Skip to main content

Bursting bubbles and the formation of gas jets and vortex rings

Abstract

Bubble bursting is important in air–sea interactions, food science, and industry, but the process of the pressurized gas escaping from inside the bursting bubble is not well understood. The fluid dynamics of gas jets and vortex rings produced by the bursting of 440 µm to 4 cm diameter smoke-filled bubbles resting at an air–water interface is investigated using high-speed stereophotogrammetry. The initial speed of the gas jet released from the bubbles increases with parent bubble size until the Bond number reaches unity and subsequently increases more slowly. The slow, low Reynolds number jets characteristic of small bubbles are attributed to high film retraction speeds which produce relatively large holes in the bubble cap, and these jets roll up into spherical, slow-growing vortex rings which travel short distances. However, the low film retraction speeds characteristic of larger bubbles produce high speed, high Reynolds number jets emitted through relatively small holes which roll up into highly oblate, fast-growing, far-traveling vortex rings. The tiniest bubbles eject only a thin stem-like jet which does not form a vortex ring. Finally, a simple scaling relationship relating the gas jet Reynolds number to the square root of the parent bubble Bond number is proposed.

Graphic abstract

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

References

  1. Afeti GM, Resch FJ (1990) Distribution of the liquid aerosol produced from bursting bubbles in sea and distilled water. Tellus B 42:378–384. https://doi.org/10.1034/j.1600-0889.1990.t01-2-00007.x

    Article  Google Scholar 

  2. Akhmetov DG (2009) Vortex rings. Springer Science & Business Media

  3. Al Shakarji R, He Y, Gregory S (2012) Acid mist and bubble size correlation in copper electrowinning. Hydrometallurgy 113–114:39–41. https://doi.org/10.1016/j.hydromet.2011.11.014

    Article  Google Scholar 

  4. Bamforth CW (1985) The foaming properties of beer. J Inst Brew 91:370–383. https://doi.org/10.1002/j.2050-0416.1985.tb04359.x

    Article  Google Scholar 

  5. Blackburn EA, Wilson L, Sparks RJ (1976) Mechanisms and dynamics of strombolian activity. J Geol Soc 132(4):429–440

    Article  Google Scholar 

  6. Blanchard DC (1963) The electrification of the atmosphere by particles from bubbles in the sea. Prog Oceanogr 1:73–202

    Article  Google Scholar 

  7. Blanchard DC (1989a) The ejection of drops from the sea and their enrichment with bacteria and other materials: a review. Estuaries 12:127–137

    Article  Google Scholar 

  8. Blanchard DC (1989b) The size and height to which jet drops are ejected from bursting bubbles in seawater. J Geophys Res 94:10999–11002

    Article  Google Scholar 

  9. Brasz CF, Bartlett CT, Walls PLL et al (2018) Minimum size for the top jet drop from a bursting bubble. Phys Rev Fluids. https://doi.org/10.1103/PhysRevFluids.3.074001

    Article  Google Scholar 

  10. Buchholz JHJ, Sigurdson LW (2000) The kinematics of the vortex ring structure generated by a bursting bubble. Phys Fluids 12:42–53. https://doi.org/10.1063/1.870283

    Article  MATH  Google Scholar 

  11. Buchholz JHJ, Sigurdson LW (2002) An apparatus to study the vortex ring structure generated by a bursting bubble. Meas Sci Technol 13:428–437. https://doi.org/10.1088/0957-0233/13/4/302

    Article  Google Scholar 

  12. Buckley MP, Veron F (2017) Airflow measurements at a wavy air–water interface using PIV and LIF. Exp Fluids 58(11):161

    Article  Google Scholar 

  13. Cheng YS, Zhou Y, Irvin CM et al (2005) Characterization of marine aerosol for assessment of human exposure to brevetoxins. Environ Health Perspect 113:638–643. https://doi.org/10.1289/ehp.7496

