Skip to main content

Segregation of a liquid mixture by a radially oscillating bubble


A theoretical formulation is proposed for forced mass transport by pressure gradients in a liquid binary mixture around a spherical bubble undergoing volume oscillations in a sound field. Assuming the impermeability of the bubble wall to both species, diffusion driven by pressure gradients and classical Fick-diffusion must cancel at the bubble wall, so that an oscillatory concentration gradient arises in the vicinity of the bubble. The Péclet number pe is generally high in typical situations and Fick diffusion cannot restore equilibrium immediately, so that an asymptotic average concentration profile may progressively build up in the liquid over large times. Such a behaviour is reminiscent of the so-called rectified diffusion problem, leading to slow growth of a gas bubble oscillating in a sound field. A rigorous method formerly proposed by Fyrillas & Szeri (J. Fluid Mech. vol. 277, 1994, p. 381) to solve the latter problem is used to solve the present one. It is based on splitting the problem into a smooth part and an oscillatory part. The smooth part is solved by a multiple scales method and yields the slowly varying average concentration field everywhere in the liquid. The oscillatory part is obtained by matched asymptotic expansions in terms of the small parameter pe−1/2: the inner solution is required to satisfy the oscillatory balance between pressure diffusion and Fick diffusion at the bubble wall, while the outer solution is required to be zero. Matching both solutions yields a unique splitting of the problem. The final analytical solution, truncated to leading order, compares successfully to direct numerical simulation of the full convection–diffusion equation. The analytical expressions for both smooth and oscillatory parts are calculated for various sets of bubble parameters: driving pressure, frequency and ambient radius. The smooth problem always yields an average depletion of the heaviest species at the bubble wall, only noticeable for large molecules or nano-particles. For driving pressures sufficiently high to yield inertial oscillations of the bubble, the oscillatory problem predicts a periodic peak excess concentration of the heaviest species at the bubble wall at each collapse, lingering on several tens of the time of the characteristic duration of the bubble rebound. The two effects may compete for large molecules and practical implications of this segregation phenomenon are proposed for various processes involving acoustic cavitation.

Hide All
Akhatov, I., Gumerov, N., Ohl, C., Parlitz, U. & Lauterborn, W. 1997 The role of surface tension in stable single bubble sonoluminescence. Phys. Rev. Lett. 78, 227230.
Archibald, W. J. 1938 The process of diffusion in a centrifugal field of force. Phys. Rev. 53, 746752.
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 1960 Transport Phenomena. John Wiley.
Brenner, M. P., Hilgenfeldt, S. & Lohse, D. 2002 Single-bubble sonoluminescence. Rev. Mod. Phys. 74, 425483.
Crum, L. A., Mason, T. J., Reisse, J. L. & Suslick, K. S., (ed.) 1999 Sonochemistry and Sonoluminescence, Proc. NATO Advanced Study Institute on Sonoluminescence and Sonoluminescence, Leavenworth, Washington, USA, 18–29 August 1997, Kluwer.
Eller, A. & Flynn, H. G. 1965 Rectified diffusion during nonlinear pulsations of cavitation bubbles. J. Acoust. Soc. Am. 37, 493503.
Fujikawa, S. & Akamatsu, T. 1980 Effects of the nonequilibrium condensation of vapour on the pressure wave produced by the collapse of a bubble in a liquid. J. Fluid Mech. 97, 481512.
Fyrillas, M. M. & Szeri, A. J. 1994 Dissolution or growth of soluble spherical oscillating bubbles. J. Fluid Mech. 277, 381407.
Fyrillas, M. M. & Szeri, A. J. 1995 Dissolution or growth of soluble spherical oscillating bubbles: the effect of surfactants. J. Fluid Mech. 289, 295314.
Fyrillas, M. M. & Szeri, A. J. 1996 Surfactant dynamics and rectified diffusion of microbubbles. J. Fluid Mech. 311, 361378.
Hickling, R. & Plesset, M. S. 1964 Collapse and rebound of a spherical bubble in water. Phys. Fluids 7, 714.
Hilgenfeldt, S., Lohse, D. & Brenner, M. P. 1996 Phase diagrams for sonoluminescing bubbles. Phys. Fluids 8 (11), 28082826.
Hilgenfeldt, S., Brenner, M. P., Grossman, S. & Lohse, D. 1998 Analysis of Rayleigh-Plesset dynamics for sonoluminescing bubbles. J. Fluid Mech. 365, 171204.
Hirschfelder, J. O., Curtiss, C. F. & Bird, R. B. 1967 Molecular Theory of Gases and Liquids. John Wiley.
Hsieh, D. Y. & Plesset, M. S. 1961 Theory of rectified diffusion of mass into gas bubbles. J. Acoust. Soc. Am. 33, 206215.
Kaschiev, D. 2000 Nucleation : Basic Theory with Applications. Butterworth–Heinemann.
Keller, J. B. & Miksis, M. 1980 Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68, 628633.
Larson, M. A. & Garside, J. 1986 Solute clustering in supersaturated solutions. Chem. Engng Sci. 41, 12851289.
Lin, H., Storey, B. D. & Szeri, A. J. 2002 Rayleigh–Taylor instability of violently collapsing bubbles. Phys. Fluids 14, 29252928.
Louisnard, O. & Gomez, F. 2003 Growth by rectified diffusion of strongly acoustically forced gas bubbles in nearly saturated liquids. Phys. Rev. E 67 (036610), 112.
Lyczko, N., Espitalier, F., Louisnard, O. & Schwartzentruber, J. 2002 Effect of ultrasound on the induction time and the metastable zone widths of potassium sulphate. Chem. Engng J. 86, 233241.
Mullin, J. W. & Leci, C. L. 1969 Evidence of molecular cluster formation in supersaturated solutions of citric acid. Phil. Mag. 19 (161), 10751077.
Plesset, M. S. & Zwick, S. A. 1952 A nonsteady heat diffusion problem with spherical symmetry. J. Appl. Phys. 23, 9598.
Prosperetti, A. 1999 Old-fashioned bubble dynamics. In Sonochemistry and Sonoluminescence (ed. Crum, L. A., Mason, T. J., Reisse, J. L. & Suslick, K. S.), Proc. NATO Advanced Study Institute on Sonoluminescence and Sonoluminescence, Leavenworth, Washington, USA, 18–29 August 1997, Kluwer.
Prosperetti, A. & Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part 1. First-order theory. J. Fluid Mech. 168, 457478.
Storey, B. D. & Szeri, A. 2000 Water vapour, sonoluminescence and sonochemistry. Proc. R. Soc. Lond. A 456, 16851709.
Storey, B. D. & Szeri, A. 2001 A reduced model of cavitation physics for use in sonochemistry. Proc. R. Soc. Lond. A 457, 16851700.
Storey, B. D. & Szeri, A. J. 1999 Mixture segregation within sonoluminescence bubbles. J. Fluid Mech. 396, 203221.
Toegel, R., Gompf, B., Pecha, R. & Lohse, D. 2000 Does water vapor prevent upscaling sonoluminescence? Phys. Rev. Lett. 85, 31653168.
Tomita, Y. & Shima, A. 1977 On the behaviour of a spherical bubble and the impulse pressure in a viscous compressible liquid. Bull. JSME 20, 14531460.
Zygmund, A. 1959 Trigonometric Series. Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed