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The history effect in bubble growth and dissolution. Part 1. Theory

Published online by Cambridge University Press:  30 June 2016

Pablo Peñas-López*
Affiliation:
Fluid Mechanics Group, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid), Spain
Miguel A. Parrales
Affiliation:
Fluid Mechanics Group, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid), Spain
Javier Rodríguez-Rodríguez
Affiliation:
Fluid Mechanics Group, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid), Spain
Devaraj van der Meer
Affiliation:
Physics of Fluids Group, Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
*
Email address for correspondence: papenasl@ing.uc3m.es

Abstract

The term ‘history effect’ refers to the contribution of any past mass transfer events between a gas bubble and its liquid surroundings towards the current diffusion-driven growth or dissolution dynamics of that same bubble. The history effect arises from the (non-instantaneous) development of the dissolved gas concentration boundary layer in the liquid in response to changes in the concentration at the bubble interface caused, for instance, by variations of the ambient pressure in time. Essentially, the history effect amounts to the acknowledgement that at any given time the mass flux across the bubble is conditioned by the preceding time history of the concentration at the bubble boundary. Considering the canonical problem of an isolated spherical bubble at rest, we show that the contribution of the history effect in the current interfacial concentration gradient is fully contained within a memory integral of the interface concentration. Retaining this integral term, we formulate a governing differential equation for the bubble dynamics, analogous to the well-known Epstein–Plesset solution. Our equation does not make use of the quasi-static radius approximation. An analytical solution is presented for the case of multiple step-like jumps in pressure. The nature and relevance of the history effect is then assessed through illustrative examples. Finally, we investigate the role of the history effect in rectified diffusion for a bubble that pulsates under harmonic pressure forcing in the non-inertial, isothermal regime.

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Papers
Copyright
© 2016 Cambridge University Press 