    Article  Google Scholar 

  14. Chojnicki KN, Clarke AB, Phillips JC, Adrian RJ (2015) Rise dynamics of unsteady laboratory jets with implications for volcanic plumes. Earth Planet Sci Lett 412:186–196

    Article  Google Scholar 

  15. Culick FEC (1960) Comments on a ruptured soap film. J Appl Phys 31:1128

    Article  Google Scholar 

  16. Dabiri JO, Gharib M (2005) Starting flow through nozzles with temporally variable exit diameter. J Fluid Mech 538:111–136

    Article  Google Scholar 

  17. Dasouqi AA, Murphy DW (2020) Gas escape behavior from bursting bubbles. Phys Rev Fluids 5:110502. https://doi.org/10.1103/PhysRevFluids.5.110502

    Article  Google Scholar 

  18. Day JA (1964) Production of droplets and salt nuclei by the bursting of air-bubble films. Q J R Meteorol Soc 90(383):72–78

    Article  Google Scholar 

  19. Day JA, Lease JC (1968) Cloud nuclei generated by bursting air bubbles at the air-sea interface. Proceedings of the International Conference on Cloud Physics, pp 20–24

  20. Deike L, Ghabache E, Liger-Belair G et al (2018) Dynamics of jets produced by bursting bubbles. Phys Rev Fluids 3:1–20. https://doi.org/10.1103/PhysRevFluids.3.013603

    Article  Google Scholar 

  21. Ehrenhauser FS, Avij P, Shu X et al (2014) Bubble bursting as an aerosol generation mechanism during an oil spill in the deep-sea environment: laboratory experimental demonstration of the transport pathway. Environ Sci Process Impacts 16:65–73. https://doi.org/10.1039/c3em00390f

    Article  Google Scholar 

  22. Fjeld AM (2006) The gas fluxing of aluminum: Mathematical modeling and experimental investigations. Dissertation, University of California, Berkeley

  23. Gekle S, Gordillo JM (2010) Generation and breakup of Worthington jets after cavity collapse. Part 1. Jet formation. J Fluid Mech 663:293

    Article  Google Scholar 

  24. Gekle S, Peters IR, Gordillo JM, van der Meer D, Lohse D (2010) Supersonic air flow due to solid-liquid impact. Phys Rev Lett 104(2):024501

    Article  Google Scholar 

  25. Ghabache E, Séon T (2016) Size of the top jet drop produced by bubble bursting. Phys Rev Fluids 1:1–7. https://doi.org/10.1103/physrevfluids.1.051901

    Article  Google Scholar 

  26. Ghabache E, Antkowiak A, Josserand C, Séon T (2014) On the physics of fizziness: how bubble bursting controls droplets ejection. Phys Fluids. https://doi.org/10.1063/1.4902820

    Article  Google Scholar 

  27. Hayami S, Toba Y (1958) Drop production by bursting of air bubbles on the sea surface (1) experiments at still sea water surface. J Oceanogr Soc Jpn 14:145–150. https://doi.org/10.1007/s11483-011-9220-5

    Article  Google Scholar 

  28. Hollingshead CL, Johnson MC, Barfuss SL, Spall RE (2011) Discharge coefficient performance of Venturi, standard concentric orifice plate, V-cone and wedge flow meters at low Reynolds numbers. J Pet Sci Eng 78(3–4):559–566

    Article  Google Scholar 

  29. Hedrick TL (2008) Software techniques for two- and three-dimensional kinematic measurements of biological and biomimetic systems. Bioinspir Biomim. https://doi.org/10.1088/1748-3182/3/3/034001

    Article  Google Scholar 

  30. Iacono MJ, Blanchard DC (1987) An investigation of vortex rings from bursting bubbles. Atmos Res 21:139–149. https://doi.org/10.1016/0169-8095(87)90004-4

