-induced liquid motions. The effect of the liquid height has previously been studied only in some special cases. Spells and Bakowski (1950)(1952) studied the bubble formation at vertical slots submerged under water (depths of immersion between 1 cm and 7 cm). In deep water, discrete bubbles similar to those of previous workers were observed. But in

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shallow water, and particularly at high gas velocities, a series of more or less similar bubbles was obtained, forming a fairly continuous connection between the slot and the water surface. Quigley et al. (1955) varied liquid depth and found no significant difference in results for depths between 76 and 305 mm. Davidson and Amick (1956) and Hayes et al. (1959) concluded that liquid depth was of

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little significance as long as it was more than 25.4 mm. Jamialahmadi et al. (2001) conducted a detailed experimental investigation and manifested that the bubble diameter remained almost

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constant as the liquid height is increased from 0.60 to 2.1 m. Recently, Stanovsky et al. (2011) found that the effect is apparent in modifying the time-scale of the system and the total bubbling period

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increased with the increase in the liquid height (≥ 60 mm). On the whole, the gas velocities in these studies were relatively low. Bubbling theory predicts that at large depths (greater than about two

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bubble diameters), bubbling characteristics become independent of depth. The discrete bubbles are

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approximately spherical. As the gas velocity increases, the bubbles become more and more distorted. When the liquid layer is relative thin (less than one bubble diameter), the growing bubbles will begin to break the free surface and then rupture before they are fully formed from the nozzle/orifice, leading to a region of “imperfect bubbling” (Muller and Prince, 1972). Nevertheless, to our best knowledge, the bubble formation dynamics in this region have not been systematically studied in the literature. Whether in deep liquid pools or thin liquid sheet, when a single bubble reaches the surface of 4

ACCEPTED MANUSCRIPT the liquid it usually rebounds back and forth with decreasing amplitude and then comes to rest with its upperpart projecting above the surface in the form of a hemispherical dome. At every bounce, a certain amount of energy of the bubble is lost in the interaction with the free surface, and the approach velocity of the bubble decreases until a critical value at which bursting occurs. If the bubble is small enough to have a terminal velocity lower than a certain threshold, the bubble coalesces with the free surface as soon as it touches the interface, and no bouncing takes place (Suñol and

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González-Cinca, 2010). The bursting of bubbles at the free liquid surface is a widespread transport mechanism in nature which is related to the capillary, gravity and viscous forces. The bubbles that are already separated from the orifice/nozzle or growing in thin liquid sheet are more likely to come into contact with the free surface and are more susceptible to burst. Woodcock et al. (1953) firstly

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studied the behavior of bubble breakup at a free surface and the subsequent jetting and identified the following three stages: (Ⅰ) the retraction and fragmentation of the top thin film, (Ⅱ) the collapse of the unstable cavity formed due to the absence of the thin film and (Ⅲ) formation and breakup of the jet. The cavity collapses through a nonlinear balance between capillary force and inertia, while the

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capillary waves travel down the side of the cavity, collide at the bottom and produce upward traveling jet (MacIntyre, 1972). There are lots of widely recognized conclusions regarding the

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characteristics of jet on bubble bursting, including critical bubble size, jet height, mean size/velocity/number of ejected droplets (Duchemin et al., 2002; Zhang et al., 2012; Walls et al.,

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2015; Ghabache et al., 2016; Gañán-Calvo, 2017; Krishnan et al., 2017). Particularly, Lee et al. (2011) find that jet formation occurs only for the bubbles larger than a critical size threshold which is

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dependent on the liquid properties and build a phase diagram for jetting and the absence of jetting.

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Qualitatively the different stages of thin-layer injection are comparable to deep-pool injection, but with some important qualitative differences, which we will explain in detail. Here, different from existing studies occurred in the deep liquid, we report an experimental study of submerged injection of gas into a quiescent thin liquid sheet with a thickness in the range of 0 ~ 5 mm, which is observed in the immersion lithography system developed in our laboratory. The occurrence of submerged gas jetting in the collection chambers is the main reason causing vibrations in the immersion lithography, which could seriously damage the exposure quality. The above-mentioned conclusions do not reveal 5

ACCEPTED MANUSCRIPT about the unique phenomenon and the underlying physics are still not very clear yet. The purpose of this paper is to systematically investigate a series of behaviors induced by the submerged gas injection and hopes to be helpful improving the collection performance of immersion lithography. The current paper is structured as follows. We provide details of our experimental setup and methods in Sec. 2. In Sec. 3, we present experimental results and discussions. The thickness of liquid sheet and gas injection velocity are varied systematically to quantify the observed phenomena and

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different flow regimes observed below the liquid surface are discussed in Sec. 3.1. We depict and analyze the complete evolution of periodic bubbling-bursting behavior in Sec. 3.2~3.5. Finally, conclusions of this work are given in Sec. 4.

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2. Experimental methods

Liquid Sheet

PC

Liquid Pool

High-speed Camera × 2

H

D

Lamp and Diffuser

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Q

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Fig. 1. Schematic of experimental setup. Different experimental results are obtained by changing liquid sheet thickness H, gas flow rate Q and orifice diameter D. Two cameras are used to capture simultaneously the side-view below and above the liquid surface

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during each experiment.

The phenomena induced by the submerged injection of gas into a thin quiescent liquid sheet

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have been observed using a high-speed video system. Fig. 1 presents a schematic view of the experimental setup and summarizes the main systematic parameters. Ultrapure water with density of 𝜌𝐿 = 997 kg/m3 from a water treatment system and atmospheric air with density of 𝜌𝐺 = 1.185 kg/m3 are employed as working fluids for all runs. To rule out the wall effects, a square transparent container with a cross section area of 250 × 250 mm2 and 100 mm in height taken sufficiently large was used. The circular micro-channel (three inner diameter of D = 0.37 mm, 0.58 mm or 1.06 mm, same outer diameter of 𝐷𝑜 = 6 ± 0.1 mm and overall length of 60 ± 1 mm) was made of 6

ACCEPTED MANUSCRIPT borosilicate glass with a static contact angle of 29 ± 1°, which was the mean value of 10 measurements over the entire surface of a flat sample whose material is same as that of the micro-channel. The air provided by an oil-free air compressor flowed through the micro-channel and then was vertically injected into the quiescent liquid sheet. The volumetric flowrates of the input gas flow Q (range: 10 ~ 300 ml/min) was adjusted by gas mass flow controller with accuracy of ±0.8 % in reading scale and ±2 % in full scale.

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The liquid layer is formed in the liquid pool. Before each experiment was performed, the liquid sheet was thinned to a required thickness H, defined as the height difference between the free liquid surface and the upper surface of the micro-channel. A three-dimensional motion stage which had micrometer size precision was employed to measure the sheet thickness. The bulk water was changed

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after each run to minimize surface contamination. Given the inner diameter of the micro-channel D, the value of gas superficial velocity 𝑈𝐺𝑆 can be computed by the equation of 𝑈𝐺𝑆 = 4

𝐷2 . In

2 addition, the gas Weber number We were modified as 𝑊𝑒 = 𝜌𝐺 𝑈𝐺𝑆 𝐷 𝜎, where the gas-liquid

surface tension 𝜎 is 0.072 N/m. Table 1 summarizes the main physical parameters and

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corresponding dimensionless numbers used in our experiments. Table 1

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Range of the control parameters (liquid sheet thickness H, gas superficial velocity 𝑈𝐺𝑆 ) and corresponding dimensionless number

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(Weber number We) used in the experiments. 𝑈𝐺𝑆 (m/s)

We

0.2 ~ 5.0

0.2 ~ 12.62

3.7 × 10−3 ~ 1.5

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H (mm)

The side-view of the observation zone is recorded by two high-speed cameras (above and below

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the free liquid surface) at a frame rate within 1000 ~ 3000 frames/s. A high-intensity lamp with a thin paper as a diffuser was used to produce non-flickering backlighting. From these recordings, we extract the gas-liquid surface and related size using a customized image-processing analysis. The upper camera is slightly mounted upwards (with an angle within 4° ~ 10°) to minimize the influence of meniscus as much as possible at the container wall that may obscure the zone close to the interface and then limit the optical imaging quality. The uncertainty of the measurements from the positioning 7

ACCEPTED MANUSCRIPT of the stage is 50 μm. All the experiments were conducted at room temperature of 25±1

and

environment pressure of 101.3 kPa. Considering that the rupture of the bubble film cap is sensitive to the concentration of contaminants at the gas-liquid interface, a super clean room is used during all the experiments in order to ensure the amount of suspended particles per cubic meter of air below a critical threshold.

