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Air sheet contraction

Published online by Cambridge University Press:  20 July 2020

Zhen Jian Open the ORCID record for Zhen Jian [Opens in a new window] ,
Peng Deng Open the ORCID record for Peng Deng [Opens in a new window]  and
Marie-Jean Thoraval Open the ORCID record for Marie-Jean Thoraval [Opens in a new window]
Zhen Jian
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures, International Center for Applied Mechanics, School of Aerospace, Xi’an Jiaotong University, Xi’an710049, PR China
Peng Deng
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures, International Center for Applied Mechanics, School of Aerospace, Xi’an Jiaotong University, Xi’an710049, PR China
Marie-Jean Thoraval*
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures, International Center for Applied Mechanics, School of Aerospace, Xi’an Jiaotong University, Xi’an710049, PR China
*
†Email address for correspondence: [email protected]
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Abstract

A two-dimensional air sheet in a surrounding liquid contracts under surface tension. We investigate numerically and analytically this contraction dynamics for a range of Ohnesorge numbers $Oh$. In a similar way as for liquid films, three contraction regimes can be identified based on the $Oh$: vortex shedding, smooth contraction and viscous regime. For $Oh\leqslant 0.02$, the rim can even pinch-off due to the rim deformations caused by the vortex shedding. In contrast with a liquid film that continuously accelerates towards the Taylor–Culick velocity when the surrounding fluid can be neglected, the air film contraction velocity first rises to a maximum value $U_{max}$ before decreasing due to the drag of the external fluid on the moving rim. This $U_{max}$ follows a capillary-inertial scaling at low $Oh$ and continuously shifts to a capillary-viscous scaling with increasing $Oh$. We demonstrate that the decreasing contraction velocity scales as $t^{-0.15}$, which is faster than the scaling $t^{-0.2}$ derived under the assumption of a constant drag coefficient. The transition between the capillary-inertial and capillary-viscous regimes can be characterised by the local time evolving Ohnesorge number $Oh_{\unicode[STIX]{x1D6FF}}$ based on the thickness of the rim. The oscillations of the rim appear at a critical local Weber number $We_{\unicode[STIX]{x1D6FF}}$. Then they follow a well-defined oscillation frequency with a characteristic Strouhal number. Beyond a local Reynolds number larger than 200, the oscillations become more irregular with more complex vortex sheddings, eventually leading to the pinch-off of the rim.

JFM classification

Drops and Bubbles: Drops and Bubbles
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 899 , 25 September 2020 , A7
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

‡

These authors contributed equally to this work.