    Article  Google Scholar 

  31. Illy E, Navarini L (2011) Neglected food bubbles: the espresso coffee foam. Food Biophys 6:335–348

    Article  Google Scholar 

  32. Jackson BE, Evangelista DJ, Ray DD, Hedrick TL (2016) 3D for the people: multi-camera motion capture in the field with consumer-grade cameras and open source software. Biol Open 5:1334–1342. https://doi.org/10.1242/bio.018713

    Article  Google Scholar 

  33. Jaw SY, Chen CJ, Hwang RR (2007) Flow visualization of bubble collapse flow. J Vis 10:21–24. https://doi.org/10.1007/BF03181797

    Article  Google Scholar 

  34. Kim D, Yi SJ, Kim HD, Kim KC (2014) Visualization study on the transient liquid film behavior and inner gas flow after rupture of a soap bubble. J Vis 17:337–344. https://doi.org/10.1007/s12650-014-0217-2

    Article  Google Scholar 

  35. Kim S, Park H, Gruszewski HA, Schmale DG, Jung S (2019) Vortex-induced dispersal of a plant pathogen by raindrop impact. Proc Natl Acad Sci 116(11):4917–4922

    Article  Google Scholar 

  36. Kobayashi T, Namiki A, Sumita I (2010) Excitation of airwaves caused by bubble bursting in a cylindrical conduit: Experiments and a model. J Geophys Res Solid Earth 115(B10). https://doi.org/10.1029/2009JB006828

  37. Krishnan S, Hopfinger EJ, Puthenveettil BA (2017) On the scaling of jetting from bubble collapse at a liquid surface. J Fluid Mech 822:791–812. https://doi.org/10.1017/jfm.2017.214

    Article  Google Scholar 

  38. Lewis ER, Lewis ER, Lewis R, Karlstrom KE, Schwartz SE (2004) Sea salt aerosol production: mechanisms, methods, measurements, and models, vol 152. American Geophysical Union

  39. Lhuissier H, Villermaux E (2011) Bursting bubble aerosols. J Fluid Mech 696:5–44. https://doi.org/10.1017/jfm.2011.418

    Article  MATH  Google Scholar 

  40. Liger-Belair G, Cilindre C, Gougeon RD et al (2009) Unraveling different chemical fingerprints between a champagne wine and its aerosols. Proc Natl Acad Sci 106:16545–16549. https://doi.org/10.1073/pnas.0906483106

    Article  Google Scholar 

  41. Lourakis MI, Argyros AA (2009) SBA: a software package for generic sparse bundle adjustment. ACM Trans Math Softw 36(1):2

    MathSciNet  Article  Google Scholar 

  42. MacIntyre F (1972) Flow patterns in breaking bubbles. J Geophys Res 77:5211–5228

    Article  Google Scholar 

  43. Mason BJ (1957) The oceans as source of cloud-forming nuclei. Geofis Pura e Appl 36:148–155

    Article  Google Scholar 

  44. Mortensen D, Hua J, Håkonsen A, Haugen T, Fagerlie JO (2017) Modelling of hydrogen removal in gas fluxing of molten aluminium. In: Light metals 2017. Springer, Cham, pp 1445–1450

  45. Murashita T, Ito A, Metoki T, Torikai H (2012) Flow visualization of extinguishing gas released from bursting soap bubbles. Vis Mech Process Int Online J 2:557–568

    Google Scholar 

  46. Murphy DW, Li C, d’Albignac V et al (2015) Splash behaviour and oily marine aerosol production by raindrops impacting oil slicks. J Fluid Mech 780:536–577. https://doi.org/10.1017/jfm.2015.431

    Article  Google Scholar 

  47. Newitt DM (1954) Liquid entrainment 1. The mechanism of drop formation from gas or vapour bubbles. Trans Inst Chem Eng 32:244–261. https://doi.org/10.1016/j.neubiorev.2012.07.005

