International Journal of Thermal Sciences 152 (2020) 106341

1290-0729/© 2020 Elsevier Masson SAS. All rights reserved.

Numerical analysis of bubble bursting at the liquid surface by wave propagation

Yongpan Cheng a,*, Yang Shen a, Dong Liu a, Jinliang Xu a,**, Yi Sui b

a Beijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy, North China Electric Power University, Beijing, 102206, China b School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom

A R T I C L E I N F O

Keywords: Bubble bursting Jet formation Capillary waves Wave propagation

A B S T R A C T

Bubble bursting at the liquid surface can generate the jet drops and satellite bubbles, which is widely encoun-tered in boiling, ocean engineering, bioengineering etc. This process is affected by quite a few parameters, such as surface capillarity, gravity, temperature, density, viscosity and bubble size. In this paper, the axisymmetric numerical model with level set method is built up to study the bubble bursting process. It is found that the bubble bursting process can be analyzed through the capillary wave propagation in terms of the tangential angle along the gas-liquid interface. The capillary wave propagation method can describe well the effect of Ohnesorge number, Marangoni number, density and viscosity ratios of gas over liquid on the bubble bursting. The critical Ohnesorge number for jet drop formation during bubble bursting with negligible gravity, gas inertia and vis-cosity is around 0.045, and it will decrease dramatically with increasing ratios of gas density or viscosity over liquid. The jet formation during bubble bursting can be suppressed by applying the negative temperature gradient across the bubble, or by increasing density and viscosity ratio of gas over liquid, mainly because they can damp the capillary waves. These findings may help get insight in the bubble bursting when the gas properties are transiting towards the liquid under pressurized atmosphere.

1. Introduction

Bubble bursting at the gas-liquid interface is a fundamental phe-nomenon in boiling, degassing distillation, ocean engineering, bioengi-neering, food engineering etc. In the oceans the bubbles can be generated from the breaking waves, when the bubbles break up at the sea surface, the droplets with nano-metric to hundred-micrometers size will be produced, carrying billions of tons of sea salt into the atmosphere every year [1]. Richter and Veron [2] found that the jet droplets, especially those with small size, could exist in the atmosphere for several days. During this period, the momentum and heat will be exchanged with the atmosphere through the suspended droplets. The exchange will become more significant on tropical cyclones, when the normal sharp air-sea interface is replaced by a thick, foamy mixture of air and water [2]. The aerosol particles from the bursting bubbles can act as the cloud condensation nuclei and scattering radiation [3]. The aerosols can also transfer the pathogens from the surf zone of coastal regions [4] and

raindrops on soil [5]. In carbonated beverages, such as sparkling wine, the bursting bubbles are desired because the generated aerosols can enhance the aroma [6–8], which can enhance the sensorial experiences greatly. Besides, in the bioreactor, the bubble bursting at free surface can destroy the bacterial cell around the aeration sites [9].

The bubble bursting was firstly studied by Woodcock et al. [10] with the high-speed photographic techniques, they observed the bubble collapse, jet formation and satellite droplet generation etc. They further found that bubble bursting time is proportional to the bubble size, each bubble can produce about five droplets, which are at the order of one tenth of the bubble size [11]. In order to reveal the mechanism behind the bubble bursting at the free surface, through experiment MacIntyre [12] found that the boundary-layer shear flow during bubble collapse is driven by the high pressure gradients due to the high surface curvature. A capillary ripple precedes the main toroidal rim, transporting mass along the surface at about 90% of its phase velocity, the axial jets are created by the convergence of this flow, around 20% of the

* Corresponding author. ** Corresponding author.

E-mail addresses: chengyp@ncepu.edu.cn (Y. Cheng), xjl@ncepu.edu.cn (J. Xu).

