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On the statics and dynamics of fully confined bubbles

  • Olivier Vincent (a1) (a2) and Philippe Marmottant (a1)

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.

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Corresponding author

Email address for correspondence: orv3@cornell.edu

References

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Alekseev, V. N. & Rybak, S. A. 1999 Gas bubble oscillations in elastic media. Acoust. Phys. 45, 535540.
Becher, H. & Burns, P. 2000 Handbook of Contrast Echocardiography. Springer.
Blake, F. G. Jr. 1949 Onset of cavitation in liquids. Thèse de doctorat, Harvard University.
Blander, M. & Katz, J. L. 1975 Bubble nucleation in liquids. AIChE J. 21 (5), 833848.
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.
Campbell, G. M. & Mougeot, E. 1999 Creation and characterisation of aerated food products. Trends Food Sci. Technol. 10, 283296.
Caupin, F. & Herbert, E. 2006 Cavitation in water: a review. C. R. Physique 7 (9–10), 10001017.
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), 15101521.
Cochard, H. 2006 Cavitation in trees. C. R. Physique 7 (9–10), 10181026.
Debenedetti, P. G. 1996 Metastable Liquids: Concepts and Principles. Princeton University Press.
Del Grosso, V. A. & Mader, C. W. 1972 Speed of sound in pure water. J. Acoust. Soc. Am. 52 (5B), 14421446.
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, 6772.
Gaudron, R., Warnez, M. & Johnsen, E. 2015 Bubble dynamics in a viscoelastic medium with nonlinear elasticity. J. Fluid Mech. 766, 5475.
Holbrook, N. M. & Zwieniecki, M. A. 1999 Embolism repair and xylem tension: Do we need a miracle? Plant Physiol. 120 (1), 710.
Hover, K. 1993 Why is there air in concrete? Concrete Construction 38 (1), 1115.
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, 137170.
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), 97105.
Landau, L. D. & Lifshitz, E. 1976 Mechanics, vol. 1. Butterworth-Heinemann.
Landau, L. D. & Lifshitz, E. 1986 Theory of Elasticity, vol. 7. Butterworth-Heinemann.
Landau, L. D. & Lifshitz, E. 1987 Fluid Mechanics, vol. 6. Butterworth-Heinemann.
Lauterborn, W. & Kurz, T. 2010 Physics of bubble oscillations. Rep. Prog. Phys. 73 (10), 106501.
Lauterborn, W. & Ohl, C.-D. 1997 Cavitation bubble dynamics. Ultrasonics Sonochemistry 4, 6575.
Leighton, T. G. 1994 The Acoustic Bubble. Academic.
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.
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), 861867.
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.
Marmottant, P., Bouakaz, A., de Jong, N. & Quilliet, C. 2011 Buckling resistance of solid shell bubbles under ultrasound. J. Acoust. Soc. Am. 129 (3), 12311239.
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), 1321.
Martynov, S., Stride, E. & Saffari, N. 2009 The natural frequencies of microbubble oscillation in elastic vessels. J. Acoust. Soc. Am. 126 (6), 29632972.
Minnaert, M. 1933 On musical air-bubbles and the sounds of running water. Phil. Mag. Ser. 7 16 (104), 235248.
Noblin, X., Rojas, N. O., Westbrook, J., Llorens, C., Argentina, M. & Dumais, J. 2012 The fern sporangium: a unique catapult. Science 335 (6074), 1322.
Og̃uz, H. N. & Prosperetti, A. 1998 The natural frequency of oscillation of gas bubbles in tubes. J. Acoust. Soc. Am. 103, 33013308.
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.
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), 22222222.
Or, D. & Tuller, M. 2002 Cavitation during desaturation of porous media under tension. Water Resour. Res. 38 (5), 1061.
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, 28062817.
Plesset, M. S. 1949 The dynamics of cavitation bubbles. J. Appl. Mech. 16, 277282.
Poivet, S., Nallet, F., Gay, C. & Fabre, P. 2003 Cavitation-induced force transition in confined viscous liquids under traction. Europhys. Lett. 62 (2), 244250.
Prosperetti, A., Crum, L. A. & Pumphrey, H. C. 1989 The underwater noise of rain. J. Geophys. Res. 94 (C3), 32553259.
Rayleigh, L. 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 9498.
Roedder, E. & Bodnar, R. J. 1980 Geologic pressure determinations from fluid inclusion studies. Annu. Rev. Earth Planet. Sci. 8, 263301.
Strasberg, M. 1953 The pulsation frequency of nonspherical gas bubbles in liquids. J. Acoust. Soc. Am. 25 (3), 536537.
Stroock, A. D., Pagay, V. V., Zwieniecki, M. A. & Michele Holbrook, N. 2014 The physicochemical hydrodynamics of vascular plants. Annu. Rev. Fluid Mech. 46, 615642.
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. 189210. Springer.
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), 15371540.
Tyree, M. T. & Sperry, J. S. 1989 Vulnerability of xylem to cavitation and embolism. Annu. Rev. Plant Phys. Mol. Bio. 40, 1938.
Versluis, M., Schmitz, B., von der Heydt, A. & Lohse, D. 2000 How snapping shrimp snap: through cavitating bubbles. Science 289 (5487), 21142117.
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.
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.
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, 14551461.
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.
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.
Wheeler, T. D. & Stroock, A. D. 2008 The transpiration of water at negative pressures in a synthetic tree. Nature 455, 208212.
Wheeler, T. D. & Stroock, A. D. 2009 Stability limit of liquid water in metastable equilibrium with subsaturated vapors. Langmuir 25 (13), 76097622.
Wilhelmsen, Ø., Bedeaux, D., Kjelstrup, S. & Reguera, D. 2014 Communication: superstabilization of fluids in nanocontainers. J. Chem. Phys. 141 (7), 071103.
Yang, X. & Church, C. C. 2005 A model for the dynamics of gas bubbles in soft tissue. J. Acoust. Soc. Am. 118 (6), 35953606.
Zwaan, E., Le Gac, S., Tsuji, K. & Ohl, C.-D. 2007 Controlled cavitation in microfluidic systems. Phys. Rev. Lett. 98 (25), 254501.
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