    Article  Google Scholar 

  48. Patrick MR (2007) Dynamics of Strombolian ash plumes from thermal video: motion, morphology, and air entrainment. J Geophys Res Solid Earth 112(B6). https://doi.org/10.1029/2006JB004387

  49. Peters IR, Gekle S, Lohse D, van der Meer D (2013) Air flow in a collapsing cavity. Phys Fluids 25(3):032104

    Article  Google Scholar 

  50. Poulain S, Villermaux E, Bourouiba L (2018) Ageing and burst of surface bubbles. J Fluid Mech 851:636–671. https://doi.org/10.1017/jfm.2018.471

    MathSciNet  Article  Google Scholar 

  51. Prather KA, Bertram TH, Grassian VH, Deane GB, Stokes MD, DeMott PJ et al (2013) Bringing the ocean into the laboratory to probe the chemical complexity of sea spray aerosol. Proc Natl Acad Sci. https://doi.org/10.1073/pnas

    Article  Google Scholar 

  52. Resch FJ, Darrozes JS (1986) Marine liquid aerosol production from bursting of air bubbles. J Geophys Res 91:1019–1029

    Article  Google Scholar 

  53. Rogers WB (1858) On the formation of rotating rings by air and liquids under certain conditions of discharge. Am J Sci Arts 26:246–258

    Google Scholar 

  54. Savva N, Bush JWM (2009) Viscous sheet retraction. J Fluid Mech 626:211–240. https://doi.org/10.1017/S0022112009005795

    MathSciNet  Article  MATH  Google Scholar 

  55. Séon T, Liger-Belair G (2017) Effervescence in champagne and sparkling wines: from bubble bursting to droplet evaporation. Eur Phys J Spec Top 226:117–156. https://doi.org/10.1140/epjst/e2017-02679-6

    Article  Google Scholar 

  56. Smith B, Neal D (2020) Particle image velocimetry. In: Johnson RW (ed) Handbook of fluid dynamics, 2nd edn. CRC Press, New York, pp 48–1–48-27

    Google Scholar 

  57. Song B, Springer J (1996) Determination of interfacial tension from the profile of a pendant drop using computer-aided image processing. J Colloid Interface Sci 184:77–91. https://doi.org/10.1006/jcis.1996.0597

    Article  Google Scholar 

  58. Spiel DE (1997) A hypothesis concerning the peak in film drop production as a function of bubble size. J Geophys Res C Ocean 102:1153–1161

    Article  Google Scholar 

  59. Spiel DE (1998) On the births of film drops from bubbles bursting on seawater surfaces. J Geophys Res Ocean 103:24907–24918. https://doi.org/10.1029/98JC02233

    Article  Google Scholar 

  60. Swinton AC, Beale E (1917) The bursting of bubbles. Nature 98:469. https://doi.org/10.1038/099005b0

    Article  Google Scholar 

  61. Taylor GI (1959) The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc R Soc Lond Ser A Math Phys Sci 253(1274):313–321

    MathSciNet  MATH  Google Scholar 

  62. Teixeira MA, Arscott S, Cox SJ, Teixeira PI (2015) What is the shape of an air bubble on a liquid surface? Langmuir 31(51):13708–13717

    Article  Google Scholar 

  63. Theriault DH, Fuller NW, Jackson BE, Bluhm E, Evangelista D, Wu Z, Betke M, Hedrick TL (2014) A protocol and calibration method for accurate multi-camera field videography. J Exp Biol 217(11):1843–1848

    Article  Google Scholar 

  64. Torikai H, Murashita T, Ito A, Metoki T (2011) Extinguishment of a laminar jet diffusion flame using a soap bubble filled with nitrogen gas. Fire Saf Sci 10:557–568. https://doi.org/10.3801/IAF

    Article  Google Scholar 

  65. Vergniolle S, Brandeis G (1996) Strombolian explosions: 1. A large bubble breaking at the surface of a lava column as a source of sound. J Geophys Res Solid Earth 101(B9):20433–20447