Contents lists available at ScienceDirect

International Journal of Thermal Sciences

journal homepage: http://www.elsevier.com/locate/ijts

https://doi.org/10.1016/j.ijthermalsci.2020.106341 Received 13 August 2019; Received in revised form 16 February 2020; Accepted 20 February 2020

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boundary-layer thickness flows into the upward jet. With the boundary integral method, Boulton-Stone and Blake [9]

simulated the bubble bursting at the free surface, the boundary layer was included to investigate the viscous effect, the vorticity development in the fluid was obtained as well as the pressure and energy dissipation rates. Through experimental research and theoretical prediction Walls et al. [13] found that at Oh > 0:037 no jet drops are generated due to the viscous damping effect, at Bo > 3 no jet drops are generated due to the gravitational effect. The bubble bursting is investigated in the inter-mediate regime when both the viscosity and gravity play the roles.

Furthermore, they also found that the equilibrium shape has great effect in the production of jet drops, and the shape is mainly determined by the gravity before the rupture of bubble. The surrounding gas has minimal influence on the jet drop formation when the viscosity ratio μg=μl ranges from 10� 3 to10� 2. Based on the numerical simulation and experiment, Mead-Hunter et al. [14] proposed a new parameter called droplet number Dn ¼ BoOh2 which governs the formation of jet drops and complements current limit Oh < 0:037 and Bo < 3. The droplet number provides the final boundary condition to the system at the values of around 1:26� 10� 4 and 3:98� 10� 4. Through scaling analysis Ga~n�an-Calvo [15] proposed scaling laws for top jet drop size and speed, which collapse the published data since 1954, the critical Oh number is 0.043 for jet droplet generation when gravity effects are negligible. With the ultrafast X-ray and optical imaging method, Lee et al. [16] studied the bursting for small bubbles (R < 100μm), and built a phase diagram for jetting and the absence of jetting, the critical Oh number was found to be 0.052 for jet formation. Zhang et al. [17] proposed the critical bubble size Dcr ¼ 100:1914σ0:5517

L ρ� 0:4830L g� 0:5170μ� 0:0688

L , above which the bursting bubble will not produce the jet drop, the drop numbers for bubble bursting are also correlated asNdr ¼ 7:9 expð � DB =ð0:338DcrÞÞ �

0:41. Duchemin [18] comprehensively studied the bursting process of

bubbles with different sizes on the free surface by numerical simulation. It is found that the fast jets are produced by the collapse of a cavity created by focusing the capillary waves, the cavity collapse can be described by a self-similar dynamics based on the balance of inertial and capillary. It is also found that the diameter of the droplets produced at the top of the jet is about one tenth of the initial diameter of the bubble. Furthermore, a tiny bubble may be formed at the base of the jet. Brasz et al. [19] found that the smallest size of bursting droplets is not necessarily produced by the smallest radius of bubbles. Through the high-speed camera imaging and numerical calculation, they found that the minimum drop size produced by the jet is determined by the inter-action of viscous, inertial and capillary forces prior and subsequent to the jet formation. Deike et al. [20] studied the jet velocity in details through both experiment and simulation, they focused on the shape of the cavity on the jet velocity and the regime of parameters for which

Fig. 1. Schematic diagram of bubble bursting at the gas-liquid interface.

Fig. 2. Comparison of the snapshots of gas-liquid interfaces during bubble bursting between current numerical simulation and experiment by Ghabache22. The time in the experiment is normalized with the capillary-inertial time scale for consistency.

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drop ejection was observed. A formula was proposed to predict the jet velocity for the full range of parameters. The entrainment of small gas bubble at the bottom of jet was found due to the fast oscillations of the cavity bottom.

With scaling laws Krishnan et al. [21] studied the jet velocity from the bubble bursting with the effects of gravity and viscosity, they found that the viscosity affects the jet velocity in two ways, one is that it can damp the precursor capillary waves, the other is that it can damp the jet formation and dynamics. Ghabache et al. [22] proposed a scaling law for the jet velocity considering the effect of bubble shape, capillary waves, gravity and liquid properties. Lai et al. [23] proposed a non-dimensional Laplace number, which controls the cavity and jet dynamics. Based on a scaling law above a critical Laplace number, a self-similar formula is presented to describe the history of the jetting process from cavity collapse to droplet formation.

