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 | References | Additional Information

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 . In between these asymptotic regimes, however, the motion is significantly different from either approximation.

**[1]**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**[2]**H. Lamb,*Hydrodynamics*, 6th ed., Dover Publications, New York, 1945**[3]**S. Chandrasekhar,*Hydrodynamics and hydromagnetic stability*, Clarendon Press, Oxford, 1961 MR**0128226****[4]**A. Prosperetti,*Viscous effects on small-amplitude surface waves*, Phys. Fluids, in press MR**0413747****[5]**S. Chandrasekhar,*The oscillations of a viscous liquid globe*, Proc. London Math. Soc.**9**, 141-149 (1959) MR**0105246****[6]**W. H. Reid,*The oscillations of a viscous liquid drop*, Quart. Appl. Math.**18**, 86-89 (1960) MR**0114449****[7]**C. A. Miller and L. E. Scriven,*The oscillations of a fluid droplet immersed in another fluid*, J. Fluid Mech.**32**, 417-435 (1968)**[8]**H. H. K. Tong and C. Y. Wong,*Vibrations of a viscous liquid sphere*, J. Phys.**A7**, 1038-1050 (1974)**[9]**M. S. Plesset,*On the stability of fluid flows with spherical symmetry*, J. Appl. Phys.**25**, 96-98 (1954) MR**0059692****[10]**G. Birkhoff,*Note on Taylor instability*, Quart. Appl. Math.**12**, 306-309 (1954);*Stability of spherical bubbles*, Quart. Appl. Math.**13**, 451-453 (1956) MR**0065316****[11]**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) MR**0079931****[12]**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)**[13]**A. Prosperetti,*Viscous and nonlinear effects in the oscillations of drops and bubbles*, Thesis, California Institute of Technology, Pasadena, California, 1974, pp. 127-150**[14]**G. K. Batchelor,*An introduction to fluid mechanics*, Cambridge University Press, Cambridge, 1971**[15]**Lord Rayleigh,*On the pressure developed in a liquid during the collapse of a spherical cavity*, Phil. Mag.**34**, 94-98 (1917)**[16]**L. Landau and E. Lifshitz,*Fluid mechanics*, Addison-Wesley, Reading, Mass., 1959 MR**0108121****[17]**M. S. Plesset,*Cavitating flows*, in*Topics in ocean engineering*, C. I. Bretschneider, ed., Gulf Publishing Co., Houston, 1969, Vol. 1, pp. 85-95**[18]**D. Y. Hsieh,*Some analytical aspects of bubble dynamics*, J. Basic Eng.**87**, 991-1005 (1965)**[19]**H. L. Dryden, F. D. Murnaghan and H. Bateman,*Hydrodynamics*, Dover Publications, New York, 1956 MR**0077307****[20]**W. Rayleigh,*The theory of sound*, 2nd Ed., Dover Publications, New York, 1945, Art. 364 MR**0016009****[21]**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 MR**0095590****[22]**D. V. Widder,*The Laplace transform*, Princeton University Press, Princeton, 1941, Chap. 5 MR**0005923****[23]**F. Durbin,*Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate's method*, Comp. J.**17**, 371-376 (1974) MR**0356449**

Additional Information

DOI:
https://doi.org/10.1090/qam/99652

Article copyright:
© Copyright 1977
American Mathematical Society