    Article  Google Scholar 

  66. Veron F (2015) Ocean spray. Annu Rev Fluid Mech 47:507–538. https://doi.org/10.1146/annurev-fluid-010814-014651

    MathSciNet  Article  Google Scholar 

  67. Yousefi K, Veron F, Buckley MP (2020) Momentum flux measurements in the airflow over wind-generated surface waves. J Fluid Mech. https://doi.org/10.1017/jfm.2020.276

    MathSciNet  Article  MATH  Google Scholar 

  68. Zhang L, Lv X, Torgerson AT, Long M (2011) Removal of impurity elements from molten aluminum: a review. Miner Process Extr Metall Rev 32:150–228. https://doi.org/10.1080/08827508.2010.483396

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge Ali Alshamrani for laboratory assistance and Ferhat Karakas for assistance with analysis.

Funding

Funding was provided by a National Academies Gulf Research Program Early Career Research Fellowship and an NSF CAREER award (Award no. 1846925) to DWM.

Author information

Affiliations

Authors

Contributions

AD and DM conceived and designed the experiment and analyzed data. AD carried out experimental work. G-SY developed the model. AD, G-SY, and DM wrote the manuscript. All authors approved the final manuscript.

Corresponding author

Correspondence to David W. Murphy.

Ethics declarations

Conflict of interest

The authors report no conflict of interest.

Availability of data and material

The authors will provide data and material upon reasonable request.

Code availability

The authors will provide code upon reasonable request.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Modeling

We use a quasi-one-dimensional nozzle model to theoretically predict the velocity of the gas jet generated by the bubble bursting. It is assumed that the shape of the bubble is spherical for small bubbles (2R < 10 mm) and hemispherical for large bubbles, and that the temperature of the gas inside of the bubble is the same as that of the surrounding air. We also assume the volume of the bubble is constant in the time interval of interest. If there is no choking in the flow, the mass flow rate of the gas, \(\dot{m}\), in steady isentropic compressible flow through a quasi-one-dimensional nozzle is given by Anderson (2003) as

$$\dot{m} = C_{\text{d}} \frac{{AP_{\text{b}} }}{{\sqrt {\Re T_{\text{b}} } }}\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{P}{{P_{\text{b}} }}} \right)^{{\frac{2}{\gamma }}} \left[ {1 - \left( {\frac{P}{{P_{\text{b}} }}} \right)^{{\frac{\gamma - 1}{\gamma }}} } \right]} ,$$
(4)

where \(P\) is the pressure, \(T\) is the temperature, \(\Re\) is the specific gas constant, \(\gamma\) is the specific heat ratio, \(A\) is the nozzle exit area (i.e., the area of the hole in the bubble), and \(C_{\text{d}}\) is the discharge coefficient. Here the subscript \({\text{b}}\) represents the properties of the air inside the bubble. In addition, the mass flow rate through the opening hole in the bubble film satisfies the following relation:

$$\dot{m} = \rho v_{\text{h}} A_{\text{h}} ,$$
(5)

where \(v_{\text{h}}\) and \(A_{\text{h}}\) are the velocity through the hole and the area of the hole, respectively, and ρ is the density of air. The initial pressure inside the bubble \(P_{{{\text{b}}0}}\) can be given from the Young–Laplace equation as

$$\Delta P = P_{{{\text{b}}0}} - P = \frac{4\sigma }{R},$$
(6)

where P = 101325 Pa is the atmospheric pressure. When the gas is released through the hole in the bubble cap, choking may occur if

$$\frac{{P_{\text{b}} }}{P} \ge \left( {\frac{2}{\gamma + 1}} \right)^{{\frac{\gamma }{1 - \gamma }}} .$$
(7)

For air \((\gamma = 1.4),\) the condition is \(\frac{{P_{\text{b}} }}{P} \ge 1.893.\) For the smallest bubble considered in this model (2R = 4.9), the pressure ratio \(\frac{{P_{\text{b}} }}{P} = 1.002\) is much smaller than the choking condition. Therefore, choking cannot occur for any of the bubbles we consider in this model.