From the review above, it can be seen that the previous studies focus on the bubble bursting when the gas properties, such as density, vis-cosity, are much lower than the liquid properties. However, in some cases, such as boiling in pressurized vessels, the properties at the gas and liquid regions are comparable, the inertia and viscosity in the gas region cannot be ignored any more, the characteristics of bubble bursting will be changed. During the heat transfer, there exits the temperature gradient near the gas-liquid surface, the uneven temperature distribu-tion may cause the thermal capillary effect, which can affect the bubble bursting definitely. Hence in this study, the bubble bursting will be investigated when the gas properties transits towards liquid with increasing ratios of density and viscosity. Furthermore, the capillary waves are often used to analyze the jet formation in the previous studies, but it is difficult to quantitatively capture the capillary waves on the deforming interface. In our paper the propagation of capillary wave will be quantified, and the universal explanation will be provided to explain the bursting behavior under different parameters.

In the following sections, the physical and numerical models will be built up first; After validation with the reliable experimental data, the effects of surface tension, viscosity, density and viscosity ratio and thermal capillarity on the bubble bursting will be investigated in details, the generation of satellite drop and bubble will be addressed; Finally, some conclusions will be drawn for bubble bursting.

2. Physical model and mathematical formulation

As shown in Fig. 1, the schematic diagram of bursting bubbles at the liquid surface is provided. Before the bubble bursts at the gas-liquid interface, the bubble rises in the liquid under the buoyant force, then reaches the interface with a thin film separated from the ambient gas. Then the film will retract and fragment, leading to the bubble bursting. In this paper the bubble bursting after the film fragment is focused with prior bubble rising process ignored. As suggested by Krishnan et al. [21], at Bo < 0:1, the gravity has negligible effect on the bubble bursting in terms of the cavity depth, hence the initial condition can be assumed as that a spherical bubble is exactly tangent to the flat gas-liquid interface with a small hole at the top. The bubble bursting is capillary-driven process with viscous effect. Due to the large curvature at the film edge at the initial condition, the generated high pressure will drive the film to retract, and the capillary waves will travel from the breaking position at the bubble top to the bottom. When the waves converge at the bottom, the jet or tiny bubble will be generated, and the jet may break into tiny droplet due to the wave instability. Due to the symmetry in r direction, the axisymmetric level set method is used to simulate the bursting process of bubble, the right region from the axis is taken as the computational domain.

The bubble radius R0, the capillary-inertial velocity u0 ¼ffiffiffiffiffiffiffiσ0

ρlR0

q, the

temperature drop ΔT across the bubble radius are defined as the

Fig. 3. Snapshots for bubble bursting process at Oh ¼ 0.035.

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characteristic length, velocity and temperature respectively, here σ0 is the surface tension, ρlis the liquid density. The following non- dimensional parameters are defined:

Reynolds number

Re¼ρlu0R0

μl(1a)

Capillary number

Ca¼μlu0

σ0(1b)

Marangoni number

Ma¼σT ΔT

σ0(1c)

Ohnesorge number

Oh¼μlffiffiffiffiffiffiffiffiffiffiffiffiffi

ρlσ0R0p (1d)

Then the normalized governing equations for bubble bursting are as follows:

Continuity equation

r ⋅ u!¼ 0 (2)

Momentum equation

∂ u!

∂tþ u!⋅r u!¼ �

1ρ*rpþ

1ρ*Re

r⋅�μ*�r u!þr u!T��

þ1 � Ma⋅Tρ*Re⋅Ca

κ n!δðφÞ

�Ma

ρ*Re⋅CarsTδðφÞ

(3)

Energy equation

∂T∂tþ u! ⋅rT ¼

1ρ*C*

pRe⋅Prr⋅ðλ*rTÞ (4)

Level Set convection equation

∂φ∂tþ u!⋅rφ ¼ 0 (5)

The normalized density, dynamic viscosity, specific heat and thermal conductivity are defined as follows

ρ*¼H þ ð1 � HÞrd (6a)

μ * ¼ H þ ð1 � HÞrμ (6b)

C*p¼H þ ð1 � HÞrCp (6c)

λ*¼H þ ð1 � HÞrλ (6d)

Here the property ratios of gas over liquid are

rd ¼ ρg�

ρl; rμ ¼ μg�

μl; rλ ¼ λg�

λl; rCp ¼ Cpg�

Cpl (17)

The volume fraction is defined with the Heaviside function

HεðφÞ¼

8>>><

>>>:

0 φ < ε12

�

1þφε þ

1π sin

�πφε

��

jφj � ε

1 φ > ε

(8)

where ε ¼ 1:5dx, δ is the Dirac delta function as. rHðφÞ

Fig. 4. Transient variations of gas-liquid interface at the axis (r ¼ 0) during bubble bursting at Oh ¼ 0.035.