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References

Barker, G. S., Jefferson, B. & Judd, S. J. 2002 The control of bubble size in carbonated beverages. Chem. Engng Sci. 57 (4), 565573.CrossRefGoogle Scholar
Birkhoff, G., Margulies, R. S. & Horning, W. A. 1958 Spherical bubble growth. Phys. Fluids 1 (3), 201204.CrossRefGoogle Scholar
Crum, L. A. & Hansen, G. M. 1982 Generalized equations for rectified diffusion. J. Acoust. Soc. Am. 72, 15861592.CrossRefGoogle Scholar
Crum, L. A. & Mao, Y. 1996 Acoustically enhanced bubble growth at low frequencies and its implications for human diver and marine mammal safety. J. Acoust. Soc. Am. 99, 28982907.CrossRefGoogle ScholarPubMed
Duda, J. L. & Vrentas, J. S. 1969 Mathematical analysis of bubble dissolution. AIChE J. 15 (3), 351356.CrossRefGoogle Scholar
Eller, A. & Flynn, H. G. 1965 Rectified diffusion during nonlinear pulsations of cavitation bubbles. J. Acoust. Soc. Am. 37 (3), 493503.CrossRefGoogle Scholar
Enríquez, O. R., Hummelink, C., Bruggert, G.-W., Lohse, D., Prosperetti, A., van der Meer, D. & Sun, C. 2013 Growing bubbles in a slightly supersaturated liquid solution. Rev. Sci. Instrum. 84 (6), 065111.CrossRefGoogle Scholar
Enríquez, O. R., Sun, C., Lohse, D., Prosperetti, A. & van der Meer, D. 2014 The quasi-static growth of CO2 bubbles. J. Fluid Mech. 741, R1.CrossRefGoogle Scholar
Epstein, P. S. & Plesset, M. S. 1950 On the stability of gas bubbles in liquid–gas solutions. J. Chem. Phys. 18 (11), 15051509.CrossRefGoogle Scholar
Fuster, D. & Montel, F. 2015 Mass transfer effects on linear wave propagation in diluted bubbly liquids. J. Fluid Mech. 779, 598621.CrossRefGoogle Scholar
Fyrillas, M. M. & Szeri, A. J. 1994 Dissolution or growth of soluble spherical oscillating bubbles. J. Fluid Mech. 277, 381407.CrossRefGoogle Scholar
Houser, D. S., Howard, R. & Ridgway, S. 2001 Can diving-induced tissue nitrogen supersaturation increase the chance of acoustically driven bubble growth in marine mammals? J. Theor. Biol. 213, 183195.CrossRefGoogle ScholarPubMed
Hsieh, D. Y. & Plesset, M. S. 1961 Theory of rectified diffusion of mass into gas bubbles. J. Acoust. Soc. Am. 33 (2), 206215.CrossRefGoogle Scholar
Ichihara, M. & Brodsky, E. E. 2006 A limit on the effect of rectified diffusion in volcanic systems. Geophys. Res. Lett. 33 (2), L02316.CrossRefGoogle Scholar
Ilinskii, Y. A., Wilson, P. S. & Hamilton, M. F. 2008 Bubble growth by rectified diffusion at high gas supersaturation levels. J. Acoust. Soc. Am. 124 (4), 19501955.CrossRefGoogle ScholarPubMed
Jones, O. C. & Zuber, N. 1978 Bubble growth in variable pressure fields. Trans. ASME J. Heat Transfer 100 (3), 453459.CrossRefGoogle Scholar
Kapodistrias, G. & Dahl, P. H. 2012 Scattering measurements from a dissolving bubble. J. Acoust. Soc. Am. 131 (6), 42434251.CrossRefGoogle ScholarPubMed
Landau, L. D. & Lifshitz, E. M. 1987 Viscous fluids. In Fluid Mechanics, 2nd edn, chap. 2, pp. 8388. Pergamon.Google Scholar
Louisnard, O. & Gomez, F. 2003 Growth by rectified diffusion of strongly acoustically forced gas bubbles in nearly saturated liquids. Phys. Rev. E 67, 036610.CrossRefGoogle ScholarPubMed
Magnaudet, J. & Legendre, D. 1998 The viscous drag force on a spherical bubble with a time-dependent radius. Phys. Fluids 10, 550554.CrossRefGoogle Scholar
Marzella, L. & Yin, A. 1994 Role of extravascular gas bubbles in spinal cord injury induced by decompression sickness in the rat. Exp. Mol. Pathol. 61 (1), 1623.CrossRefGoogle ScholarPubMed
Payvar, P. 1987 Mass transfer-controlled bubble growth during rapid decompression of a liquid. Intl J. Heat Mass Transfer 30 (4), 699706.CrossRefGoogle Scholar
Peñas López, P., Parrales, M. A. & Rodríguez-Rodríguez, J. 2015 Dissolution of a spherical cap bubble adhered to a flat surface in air-saturated water. J. Fluid Mech. 775, 5376.CrossRefGoogle Scholar
Plesset, M. S. & Zwick, S. A. 1954 The growth of vapor bubbles in superheated liquids. J. Appl. Phys. 25 (4), 493500.CrossRefGoogle Scholar
Prosperetti, A. 1977 Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids. J. Acoust. Soc. Am. 61, 1727.CrossRefGoogle Scholar
Rosner, D. E. & Epstein, M. 1972 Effects of interface kinetics, capillarity and solute diffusion on bubble growth rates in highly supersaturated liquids. Chem. Engng Sci. 27 (1), 6988.CrossRefGoogle Scholar
Safar, M. H. 1968 Comment on papers concerning rectified diffusion of cavitation bubbles. J. Acoust. Soc. Am. 43 (5), 11881189.CrossRefGoogle Scholar
Scriven, L. E. 1959 On the dynamics of phase growth. Chem. Engng Sci. 10 (1), 113.CrossRefGoogle Scholar
Shim, S., Wan, J., Hilgenfeldt, S., Panchal, P. D. & Stone, H. A. 2014 Dissolution without disappearing: multicomponent gas exchange for CO2 bubbles in a microfluidic channel. Lab on a Chip 14, 24282436.CrossRefGoogle Scholar
Stepanyants, Y. A. & Yeoh, G. H. 2009 Particle and bubble dynamics in a creeping flow. Eur. J. Mech. (B/Fluids) 28 (5), 619629.CrossRefGoogle Scholar
Szekely, J. & Martins, G. P. 1971 Non-equilibrium effects in the growth of spherical gas bubbles due to solute diffusion. Chem. Engng Sci. 26 (1), 147159.CrossRefGoogle Scholar
Tao, L. N. 1978 Dynamics of growth or dissolution of a gas bubble. J. Chem. Phys. 69, 41894194.CrossRefGoogle Scholar
Theofanous, T., Biasi, L., Isbin, H. S. & Fauske, H. 1969 A theoretical study on bubble growth in constant and time-dependent pressure fields. Chem. Engng Sci. 24 (5), 885897.CrossRefGoogle Scholar
Tisato, N., Quintal, B., Chapman, S., Podladchikov, Y. & Burg, J.-P. 2015 Bubbles attenuate elastic waves at seismic frequencies: first experimental evidence. Geophys. Res. Lett. 42 (10), 38803887.CrossRefGoogle Scholar
Webb, I. R., Arora, M., Roy, R. A., Payne, S. J. & Coussios, C.-C. 2010 Dynamics of gas bubbles in time-variant temperature fields. J. Fluid Mech. 663, 209232.CrossRefGoogle Scholar
Weinberg, M. C. & Subramanian, R. S. 1980 Dissolution of multicomponent bubbles. J. Am. Ceram. Soc. 63 (9‐10), 527531.CrossRefGoogle Scholar
Zhang, Y. & Li, S. 2014a A general approach for rectified mass diffusion of gas bubbles in liquids under acoustic excitation. Trans. ASME J. Heat Transfer 136, 042001.CrossRefGoogle Scholar
Zhang, Y. & Li, S. 2014b Mass transfer during radial oscillations of gas bubbles in viscoelastic mediums under acoustic excitation. Intl J. Heat Mass Transfer 69, 106116.CrossRefGoogle Scholar
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