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The instability of a moving viscous drop

Published online by Cambridge University Press:  26 April 2006

C. Pozrikidis
C. Pozrikidis
Affiliation:
Department of Applied Mechanics and Engineering Sciences, B-Oil, University of California at San Diego, La Jolla, CA 92093, USA
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Abstract

The deformation of a moving spherical viscous drop subject to axisymmetric perturbations is considered. The problem is formulated using two different variations of the boundary integral method for Stokes flow, one due to Rallison & Acrivos, and the other based on an interfacial distribution of Stokeslets. An iterative method for solving the resulting Fredholm integral equations of the second kind is developed, and is implemented for the case of axisymmetric motion. It is shown that in the absence of surface tension, a moving spherical drop is unstable. Prolate perturbations cause the ejection of a tail from the rear of the drop, and the entrainment of a thin filament of ambient fluid into the drop. Oblate perturbations cause the drop to develop into a nearly steady ring. The viscosity ratio plays an important role in determining the timescale and the detailed pattern of deformation. Filamentation of the drop emerges as a persistent but secondary mechanism of evolution for both prolate and oblate perturbations. Surface tension is not capable of suppressing the growth of perturbations of sufficiently large amplitude, but is capable of preventing filamentation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 210 , January 1990 , pp. 1 - 21
Copyright
© 1990 Cambridge University Press

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References

Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bergman, S. & Schiffer, M. 1953 Kernell Functions and Elliptic Differential Equations in Mathematical Physics. Academic.
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic.
Delves, L. M. & Mohamed, J. L. 1985 Computational Methods for Integral Equations. Cambridge University Press.
Gradshteyn, I. S. & Ryshik, I. M. 1980 Table of Integrals, Series, and Products. Academic.
Griffiths, R. W. 1986 Thermals in extremely viscous fluids, including the effects of temperature-dependent viscosity. J. Fluid Mech. 166, 115–138.Google Scholar
Harper, J. F. 1972 The motion of bubbles and drops through liquids. Adv. Appl. Mech. 12, 59–129.Google Scholar
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27, 19–32.Google Scholar
Korn, G. A. & Korn, T. M. 1968 Mathematical Handbook for Scientists and Engineers. McGraw-Hill.
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow (2nd edn.) Gordon & Breach.
Power, H. 1987 On the Rallison and Acrivos solution for the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 185, 547–550.Google Scholar
Rallison, J. M. 1981 A numerical study of the deformation and burst of a viscous drop in general shear flows. J. Fluid Mech. 109, 465–482.Google Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid Mech. 16, 45–66.Google Scholar
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–200.Google Scholar
Stone, H. A. & Leal, L. G. 1989 Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399–427.Google Scholar
Zinemanas, D. & Nir, A. 1988 On the viscous deformation of biological cells under anisotropic surface tension. J. Fluid Mech. 193, 217–241.Google Scholar