Fig. 5. Wave propagation along gas-liquid interface in terms of tangential angle at Oh ¼ 0.035.

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Fig. 6. Bubble bursting process under different Oh numbers (Jet drop at Oh ¼ 0.01 and 0.035; No jet drop at Oh ¼ 0.04 and 0.08).

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2.1. The normal vector at the interface

n!¼rφjrφj

(8a)

The curvature

κ¼r⋅rφjrφj

(8b)

The finite volume method is adopted to discretize the governing

equations on a staggered marker-and-cell (MAC) mesh. The standard projection method is adopted to solve the coupling between the conti-nuity equation and momentum equation. In order to improve the ac-curacy, the advection term is solved with the Adams-Bashforth scheme, the diffusion term is solved with the Crank-Nicolson scheme [24]. For the level set advection equation, the advection term is discretized with a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme, then the re-initialization algorithm is adopted to keep φ as the algebraic distance to the interface [25]. All the spatial discretizations are per-formed with the second-order central difference scheme.

After the sensitivity study the computational domain is of the size of 4� 8, the left boundary r ¼ 0 is symmetric, other boundaries are free boundary conditions, including lower boundary (z ¼ 0), the right boundary (r ¼ 4) and upper boundaries (z ¼ 8). The grid size 201� 401 is adopted, and the time step is set as 1� 10� 5. The radius of initial hole for bubble bursting is 0.26, corresponding to the tangent angle is 165o. The default parameters for bubble bursting are follows: Oh ¼ 0:035, Ma ¼ 0, the property ratio rd ¼ rμ ¼ 0:02rλ ¼ 0:043,rCp ¼ 0:012.

3. Results and discussion

3.1. Numerical validation

In order to prove the reliability of current numerical mode, the nu-merical results are compared with the experiment by Ghabache et al. [22], as shown in Fig. 2. For the convenience of comparison, the bursting time is normalized with the characteristic time, it can be seen that the snapshots of the bubble profiles in our numerical simulation agree quite well with the experimental observations. The feasibility of current model lays the solid foundation for the following simulations.

3.2. Bubble bursting process

Fig. 3 shows the snapshots for bubble bursting process at the gas-

Fig. 7. Wave propagation along gas-liquid interface in terms of tangential angle under different Oh numbers att ¼ 0:3.

Fig. 8. Velocity and temperature fields at Ma ¼ 0:1 att ¼ 0:3.

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Fig. 9. Bubble bursting process under different Ma numbers at Oh ¼ 0:035.

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liquid interface at Oh ¼ 0:035. At the initial stage t ¼ 0, the hole is assumed at the top of the bubble, due to the high curvature at the edge of bubble, the generated high capillary pressure will drive the liquid film to retract, the capillary waves will be generated and propagate along both the horizontal direction and the downward direction. The wave propa-gation in the horizontal gas-liquid interface has little effect on the bubble bursting, hence the wave propagation along the bubble interface is focused. With the wave propagation, the bubble is deforming with the rim at the edge of bubble becoming larger. It is noted that in the un-disturbed region the bubble still keeps spherical. At around t ¼ 0:51, the capillary waves reach the bottom of the bubble, then a jet is formed and moves rapidly upward. In the meanwhile, a tiny bubble is generated below the jet, and remains in the liquid. During the jet formation the satellite droplet will be pinched off due to the instability of the jet interface, then the remaining jet will retract back to the liquid. There-fore, during the bubble bursting both the capillary force and viscous force play the important role, their interplay determines the bursting characteristics of the bubble.

