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Inviscid drops with internal circulation

Published online by Cambridge University Press:  26 April 2006

C. Pozrikidis
C. Pozrikidis
Affiliation:
Department of Applied Mechanics and Engineering Sciences, R-011, University of California at San Diego, La Jolla, CA 92093, USA
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Abstract

The shape of a moving inviscid axisymmetric drop is considered as a function of surface tension and of the intensity of the internal circulation. In a frame of reference moving with the drop, the drop is modelled as a region of diffused vorticity which is bounded by a vortex sheet, and is imbedded in streaming flow. First, an asymptotic analysis is performed for a slightly non-spherical drop whose circulation is very close to that required for the spherical shape. The results indicate that steady drop shapes may exist at all but a number of distinct values of the Weber number, the lowest two of which are 4.41 and 6.15. For highly deformed drops, the problem is formulated as an integral equation for the shape of the drop, and for the strength of the bounding vortex sheet. A numerical procedure is developed for solving this equation, and numerical calculations are performed for Weber numbers between 0 and 4.41. Limiting members in the computed family of solutions contain spherical drops, and inviscid bubbles with vanishing circulation. Computed new shapes include saucer-like shapes with a rounded main body and an elongated tip. The relationship between inviscid drops and drops moving at large Reynolds numbers is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 209 , December 1989 , pp. 77 - 92
Copyright
© 1989 Cambridge University Press

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References

Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190.Google Scholar
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves on deep water-I. Weakly nonlinear waves. Stud. Appl. Maths 60, 183–210.Google Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity-capillary waves ondeep water-II. Numerical results for finite amplitude. Stud. Appl. Maths 62, 95–111.Google Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.
Dandy, D. S. & Leal, L. G. 1986 Boundary layer separation from a smooth slip surface. Phys. Fluids 29, 1360–1366.Google Scholar
Dandy, D. S. & Leal, L. G. 1989 Buoyancy-driven motion of a deformable drop through a quiescent liquid at intermediate Reynolds numbers. J. Fluid Mech. 208, 161–192.Google Scholar
ElSawi, M. 1974 Distorted gas bubbles at large Reynolds number. J. Fluid Mech. 62, 163–183.Google Scholar
Harper, J. F. 1972 Motion of bubbles and drops through liquids. Adv. Appl. Mech. 12, 59–129.Google Scholar
Harper, J. F. 1982 Surface activity and bubble motion. Appl. Sci. Res. 38, 343–352.Google Scholar
Harper, J. F. & Moore, D. W. 1968 The motion of a spherical liquid drop at high Reynolds number. J. Fluid Mech. 32, 367–391.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.
Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1981 Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–100.Google Scholar
Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1982 Rising bubbles. J. Fluid Mech. 123, 31–41.Google Scholar
Moffatt, H. K. & Moore, D. W. 1978 The response of Hill's spherical vortex to a small axisymmetric disturbance. J. Fluid Mech. 87, 749–760.Google Scholar
Moore, D. W. 1959 The rise of a gas bubble in a viscous fluid. J. Fluid Mech. 6, 113–130.Google Scholar
Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.Google Scholar
Moore, D. W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.Google Scholar
Moore, D. W., Saffman, P. G. & Tanveer, S. 1988 The calculation of some Batchelor flows: the Sadovskii vortex and rotational corner flow. Phys. Fluids 31, 978–990.Google Scholar
Oliver, D. L. R. & Chung, J. N. 1987 Flow about a fluid sphere at low to moderate Reynolds numbers. J. Fluid Mech. 177, 1–18.Google Scholar
Pozrikidis, C. 1986 The nonlinear instability of Hill's vortex. J. Fluid Mech. 168, 337–367.Google Scholar
Rivkind, V. Y. & Ryskin, G. M. 1976 Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers. Fluid Dyn. 1, 8–15.Google Scholar
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescen liquid. J. Fluid Mech. 148, 19–35.Google Scholar
Winnikow, S. & Chao, B. T. 1966 Droplet motion in purfied systems. Phys. Fluids 9, 50–61.Google Scholar