Ⅰ

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3.96

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H (mm)

3.1 Flow regimes

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3. Results and discussion

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Fig. 2. As the liquid sheet thickness H decreases, three types of flow regime are observed below the free liquid surface. From top to bottom: (Ⅰ) multi-bubble coalescence regime ( 𝑈𝐺𝑆 = 3.16 m s ), (Ⅱ) periodic

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bubbling-bursting regime (𝑈𝐺𝑆 = 3.16 m s), and (Ⅲ) pulsating jet regime (𝑈𝐺𝑆 = 7.75 m s). The schematic (right) is depicted according to the experimental results.

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Visual observations showed a number of regimes determined by liquid sheet thickness H and

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gas superficial velocity 𝑈𝐺𝑆 through the micro-channel. As the liquid sheet thickness H decreases at low gas velocities, three different behaviors can be observed under the free water surface, as illustrated in Fig. 2. At high liquid thickness within the range studied the common phenomenon observed is the continuous coalescence between two successive bubbles. Aperiodic bubbling occurs with the following bubble coalescing with the previous detached bubble during its growth stage. This behavior is denoted as multi-bubble coalescence (Ⅰ). The coalescence process (the tendency for two 8

ACCEPTED MANUSCRIPT touching bubbles to form a single larger bubble) occurs as a result of the surface energy minimization. It is generally accepted that the bubble coalescence mechanism mainly includes three steps: (1) the approach (collision) of two bubbles to within a critical distance; (2) further thinning of the liquid film; and (3) rupture of the thin liquid film via an instability mechanism (Liao and Lucas, 2010). We can note that the growing bubble B soon collides with bubble A that has detached, trapping a small amount of water between them. In this confined space, the upper bubble A becomes more and more

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distorted and shows approximately equivalent cylindrical shape. They keep in contact till the liquid film drains out to a critical rupture thickness. During the merging process, two sets of capillary waves generate. Then the two sets of capillary waves travel in opposite directions from the initial collision location, and meanwhile the surface amplitude increases greatly. When the waves reach the

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opposite ends, the upward group converges at the bubble A apex to produce a sharp bulge and the downward one converges at the bubble B base to pinch the tail off. Afterwards, these waves backtrack to the middle of the daughter bubble C and fluctuate along the bubble interface. Consequently, the daughter bubble C, floating on the free surface, gets bigger and bigger by continually coalescence with the growing bubbles until bubble bursting occurs. The size ratio of

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parent bubbles 𝐷𝐴 ⁄𝐷𝐵 also increases. Moreover, the turbulent wake formed behind the previous

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bubbles attracts the trailing bubbles. This may promote binary coalescence if the lower bubble has severed from the gas source, or stem coalescence if the new bubble is still attached to the orifice.

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Local hydrodynamic interactions between leading and trailing bubbles caused by the bubble wakes become important at high gas injection velocities (small bubble spacing). Finally, chaotic bubbling

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becomes dominant when 𝑈𝐺𝑆 reaches a high level where the coalescence events are irregular and

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bubble deformations are more severe. Second, decreasing the liquid depth now at constant velocity will significantly inhibit the

coalescence process between bubbles, leading to the bubbling instability and finally the appearance of the periodic bubbling-bursting regime which is the most dominant regime. As shown in Fig. 2 (Ⅱ), in this regime, bubble can be generated at a regular interval of time. Then, the bubble ruptures after a few milliseconds followed by an inertial liquid jet that exceeds the free surface. This behavior will be further studied below in Sec. 3.2 ~ 3.5. 9

ACCEPTED MANUSCRIPT As the liquid thickness is further reduced, the thin liquid layer ruptures immediately and then a pulsating jet (Ⅲ) is produced under a suitable range of gas jet velocities. In this regime, the effect of gravity is negligible for the films that are only several hundred micrometers thick and, hence the instability is governed by a competition between the various interfacial tensions involved. Depending on the balance between the pressure forces induced by the gas flow acceleration and the capillary forces acting at the interface, the liquid can’t flood back and close the hole. An important observation

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is that the triple-phase contact line does not reside on the rim of the orifice but stays below it. Besides, the gas jet not only builds a continuous gas path through the liquid but also disturbs it frequently. Accompanied by an interfacial oscillation, a surface wave was generated accordingly and then radially propagated away. The wave generation and propagation would be endless for the

0

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continuous interface oscillations. 5

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0.8

UGS = 7.75 m/s T

0.4 0.2

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UGS = 9.31 m/s

0.2

dneck

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fs = 322 Hz

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dneck

0.6

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fs = 264 Hz

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UGS = 10.86 m/s T

fs = 1250 Hz

0.4 0.2 0

5

10

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20

25

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∗ Fig. 3. Time dependence of the dimensionless neck diameter 𝑑𝑛𝑒𝑐𝑘 obtained experimentally for three different gas

superficial velocities (H = 0.42 mm, D = 0.37 mm).

In the generation process, the hole neck oscillated radially like a spring, which expanded and 10

ACCEPTED MANUSCRIPT ∗ shrunk within a certain range. We adopted the diameter of the neck 𝑑𝑛𝑒𝑐𝑘 , which was represented in ∗ dimensionless forms of 𝑑𝑛𝑒𝑐𝑘 = 𝑑𝑛𝑒𝑐𝑘 ⁄𝐷, to evaluate the deformation of the horizontal gas-liquid ∗ interface during the dynamic oscillations. The time dependence of 𝑑𝑛𝑒𝑐𝑘 for various 𝑈𝐺𝑆 is plotted

in Fig. 3. Here, time 𝜏 has been made dimensionless with the inertia-capillary time scale, 𝜏 = 𝑡⁄√𝜌𝐿 𝐻 3 𝜎 . It exhibits that these neck diameters keep periodic variation with different oscillatory intensity. Consequently, in the pulsating jet regime, increase in 𝑈𝐺𝑆 causes shorter period

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T (higher frequency 𝑓𝑠 ) and weaker oscillatory intensity. Moreover, as 𝑈𝐺𝑆 increases further, the position of the neck no longer oscillates vertically along the interface, but is fixed at a certain position below the rim of the orifice.

A convenient way to characterize these flow regimes is to quantify them by transition

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boundaries determined by the characteristic dimensionless parameters. The dynamics of transition in our experiments depends on the coupling of inertia of the gas flow, surface tension and gravitational forces on the liquid above the orifice. Hence, for all the conducted experiments, we plot our experimental data in a phase diagram, as shown in Fig. 4, in terms of the gas Weber number We (the

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surface tension), which is defined as

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ratio of gas inertia force to surface tension) and Bond number Bo (the ratio of liquid gravity to

𝐵𝑜 = 𝜌𝐿 𝑔𝐻 2 ⁄𝜎.