Fig. 4 shows the transient variations of interface location and ve-locity at the axis (r ¼ 0) during bubble bursting. It can be seen that at the initial stage the interface at the bubble bottom moves slowly, at point A the capillary waves from the top of the bubble reach the bottom of the bubble, then at t ¼ 0:54 the jet is formed as well as the tiny bubble. The jet velocity at the tip increases rapidly to as high as 27, then slows down due to the viscous dissipation inside the jet. At t ¼ 0:77 for point B, the satellite drop is pinched off from the tip of the jet, and starts to move upward at the steady velocity around 5.0. The gas-liquid interface will retract and stabilize to the location at height z ¼ 4:0.

The capillary wave along the interface around the bubble has sig-nificant effect on the bubble bursting, but it is difficult to quantify the wave propagation along the interface, because the interface is deforming when the wave is propagating along it. Here the tangential angle along the interface is quantified the propagating waves, as defined in Fig. 1. The origin is defined at the bottom of the bubble, l is the arc length from the origin. At t ¼ 0 the bubble is spherical, then the tangential angle will increase linearly with the arc length along the bubble, as shown in Fig. 5. When the wave is generated and propagates along the interface, the rim is formed with the increasing tangential angle before it and decreasing tangential angle behind it. The peak position with the largest value of tangential angle can be considered roughly as the wave crest. For the undisturbed region the tangential angle remains the same with the original state, i.e. increasing linearly with the arc length. Fig. 5 shows the transient variations of tangential angles along the arc length. It can

be found that the wave will propagate from the top to the bottom of the bubble. At the initial stage the variation of tangential angle near the crest is sharp, indicating the shorter wave, then it will become larger due to the mild variation at the crest. By tracking the front edge of the wave, the wave propagation velocity can also be obtained. Therefore, it is a feasible method to adopt the tangential angle along the gas-liquid interface to track the wave propagation.

3.3. Effect of Oh number

The bubble bursting is determined by the interplay of capillary force and viscous force, which can be characterized with Ohnesorge number, as defined in Eq. (1d). Fig. 6 shows the snapshots of bubble profiles during bursting at Oh ¼ 0:01, 0.035, 0.040 and 0.08. It is obvious that with the increasing Oh number, the viscous effect is increased, the bubble bursting can be suppressed. Taking t ¼ 0:57 as an example, at Oh ¼ 0:01 a vertical jet has been formed and the satellite drop is going to pinch off. At Oh ¼ 0:035 and 0.04, the jet has just be formed, and starts to move upward. But it is noted that there is satellite drop from the jet at Oh ¼ 0:035, while no any drop at Oh ¼ 0:04. At Oh ¼ 0:08 the capillary waves just converge at the bubble bottom without jet formation. It can also be observed that for the intermediate Ohnesorge number (0.035 and 0.04), the tiny bubbles can be generated at the bottom of the jet, while at lower Ohnesorge number or larger Ohnesorge number, no bubbles are generated.

From Fig. 6 it can be found that the wave at low Ohnesorge number can propagate faster than that at high Ohnesorge number. In order to quantify the wave propagation along the interface, the tangential angles along the gas-liquid interface around the bubble are provided at the typical time t ¼ 0:3 at which the waves have not reached the bubble bottom. It is proved that the wave can propagate faster at low Ohnesorge number. Although the wave at Oh ¼ 0:035 propagates only a little faster than that at Oh ¼ 0:04, there is much difference in bursting the satellite drop. Prior to the main wave at Oh ¼ 0:01, a weaker wave is traveling along the surface, which can also be found from Fig. 7. At higher Ohnesorge number, the viscous force will damp the weak wave, so it is not easy to be observed. With the help of obtained Ohnesorge number for different bursting performance, the satellite drop can be suppressed by controlling the corresponding liquid physical properties and bubble size, according to Eq. (1d).

3.4. Effect of thermo-capillarity

When the liquid and gas are of different temperatures, there will be temperature gradient during bubble bursting at the interface. As the surface tension depends greatly on the temperature, the tangential thermal capillary force may be generated due to the uneven temperature distribution along the surface, then it will affect the bubble bursting. As seen in Fig. 8, for simplicity the uniform vertical temperature gradient is assumed across the liquid and gas regions. Along the gas-liquid interface around the bubble, for the downward temperature gradient, the upward thermal capillary force will be generated, which may impede the wave propagation along the interface, hence affect the bubble bursting. In contrast, for the upward temperature gradient the downward thermal capillary force can promote the wave propagation.

