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Drop formation in a one-dimensional approximation of the Navier–Stokes equation

Published online by Cambridge University Press:  26 April 2006

Jens Eggers  and
Todd F. Dupont
Jens Eggers
Affiliation:
Department of Mathematics and the James Franck Institute, the University of Chicago, 5734 University Avenue, Chicago, IL 60637, USA Present address: Universität GH Essen, FB7, 45117 Essen, Germany.
Todd F. Dupont
Affiliation:
Department of Computer Science, the University of Chicago, 1100 East 58th Street, Chicago, IL 60637, USA
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Abstract

We consider the viscous motion of a thin axisymmetric column of fluid with a free surface. A one-dimensional equation of motion for the velocity and the radius is derived from the Navier–Strokes equation. We compare our results with recent experiments on the breakup of a liquid jet and on the bifurcation of a drop suspended from an orifice. The equations form singularities as the fluid neck is pinching off. The nature of the singularities is investigated in detail.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 262 , 3 October 1994 , pp. 205 - 221
Copyright
© 1994 Cambridge University Press

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References

Becker, E., Hiller, W. J. & Kowalewski, T. A. 1991 Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets. J. Fluid Mech. 231, 189–210.Google Scholar
Bogy, D. B. 1979 Drop formation in a circular liquid jet. A. Rev. Fluid Mech. 11, 207–228.Google Scholar
Carmo, M. P. do 1976 Differential Geometry of Curves and Surfaces. Prentice-Hall.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, Chap. 12. Clarendon.
Chaudhary, K. C. & Maxworthy, T. 1980 The nonlinear capillary instability of a liquid jet. Part 2. Experiments on jet behaviour before droplet formation. J. Fluid Mech. 96, 275–286.Google Scholar
Chaudhary, K. C. & Redekopp, L. G. 1980 The nonlinear instability of a liquid jet. Part 1. Theory. J. Fluid Mech. 96, 257–274.Google Scholar
Constantin, P., Dupont, T. F., Goldstein, R. E., Kadanoff, L. P., Shelley, M. & Zhou, S. M. 1993 Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E 47, 4169–4181.Google Scholar
Cram, L. E. 1984 A numerical model of droplet formation. In Computational Techniques and Applications: CTAC-83 (ed. J. Noye & C. Fletcher). Elsevier.
Johnson, M., Kamm, R. D., Ho, L. W., Shapiro, A. & Pedley, T. J. 1991 The nonlinear growth of surface-tension-driven instabilities of a thin annular film. J. Fluid Mech. 233, 141–156.Google Scholar
Keller, J. B. & Miksis, M. J. 1983 Surface tension driven flows. SIAM J. Appl. Maths 43, 268–277.Google Scholar
Keller, J. B., Rubinow, S. I. & Tu, Y. O. 1973 Spatial instability of a jet. Phys. Fluids 16, 2052–2055.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1984 Fluid Mechanics. Pergamon.
Lee, H. C. 1974 Drop formation in a liquid jet. IBM J. Res. Dev. 18, 364–369.Google Scholar
Leib, S. J. & Goldstein, M. E. 1986 The generation of capillary instabilities on a liquid jet. J. Fluid Mech. 168, 479–500.Google Scholar
Majda, A. 1986 Vorticity and the mathematical theory of incompressible fluid flow. Commun. Pure Appl. Maths. 39, S187–S220.Google Scholar
Meseguer, J. 1983 The breaking of axisymmetric slender liquid bridges. J. Fluid Mech. 130, 123–151.Google Scholar
Michael, D. H. & Williams, P. G. 1976 The equilibrium and stability of axisymmetric pendant drops. Proc. R. Soc. Land. A 351, 117–127.Google Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximation behind them. In Waves on Beaches (ed. R. E. Meyer). Academic.
Peregrine, D. H., Shoker, G. & Symon, A. 1990 The bifurcation of liquid bridges. J. Fluid Mech. 212, 25–39.Google Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 4, 10.Google Scholar
Shi, X. D. & Nagel, S. R. 1992 Droplet formation (unpublished).
Smoller, J. 1983 Shock Waves and Reaction-Diffusion Equations, Chap. 15. Springer.
Weast, R. C. (ed.) 1978 Handbook of Chemistry and Physics. CRC Press.