Viscous effects on perturbed spherical flows
Author:
Andrea Prosperetti
Journal:
Quart. Appl. Math. 34 (1977), 339-352
DOI:
https://doi.org/10.1090/qam/99652
MathSciNet review:
QAM99652
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Abstract: The problem of two viscous, incompressible fluids separated by a nearly spherical free surface is considered in general terms as an initial-value problem to first order in the perturbation of the spherical symmetry. As an example of the applications of the theory, the free oscillations of a viscous liquid drop and of a bubble in a viscous liquid are studied in some detail. It is shown that the oscillations are initially describable in terms of an irrotational approximation, and that the normal-mode results are recovered as $t \to \infty$. In between these asymptotic regimes, however, the motion is significantly different from either approximation.
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G. G. Stokes, On waves, Cambridge and Dublin Math. J. 4, 219 (1849), reprinted in Mathematical and physical papers, Cambridge University Press, Cambridge, Vol. II, pp. 220–242
H. Lamb, Hydrodynamics, 6th ed., Dover Publications, New York, 1945
S. Chandrasekhar, Hydrodynamics and hydromagnetic stability, Clarendon Press, Oxford, 1961
A. Prosperetti, Viscous effects on small-amplitude surface waves, Phys. Fluids, in press
S. Chandrasekhar, The oscillations of a viscous liquid globe, Proc. London Math. Soc. 9, 141–149 (1959)
W. H. Reid, The oscillations of a viscous liquid drop, Quart. Appl. Math. 18, 86–89 (1960)
C. A. Miller and L. E. Scriven, The oscillations of a fluid droplet immersed in another fluid, J. Fluid Mech. 32, 417–435 (1968)
H. H. K. Tong and C. Y. Wong, Vibrations of a viscous liquid sphere, J. Phys. A7, 1038–1050 (1974)
M. S. Plesset, On the stability of fluid flows with spherical symmetry, J. Appl. Phys. 25, 96–98 (1954)
G. Birkhoff, Note on Taylor instability, Quart. Appl. Math. 12, 306–309 (1954); Stability of spherical bubbles, Quart. Appl. Math. 13, 451–453 (1956)
M. S. Plesset and T. P. Mitchell, On the stability of the sperical shape of a vapor cavity in a liquid, Quart. Appl. Math. 13, 419–430 (1956)
R. B. Chapman and M. S. Plesset, Nonlinear effects in the collapse of a nearly spherical cavity in a liquid, J. Basic Eng. 94, 142–146 (1972)
A. Prosperetti, Viscous and nonlinear effects in the oscillations of drops and bubbles, Thesis, California Institute of Technology, Pasadena, California, 1974, pp. 127–150
G. K. Batchelor, An introduction to fluid mechanics, Cambridge University Press, Cambridge, 1971
Lord Rayleigh, On the pressure developed in a liquid during the collapse of a spherical cavity, Phil. Mag. 34, 94–98 (1917)
L. Landau and E. Lifshitz, Fluid mechanics, Addison-Wesley, Reading, Mass., 1959
M. S. Plesset, Cavitating flows, in Topics in ocean engineering, C. I. Bretschneider, ed., Gulf Publishing Co., Houston, 1969, Vol. 1, pp. 85–95
D. Y. Hsieh, Some analytical aspects of bubble dynamics, J. Basic Eng. 87, 991–1005 (1965)
H. L. Dryden, F. D. Murnaghan and H. Bateman, Hydrodynamics, Dover Publications, New York, 1956
W. Rayleigh, The theory of sound, 2nd Ed., Dover Publications, New York, 1945, Art. 364
M. Onoe, Tables of the modified quotients of Bessel functions of the first kind for real and imaginary arguments, Columbia University Press, New York, 1958
D. V. Widder, The Laplace transform, Princeton University Press, Princeton, 1941, Chap. 5
F. Durbin, Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method, Comp. J. 17, 371–376 (1974)
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Article copyright:
© Copyright 1977
American Mathematical Society