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On the statics and dynamics of fully confined bubbles

Published online by Cambridge University Press:  18 August 2017

Olivier Vincent Open the ORCID record for Olivier Vincent [Opens in a new window]  and
Philippe Marmottant
Olivier Vincent*
Affiliation:
CNRS/Université Grenoble-Alpes, LIPhy UMR 5588, Grenoble, F-38401, France Cornell University, Robert Frederick Smith School of Chemical and Biomolecular Engineering, 120 Olin Hall, Ithaca, NY 14850, USA
Philippe Marmottant
Affiliation:
CNRS/Université Grenoble-Alpes, LIPhy UMR 5588, Grenoble, F-38401, France
*
†Email address for correspondence: [email protected]
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Abstract

We investigate theoretically the statics and dynamics of bubbles in fully confined liquids, i.e. in liquids surrounded by solid walls in all directions of space. This situation is found in various natural and technological contexts (geological fluid inclusions, plant cells and vessels, soil tensiometers, etc.), where such bubbles can pre-exist in the trapped liquid or appear by nucleation (cavitation). We focus on volumetric deformations and first establish the potential energy of fully confined bubbles as a function of their radius, including contributions from gas compressibility, surface tension, liquid compressibility and elastic deformation of the surrounding solid. We evaluate how the Blake threshold of unstable bubble growth is modified by confinement and we also obtain an original bubble stability phase diagram with a regime of liquid superstability (spontaneous bubble collapse) for strong confinements. We then calculate the liquid velocity field associated with radial deformations of the bubble and strain in the solid, and we predict large deviations in the kinematics compared to bubbles in extended liquids. Finally, we derive the equations governing the natural oscillation dynamics of fully confined bubbles, extending Minnaert’s formula and the Rayleigh–Plesset equation, and we show that the compressibility of the liquid as well as the elasticity of the walls can result in ultra-fast bubble radial oscillations and unusually quick damping. We find excellent agreement between the predictions of our model and recent experimental results.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Cavitation Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 827 , 25 September 2017 , pp. 194 - 224
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© 2017 Cambridge University Press 

