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Time-dependent viscous deformation of a drop in a rapidly rotating denser fluid

Published online by Cambridge University Press:  26 April 2006

J. R. Lister  and
H. A. Stone
J. R. Lister
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK
H. A. Stone
Affiliation:
Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA
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Abstract

Viscous stretching of a cigar-shaped drop due to the centrifugal pressure field in a surrounding rapidly rotating denser fluid is analysed. Scaling arguments are used to examine the various contributions to the viscous stresses resisting deformation, and a number of asymptotic regimes are identified which are delineated by the relative magnitudes of the aspect ratio, the viscosity ratio and unity. These asymptotic regimes may usefully be described as the bubble, pipe, sliding-rod and toffee-strand limits. Detailed analysis based upon a slenderness assumption combined with an integral representation of Stokes equations is used to derive evolution equations for the shape of the drop as a function of time in the different regimes. In the limit that interfacial-tension effects are negligible, similarity solutions are developed in which the length of the drop is found to increase as t2/5, t1/4, (t ln t)1/4 and t. The analytical results are in good agreement with numerical simulations based upon a boundary-integral solution to the full viscous flow equations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 317 , 25 June 1996 , pp. 275 - 299
Copyright
© 1996 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1977 Dynamics of Polymeric Liquids, Vol. 1 (1st edn). John Wiley.
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General Theory. J. Fluid Mech. 44, 791–810.Google Scholar
Davis, R. H., Schonberg, J. A. & Rallison, J. M. 1989 The lubrication force between two viscous drops. Phys. Fluids A 1, 77–81.Google Scholar
Eggers, J. 1995 Theory of drop formation. Phys. Fluids A 7, 941–953.Google Scholar
Elmendorp, J. J. & De Vos, G. 1986 Polym. Engng Sci. 26, 415–417.
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.
Hsu, J. C. & Flumerfelt, R. W. 1975 Rheological applications of a drop elongation experiment. Trans. Soc. Rheol. 19, 523–540.Google Scholar
Hu, H. H. & Joseph, D. D. 1994 Evolution of a liquid drop in a spinning drop tensiometer. J. Colloid Interface Sci. 162, 331–339.Google Scholar
Huppert, H. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.Google Scholar
Joseph, D. D., Arney, M.S., Gillberg, G., Hu, H., Hultman, D., Verdier, C. & Vinagre, T. M. 1992 A spinning-drop tensioextensometer. J. Rheol. 36, 621–662.Google Scholar
Koch, D. M. & Koch, D. L. 1995 Numerical and theoretical solutions for a drop spreading below a free fluid surface. J. Fluid Mech. 287, 251–278.Google Scholar
Lister, J. R. & Kerr, R. C. 1989 The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface. J. Fluid Mech. 203, 215–249.Google Scholar
Lister, J. R., Stone, H. A. & Brenner, M. P. 1996 Capillary breakup of a viscous thread in the presence of an external fluid (in preparation).
Manga, M. & Stone, H. A. 1993 Buoyancy-driven interactions between two deformable viscous drops. J. Fluid Mech. 256, 647–683.Google Scholar
Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids A 7, 1529–1544.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods For Linearized Viscous Flow. Cambridge University Press.
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
Rosenthal, D. K. 1962 The shape and stability of a bubble at the axis of a rotating liquid. J. Fluid Mech. 12, 358–366.Google Scholar
Stone, H. A. & Bush, J. W. M. 1996 Time-dependent drop deformation in a rotating high viscosity fluid. Q. Appl. Maths (in press).Google Scholar
Stone, H. A. & Leal, L. G. 1990 The effect of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161–186.Google Scholar
Tanzosh, J., Manga, M. & Stone, H. A. 1992 Boundary integral methods for viscous free-boundary problems: Deformation of single and multiple fluid-fluid interfaces. In Proc. Boundary Element Technologies VII (ed. C. A. Brebbia & M. S. Ingber), pp. 19–39. Computational Mechanics Publications.
Taylor, G.I. 1964 Conical free surfaces and fluid interfaces. In Proc. 11th Intl Cong. Appl. Mech., Munich, pp. 790–796.
Vonnegut, B. 1942 Rotating bubble method for the determination of surface and interfacial tension. Rev. Sci. Instrum. 13, 6–9.Google Scholar