If we assume that both the bubble volume Vb and the pressure inside the bubble remain constant over the time period of interest during which the gas is initially released (\(P_{\text{b}} = P_{{{\text{b}}0}}\) and Vb = Vb0), then using Eqs. (4) and (5) with the assumption \(T \approx T_{\text{b}}\) we can obtain a “constant P and V” model for the velocity at the hole:

$$\left. {v_{\text{h}} } \right|_{{P_{\text{const}} V_{\text{const}} }} = C_{\text{d}} \sqrt {\Re T_{\text{b}} } \left( {\frac{{P_{{{\text{b}}0}} }}{P}} \right)\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{{P_{{{\text{b}}0}} }}{P}} \right)^{{ - \frac{2}{\gamma }}} \left[ {1 - \left( {\frac{{P_{{{\text{b}}0}} }}{P}} \right)^{{\frac{1 - \gamma }{\gamma }}} } \right]} .$$
(8)

In reality, both the pressure inside the bubble and the bubble volume can be expected to decrease as the gas is released. We now consider an alternate “constant V” model in which the pressure in the bubble is allowed to decrease but the bubble volume remains constant over the time of interest. The mass reduction rate of the gas in the bubble then can be written using the equation of state of an ideal gas \(P_{\text{b}} = \rho_{\text{b}} \Re T_{\text{b}}\) and the assumption \(T \approx T_{\text{b}}\) as

$$\frac{{{\text{d}}m_{\text{b}} }}{{{\text{d}}t}} = \frac{{{\text{d}}\left( {\rho_{\text{b}} V_{\text{b}} } \right)}}{{{\text{d}}t}} \approx \frac{{V_{\text{b}} }}{\Re T}\frac{{{\text{d}}P_{\text{b}} }}{{{\text{d}}t}}.$$
(9)

From Eqs. (4) and (9), we can obtain the following equation for the pressure ratio \(\frac{{P_{\text{b}} }}{P}\):

$$\frac{{{\text{d}}\left( {P_{\text{b}} /P} \right)}}{{{\text{d}}t}} = - C_{\text{d}} \sqrt {\Re T_{\text{b}} } \frac{{A_{\text{h}} }}{{V_{\text{b}} }}\left( {\frac{{P_{\text{b}} }}{P}} \right)\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{{P_{\text{b}} }}{P}} \right)^{{ - \frac{2}{\gamma }}} \left[ {1 - \left( {\frac{{P_{\text{b}} }}{P}} \right)^{{\frac{1 - \gamma }{\gamma }}} } \right]} .$$
(10)

The above equation can be numerically integrated using an ordinary differential equation solver such as the fourth-order Runge–Kutta scheme. The equation of the velocity at the hole for this “constant V” model is then given as

$$\left. {v_{\text{h}} } \right|_{{V_{\text{const}} }} = C_{\text{d}} \sqrt {\Re T_{\text{b}} } \left( {\frac{{P_{\text{b}} }}{P}} \right)\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{{P_{\text{b}} }}{P}} \right)^{{ - \frac{2}{\gamma }}} \left[ {1 - \left( {\frac{{P_{\text{b}} }}{P}} \right)^{{\frac{1 - \gamma }{\gamma }}} } \right]} .$$
(11)