References

Beilharz, D., Guyon, A., Li, E. Q., Thoraval, M.-J. & Thoroddsen, S. T. 2015 Antibubbles and fine cylindrical sheets of air. J. Fluid Mech. 779, 87–115.CrossRefGoogle Scholar
Brenner, M. P. & Gueyffier, D. 1999 On the bursting of viscous films. Phys. Fluids 11 (3), 737–739.CrossRefGoogle Scholar
Castrejón-Pita, A. A., Castrejón-Pita, J. R. & Hutchings, I. M. 2012 Experimental observation of von Kármán vortices during drop impact. Phys. Rev. E 86 (4), 045301(R).Google Scholar
Chan, W. H. R., Mirjalili, S., Jain, S. S., Urzay, J., Mani, A. & Moin, P. 2018 Video: Birth of microbubbles in turbulent breaking waves. In 71th Annual Meeting of the APS Division of Fluid Dynamics – Gallery of Fluid Motion, V0027. American Physical Society.Google Scholar
Chan, W. H. R., Mirjalili, S., Jain, S. S., Urzay, J., Mani, A. & Moin, P. 2019 Birth of microbubbles in turbulent breaking waves. Phys. Rev. Fluids 4 (10), 100508.CrossRefGoogle Scholar
Chebil, M. S., McGraw, J. D., Salez, T., Sollogoub, C. & Miquelard-Garnier, G. 2018 Influence of outer-layer finite-size effects on the dewetting dynamics of a thin polymer film embedded in an immiscible matrix. Soft Matt. 14 (30), 6256–6263.CrossRefGoogle Scholar
Culick, F. E. C. 1960 Comments on a Ruptured Soap Film. J. Appl. Phys. 31 (6), 1128–1129.CrossRefGoogle Scholar
Czerski, H., Twardowski, M., Zhang, X. & Vagle, S. 2011 Resolving size distributions of bubbles with radii less than 30 μm with optical and acoustical methods. J. Geophys. Res. 116, C00H11.CrossRefGoogle Scholar
Debrégeas, G., de Gennes, P.-G. & Brochard-Wyart, F. 1998 The life and death of “bare” viscous bubbles. Science 279 (5357), 1704–1707.Google Scholar
Debrégeas, G., Martin, P. & Brochard-Wyart, F. 1995 Viscous bursting of suspended films. Phys. Rev. Lett. 75 (21), 3886–3889.CrossRefGoogle ScholarPubMed
Deike, L., Melville, W. K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.CrossRefGoogle Scholar
Derby, B. 2010 Inkjet printing of functional and structural materials: fluid property requirements, feature stability, and resolution. Annu. Rev. Mater. Res. 40, 395–414.CrossRefGoogle Scholar
Dorbolo, S., Caps, H. & Vandewalle, N. 2003 Fluid instabilities in the birth and death of antibubbles. New J. Phys. 5, 161.CrossRefGoogle Scholar
Dorbolo, S., Reyssat, E., Vandewalle, N. & Quéré, D. 2005 Aging of an antibubble. Eur. Phys. Lett. 69 (6), 966–970.CrossRefGoogle Scholar
Dorbolo, S., Terwagne, D., Delhalle, R., Dujardin, J., Huet, N., Vandewalle, N. & Denkov, N. 2010 Antibubble lifetime: influence of the bulk viscosity and of the surface modulus of the mixture. Colloids Surf. A 365 (1–3), 43–45.CrossRefGoogle Scholar
Eggers, J., Lister, J. R. & Stone, H. A. 1999 Coalescence of liquid drops. J. Fluid Mech. 401, 293–310.CrossRefGoogle Scholar
Esmailizadeh, L. & Mesler, R. 1986 Bubble entrainment with drops. J. Colloid Interface Sci. 110 (2), 561–574.CrossRefGoogle Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.CrossRefGoogle Scholar
Gordillo, L., Agbaglah, G., Duchemin, L. & Josserand, C. 2011 Asymptotic behavior of a retracting two-dimensional fluid sheet. Phys. Fluids 23 (12), 122101.CrossRefGoogle Scholar
Hicks, P. D. & Purvis, R. 2011 Air cushioning in droplet impacts with liquid layers and other droplets. Phys. Fluids 23 (6), 062104.CrossRefGoogle Scholar
Jian, Z., Channa, M. A., Kherbeche, A., Chizari, H., Thoroddsen, S. T. & Thoraval, M.-J. 2020 To split or not to split: dynamics of an air disk formed under a drop impacting on a pool. Phys. Rev. Lett. 124 (18), 184501.CrossRefGoogle ScholarPubMed
Josserand, C., Ray, P. & Zaleski, S. 2016 Droplet impact on a thin liquid film: anatomy of the splash. J. Fluid Mech. 802, 775–805.CrossRefGoogle Scholar
Josserand, C. & Thoroddsen, S. T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48, 365–391.CrossRefGoogle Scholar
Kiger, K. T. & Duncan, J. H. 2012 Air-entrainment mechanisms in plunging jets and breaking waves. Annu. Rev. Fluid Mech. 44, 563–596.CrossRefGoogle Scholar
Langley, K. R. & Thoroddsen, S. T. 2019 Gliding on a layer of air: impact of a large-viscosity drop on a liquid film. J. Fluid Mech. 878, R2.CrossRefGoogle Scholar
Legendre, D., Lauga, E. & Magnaudet, J. 2009 Influence of slip on the dynamics of two-dimensional wakes. J. Fluid Mech. 633, 437–447.CrossRefGoogle Scholar
Mills, B. H., Saylor, J. R. & Testik, F. Y. 2012 An experimental study of Mesler entrainment on a surfactant-covered interface: the effect of drop shape and Weber number. AIChE J. 58 (1), 46–58.CrossRefGoogle Scholar
Mirjalili, S., Chan, W. H. R. & Mani, A. 2018 High Fidelity simulations of micro-bubble shedding from retracting thin gas films in the context of liquid–liquid impact. In 32nd Symposium on Naval Hydrodynamics, Hamburg, Germany, arXiv:1811.12352.Google Scholar