(1)

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In this diagram, we propose the appropriate transition lines fitted to the relevant data and recommended for regime transition predictions. The aforementioned three regimes can be

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distinguished by the critical values of (We, Bo). The zone of 𝐵𝑜 ≥ 1.72 describes the multi-bubble coalescence case, whereas the zone of 0.1 ≤ 𝐵𝑜 < 1.72 indicates the periodic bubbling-bursting

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case, which occupies the largest region in this diagram. Pulsating jet appears only when We exceeds a critical value, which is suggested to be around 0.22 for our experiments. Our results show that at high gas velocities the main factor in determining the transition of the regimes is the liquid thickness H. In other words, a reduction in liquid depth at high gas velocities results in transition from the multi-bubble coalescence to periodic bubbling-bursting to the pulsating jet. However, there is a transition regime between periodic bubbling-bursting regime and multi-bubble coalescence regime. In this region, both flow regimes can be observed. As a consequence, this region is highly sensitive 11

ACCEPTED MANUSCRIPT to tiny variations in H or 𝑈𝐺𝑆 . When 𝑈𝐺𝑆 is low, the following bubble does not grow immediately after the previous bubble detached from the orifice and the waiting time between them has to be considered. Therefore the data range of 𝑊𝑒 < 0.03 is not considered in this phase diagram. In the following sections (Sec. 3.2 ~ 3.5), we mainly study the behavior of periodic bubbling-bursting.

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(I) Bo = 1.72

1

Bo

(II) Bo = 0.1

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0.1

0.01 0.01

We = 0.22

Pulsating jet Periodic bubbling-bursting Transition regime Multi-bubble coalescence Transition lines

(III)

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0.1

1

We

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Fig. 4. Phase diagram of the flow behaviors in thin liquid as a function of the Weber number We and Bond number

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Bo. These dashed lines represent the boundaries of the transitions between different regimes.

3.2 Typical features of periodic bubbling-bursting phenomenon

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Fig. 5 illustrates a typical periodic bubbling-bursting event involving all the stages that take place in a period. The time sequences can be divided into six stages, involving initial bubble

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formation, liquid film cap drainage, bubble bursting, bubble pinch-off, collapse and resulting liquid jet. The top two sequences show the underwater dynamics while the bottom one displays the free surface view. During the bubble formation stage (𝑡 < 0 ms), the bubble emerges from the orifice and grows in the shape of spherical segment with foot remaining fixed to the rim of the orifice. After the growing bubble reaches the free surface, the thin liquid film separating the bubble from the atmosphere drains until a critical thickness, which falls within the range of 100 nm ~ 10 μm, and then disintegrates rapidly, leaving an opened hemispherical hole (Duchemin et al., 2002; Lhuissier and 12

ACCEPTED MANUSCRIPT Villermaux, 2012; Deike et al., 2018). Afterwards, the rapid retraction of the liquid film results in an instantaneous deformation of the bubble interface. In Fig. 5, the middle sequence (0 ms ≤ 𝑡 ≤ 2.8 ms) displays capillary waves propagating along the bubble and focusing at the bottom. These waves carry enough momentum to significantly distort the bubble and then pinch off the root in the vicinity of the orifice. Then, the closure of the cavity starts and these collapsing waves give rise to a vertical or nearly vertical jet shooting out the free surface, as observed on the bottom sequence. The

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rising jet then fragments into several droplets due to Rayleigh-Plateau destabilization. In the meantime (right after the detachment of the previous bubble), the second bubble nucleates and emerges from the orifice under constant gas injection velocity. 1 mm

-12 ms

-9.6 ms

0 ms

0.4 ms

1.2 ms

0 ms

4.0 ms

5.3 ms

-4.8 ms

-2.4 ms

2 ms

2.8 ms

3.6 ms

6.7 ms

8.0 ms

9.3 ms

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2 mm

Ejected droplets

-7.2 ms

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-14.4 ms

Fig. 5. Experimental time sequence of a typical case of periodic bubbling-bursting behavior following gas injection

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from the micro-channel. Inner diameter of the micro-channel, gas superficial velocity and liquid sheet thickness are 𝐷 = 0.58 mm, 𝑈𝐺𝑆 = 3.16 m s and 𝐻 = 2.54 mm, respectively. The top two sequences show the bubble

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evolution under the free surface during the bubble formation and collapse, while the bottom sequence displays the

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upward jet evolution above the free surface. 𝑡 = 0 ms marks the final instant previous to burst, and the whole process until the jet emergence above the free surface takes about 4.5 ms.

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In this regime, bubbles are formed after every strictly identical time interval at a certain gas superficial velocity. To observe the evolution of bubbling-bursting frequency 𝑓𝑏 under different experimental conditions, we make a reciprocal transformation of the bubbling-bursting period 𝑇𝑏 , which is defined as the time interval between the pinch-off moment of a bubble and that of the next bubble. Unlike bubbling in deep pools, the bubbling-bursting period 𝑇𝑏 in this regime consists of three stages: growth stage, collapse stage, and waiting stage. Fig. 6 shows the dependence of the bubbling-bursting frequency 𝑓𝑏 on the gas superficial velocity 𝑈𝐺𝑆 for two different values of the 13

ACCEPTED MANUSCRIPT orifice diameter (D = 0.58 mm in Fig. 6(a), D = 1.06 mm in Fig. 6(b)). It can be concluded that, for different liquid depth, the evolution of 𝑓𝑏 with increasing 𝑈𝐺𝑆 shows complicated tendencies. Regardless of some slight differences, the overall frequency 𝑓𝑏 decreases (the corresponding bubbling-bursting period increases) as H increases until reaching the critical height marked a transition from periodic bubbling-bursting regime to transition regime (as shown in Fig. 6(a)). In transition region, the lack of bursting stage and collapse stage increases dramatically the bubbling

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frequency. The differences of the results between Fig. 6(a) and Fig. 6(b) over the same 𝑈𝐺𝑆 range allow us to draw following conclusion: increase in micro-channel diameter leads to bigger bubble size and lower bubbling-bursting frequency. 80

70

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H = 1.43 mm, H/D = 2.45

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H = 2.50 mm, H/D = 2.36

UGS (m/s)

3

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H = 3.94 mm, H/D = 3.72

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UGS (m/s)

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H = 2.04 mm, H/D = 1.92

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H = 3.10 mm, H/D = 5.32

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fb (Hz)

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UGS (m/s)

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H = 2.54 mm, H/D = 4.36

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fb (Hz)

(a)

4

10

1

2

3

4

UGS (m/s)

Fig. 6. Bubbling-bursting frequency 𝑓𝑏 as a function of gas superficial velocity 𝑈𝐺𝑆 for different values of liquid sheet thickness H, and (a) D = 0.58 mm, (b) D = 1.06 mm. The solid lines are guides to the eye. The dashed lines indicate the turning point of the evolution.

Generally, for each liquid depth H there is a critical value of the gas velocity 𝑈𝐺𝑆𝐶 above which the frequency 𝑓𝑏 would decrease or stabilize at a constant value, as indicated by the dashed lines in 14

ACCEPTED MANUSCRIPT Fig. 6. This decrease or stabilization (when 𝑈𝐺𝑆 > 𝑈𝐺𝑆𝐶 ) can be mainly attributed to the irregular growth of the bubbles with deformed interface (above and below the free liquid surface) and volume-oscillation. It has also been demonstrated that, the rate of drainage is not determined by the flow over the whole bubble cap but is instead prescribed by the conditions at the foot of the bubble cap, where the cap connects to the meniscus (Lhuissier and Villermaux, 2012). As a result, the drainage of the liquid cap companied with pressure fluctuation around the bubble is retarded and

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consequently the rupture occurs more slowly. The effect of gas momentum has become insignificant in this range, in particular for H = 2.50 mm and 3.94 mm in Fig. 6(b). On the other hand, we observe that for 𝑈𝐺𝑆 < 𝑈𝐺𝑆𝐶 , the frequency fb is approximately proportional to 𝑈𝐺𝑆 . This is because the higher gas velocities in this region, which means higher gas momentum, result in faster rupture and

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collapse of the bubbles.