However, the jet speed at the hole could not be measured from experiments whereas the speed of the jet front could be measured. Thus, to compare model and experimental results, the velocity at the jet front \(V_{\text{jet}}\) can be derived from the total momentum conservation relation once the velocity at the hole is obtained. Thus, consider the gas jet at \(t = t_{\text{jet}}\) of Fig. 2c. The total momentum of this gas jet is

$${\mathbf{M}}_{\text{jet}} = \mathop \int \limits_{0}^{{L_{\text{jet}} }} v\rho A{\text{d}}l,$$
(12)

where \(L_{\text{jet}}\) is the length of the gas jet from the bubble film to the jet front. Since the gas jet can be considered as incompressible flow, using the mass conservation relation \(\rho vA = \rho V_{\text{jet}} A_{\text{jet}} ,\) Eq. (12) can be rewritten as

$${\mathbf{M}}_{\text{jet}} = \rho V_{\text{jet}} A_{\text{jet}} L_{\text{jet}} .$$
(13)

Here, as shown in Fig. 2c, Ajet is the cross sectional area of the jet front. Because \({\mathbf{M}}_{\text{jet}}\) is equal to the total amount of the momentum incoming through the hole from \(t = 0\) to \(t_{\text{jet}} ,\)

$${\mathbf{M}}_{\text{in}} = \mathop \int \limits_{0}^{{t_{\text{jet}} }} \dot{m}v_{\text{h}} {\text{d}}t = \rho \mathop \int \limits_{0}^{{t_{\text{jet}} }} A_{\text{h}} v_{\text{h}}^{2} {\text{d}}t,$$
(14)

the velocity at the jet front can be given as

$$V_{\text{jet}} = \frac{1}{{A_{\text{jet}} L_{\text{jet}} }}\mathop \int \limits_{0}^{{t_{\text{jet}} }} A_{\text{h}} v_{\text{h}}^{2} {\text{d}}t.$$
(15)

To compute values of Vjet from both the “constant P and V” and “constant V” versions of the model, values of Ljet, Ajet, Ah, and Cd are required. The jet length Ljet, jet frontal diameter Djet, and the opening hole diameter Dh were measured for bubble diameters of 2R = 4.90, 6.22, 11.2, 15.1, 16.8, 21.1, 22.9, 26.5, 28.2, and 41.0 mm. Smaller bubbles were not modeled because the bubble cap had completely disintegrated by this time point. The parameters Ah and Ajet were then calculated assuming a circular cross section of the expanding hole in the bubble film and of the jet, respectively.

To integrate these equations over time from t = 0 ms to tjet = 0.4 ms, it was necessary to determine Ah as a function of time. Thus, for bubbles with 2R > 10 mm, the diameter of the hole in the bubble cap film \(D_{\text{h}}\) is interpolated with a quadratic curve as a function of time, using a diameter of zero at t = 0 ms and measured values of \(D_{\text{h}}\) at t = 0.2 and 0.4 ms. For bubbles with 2R < 10 mm, Dh was linearly interpolated between t = 0 ms and t = 0.2 ms and again between t = 0.2 ms and t = 0.4 ms because a quadratic fit resulted in negative values of Dh. Then, the hole area is readily obtained as \(A_{\text{h}} = \frac{\pi }{4}D_{\text{h}}^{2}\). Finally, the value of the discharge coefficient is obtained by fitting the data of Hollingshead (2011) for a Venturi meter (smooth beta in the range of 100 < Re < 1000: \(C_{\text{d}} = - 0.1413(\log Re_{\text{jet}} )^{2} + 0.9085\log Re_{\text{jet}} - 0.542).\) Values of Cd ranged from Cd = 0.75051 for a 2R = 4.9 mm bubble up to Cd = 0.89278 for a 2R = 41.0 mm bubble. The model was solved using MATLAB software.