The thermal capillary effect is defined with Marangoni number in Eq. (1c), which is positive for downward temperature gradient and negative for upward temperature gradient. Fig. 9 shows the snapshots during bubble bursting processes at different Marangoni numbers. It can be found that when Marangoni number is negative, which corresponds to the positive temperature gradient, the jet formation at Ma ¼ � 0:2 is earlier than that at Ma ¼ � 0:1, and both are earlier than that at Ma ¼ 0 without thermal capillary effect. When Marangoni number is positive, which corresponds to the negative temperature gradient, there is no jet formation for both Ma ¼ 0:1 and 0.2, furthermore, the bubble profile evolution during bubble bursting at Ma ¼ 0:2 is slower than that at

Fig. 10. Wave propagation along gas-liquid interface in terms of tangential angle under different Ma numbers at t ¼ 0:3.

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Fig. 11. Bubble bursting process under different gas-liquid density ratios at Oh ¼ 0.035. (Jet drop atrd ¼ 0:02; Without jet drop at rd ¼ 0:04rd ¼ 0:3rd ¼ 1:0).

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Ma ¼ 0:1. This can be explained from the wave propagation along the gas-liquid interface. Fig. 10 shows the wave propagation under different Marangoni numbers in terms of tangential angles over arc length at t ¼0:3. It is proved that with the increasing Marangoni number, the wave propagation can be impeded, thus the jet formation can be suppressed. It is obvious that for the condensation heat transfer, corresponding to positive temperature and negative Marangoni number, the larger jet is easier to be generated, while for the boiling heat transfer, the jet is easier to be suppressed.

3.5. Effect of gas-liquid density ratio

It is well known that the liquid density has great effect on the bubble bursting, but the density ratio of gas over liquid has also great effect. For example, under the pressurized atmosphere, the gas density may be comparable with the liquid density. Zhang et al. [26] investigated the bubble coalescence under different pressures through both experiment and simulation, and found that the density variation caused by the pressure in the chamber has great effect on the bubble coalescence, but the effect of density ratio on the bubble bursting has seldom be addressed so far. Fig. 11 shows the snapshots of bubble bursting at Oh ¼0:035 under density ratio from 0.02 to 1.0. It is obvious that when the density ratio is increased from 0.02 to 0.04, there is no satellite drop bursting from the jet. If the density ratio is increased further to 0.3 and 1.0, no upward jet is produced, because the capillary waves do not converge at the bottom of the bubble, then the satellite bubble is pinched off from the original bubble. With the increasing density ratio, the sat-ellite bubble can become larger, even with the size of half over the original bubble.

The effect of density ratio on the bubble bursting can be attributed to the difference in propagating capillary waves, as shown in Fig. 12. It is found that the large density ratio can impede the wave propagation along the bubble surface. Take the location of wave crest as example. At time t ¼ 0:3, the wave crest for rd ¼ 0:02 is at the arc length of 1.18, while that for rd ¼ 1:0 is at the arc length of 1.79. It is interesting to find that with large density ratio, the inertia in the gas region cannot be ignored, it can not only suppress the jet formation above the original bubble, but also promote the satellite bubble formation below the original bubble. This may be used to generate new bubbles by adjusting the density ratio of gas over liquid.

