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The oscillations of a fluid droplet immersed in another fluid

Published online by Cambridge University Press:  28 March 2006

C. A. Miller  and
L. E. Scriven
C. A. Miller
Affiliation:
Department of Chemical Engineering, University of Minnesota, Minneapolis, Minnesota 55455
L. E. Scriven
Affiliation:
Department of Chemical Engineering, University of Minnesota, Minneapolis, Minnesota 55455
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Abstract

From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.

When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 32 , Issue 3 , 24 May 1968 , pp. 417 - 435
Copyright
© 1968 Cambridge University Press

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References

Benjamin, T. B. 1962 Non-spherical motions of cavities. In Davies, R. 1964 (ed.), Cavitation in Real Liquids, Proceedings of 1962 Symposium. Amsterdam: Elsevier, pp. 164–180.
Bupara, S. S. 1964 Spontaneous Movements of Small Round Bodies in Viscous Fluids. Ph.D. Thesis, Department of Chemical Engineering, University of Minnesota.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, p. 466–466. Oxford: Clarendon Press.
Eliassen, J. D. 1963 Interfacial Mechanics. Ph.D. Thesis, Department of Chemical Engineering, University of Minnesota.
Goodrich, F. C. 1961 Mathematical theory of capillarity. Proc. Roy. Soc A 260, 481–509.Google Scholar
Hansen, R. S. & Mann, J. A. 1964 Propagation characteristics of capillary ripples. I. The theory of velocity dispersion and amplitude attenuation of plane capillary waves on viscoelastic films J. Appl. Phys. 35, 152–158.CrossRefGoogle Scholar
Harrison, W. J. 1908 The influence of viscosity on the oscillations of superposed fluids. Proc. Lond. Math. Soc. Second Series 6, 396–405.Google Scholar
Lamb, H. 1932 Hydrodynamics, sixth edition. Cambridge University Press. Reprinted by Dover, New York, 1945.
Landau, L. D. & Lifshitz, E. J. 1959 Fluid Mechanics. London: Pergamon Press.
Levich, V. G. 1962 Physicochemical Hydrodynamics. Englewood Cliffs, N. J. Prentice-Hall.
Oldroyd, J. G. 1955 The effect of interfacial stabilizing films on the elastic and viscous properties of emulsions. Proc. Roy. Soc A 232, 567–577.Google Scholar
Reid, W. H. 1960 The oscillations of a viscous liquid drop Quart. Appl. Math. 18, 86–89.Google Scholar
Sani, R. L. 1963 Convective Instability. Ph.D. Thesis, Department of Chemical Engineering, University of Minnesota.
Scriven, L. E. 1960 Dynamics of a fluid interface. Equations of motion for Newtonian surface fluids Chem. Engng. Sci. 12, 98–108.Google Scholar
Scriven, L. E. & Sterling, C. V. 1964 On cellular convection driven by surface tension gradients: Effects of mean surface tension and surface viscosity. J. Fluid Mech. 19, 321–340.CrossRefGoogle Scholar
Valentine, R. S., Sather, N. F. & Heideger, W. J. 1965 The motion of drops in viscous media Chem. Engng. Sci. 20, 719–728.Google Scholar
Weatherburn, C. E. 1927 Differential Geometry of Three Dimensions, vol. 1. Cambridge University Press.
Weatherburn, C. E. 1930 Differential Geometry of Three Dimensions, vol. 2. Cambridge University Press.
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves, in Handbuch der Physik, pp. 446–778. Vol. IX. Berlin: Springer Verlag.
Willson, A. J. 1965 On the stability of two superposed fluids Proc. Camb. Phil. Soc. 61, 595–607.Google Scholar