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References

Alekseev, V. N. & Rybak, S. A. 1999 Gas bubble oscillations in elastic media. Acoust. Phys. 45, 535–540.Google Scholar
Becher, H. & Burns, P. 2000 Handbook of Contrast Echocardiography. Springer.Google Scholar
Blake, F. G. Jr. 1949 Onset of cavitation in liquids. Thèse de doctorat, Harvard University.Google Scholar
Blander, M. & Katz, J. L. 1975 Bubble nucleation in liquids. AIChE J. 21 (5), 833–848.CrossRefGoogle Scholar
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.CrossRefGoogle Scholar
Campbell, G. M. & Mougeot, E. 1999 Creation and characterisation of aerated food products. Trends Food Sci. Technol. 10, 283–296.CrossRefGoogle Scholar
Caupin, F. & Herbert, E. 2006 Cavitation in water: a review. C. R. Physique 7 (9–10), 1000–1017.CrossRefGoogle Scholar
Church, C. C. 1995 The effects of an elastic solid surface layer on the radial pulsations of gas bubbles. J. Acoust. Soc. Am. 97 (3), 1510–1521.CrossRefGoogle Scholar
Cochard, H. 2006 Cavitation in trees. C. R. Physique 7 (9–10), 1018–1026.CrossRefGoogle Scholar
Debenedetti, P. G. 1996 Metastable Liquids: Concepts and Principles. Princeton University Press.Google Scholar
Del Grosso, V. A. & Mader, C. W. 1972 Speed of sound in pure water. J. Acoust. Soc. Am. 52 (5B), 1442–1446.CrossRefGoogle Scholar
Fourest, T., Laurens, J.-M., Deletombe, E., Dupas, J. & Arrigoni, M. 2015 Confined Rayleigh–Plesset equation for hydrodynamic ram analysis in thin-walled containers under ballistic impacts. Thin-Walled Struct. 86, 67–72.CrossRefGoogle Scholar
Gaudron, R., Warnez, M. & Johnsen, E. 2015 Bubble dynamics in a viscoelastic medium with nonlinear elasticity. J. Fluid Mech. 766, 54–75.CrossRefGoogle Scholar
Holbrook, N. M. & Zwieniecki, M. A. 1999 Embolism repair and xylem tension: Do we need a miracle? Plant Physiol. 120 (1), 7–10.CrossRefGoogle Scholar
Hover, K. 1993 Why is there air in concrete? Concrete Construction 38 (1), 11–15.Google Scholar
Hsiao, C.-T., Choi, J.-K., Singh, S., Chahine, G., Hay, T., Ilinskii, Y. A., Zabolotskaya, E., Hamilton, M., Sankin, G., Yuan, F. et al. 2013 Modelling single-and tandem-bubble dynamics between two parallel plates for biomedical applications. J. Fluid Mech. 716, 137–170.CrossRefGoogle ScholarPubMed
Kell, G. S. 1975 Density, thermal expansivity, and compressibility of liquid water from 0 to 150 °C: correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. J. Chem. Engng Data 20 (1), 97–105.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. 1976 Mechanics, vol. 1. Butterworth-Heinemann.Google Scholar
Landau, L. D. & Lifshitz, E. 1986 Theory of Elasticity, vol. 7. Butterworth-Heinemann.Google Scholar
Landau, L. D. & Lifshitz, E. 1987 Fluid Mechanics, vol. 6. Butterworth-Heinemann.Google Scholar
Lauterborn, W. & Kurz, T. 2010 Physics of bubble oscillations. Rep. Prog. Phys. 73 (10), 106501.CrossRefGoogle Scholar
Lauterborn, W. & Ohl, C.-D. 1997 Cavitation bubble dynamics. Ultrasonics Sonochemistry 4, 65–75.CrossRefGoogle ScholarPubMed
Leighton, T. G. 1994 The Acoustic Bubble. Academic.Google Scholar
Leroy, V., Bretagne, A., Fink, M., Willaime, H., Tabeling, P. & Tourin, A. 2009 Design and characterization of bubble phononic crystals. Appl. Phys. Lett. 95 (17), 171904–3.CrossRefGoogle Scholar
Lura, P., Couch, J., Jensen, O. M. & Weiss, J. 2009 Early-age acoustic emission measurements in hydrating cement paste: evidence for cavitation during solidification due to self-desiccation. Cement Concrete Res. 39 (10), 861–867.CrossRefGoogle Scholar
Macdowell, L. G., Shen, V. K. & Errington, J. R. 2006 Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems. J. Chem. Phys. 125 (3), 034705–15.CrossRefGoogle ScholarPubMed
Marmottant, P., Bouakaz, A., de Jong, N. & Quilliet, C. 2011 Buckling resistance of solid shell bubbles under ultrasound. J. Acoust. Soc. Am. 129 (3), 1231–1239.CrossRefGoogle ScholarPubMed
Marti, D., Krüger, Y., Fleitmann, D., Frenz, M. & Rička, J. 2012 The effect of surface tension on liquid–gas equilibria in isochoric systems and its application to fluid inclusions. Fluid Phase Equilib. 314 (0), 13–21.CrossRefGoogle Scholar
Martynov, S., Stride, E. & Saffari, N. 2009 The natural frequencies of microbubble oscillation in elastic vessels. J. Acoust. Soc. Am. 126 (6), 2963–2972.CrossRefGoogle ScholarPubMed
Minnaert, M. 1933 On musical air-bubbles and the sounds of running water. Phil. Mag. Ser. 7 16 (104), 235–248.CrossRefGoogle Scholar
Noblin, X., Rojas, N. O., Westbrook, J., Llorens, C., Argentina, M. & Dumais, J. 2012 The fern sporangium: a unique catapult. Science 335 (6074), 1322.CrossRefGoogle ScholarPubMed