When the gas superficial velocity 𝑈𝐺𝑆 keeps rising, the aperiodic characteristics gradually become distinct. In addition, the volume of the bubbles cannot be obtained from the bubbling-bursting frequency 𝑓𝑏 as

⁄𝑓𝑏 because of the collapse stage that gas is injected directly

M

into the atmosphere without bubbles formed. The bubbling-bursting frequency is not sufficient to

z

Liquid film

PT

3.3 Bubble dynamics

ED

describe the bubbling dynamics and more detailed analysis is required.

CE

Meniscus

AC

Interior interface

O2

zc

Free liquid surface

rc

H

Rct O1

D



r

Fig. 7. Coordinate system and geometry for a bubble before rupture. 𝑧𝑐 is the centroid height of the bubble and 𝑟𝑐 is the measured horizontal radius at the equator. 𝜃 and 𝑅𝑐𝑡 are the instantaneous contact angle and the instantaneous radius of the contact line during bubble formation, respectively.

In this section, the dynamics of bubble formation and deformation before bursting are discussed, 15

ACCEPTED MANUSCRIPT respectively. We consider the bubble formation in detail referred to the shape and coordinate system defined in Fig. 7. These parameters are varied systematically while keeping the orifice diameter D constant. Fig. 8 shows a temporal evolution of the bubble shape just before rupture for three different liquid sheet thickness and same gas injection velocity. The differences in shape are evident by visual examination. As 𝑈𝐺𝑆 is kept constant, decreasing H (which mean decreasing the effect of buoyancy) promotes deformation, as expected, which in turn changes spherical to oblate bubbles. These

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sequences also show that by increasing H, results in an increase in bubble formation time, correspondingly to larger bubble volume. For most of the previous studies on bubble formation in deep pool, the process can be divided into three stages: nucleation, expansion and pinch-off. However, in thin liquid, the bubble formation only experienced the first two stages. At early times,

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the shape of these bubbles almost exhibits same characteristics and is that of a spherical segment. The contact line expands rapidly in the micro-channel and reaches the rim of the orifice (𝑡 ≅ 𝑡0). Similarly, the shape of bubble oscillates due to the interaction between inertial and surface tension forces. Considering that the formation of a bubble is often accompanied with the cavity collapse

M

simultaneously, the influence from the wake of the previous cavity collapse cannot be negligible in this study. During the expansion stage, the contact line always sticks on the orifice rim (2𝑅𝑐𝑡 ≅ 𝐷),

ED

as shown in Fig. 8(b) and (c). Once the liquid thickness H is thinned to a threshold, the bubble instantaneous contact line will cross the rim of the orifice and spread outward gradually on the upper

PT

surface of the micro-channel, as shown in Fig. 8(a). During the growth stage, it reached the maximum volume in a very short time and deformed significantly. However, in deeper water, it was

CE

found that the bubble foot diameter remained fairly constant during the bubble formation lifetime.

z-position (mm)

z-position (mm)

AC

(a)

(c)

t0+8 ms

0.8

t0+6 ms

t0+28.7 ms t0+23.9 ms

t0+4 ms

0.4

3

t0+2 ms

H = 1.43 mm

t0 ms

(b) 2

1.5 1.0 0.5

t0+12.8 ms

-1

t0+13.5 ms

t0+6.4 ms

t0+8.3 ms

t0+3.2 ms

H = 2.54 mm -2

t0+18.7 ms

1

t0+9.6 ms

t0+16.0 ms

0

1

t0+3.0 ms

H = 50 mm

t0 ms

2

-2

t0 ms

-1

0

1

r-position (mm)

r-position (mm) 16

2

ACCEPTED MANUSCRIPT Fig. 8. Experimental bubble surface profiles evolution before rupture or pinching-off for three different values of liquid sheet thickness (a) 𝐻 = 1.43 mm, (b) 𝐻 = 2.54 mm, (c) 𝐻 = 50 mm and same 𝑈𝐺𝑆 = 2.52 m s. The inner diameter of the micro-channel is D = 0.58 mm. The initial bubble profiles (𝑡 ≅ 𝑡0) are identified just after the nucleation stage, in which the contact line expands rapidly and reaches the inner rim of the micro-channel.

Next, we further clarify the effect of gas superficial velocity 𝑈𝐺𝑆 on the transient behaviors of bubble formation. From image processing, some parameters, including the transient behaviors of the

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moving contact line radius 𝑅𝑐𝑡 and contact angle 𝜃, were determined to characterize the bubble formation process. For example, as shown in Fig. 9, the contact line radius 𝑅𝑐𝑡 increases with time, while the contact angle 𝜃 decreases. As 𝑈𝐺𝑆 increases, the temporal variations of 𝑅𝑐𝑡 show different characteristics. For 𝑈𝐺𝑆 = 1.89 m s, the contact line radius 𝑅𝑐𝑡 monotonously increases,

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whereas for the higher gas velocity of 3.16 m/s, the 𝑅𝑐𝑡 firstly reaches a maximum value with a sharp peak appeared and then shrinks slowly until rupture occurs. In other words, the peak of the radius evolution profile, which was induced by a oscillation in the shape of the bubble, becomes more pronounced with the increase in 𝑈𝐺𝑆 . Visual observations demonstrate that the unsteady

M

oscillations are related to the formation of the liquid cap with thickness thinning. It can also be noticed in Fig. 9(b) that, for a higher gas velocity, the initial bubble is more flattened which leads to a

ED

smaller initial contact angle. However, the overall tendency of contact angle during the process of bubble formation is similar for increased gas velocities. After the initiation stage, these contact

PT

angles decrease rapidly and change from obtuse angle to a same acute angle of about 𝜃 = 22°, which is slightly smaller than the static contact angle of the micro-channel. For bubble formation in

CE

deep liquid, previous reports (Chen et al., 2009; Chen et al., 2013) demonstrate that the transient

AC

contact angle decreases in the beginning stage to a minimum value, then increases till the bubble detaches. Generally, a strong increase in the contact angle is seen during the necking stage. For the present experiments, this result naturally suggests that the necking stage is far from being reached before bursting.

17

ACCEPTED MANUSCRIPT 0.6

175

(a) 140

0.5

H/D = 2.45, UGS = 1.89 m/s H/D = 2.45, UGS = 2.52 m/s H/D = 2.45, UGS = 3.16 m/s

105

0.4

()

Rct (mm)

(b)

0.3

H/D = 2.45, UGS = 1.89 m/s

70

= 22

35

H/D = 2.45, UGS = 2.52 m/s H/D = 2.45, UGS = 3.16 m/s

0.2

0

0

2

4

6

8

10

0

2

t-t0 (ms)

4

6

8

10

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t-t0 (ms)

Fig. 9. Variation of the characteristic parameters during bubble formation before rupture. (a) Time evolution of the contact radius 𝑅𝑐𝑡 and (b) time evolution of the contact angle 𝜃 for three different gas superficial velocities 𝑈𝐺𝑆 , same liquid sheet thickness H = 1.43 mm and same inner diameter D = 0.58 mm.

