Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-27T02:55:07.266Z Has data issue: false hasContentIssue false
  • English
  • Français

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: mjthoraval@xjtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

Download Jian et al. supplementary movie 1(Video)
Video 610.4 KB

Jian et al. supplementary movie 2

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

Download Jian et al. supplementary movie 2(Video)
Video 411.4 KB

Jian et al. supplementary movie 3

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

Download Jian et al. supplementary movie 3(Video)
Video 285.1 KB

Jian et al. supplementary movie 4

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

Download Jian et al. supplementary movie 4(Video)
Video 1.1 MB
Supplementary material: PDF

Jian et al. supplementary material

Supplementary data

Download Jian et al. supplementary material(PDF)
PDF 3.1 MB