3.6. Effect of gas-liquid viscosity ratio

The viscosity can affect the bubble bursting in two ways, one is that it can damp the precursor capillary waves, the other is that it can damp the jet formation and dynamics. Not only the liquid viscosity, but also the gas viscosity can affect the bubble bursting. Here the effect of gas-liquid viscosity ratio on the bubble bursting is provided at Oh ¼ 0:035, as shown in Fig. 13. It can be seen that when the viscosity ratio is increased from 0.02 to 0.2, there is no satellite drop generation; if the viscosity is increased further to 0.4, the jet formation is greatly suppressed, and there is almost no jet formation when the ratio is 0.8, also there is no trapped tiny bubble at the bottom of the bubble. The difference under different viscosity ratios can be explained by the capillary wave prop-agation, as seen in Fig. 14. It is found that with increasing viscosity ratio, due to the damping effect the wave will become weaker and propagate slower, hence leading to the suppression of jet formation and jet drop. For example, when the viscosity ratio increases from 0.02 to 0.8, the location of wave crest will increase from 1.13 to 1.41 in the arc length. It is obvious that viscosity in gas region can affect the jet formation during bubble bursting; hence the bubble bursting can be suppressed under the pressurized atmosphere.

3.7. The critical Oh number

The density ratio and viscosity ratio have great effect on the jet formation during bubble bursting, because they can affect the wave propagation along the gas-liquid interface. Here the critical Oh number on jet formation is studied under different density ratio and viscosity ratio. After sensitivity check on the density ratio and viscosity ratio, it is found that when the density ratio and viscosity ratio are below 0.01, the ratios have little effect on the critical Oh number, which is fixed at 0.045. It is quite close to the value 0.037 by Walls et al. [13], 0.043 by Ga~n�an-Calvo [15], 0.052 by Lee et al. [16]. It is noted that in the study in Fig. 15, when the density ratio is varied, the viscosity ratio is fixed at 0.02, and when the viscosity ratio is varied, the density ratio is fixed at 0.02. From Fig. 15 it is found that with the increasing density ratio or viscosity ratio, the critical Oh on jet formation is decreased dramatically, especially for density ratio. When the density ratio is increased from 0.01 to 0.06, the critical Oh will decrease from 0.044 to 0.011, the effect of viscosity ratio is not so significant as density ratio. When the viscosity ratio is increased from 0.01 to 0.2, the critical Oh will decrease from 0.041 to 0.026.

4. Conclusion

The bubble bursting at the liquid surface is a fundamental phe-nomenon in ocean engineering, bioreactor, boiling etc. In this study the axisymmetric numerical model with a level set method is built up to study the bubble bursting process with negligible gravity effect. The effect of surface tension, thermocapillarity, density and viscosity ratio on the bubble bursting is provided, the following results are found:

1. The tangential angle along the gas-liquid interface can be adopted to quantify the capillary wave propagation.

2. The positive temperature gradient across the bubble can promote the bubble bursting, while the negative temperature gradient can hinder the bubble bursting.

3. The gas density and viscosity can damp the capillary wave propa-gation and jet formation. The critical Oh number for jet drop for-mation during bubble bursting with negligible gravity, gas inertia and viscosity is 0.045. With the increasing density ratio or viscosity ratio, the critical Oh number for jet formation will be decreased.

4. The increasing density ratio of gas over liquid can suppress the jet formation, but can promote the satellite bubble formation below the jet during bubble bursting. The increasing viscosity ratio can also suppress the jet formation.

Fig. 12. Wave propagation along gas-liquid interface in terms of tangential angle under different gas-liquid density ratios at Oh ¼ 0.035 and t ¼ 0:3.

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Fig. 13. Bubble bursting process under different gas-liquid viscosity ratio at Oh ¼ 0.035 (Jet drop at rv ¼ 0:02; Without jet drop at rv ¼ 0:2rv ¼ 0:4rv ¼ 0:8).

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The above results can be quite useful for getting insight on the bubble bursting process at the gas-liquid interface when the gas properties transit towards liquid, and they can guide the active manipulation on jet formation during bubble bursting.

Acknowledgement

This work was financially supported by the Natural Science Foun-dation of China (Grant No: 51436004) and Marie Curie European Fellowship (Grant No. 658437).

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Fig. 15. The critical Oh number for jet droplet formation under (a) different density ratios (viscosity ratio is fixed at rv ¼ 0:02) and (b) different viscosity ratios (density ratio is fixed at rd ¼ 0:02).

Fig. 14. Wave propagation along gas-liquid interface in terms of tangential angle under different gas-liquid viscosity ratios at Oh ¼ 0.035 and t ¼ 0:3.

Y. Cheng et al.