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From the above examples, the effect of the gas superficial velocities 𝑈𝐺𝑆 manifests itself through the different contact line behavior and thus on the bubble shape and rupture size. In the confined space, both the apparent contact angle and the moving contact line vary in a complex manner as the bubble grows. Consequently, there is no universal pattern of the time-history of the

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contact angle or contact radius. 3.5

ED

3.0

1.8

UGS = 2.52 m/s

1.6

UGS = 3.16 m/s

1.4

UGS = 3.79 m/s

1.5

1.0 0.8

UGS = 2.52 m/s

0.6

UGS = 3.16 m/s

0.4

UGS = 3.79 m/s

0.2 0

2

4

CE

e

2.0

1.2

rc (mm)

PT

2.5

6

8

10

12

t-t0 (ms)

1.0

14

16

18

te ~ UGS

AC

0.5 0.0 0

2

4

6

8

10

12

14

16

18

t-t0 (ms)

Fig. 10. Time evolution of the bubble eccentricity during bubble formation for three different gas superficial velocities 𝑈𝐺𝑆 (H = 2.54 mm, D = 0.58 mm). Top right inset: time evolution of the horizontal radius 𝑟𝑐 . The arrows indicate a linear law behavior for the time at which the minimum occurs.

18

ACCEPTED MANUSCRIPT When a bubble attached to the orifice arrives at the free surface, the liquid film emerges above the surface level to form a cap, as seen in Fig. 5 (t = 0 ms). The bubble deformation directly influences the film drainage through both the film orientation with respect to the direction of gravity, and capillary pressure induced by the interface curvatures. In order to obtain a quantitative measurement of the bubble deformation roughly, we define the parameter of bubble eccentricity, as follows,

𝑐

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𝑟

𝑒 = √|(𝑧𝑐 )2 − 1| ,

(2)

where 𝑧𝑐 is defined as the height of the bubble centroid from the upper surface of the micro-channel and 𝑟𝑐 is the measured horizontal radius at the equator, as shown in Fig. 7.

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The bubble eccentricity 𝑒 through the bubble growth keeps changing due to the complicated effect of inertia, gravity and surface tension. The variation of the eccentricity with time for three different gas velocities and same liquid thickness (H = 2.54 mm) is given in Fig. 10, where it can be seen that the trends are similar for three cases. When the bubble forms on top of the micro-channel 𝑒

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decreases sharply at the onset of bubble growth and then reaches the minimum ( 𝑟𝑐 ≈ 𝑧𝑐 ), corresponding to a critical time 𝑡𝑒 . After this point, it increases slowly until the bubble finally bursts.

ED

As the gas velocity increases, it takes longer to reach this turning point, as indicated by the arrows in Fig. 10. The times at which the minima occurs correspond to a linear law 𝑡𝑒 ~𝑈𝐺𝑆 . This can be

PT

attributed to the higher gas momentum force which arises from the incoming flow of gas through the orifice that imparts a force in the direction of the gas flow. It plays an important role in bubble

CE

morphology during its formation and growth. Additionally, higher gas velocities tend to cause larger horizontal radius 𝑟𝑐 , as shown in the top right inset of Fig. 10. It is observed that the trends are also

AC

similar for different gas velocities. The rate of change of the radius 𝑟𝑐 (the expansion rate in horizontal direction) decreases gradually to remain at a low value until the bubble bursting. The variation of the effective forces through the bubble growth explains the dynamics of variation of these bubble characteristics. As the bubble evolves from a hemispherical to a spherical shape (0~𝑡𝑒 ), a larger portion of the bubble is acted upon by buoyancy. For (𝑡 − 𝑡0 ) > 𝑡𝑒 , with the thin liquid film draining the equivalent radius of curvature at the apex increases drastically and thus the force due to 19

ACCEPTED MANUSCRIPT Laplace pressure decreases monotonically to burst. At the same time the bubble expansion rate and the downward inertial force decrease. As a result, the portion of the bubble below the centroid 𝑂2 becomes progressively more elongated as these forces tend to lift it with the bubble foot remaining fixed to the orifice mouth. At the final stage of bubble formation, the increase in the eccentricity of the bubble (> 𝑡𝑒 ) is mainly induced by the increase in 𝑧𝑐 . 2.5 H = 1.43 mm H = 2.54 mm H = 3.10 mm

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2.0

em

1.5 1.0

0.0 1.5

2.0

2.5

AN US

0.5

3.0

3.5

4.0

4.5

5.0

UGS (m/s)

M

Fig. 11. Variation of the eccentricity of the bubble which is about to burst 𝑒𝑚 with the gas superficial velocity 𝑈𝐺𝑆 for different liquid sheet thickness H and same inner diameter D = 0.58 mm.

ED

In particular, determining the morphology of a bubble, which is about to rupture, is important for understanding the following bubble collapse dynamics. The shape of the bubble formed in deep

PT

liquid pool gradually approximates a sphere as the gas velocity increases. However, the boundary

CE

conditions of the bubbles in thin liquid are very complex and variable so that their shape becomes more distorted and irregular. Here, we mainly focus on the eccentricity of the bubbles on the final

AC

instants previous to rupture, which is expressed by 𝑒𝑚 . When a bubble volume reaches the maximum, the transient parameters of 𝑟𝑐𝑚 (maximum radius in the horizontal direction) and 𝑧𝑐𝑚 (maximum centroid height) were measured under different experimental conditions. The resulting eccentricity 𝑒𝑚 also can be obtained, as shown in Fig. 11. The effect of 𝑈𝐺𝑆 becomes more pronounced in thin liquid sheet, for which the bubbles are smaller and, therefore, less affected by the buoyancy. The amplitude of deformation increases significantly with increasing gas velocities. However, there is no obvious difference between H = 2.54 mm and H = 3.10 mm, although the values of 𝑒𝑚 for 3.1 mm 20

ACCEPTED MANUSCRIPT thick are somewhat higher than that for 2.54 mm when 𝑈𝐺𝑆 is relatively low. For these two cases, slight turbulence can have a significant effect on the morphology of the bubbles which are about to burst. As a result, the values of 𝑒𝑚 change irregularly with a slight decrease towards the minimum 𝑒𝑚 = 0. 3.4 Scaling for the rupture radius 2.0

9

(a)

(b)

7

Rb/(Bo1/2 D) 

Rb (mm)

1.5 4

2

3

Rb/D

1

2

1.0

0.1

0.2

0.3

1

2

3

6

f(We)∝ We3/20

5

0.4

AN US

Bo

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8 1.89 m/s 2.52 m/s 3.16 m/s 3.79 m/s 4.42 m/s

4

5

H (mm)

4 0.01

0.1

1

We

Fig. 12. (a) Rupture radius 𝑅𝑏 as a function of the liquid sheet thickness H for different gas superficial velocities

M

𝑈𝐺𝑆 . In the inset, the dimensionless rupture radius 𝑅𝑏 𝐷 is plotted as a function of the Bond number 𝐵𝑜∗ built on the liquid sheet thickness for the same velocities. The dashed lines represent 𝑅𝑏 ∝ 𝐻1 1 2

𝑅

1 2

in the inset. (b) ( 𝑏 ) 𝐵𝑜∗ 𝐷

𝑅

1 2

experimental data plotted with closed symbols, following the trend ( 𝐷𝑏 ) 𝐵𝑜∗

PT

in the graph and

as a function of the Weber number We. The dashed line fits the

ED

𝑅𝑏 𝐷 ∝ 𝐵𝑜∗

2

exponent is 1/200 and the two bounds 𝑓(We) ∝ We3

20±1 200

∝ 𝑊𝑒 3

20

. The error bar on the

are plotted on the graph with gray dashed line. The

CE

inner diameter of the micro-channel used for all experiments is D = 0.58 mm.