Oguz, H. N. & Prosperetti, A. 1989 Surface-tension effects in the contact of liquid surfaces. J. Fluid Mech. 203, 149–171.CrossRefGoogle Scholar
Pandit, A. B. & Davidson, J. F. 1990 Hydrodynamics of the rupture of thin liquid films. J. Fluid Mech. 212, 11–24.CrossRefGoogle Scholar
Popinet, S.2019 Basilisk. http://basilisk.fr.Google Scholar
Reyssat, É. & Quéré, D. 2006 Bursting of a fluid film in a viscous environment. Eur. Phys. Lett. 76 (2), 236–242.CrossRefGoogle Scholar
Savva, N. & Bush, J. W. M. 2009 Viscous sheet retraction. J. Fluid Mech. 626, 211–240.CrossRefGoogle Scholar
Saylor, J. R. & Bounds, G. D. 2012 Experimental study of the role of the Weber and capillary numbers on Mesler entrainment. AIChE J. 58 (12), 3841–3851.CrossRefGoogle Scholar
Scheid, B., Dorbolo, S., Arriaga, L. R. & Rio, E. 2012 Antibubble dynamics: the drainage of an air film with viscous interfaces. Phys. Rev. Lett. 109 (26), 264502.CrossRefGoogle ScholarPubMed
Scheid, B., Zawala, J. & Dorbolo, S. 2014 Gas dissolution in antibubble dynamics. Soft Matt. 10 (36), 7096–7102.CrossRefGoogle ScholarPubMed
Sigler, J. & Mesler, R. 1990 The behavior of the gas film formed upon drop impact with a liquid surface. J. Colloid Interface Sci. 134 (2), 459–474.CrossRefGoogle Scholar
Sob’yanin, D. N. 2015 Theory of the antibubble collapse. Phys. Rev. Lett. 114 (10), 104501.CrossRefGoogle ScholarPubMed
Song, M. & Tryggvason, G. 1999 The formation of thick borders on an initially stationary fluid sheet. Phys. Fluids 11 (9), 2487–2493.CrossRefGoogle Scholar
Sünderhauf, G., Raszillier, H. & Durst, F. 2002 The retraction of the edge of a planar liquid sheet. Phys. Fluids 14 (1), 198–208.CrossRefGoogle Scholar
Taylor, G. 1959 The dynamics of thin sheets of fluid. II. Waves on fluid sheets. Proc. R. Soc. Lond. A 253 (1274), 296–312.Google Scholar
Thoraval, M.-J., Takehara, K., Etoh, T. G., Popinet, S., Ray, P., Josserand, C., Zaleski, S. & Thoroddsen, S. T. 2012 von Kármán vortex street within an impacting drop. Phys. Rev. Lett. 108 (26), 264506.CrossRefGoogle Scholar
Thoraval, M.-J., Takehara, K., Etoh, T. G. & Thoroddsen, S. T. 2013 Drop impact entrapment of bubble rings. J. Fluid Mech. 724, 234–258.CrossRefGoogle Scholar
Thoraval, M.-J. & Thoroddsen, S. T. 2013 Contraction of an air disk caught between two different liquids. Phys. Rev. E 88 (6), 061001(R).Google ScholarPubMed
Thoroddsen, S. T., Etoh, T. G. & Takehara, K. 2003 Air entrapment under an impacting drop. J. Fluid Mech. 478, 125–134.CrossRefGoogle Scholar
Thoroddsen, S. T., Etoh, T. G., Takehara, K., Ootsuka, N. & Hatsuki, Y. 2005 The air bubble entrapped under a drop impacting on a solid surface. J. Fluid Mech. 545, 203–212.CrossRefGoogle Scholar
Thoroddsen, S. T., Thoraval, M.-J., Takehara, K. & Etoh, T. G. 2012 Micro-bubble morphologies following drop impacts onto a pool surface. J. Fluid Mech. 708, 469–479.CrossRefGoogle Scholar
Tomiyama, A., Kataoka, I., Zun, I. & Sakaguchi, T. 1998 Drag coefficients of single bubbles under normal and micro gravity conditions. JSME Intl J. B 41 (2), 472–479.CrossRefGoogle Scholar
Tran, T., de Maleprade, H., Sun, C. & Lohse, D. 2013 Air entrainment during impact of droplets on liquid surfaces. J. Fluid Mech. 726, R3.CrossRefGoogle Scholar
Vandewalle, N., Terwagne, D., Gilet, T., Caps, H. & Dorbolo, S. 2009 Antibubbles, liquid onions and bouncing droplets. Colloids Surf. A 344 (1–3), 42–47.CrossRefGoogle Scholar
Wanninkhof, R., Asher, W. E., Ho, D. T., Sweeney, C. & McGillis, W. R. 2009 Advances in quantifying air–sea gas exchange and environmental forcing. Annu. Rev. Marine Sci. 1, 213–244.CrossRefGoogle ScholarPubMed
Williamson, C. H. K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413–455.CrossRefGoogle Scholar
Yan, X., Jia, Y., Wang, L. & Cao, Y. 2017 Drag coefficient fluctuation prediction of a single bubble rising in water. Chem. Engng J. 316, 553–562.CrossRefGoogle Scholar
Yarin, A. L. 2006 Drop impact dynamics: splashing, spreading, receding, bouncing… Annu. Rev. Fluid Mech. 38, 159–192.CrossRefGoogle Scholar
Zou, J., Ji, C., Yuan, B., Ruan, X. & Fu, X. 2013 Collapse of an antibubble. Phys. Rev. E 87 (6), 061002(R).Google ScholarPubMed

Jian et al. supplementary movie 1

Figure 4(a,d,g): Oh = 0.05, t* from 0 to 500

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Jian et al. supplementary movie 2

Figure 4(b,e,h): Oh = 0.3, t* from 0 to 500

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Jian et al. supplementary movie 3

Figure 4(c,f,i): Oh = 7, t* from 0 to 500

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Jian et al. supplementary movie 4

Figure 5(a,c-e): Oh = 0.01, t* from 0 to 500

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Supplementary material: PDF

Jian et al. supplementary material

Supplementary data

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