Lagrangian formulation of bubble dynamics
Author:
D. Y. Hsieh
Journal:
Quart. Appl. Math. 33 (1975), 115-130
MSC:
Primary 76.53
DOI:
https://doi.org/10.1090/qam/446078
MathSciNet review:
446078
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Abstract: The dynamical problem of two ideal fluids separated by an interface is formulated in terms of general coordinates in Lagrangian variables. The same problem is shown to be equivalent to a Hamiltonian variational principle which takes into account explicitly the surface energy of the interface between the two fluids. The formulation is applied to the motion of slightly nonspherical bubbles. It is shown that, although we start from an entirely different set of differential equations, the results of the linear stability analysis in Eulerian formulation are recovered.
T. B. Benjamin, Pressure waves from collapsing cavities, 2nd Symposium on Naval Hydrodynamics, ONR/ACR-38, 207–233 (1958)
R. Hickling and M. S. Plesset, Collapse and rebound of a spherical bubble in water, Phys. Fluids 7, 7–14 (1964)
- M. S. Plesset and T. P. Mitchell, On the stability of the spherical shape of a vapor cavity in a liquid, Quart. Appl. Math. 13 (1956), 419–430. MR 79931, DOI https://doi.org/10.1090/S0033-569X-1956-79931-7
C. F. Naudé and A. T. Ellis, On the mechanism of cavitation damage by nonhemispherical cavities collapsing in contact with a solid boundary, Trans. ASME 83D, 648–656 (1961)
W. Lauterborn, Laser-induced cavitation, J. Acoust. Soc. Am. 52, 151 (1972)
T. M. Mitchell and F. G. Hammitt, Asymptotic cavitation bubble collapse, Trans. ASME 95I, 29–37 (1973)
D. L. Storm, Thresholds for surface waves and subharmonics associated with a single bubble, J. Acoust. Soc. Am. 52, 152 (1972)
D. Y. Hsieh, On thresholds for surface waves and subharmonics of an oscillating bubble, J. Acoust. Soc. Am. 56, 392–393 (1974)
R. B. Chapman and M. S. Plesset, Nonlinear effects in the collapse of a nearly spherical cavity in a liquid, Trans. ASME 94D, 142–146 (1972)
D. Y. Hsieh, On the dynamics of nonspherical bubbles, Trans. ASME 94D, 655-665 (1972)
D. Y. Hsieh, Variational methods and dynamics of nonspherical bubbles and liquid drops, in Proc. Symposium on Finite-Amplitude Effects in Fluids, IPC Science and Tech. Press, Surrey, England, 1974
D. Y. Hsieh, Dynamics and oscillation of nonspherical bubbles, J. Acoust. Soc. Am. 52, 151 (1972)
R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics, Proc. Roy. Soc. London A305, 1–25 (1968)
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D. Y. Hsieh, Some analytical aspects of bubble dynamics, Trans. ASME 87D, 991–1005 (1965)
- Barry Spain, Tensor Calculus, Oliver and Boyd, Edinburgh and London; Interscience Publishers, Inc., New York, 1953. MR 0055027
T. B. Benjamin, Pressure waves from collapsing cavities, 2nd Symposium on Naval Hydrodynamics, ONR/ACR-38, 207–233 (1958)
R. Hickling and M. S. Plesset, Collapse and rebound of a spherical bubble in water, Phys. Fluids 7, 7–14 (1964)
M. S. Plesset and T. P. Mitchell, “On the stability of the spherical shape of a vapor cavity in a liquid", Quart. Appl. Math. 8, 419–430 (1956)
C. F. Naudé and A. T. Ellis, On the mechanism of cavitation damage by nonhemispherical cavities collapsing in contact with a solid boundary, Trans. ASME 83D, 648–656 (1961)
W. Lauterborn, Laser-induced cavitation, J. Acoust. Soc. Am. 52, 151 (1972)
T. M. Mitchell and F. G. Hammitt, Asymptotic cavitation bubble collapse, Trans. ASME 95I, 29–37 (1973)
D. L. Storm, Thresholds for surface waves and subharmonics associated with a single bubble, J. Acoust. Soc. Am. 52, 152 (1972)
D. Y. Hsieh, On thresholds for surface waves and subharmonics of an oscillating bubble, J. Acoust. Soc. Am. 56, 392–393 (1974)
R. B. Chapman and M. S. Plesset, Nonlinear effects in the collapse of a nearly spherical cavity in a liquid, Trans. ASME 94D, 142–146 (1972)
D. Y. Hsieh, On the dynamics of nonspherical bubbles, Trans. ASME 94D, 655-665 (1972)
D. Y. Hsieh, Variational methods and dynamics of nonspherical bubbles and liquid drops, in Proc. Symposium on Finite-Amplitude Effects in Fluids, IPC Science and Tech. Press, Surrey, England, 1974
D. Y. Hsieh, Dynamics and oscillation of nonspherical bubbles, J. Acoust. Soc. Am. 52, 151 (1972)
R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics, Proc. Roy. Soc. London A305, 1–25 (1968)
C. Truesdell and R. A. Toupin, The classical field theories, in Handbuch der Physik, edited by S. Flügge, Vol. III/1, Springer-Verlag, Berlin (1960)
D. Y. Hsieh, Some analytical aspects of bubble dynamics, Trans. ASME 87D, 991–1005 (1965)
B. Spain, Tensor calculus, Oliver and Boyd, Edinburgh (1953)
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Article copyright:
© Copyright 1975
American Mathematical Society