Just before the bubble bursts at the free surface, its rupture radius (equivalent radius after bubble

AC

formation) plays an important role in the following collapse dynamics that needs to be determined. With the bubble parameters of 𝑟𝑐𝑚 and 𝑧𝑐𝑚 extracted from the experimental images under different experimental conditions, the corresponding rupture radius 𝑅𝑏 can be defined as 3

2 𝑧 . 𝑅𝑏 = √𝑟𝑐𝑚 𝑐𝑚

(3)

We now investigate how the bubble rupture radius 𝑅𝑏 depends on the natural control parameters of gas superficial velocity 𝑈𝐺𝑆 and liquid sheet thickness H. Intending to obtain a robust 21

ACCEPTED MANUSCRIPT scaling law for the rupture radius of the bubbles, we have so far identified seven relevant parameters that directly govern the bubble formation. The relationship between the rupture radius 𝑅𝑏 and these parameters can be expressed as follows 𝑅𝑏 = Π(𝜌𝐺 , 𝜌𝐿 , 𝑈𝐺𝑆 , 𝐻, 𝐷, 𝜎, 𝑔).

(4)

In the scaling analysis, gas density (𝜌𝐺 ), surface tension (𝜎) and orifice diameter (D) are taken to be the repeating variables (using the MLT system), while other parameters are considered to be the

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non-repeating variables. Obviously, each repeating variable is dimensionally independent of the others. Then, using dimensional arguments, this equation becomes a relation between five dimensionless parameters (𝑅𝑏 𝐷, 𝜌𝐿 𝜌𝐺 , 𝐻 𝐷, 𝑔𝜌𝐺 𝐷2 𝜎, We) fully describing the rupture radius selection. Among them, the three parameters of 𝜌𝐿 𝜌𝐺 , 𝐻 𝐷 and 𝑔𝜌𝐺 𝐷2 𝜎 can be multiplied to

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form the modified Bond number 𝐵𝑜∗ = 𝜌𝐿 𝑔𝐷𝐻 𝜎. It should be noted that the density ratio 𝜌𝐿 𝜌𝐺 is constant in the current study. Therefore, in this regime, this yields the following form primarily relating the bubble rupture radius (𝑅𝑏 ⁄𝐷 ), the gas velocity (𝑊𝑒) and the liquid thickness (𝐵𝑜∗ ): 𝑅𝑏 ⁄𝐷 = 𝐹(𝑊𝑒, 𝐵𝑜∗ ),

(5)

M

where 𝑊𝑒 compares the effect of inertia and capillarity on the bubble and 𝐵𝑜∗ describes the

ED

gravity effects. We now plot, in Fig. 12(a), the variation of the rupture radius 𝑅𝑏 as a function of the liquid sheet thickness H for different values of the gas superficial velocity 𝑈𝐺𝑆 . One experimental

PT

fact directly emerges from the picture: 𝑅𝑏 becomes larger with increasing 𝑈𝐺𝑆 and H, respectively. Moreover, it is instructive to note that, regardless of the gas velocities considered in this graph, the

CE

rupture radius increases with liquid sheet thickness following roughly the same variation for all the curves, 𝑅𝑏 ∝ 𝐻1 2 , shown with dashed lines on the graph. In the inset of Fig. 12(a), the variation of

AC

the dimensionless rupture radius is plotted as a function of the liquid sheet thickness Bond number 1 2

for the same experimental conditions as in Fig. 12(a). The variation 𝑅𝑏 ⁄𝐷 ∝ 𝐵𝑜∗

is also plotted

with dashed lines. This power law, independent of the gas velocities, still works reasonably well, allowing us to write the scaling law 1 2

𝑅𝑏 ⁄𝐷 = 𝐵𝑜∗ 𝑓(𝑊𝑒). 22

(6)

ACCEPTED MANUSCRIPT In order to estimate the dependence of the bubble rupture radius with the gas superficial velocities, namely, 𝑓(𝑊𝑒),

𝑅𝑏 𝐷

1 2

𝐵𝑜∗

is plotted as a function of the Weber number, as shown in Fig.

12(b). We observe that the data generally gather along a line. This line is properly fitted by 𝑓(𝑊𝑒) = 𝛼𝑊𝑒 3

20

with 𝛼 = 8.605. We then estimate the error bar of the results by fitting the data

of lower and upper bounds, respectively, and find 𝑓(𝑊𝑒) ∝ 𝑊𝑒 3

20±1 200

. Both two bounds are

plotted on the graph with grey dashed lines. It can be seen that the results are just slightly scattered.

follows, 𝑅𝑏 ⁄𝐷 = 𝛼𝑊𝑒 3 20

1 2

𝐵𝑜∗

1 2

𝐵𝑜∗ .

(7)

for the experimental data, and the theoretical

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The relation between the 𝑅𝑏 ⁄𝐷 and 𝑊𝑒 3

20

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Consequently, in the context of bubble formation, we establish a scaling law for the rupture radius, as

prediction curve given by Eq. (7) are illustrated in Fig. 13. The collapse of all the experimental data on the single master curve with a fixed slope is remarkable, thereby capturing the relation between the rupture radius and the initial experimental conditions. It should be noted that although the Eq. (7) is

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based on dimensionless analysis, the current form is only verified for periodic bubbling-bursting regime. The range of the liquid thickness in which this scaling law is valid varies with the change in the

ED

inner diameter of the micro-channel. For instance, for the micro-channel with inner diameter of D = 0.58 mm, the proposed scaling law is valid in the range of 0.86 mm < H < 3.56 mm according to the Fig.

AC

CE

PT

4.

23

ACCEPTED MANUSCRIPT 5

Exper. data Present scaling law

4

Rb/D

3 2

0 0.2

0.3

We3/20Bo1/2 

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1

0.4

(7).

3.5 Bubble collapse and jet formation

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Fig. 13. Scaling of the experimental results according to the proposed scaling law. The solid line corresponds to Eq.

As soon as the height of the growing bubble exceeds the liquid thickness, the thin liquid film

M

begins to drain and thin. Then, the growing bubble bursts by nucleating a hole in their cap, for most cases, in the vicinity of the cap foot. The initial hole extends circularly under the action of surface

ED

tension at an approximately constant velocity tangential to the film given by the Taylor-Culick velocity (Culick, 1960). During the retraction of the film, the rim suffers an inertial destabilization of

PT

a Rayleigh–Taylor type, which leads to the formation of ligaments. Ligaments are then stretched out by centrifugation, producing disjointed droplets by a Plateau–Rayleigh destabilization. Because of

CE

the relatively small bubble size ( 𝑅𝑏 ≤ 2 mm) generated in our experiments, the resulting fragmentation dynamics from the negligible film cap is very fast that we can hardly observe it. We

AC

consider this stage as the starting point of the bubble collapse. Different from the collapse induced by a bursting bubble floating at the free surface, there is a stage of pinching-off involved in the collapse process for our experiments.

24

ACCEPTED MANUSCRIPT 2.5

z-position (mm)

2.0

Rw

1.5

vw

1.0

vw

0.5

-2

-1

0

1

r-position (mm)

v w

CR IP T

2.8 ms 2.4 ms 2.0 ms 1.6 ms 1.2 ms 0.8 ms 0.4 ms 0 ms

2

Fig. 14. Time evolution of experimental bubble collapse profiles following bubble bursting (D = 0.58 mm, H = 3.10

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mm, 𝑈𝐺𝑆 = 3.16 m s). t = 0 ms marks the final instant previous to burst.

After the bubble bursting, the incoming gas communicates with atmosphere and therefore a through-hole appears in the liquid. The rim or the juncture between the film and the bulk liquid develops a small crest, which moves down the bubble (Zhang and Thoroddsen, 2008). Fig. 14

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displays a typical bubble collapse sequence in water after bursting. It is observed that a train of

ED

capillary waves, which are generated by the crest, pass along the collapsing bubble surface adopt invariably a self-similar behavior that through a nonlinear balance between capillary force and inertia.

