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Buoyancy-driven interactions between two deformable viscous drops

Published online by Cambridge University Press:  26 April 2006

Michael Manga  and
H. A. Stone
Michael Manga
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
H. A. Stone
Affiliation:
Division of Applied Sciences, Harvard University, Cambridge MA 02138, USA
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Abstract

Time-dependent interactions between two buoyancy-driven deformable drops are studied in the low Reynolds number flow limit for sufficiently large Bond numbers that the drops become significantly deformed. The first part of this paper considers the interaction and deformation of drops in axisymmetric configurations. Boundary integral calculations are presented for Bond numbers ℬ = Δρga2/σ in the range 0.25 ≤ ℬ < ∞ and viscosity ratios λ in the range 0.2 ≤ λ ≤ 20. Specifically, the case of a large drop following a smaller drop is considered, which typically leads to the smaller drop coating the larger drop for ℬ [Gt ] 1. Three distinct drainage modes of the thin film of fluid between the drops characterize axisymmetric two-drop interactions: (i) rapid drainage for which the thinnest region of the film is on the axis of symmetry, (ii) uniform drainage for which the film has a nearly constant thickness, and (iii) dimple formation. The initial mode of film drainage is always rapid drainage. As the separation distance decreases, film flow may change to uniform drainage and eventually to dimpled drainage. Moderate Bond numbers, typically ℬ = O(10) for λ = O(1), enhance dimple formation compared to either much larger or smaller Bond numbers. The numerical calculations also illustrate the extent to which lubrication theory and analytical solutions in bipolar coordinates (which assume spherical drop shapes) are applicable to deformable drops.

The second part of this investigation considers the 'stability’ of axisymmetric drop configurations. Laboratory experiments and two-dimensional boundary integral simulations are used to study the interactions between two horizontally offset drops. For sufficiently deformable drops, alignment occurs so that the small drop may still coat the large drop, whereas for large enough drop viscosities or high enough interfacial tension, the small drop will be swept around the larger drop. If the large drop is sufficiently deformable, the small drop may then be ‘sucked’ into the larger drop as it is being swept around the larger drop. In order to explain the alignment process, the shape and translation velocities of widely separated, nearly spherical drops are calculated using the method of reflections and a perturbation analysis for the deformed shapes. The perturbation analysis demonstrates explicitly that drops will tend to be aligned for ℬ > O(d/a) where d is the separation distance between the drops.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 256 , November 1993 , pp. 647 - 683
Copyright
© 1993 Cambridge University Press