Model results

Figure 14 shows the comparison of Vjet from the “constant P and V” and “constant V” theoretical models with the experimental results at t = 0.4 ms. Measured values of the model parameters Ljet, Djet, and Dh and calculated values of Rejet at t = 0.4 ms are shown in the inset of Fig. 14. In general, the “constant P and V” model overestimates the experimental data. The model most closely matches the experimental data for the largest bubble size (2R = 41 mm), where the percent difference between experimental and model values is 12.7%. However, with decreasing bubble size, the model and experimental values substantially diverge, with the model overestimating the experimental value by approximately six fold for the bubble with 2R = 4.9 mm. In addition, Vjet decreases with increasing bubble diameter in the model whereas Vjet shows the opposite trend in the experimental data. This “constant P and V” model unrealistically assumes that the bubble volume and the pressure in the bubble do not change as the gas inside is released in a jet. The constant volume assumption is most realistic for the larger bubble sizes. As seen in Fig. 3 for a large bubble (2R = 41 mm), the bubble volume changes negligibly from time of bursting to t = 0.4 ms. In contrast, the bubble cap films of small bubbles are largely retracted or even destroyed by t = 0.4 ms (e.g., Fig. 5), leading to large reductions in bubble volume. In addition, the pressure decreases substantially as the gas is released, but this decrease in pressure likely depends on bubble size. If the bubble is small, the pressure reduction in the bubble is relatively large because the amount of the released gas is large compared to the initial bubble volume. However, this ratio of the released gas to the initial bubble volume decreases as the bubble size increases. The pressure reduction in the bubble thus decreases accordingly for larger bubbles. The “constant P and V” model thus matches the experimental data most closely for the largest bubble size with the smallest hole opening speed for which this condition most closely holds (e.g., Fig. 3) and for which the bubble volume does not change much. In contrast, the “constant P and V” model fails for small bubbles with the largest hole opening speeds (e.g., Fig. 5) and large changes in bubble volume.

Fig. 14
figure14

Comparison of measured gas jet front speeds and theoretical gas jet speeds from “constant pressure and volume” and “constant volume” models as a function of bubble equivalent diameter 2R at t = 0.4 ms. The inset shows measured values of the hole diameter Dh, jet length Ljet, and jet frontal diameter Djet and calculated values of jet Reynolds number Rejet at t = 0.4 ms as a function of bubble equivalent diameter 2R

In contrast to the “constant P and V” model, the “constant V” model consistently underestimates Vjet, generating similar values (of 0.6–1.2 m/s) regardless of the bubble size. It is believed that the main reason for this underestimation is that the pressure drop in the bubble in the model occurs much faster than in reality. For example, the “constant P and V” model predicts that the pressure inside the bubble with 2R = 21.1 mm decreases to the atmospheric pressure after about t = 0.2 ms and the jet velocity at the hole becomes zero accordingly. However, Fig. 4 shows that, in a similarly sized bubble, the gas continues to be released well after that time. The faster pressure drop in the model thus results in jet speeds slower than those measured in the experiments. One possible reason for this discrepancy is that, while the gas jet is being released, a bubble shrinks slightly, which can lead to an increase in pressure inside the bubble or at least slow the decrease in pressure over time. For example, according to the model proposed by Lhuissier and Villermaux (2011), the pressure inside the bubble increases with decreasing bubble size as it “deflates”. However, this assumption leads to jet speeds which increase with time, which contradicts the jet speeds measured here. Two other reasons may contribute to the mismatch between experiments and theory. First, the gas jet is actually a highly unsteady flow in contrast to the steady flow assumption. Second, it is assumed that the pressure inside the bubble has a constant pressure of \(P_{\text{b}}\) spatially, but in reality, the pressure inside the bubble varies depending on the location. It is noted that the computed jet velocities of both models are inversely proportional to the jet area and the jet length as shown in Eq. (15). The hole area linearly increases with the jet velocity for the “constant P and V” model while it has little effect on the jet velocity for the “constant V” model due to a fast depressurization. Finally, the two models can be considered as the upper and the lower bounds of the experimental results because it is expected that the actual pressure drop inside the bubble is between those of the models.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dasouqi, A.A., Yeom, GS. & Murphy, D.W. Bursting bubbles and the formation of gas jets and vortex rings. Exp Fluids 62, 1 (2021). https://doi.org/10.1007/s00348-020-03089-0

Download citation