PT

(Ghabache et al., 2014). Previous researches have confirmed that the collapse is often perturbed by the presence of the small capillary waves. This rapid propagation results in an instantaneous

CE

deformation of the bubble shape, which is similar to a cone shape after pinching-off (t = 2.8 ms). It is noteworthy that there are precursor capillary waves ahead of the kink resulting from the change of

AC

the curvature of the bubble morphology from convex to concave. The extent of the convex part decreases with time and the radius of the concave part 𝑅𝑤 increases till the cavity surface becomes approximately conical. As we can see that the amplitude of capillary waves is gradually enhanced with time, leading to an increase in wavelength which can be clearly identified. In other words, the wavelength decreases away from the free water level, in accordance with the dispersion relation for capillary waves, where shorter waves travel faster (Lighthill, 1967). Particularly, during the focusing these dispersive waves with smaller wavelength cause very fast perturbations and go first to thin the 25

ACCEPTED MANUSCRIPT bubble root with a neck formed, after which the arrival of the kink pinches the neck off drastically. The number of capillary waves coming from the expansion of the initial rim formed after the film cap bursting increases correspondingly with the increase in the thickness of the liquid sheet, which determines the bubble height. We regard this as the end of the stage of bubbling-bursting or bubble collapse with an unstable conical cavity formed. 3.0

1.2

(a)

z-position (mm)

(b)

H = 2.54 mm H = 3.10 mm

2.0

v'w (m/s)

1.1

1.5

t 1.0 0.5 0.0 0.5

1.0

1.5

2.0

vw (m/s)

1.0

0.9 2.0

AN US

H = 2.54 mm H = 3.10 mm

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2.5

2.5

3.0

3.5

4.0

UGS (m/s)

Fig. 15. (a) Variation of the instantaneous velocity of the capillary wave 𝑣𝑤 with z-position during bubble collapse for different liquid thickness H and same gas superficial velocity 𝑈𝐺𝑆 = 3.16 m s. (b) Mean propagation velocity

M

of the capillary wave 𝑣𝑤′ during bubble collapse as a function of the gas superficial velocity 𝑈𝐺𝑆 for two different

ED

liquid sheet thickness. Error bars come from a standard deviation derived from measurement errors. The inner diameter of the micro-channel used for all experiments is D = 0.58 mm.

PT

We now look at the dependence of the wave velocity on the initial conditions during bubble collapse. The displacement of same capillary wave between two successive images can be measured

CE

by resolving into the horizontal and vertical directions to the free water surface, as shown in the coordinate system of Fig. 7. This displacement was further processed to obtain the instantaneous

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velocity 𝑣𝑤 , the corresponding horizontal component 𝑣𝑤∥ and the vertical component 𝑣𝑤⊥ , as shown in Fig. 14. 𝑣𝑤∥ is found to be increasing until it reaches a maximum value near the bubble neck and remains almost constant thereafter during the bubble collapse. However, 𝑣𝑤⊥ decreases monotonically as the capillary waves propagate over the bubble surface, and so does the total velocity 𝑣𝑤 . This decline could be due to a portion of the momentum loss used for the neck thinning. Fig. 15(a) shows the wave velocity evolution for two bubble collapses with different liquid sheet 26

ACCEPTED MANUSCRIPT thickness and same gas injection velocity. We can see that, at the same depth above the micro-channel (same z-position), the wave velocity 𝑣𝑤 in deep water is always less than that in shallow water. However, these two cases end up with almost same velocity, unexpectedly, at almost the same position of pinching-off. To extend these results, we recorded the time difference between the emergence of the crest and the moment of pinch-off for each collapse. The travelling distance of waves to reach the bubble base is also measured. Therefore, the mean wave velocity 𝑣𝑤′ under

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different experimental conditions can be obtained, as shown in Fig. 15(b). It can be noted that 𝑣𝑤′ increases with 𝑈𝐺𝑆 at fixed H and decreases with H at fixed 𝑈𝐺𝑆 . Higher 𝑈𝐺𝑆 tends to accelerate the evolution of the traveling capillary waves converging at the bubble base. Theoretically, for capillary waves with a wavelength much smaller than the bubble radius, the dependence of the

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overall speed almost follow the relation 𝑣𝑐 ~ (𝜎𝑘 𝜌𝐿 )1⁄2 , while the dominant wavenumber 𝑘 should be proportional to 𝑅𝑜−1 (𝑅𝑜 , the equivalent radius before bubble bursts). (Blanchette and Bigioni, 2006; Gañán-Calvo, 2017) This conclusion suggests that thinner liquid leads to the formation of smaller bubble (smaller value of the rupture radius 𝑅𝑏 ) and therefore induces the

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increase in the mean wave velocity 𝑣𝑤′ .

When the capillary wavefront collapses onto the bubble neck, the interaction of the complex

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vortical structure with the interface and the complexity of the nonlinear capillary wavefront can give rise to complex geometries in the local liquid surface, leading to either single or multiple gas

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engulfment in the form of tiny bubbles just before the formation of the liquid jet. Then, this millimetric conical cavity remaining after the bubble departure relaxes under the coupled effect of

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surface tension, buoyancy and gas momentum forces. The vertical retraction of the liquid drives a

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significant partial gas ejection from the conical cavity into the atmosphere, leading to a large volume shrinkage. The partial gas ejection can be significantly enhanced as the pressure difference (∆𝑝 = 2𝜎⁄𝑟𝑗𝑒𝑡 , where 𝑟𝑗𝑒𝑡 is the local equivalent radius of the cavity) increases. Since then, a liquid jet is expected to be initiated from the dynamically balanced flow structure.

27

ACCEPTED MANUSCRIPT (a) H = 1.43 mm 1 mm

0 ms

0.67 ms

1.34 ms

2.68 ms

2.01 ms

(b) H = 2.54 mm

1 mm

0 ms

1.34 ms

2.68 ms

3.35 ms

4.02 ms

6.03 ms

1 mm

0 ms

2 ms

3 ms

4 ms

5 ms

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(c) H = 3.10 mm

6 ms

7 ms

9 ms

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Fig. 16. Time evolution of the structure of liquid jet for three different liquid sheet thickness (D = 0.58 mm, 𝑈𝐺𝑆 = 3.79 m s).

Fig. 16 displays three sequences of jet events following the cavity reversal at the free surface. The jet grows as it rises out of cavity while simultaneously thinning at its tip. In general, jet formation occurs through the convergence of capillary waves along the liquid-air interface (MacIntyre, 1968). Surface tension causes the liquid thread thin, eventually pinching off to form

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liquid droplets from the tip of the jet. Depending on their mass and initial velocity, the droplets will

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either fall back into the water or evaporate. After that, the jet gradually descends under the effects of surface tension and gravity with a smoother top surface. In this study, the radius of the droplet

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detached falls within the range of 10 ~ 200 μm. As can be seen, with an increase of the liquid film thickness, the jet morphology undergoes a neat qualitative change: the jet first becomes fat and small

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and then ends up thinner and higher, detaching more and smaller droplets due to Rayleigh-Plateau instability. As mentioned earlier, the size of the bubbles which are about to burst becomes smaller

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and smaller with decreasing liquid thickness. Only when the bubble size is sufficiently large for the energetic waves to collapse at the bubble base, a jet is formed. The thickness threshold for jetting is crucial to understanding the jetting process and deserves to be further studied. Similarly, not every liquid jet leads the observation of ejected droplets, which could mean that no droplets are pinched off from the jets, or that droplets remain trapped inside the cavity. For thin liquid below a critical thickness, no drop is detached. Both the formation of the jet and its further breakup require a balance between capillary, viscous and surface tension forces (Castillo-Orozco, 2015). For example, in the 28

