Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-09T12:36:55.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field

Published online by Cambridge University Press:  26 April 2006

Nikolaos A. Pelekasis  and
John A. Tsamopoulos
Nikolaos A. Pelekasis
Affiliation:
Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
John A. Tsamopoulos
Affiliation:
Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The motion of two gas bubbles in response to an oscillatory disturbance in the ambient pressure is studied. It is shown that the relative motion of bubbles of unequal size depends on the frequency of the disturbance. If this frequency is between the two natural frequencies for volume oscillations of the individual bubbles, the two bubbles are seen to move away from each other; otherwise attractive forces prevail. Bubbles of equal size can only attract each other, irrespective of the oscillation frequency. When the Bond number, Bo (based on the average acceleration) lies above a critical region, spherical-cap shapes appear with deformation confined on the side of the bubbles facing away from the direction of acceleration. For Bo below the critical region shape oscillations spanning the entire bubble surface take place, as a result of subharmonic resonance. The presence of the oscillatory acoustic field adds one more frequency to the system and increases the possibilities for resonance. However, only subharmonic resonance is observed because it occurs on a faster timescale, O(1/ε), where ε is the disturbance amplitude. Furthermore, among the different possible periodic variations of the volume of each bubble, the one with the smaller period determines which Legendre mode will be excited through subharmonic resonance. Spherical-cap shapes also occur on a timescale O(1/ε). When the bubbles are driven below resonance and for quite large amplitudes of the acoustic pressure, ε ≈ 0.8, a subharmonic signal at half the natural frequency of volume oscillations is obtained. This signal is primarily associated with the zeroth mode and corresponds to volume expansion followed by rapid collapse of the bubbles, a behaviour well documented in acoustic cavitation experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 254 , September 1993 , pp. 501 - 527
Copyright
© 1993 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.)

References

Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Benjamin, T. B. & Ellis, T. A. 1990 Self-propulsion of asymmetrically vibrating bubbles. J. Fluid Mech. 212, 65–80.Google Scholar
Benjamin, T. B. & Strasberg, M. 1958 Excitation of oscillations in the shape of pulsating gas bubbles; Experimental work. J. Acoust. Soc. Am. 30, 697.Google Scholar
Bjerknes, V. F. K. 1906 Fields of force. Columbia University Press.
Bjerknes, V. F. K. 1909 Die Craftfelder. Vieweg.
Blake, J. R. & Cerone, P. 1982 A note on the impulse due to a vapour bubble near a boundary. J. Austral. Math. Soc. B 23, 383–393.Google Scholar
Blake, J. R. & Gibson, D. C. 1981 Growth and collapse of a vapour cavity near a free surface. J. Fluid Mech. 111, 123–140.Google Scholar
Blake, J. R., Taib, B. B. & Doherty, G. 1986 Transient cavities near boundaries. Part I. Rigid boundary. J. Fluid Mech. 170, 479–497.Google Scholar
Crum, L. A. 1974 Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc. Am. 57, 1363–1370.Google Scholar
Crum, L. A. & Nordling, D. A. 1972 Velocity of transient cavities in an acoustic stationary wave. J. Acoust. Soc. Am. 52, 294–301.Google Scholar
Davies, R. M. & Taylor, G. I. 1950 The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. Lond. A 200, 375–390.Google Scholar
Eller, A. I. & Crum, L. A. 1970 Instability of the motion of a pulsating bubble in a sound field. J. Acoust. Soc. Am. 47, 762–767.Google Scholar
Esche, R. 1952 Untersuchung der Schwingungskavitation in Flüssigkeiten. Acustica 2, 208–218.Google Scholar
Flynn, H. G. 1964 Physical Acoustics (ed. W. P. Mason), vol. 1b, chap. 9, pp. 57–172. Academic.
Hall, P. & Seminara, G. 1980 Nonlinear oscillations of non-spherical cavitation bubbles in acoustic fields. J. Fluid Mech. 101, 423–444.Google Scholar
Hartunian, R. A. & Sears, W. R. 1957 On the instability of small gas bubbles moving uniformly in various liquids. J. Fluid Mech. 3, 21–41.Google Scholar
Kornfeld, M. & Suvorov, L. 1944 On the destructive action of cavitation. J. Appl. Phys. 15, 495–506.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lauterborn, W. 1976 Numerical investigation of nonlinear oscillations of gas bubbles in liquids. J. Acoust. Soc. Am. 59, 283–293.Google Scholar
Lauterborn, W. & Cramer, E. 1981 Subharmonic route to chaos observed in acoustics. Phys. Rev. Let. 47, 1445–1448.Google Scholar
Neppiras, E. A. 1969 Subharmonic and other low-frequency emission from bubbles in sound-irradiated liquids. J. Acoust. Soc. Am. 46, 587–601.Google Scholar
Pelekasis, N. A. 1991 A study on drop and bubble dynamics via a hybrid boundary element-finite element methodology. PhD thesis, SUNY at Buffalo.
Pelekasis, N. A. & Tsamopoulos, J. A. 1993 Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure. J. Fluid Mech. 254, 467–499.Google Scholar
Pelekasis, N. A., Tsamopoulos, J. A. & Manolis, G. D. 1991 Nonlinear oscillations of liquid shells in zero gravity. J. Fluid Mech. 230, 541–582.Google Scholar
Pelekasis, N. A., Tsamopoulos, J. A. & Manolis, G. D. 1992 A hybrid finite-boundary element method for inviscid flows with free surface. J. Comput. Phys. 101, 231–251.Google Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Ann. Rev. Fluid Mech. 9, 145–185.Google Scholar
Saffman, P. G. 1956 On the rise of small air bubbles in water. J. Fluid Mech. 1, 249–275.Google Scholar
Saffman, P. G. 1967 The self-propulsion of a deformable body in a perfect fluid. J. Fluid Mech. 28, 385–389.Google Scholar