Og̃uz, H. N. & Prosperetti, A. 1998 The natural frequency of oscillation of gas bubbles in tubes. J. Acoust. Soc. Am. 103, 3301–3308.CrossRefGoogle Scholar
Obreschkow, D., Kobel, P., Dorsaz, N., de Bosset, A., Nicollier, C. & Farhat, M. 2006 Cavitation bubble dynamics inside liquid drops in microgravity. Phys. Rev. Lett. 97, 094502.CrossRefGoogle ScholarPubMed
Ohl, S.-W., Tandiono, T., Klaseboer, E., Ow, D., Choo, A. & Ohl, C.-D. 2015 Intense cavitation in microfluidics for bio-technology applications. J. Acoust. Soc. Am. 137 (4), 2222–2222.CrossRefGoogle Scholar
Or, D. & Tuller, M. 2002 Cavitation during desaturation of porous media under tension. Water Resour. Res. 38 (5), 1061.CrossRefGoogle Scholar
Pagay, V., Santiago, M., Sessoms, D. A., Huber, E. J., Vincent, O., Pharkya, A., Corso, T. N., Lakso, A. N. & Stroock, A. D. 2014 A microtensiometer capable of measuring water potentials below - 10 MPa. Lab on a Chip 14, 2806–2817.CrossRefGoogle ScholarPubMed
Plesset, M. S. 1949 The dynamics of cavitation bubbles. J. Appl. Mech. 16, 277–282.CrossRefGoogle Scholar
Poivet, S., Nallet, F., Gay, C. & Fabre, P. 2003 Cavitation-induced force transition in confined viscous liquids under traction. Europhys. Lett. 62 (2), 244–250.CrossRefGoogle Scholar
Prosperetti, A., Crum, L. A. & Pumphrey, H. C. 1989 The underwater noise of rain. J. Geophys. Res. 94 (C3), 3255–3259.CrossRefGoogle Scholar
Rayleigh, L. 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98.CrossRefGoogle Scholar
Roedder, E. & Bodnar, R. J. 1980 Geologic pressure determinations from fluid inclusion studies. Annu. Rev. Earth Planet. Sci. 8, 263–301.CrossRefGoogle Scholar
Strasberg, M. 1953 The pulsation frequency of nonspherical gas bubbles in liquids. J. Acoust. Soc. Am. 25 (3), 536–537.CrossRefGoogle Scholar
Stroock, A. D., Pagay, V. V., Zwieniecki, M. A. & Michele Holbrook, N. 2014 The physicochemical hydrodynamics of vascular plants. Annu. Rev. Fluid Mech. 46, 615–642.CrossRefGoogle Scholar
Tarantino, A. & Mongiovì, L. 2001 Experimental procedures and cavitation mechanisms in tensiometer measurements. In Unsaturated Soil Concepts and Their Application in Geotechnical Practice (ed. Toll, D.), pp. 189–210. Springer.CrossRefGoogle Scholar
Tas, N. R., Mela, P., Kramer, T., Berenschot, J. W. & van den Berg, A. 2003 Capillarity induced negative pressure of water plugs in nanochannels. Nano Lett. 3 (11), 1537–1540.CrossRefGoogle Scholar
Tyree, M. T. & Sperry, J. S. 1989 Vulnerability of xylem to cavitation and embolism. Annu. Rev. Plant Phys. Mol. Bio. 40, 19–38.CrossRefGoogle Scholar
Versluis, M., Schmitz, B., von der Heydt, A. & Lohse, D. 2000 How snapping shrimp snap: through cavitating bubbles. Science 289 (5487), 2114–2117.CrossRefGoogle ScholarPubMed
Vidal, V., Ripepe, M., Divoux, T., Legrand, D., Géminard, J.-C. & Melo, F. 2010 Dynamics of soap bubble bursting and its implications to volcano acoustics. Geophys. Res. Lett. 37 (7), L07302.CrossRefGoogle Scholar
Vincent, O.2012 Dynamique de bulles de cavitation dans de l’eau micro-confinée sous tension: application à l’étude de l’embolie dans les arbres. Thèse de doctorat, Univ. Joseph Fourier, Grenoble, France.Google Scholar
Vincent, O., Marmottant, P., Gonzalez-Avila, S. R., Ando, K. & Ohl, C.-D. 2014a The fast dynamics of cavitation bubbles within water confined in elastic solids. Soft Matt. 10, 1455–1461.CrossRefGoogle ScholarPubMed
Vincent, O., Marmottant, P., Quinto-Su, P. A. & Ohl, C.-D. 2012 Birth and growth of cavitation bubbles within water under tension confined in a simple synthetic tree. Phys. Rev. Lett. 108 (18), 184502.CrossRefGoogle Scholar
Vincent, O., Sessoms, D. A., Huber, E. J., Guioth, J. & Stroock, A. D. 2014b Drying by cavitation and poroelastic relaxations in porous media with macroscopic pores connected by nanoscale throats. Phys. Rev. Lett. 113 (13), 134501.CrossRefGoogle ScholarPubMed
Wheeler, T. D. & Stroock, A. D. 2008 The transpiration of water at negative pressures in a synthetic tree. Nature 455, 208–212.CrossRefGoogle Scholar
Wheeler, T. D. & Stroock, A. D. 2009 Stability limit of liquid water in metastable equilibrium with subsaturated vapors. Langmuir 25 (13), 7609–7622.CrossRefGoogle ScholarPubMed
Wilhelmsen, Ø., Bedeaux, D., Kjelstrup, S. & Reguera, D. 2014 Communication: superstabilization of fluids in nanocontainers. J. Chem. Phys. 141 (7), 071103.CrossRefGoogle ScholarPubMed
Yang, X. & Church, C. C. 2005 A model for the dynamics of gas bubbles in soft tissue. J. Acoust. Soc. Am. 118 (6), 3595–3606.CrossRefGoogle Scholar
Zwaan, E., Le Gac, S., Tsuji, K. & Ohl, C.-D. 2007 Controlled cavitation in microfluidic systems. Phys. Rev. Lett. 98 (25), 254501.CrossRefGoogle ScholarPubMed