ACCEPTED MANUSCRIPT case of Fig. 16(a), not every jet produces a detached droplet. This can be explained that, when the jet moves upward, transverse oscillations in the jet lead to some oscillations in the velocity and pressure within the pinched region, but are not strong enough to lead to drop detachment (Deike et al., 2018). These oscillations are related to capillary waves under the influence of surface tension and the pinching is avoided. In order to grasp the mechanisms leading to such a particular jet dynamics, we now turn to the

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jet formation by focusing on the cavity collapse. At some instant just after the pinch-off, a conical cavity was formed with a high curvature or a cusp at its bottom. For each cavity, the opening angle 𝜃𝑐 is measured to accurately describe the shape. This cone with high kinetic energy inverts and forms a jet that rises out of the collapsing cavity. In our experiments, typical cavity dimensions

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(depths and widths) are less than a few millimeters, so that both gravity and capillarity play a role in the relaxation process to bring it back to a flat equilibrium. Gravity determines the cavity depth and shape. The inset in Fig. 17(a) compares the shape of the cavity for different H. These three images (Ⅰ), (Ⅱ), and (Ⅲ) are the cavity shapes leading to the three jets (a), (b), and (c) displayed in Fig. 16.

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As H increases, the cusps of the cavity become more sharp and longer, and the opening angle of the cavity 𝜃𝑐 decreases linearly with increasing H, as shown in Fig. 17(a). It also can be concluded that

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with higher H, the effective size of the cavity following the bubble pinch-off for jet development is bigger. Then, the jet is strongly enhanced for more released droplets, higher jet height and velocity.

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Typically, for a given liquid sheet thickness, as 𝑈𝐺𝑆 increases the jet gets thinner and taller that

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produces more droplets with reduced size. Besides, there is a critical height that the jet must reach for breakup to occur with droplets stretched vertically by the capillary waves. The jet size is directly

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related to the bubble size and shape which is about to burst. In the cases where no drop detaches, jets are displayed at their maximum height. There is a close relationship between the bubble size, the shape of the cavity and the resultant jet. In Fig. 17(b), we present how the opening angle 𝜃𝑐 and the normalized maximum jet height 𝑕𝑚𝑎𝑥 ⁄𝐻 (without droplets detached) measured when the top of the jet reaches the maximum height, depend on the bubble rupture radius 𝑅𝑏 . With increasing bubble radius, the variation in 𝜃𝑐 and 𝑕𝑚𝑎𝑥 ⁄𝐻 shows the same trend. Therefore, it is also reasonable to conclude that the probability of jet breakup increases with bubble size, more specifically, with gas 29

ACCEPTED MANUSCRIPT superficial velocity 𝑈𝐺𝑆 and liquid thickness H. 125

130

(a)

c

1 mm

H

120

1.8

Opening angle of the cavity, c

(b)

Ⅰ Ⅱ

Dimensionless maximum jet height, hmax/ H

120

1.5

100

c (°)

c()

Ⅲ

1 mm

1.2

115

hmax/ H

1 mm

110

0.9 110

80 1.0

Exper. data Fitting 1.5

0.6

2.0

2.5

3.0

3.5

4.0

105 0.9

1.0

1.1

1.2

1.3

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90

1.4

Rb (mm)

H (mm)

Fig. 17. (a) Experimental relation between the opening angle of the cavity 𝜃𝑐 and the liquid sheet thickness H (D = 0.58 mm, 𝑈𝐺𝑆 = 3.79 m s). The solid line represents the best linear fit of the data (𝜃𝑐 ∝ (−𝐻)). Top right inset:

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snapshots of the final moment of the bubble collapse before onset of ejection. (Ⅰ), (Ⅱ), and (Ⅲ) correspond, respectively, to the jets (a), (b) and (c) of Fig. 16. (b) Opening angle of the cavity 𝜃𝑐 and non-dimensional maximum jet height 𝑕𝑚𝑎𝑥 ⁄𝐻 (without ejected droplet) as a function of the rupture radius 𝑅𝑏 . Error bars come from a standard deviation derived from measurement errors.

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4. Conclusions

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An experimental study about submerged gas injection into a quiescent thin liquid sheet has been reported. We systematically performed quantitative experiments where gas was injected through a

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micro-channel into a thin liquid sheet over a wide range of gas superficial velocities and liquid thickness. High-speed photography was used to capture the bubbling/collapse (below the free surface)

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and the resulting jet (above the free surface) synchronously. For the experimentally available range of initial parameters, general flow regimes, including the multi-bubble coalescence, periodic

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bubbling-bursting, and pulsating jet, have been delineated. A phase diagram for the flow regime transition was obtained in terms of the Weber number We and the Bond number Bo. Then, what we investigated mainly focus on the behavior of periodic bubbling-bursting under

continuous gas injection. We analyzed the bubble formation, bubble bursting, pinch-off, collapse and the resulting jet in detail. Specific conclusions about the bubble formation in the thin liquid sheet can be summarized as follows. 30

ACCEPTED MANUSCRIPT (1) On the macroscopic level, the variation of the bubbling-bursting frequency upon gas superficial velocity under a broad range of liquid thickness was compiled for two different orifice diameters. The frequency evolution curves show similar trends. For each liquid thickness, there is a critical gas velocity above which the bubbling-bursting frequency would decrease or stabilize at a constant value. (2) Under the prescribed experimental conditions, by increasing the thickness of the liquid sheet,

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results in an increase in the bubble formation time, corresponding to a larger bubble volume. (3) We characterize the confined bubble growth (shape and size) with the transient apparent contact angle, the contact line movement and the bubble eccentricity. The effect of gas superficial velocities manifests itself through the different contact line behavior and thus on the bubble shape

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and rupture size. Generally, an increase in gas superficial velocity significantly increases the bubble contact line radius with higher amplitude of oscillation, and decreases the transient contact angle.

We proposed that the bubble shape and size just before rupture have a significant effect on the

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behaviors of subsequent liquid jet. Bubble deformation is strongly affected by the gas superficial

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velocity when the liquid thickness is below a critical value. Moreover, we formulated an analytical argument for the dimensionless rupture radius and provided experimentally a scaling law giving the

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rupture radius as a function of the initial parameters, expressed in terms of the modified Bond number and Weber number.

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Following bubble bursting, an unstable conical cavity forms with a train of capillary waves converging to the bottom of the bubble and pinching it. The dynamic characteristics of capillary

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wave velocity during bubble collapse are presented and analyzed in detail. It should be noted that the transient wave velocity keeps decreasing until the pinch-off occurs. Both the increased gas superficial velocity and decreased liquid thickness contribute to the enhancement of the average propagation velocity of the waves. With increase of the liquid thickness, the jet morphology undergoes a neat qualitative change: the jet first becomes fat and small and then ends up thinner, detaching more and smaller droplets due to Rayleigh-Plateau instability. Lastly, we demonstrated the 31

ACCEPTED MANUSCRIPT importance of the cavity shape, characterized with an opening angle, in determining the evolution of the jet structure. Our study provides the first quantitative and qualitative overview of the events following submerged injection of gas into a thin quiescent liquid sheet. We hope that the results herein can provide a helpful guidance and supplement to the physical problem of submerged gas injection. Nevertheless, we also note that it is by no means complete, and there are still many problems waiting

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for us to further explore, e.g., the jet characteristics and the effect of liquid viscosity on it.

Acknowledgments

This work was supported by the National Natural Science Foundation of China [grant number 51575476], and the Science Funding for Creative Research Groups of the National Natural Science

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