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References

Acrivos, A. 1983 The breakup of small drops and bubbles in shear flows. 4th Intl Conf. on Physicochemical Hydrodynamics, Ann. N.Y. Acad. Sci. 404, 1–11.
Barnocky, G. & Davis, R. H. 1989 The lubrication force between spherical drops, bubbles and rigid particles in a viscous fluid. Intl J. Multiphase Flow 15, 627–638.Google Scholar
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Boor, C. de 1978 A Practical Guide to Splines. Springer.
Bruijn, R. A. de 1989 Deformation and breakup of drops in simple shear flows. PhD thesis, Technical University at Eindhoven.
Chi, B. K. & Leal, L. G. 1989 A theoretical study of the motion of a viscous drop toward a fluid interface at low Reynolds number. J. Fluid Mech. 201, 123–146.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
Griffiths, R. W. & Campbell, I. H. 1990 Stirring and structure in mantle starting plumes. Earth Planet. Sci. Lett. 99, 66–78.Google Scholar
Griffiths, R. W., Gurnis, M. & Eitelberg, G. 1989 Holographic measurements of surface topography in laboratory models of mantle hotspots. Geophys. J. 96, 477–495.Google Scholar
Haber, S., Hetsroni, G. & Solan, A. 1973 Low Reynolds number motion of two drops submerged in an unbounded arbitrary velocity field. Intl J. Multiphase Flow 4, 627–638.Google Scholar
Haj-Hariri, H., Nadim, A. & Borhan, A. 1990 Effect of inertia on the thermocapillary velocity of a drop. J. Colloid Interface Sci. 140, 277–286.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Honda, R., Mizutani, H. & Yamamoto, T. 1993 Numerical simulations of Earth's core formation. J. Geophys. Res. 98, 2075–2089.Google Scholar
Jaupart, C. & Vergniolle, S. 1989 The generation and collapse of a foam layer at the roof of basaltic magma chamber. J. Fluid Mech. 203, 347–380.Google Scholar
Jeong, J.-T. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 1–22.Google Scholar
Joseph, D. D., Nelson, J., Renardy, M. & Renardy, Y. 1991 Two-dimensional cusped interfaces. J. Fluid Mech. 223, 383–409.Google Scholar
Kim, S. & Karrila, S.J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heineman.
Koch, D. L. & Shaqfeh, E. S. G. 1989 The instability of a dispersion of sedimenting spheroids. J. Fluid Mech. 209, 521–542.Google Scholar
Koch, D. M. 1993 A spreading drop model for mantle plumes and volcanic features on Venus. PhD thesis, Yale University.
Koh, C.J. & Leal, L. G. 1989 The stability of drop shapes for translation at zero Reynolds number through a quiescent fluid. Phys. Fluids A 1, 1309–1313.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
Leal, L. G. 1992 Laminar Flow and Convective Transport Processes. Butterworth-Heineman.
Lee, S. H. & Leal, L. G. 1982 The motion of a sphere in the presence of a deformable interface. II. A numerical study of the translation of a sphere normal to an interface. J. Colloid Interface Sci. 87, 81–106.Google Scholar
Manga, M., Stone, H. A. & O’Connell, R. J. 1993 The interaction of plume heads with compositional discontinuities in the Earth's mantle. J. Geophys. Res., in press.Google Scholar
Newhouse, L. A. & Pozrikidis, C. 1990 The Rayleigh–Taylor instability of a viscous liquid layer resting on a plane wall. J. Fluid Mech. 217, 615–638.Google Scholar
Olson, P. & Singer, H. 1985 Creeping plumes. J. Fluid Mech. 158, 511–531.Google Scholar
Power, H. 1993 Low Reynolds number deformation of compound drops in shear flow. Math. Meth. Appl Sci. 16, 61–74.Google Scholar
Pozrikidis, C. 1990 The instability of a moving viscous drop. J. Fluid Mech. 210, 1–21.Google Scholar
Pozrikidis, C. 1992a The buoyancy-driven motion of a train of viscous drops within a cylindrical tube. J. Fluid Mech. 237, 627–648.Google Scholar
Pozrikidis, C. 1992b 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
Richards, M. A., Duncan, R. A. & Courtillot, V.E. 1989 Flood basalts and hotspot tracks: Plume heads and tails. Science 246, 103–107.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
Stone, H. A. & Leal, L. G. 1990 Breakup of concentric double emulsion droplets in linear flows. J. Fluid Mech. 211, 123–156.Google Scholar
Tanzosh, J., Manga, M. & Stone, H. A. 1992 Boundary element methods for viscous free-surface flow problems: Deformation of single and multiple fluid-fluid interfaces. In Boundary Element Technologies (ed. C.A. Brebbia & M.S. Ingber), pp. 19–39. Computational Mechanics Publications and Elsevier Applied Science, Southampton.
Taylor, G.I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 41–48.Google Scholar
Unverdi, S. O. & Tryggvason, G. 1992 A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100, 25–37.Google Scholar
Yiantsios, S. G. & Davis, R. H. 1990 On the buoyancy-driven motion of a drop towards a rigid or deformable interface. J. Fluid Mech. 217, 547–573.Google Scholar
Yiantsios, S. G. & Davis, R. H. 1991 Close approach and deformation of two viscous drops due to gravity and van der Waals forces. J. Colloid Interface Sci. 144, 412–433.Google Scholar