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Analytical description of the breakup of liquid jets

Published online by Cambridge University Press:  26 April 2006

Demetrios T. Papageorgiou
Demetrios T. Papageorgiou
Affiliation:
Department of Mathematics and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
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Abstract

A viscous or inviscid cylindrical jet with surface tension in a surrounding medium of negligible density tends to pinch owing to the mechanism of capillary instability. We construct similarity solutions which describe this phenomenon as a critical time is encountered, for three distinct cases: (i) inviscid jets governed by the Euler equations, (ii) highly viscous jets governed by the Stokes equations, and (iii) viscous jets governed by the Navier-Stokes equations. We look for singular solutions of the governing equations directly rather than by analysis of simplified models arising from slender-jet theories. For Stokes jets implicitly defined closed-form solutions are constructed which allow the scaling exponents to be fixed. Navier-Stokes pinching solutions follow rationally from the Stokes ones by bringing unsteady and nonlinear terms into the momentum equations to leading order. This balance fixes a set of universal scaling functions for the phenomenon. Finally we show how the pinching solutions can be used to provide an analytical description of the dynamics beyond breakup.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 301 , 25 October 1995 , pp. 109 - 132
Copyright
© 1995 Cambridge University Press

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References

Bogy, D. B. 1979 Drop formation in a circular liquid jet. Ann. Rev. Fluid Mech. 11, 207–228.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.
Chaudhary, K. C. & Maxworthy, T. 1980a 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. & Maxworthy, T. 1980b The nonlinear capillary instability of a liquid jet. Part 3. Experiments on satellite drop formation and control. J. Fluid Mech. 96, 287–297.Google Scholar
Chaudhary, K. C. & Redekopp, L. G. 1980 The nonlinear capillary instability of a liquid jet. Part 1. Theory. J. Fluid Mech. 96, 257–274.Google Scholar
Donnelly, R. J. & Glaberson, W. 1966 Experiments on the capillary instability of a liquid jet. Proc. R. Soc. Lond. A 290, 547–556.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier-Stokes equation. J. Fluid Mech. 262, 205–221.Google Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458–3460.Google Scholar
Eggers, J. 1995 Theory of drop formation. Phys. Fluids 7, 941–953.Google Scholar
Goedde, E. F. & Yuen, M. C. 1970 Experiments on liquid jet instability. J. Fluid Mech. 40, 495–511.Google Scholar
Keller, J. B. 1983 Breaking of liquid films and threads. Phys. Fluids 26, 3451–3453.Google Scholar
Keller, J. B., King, A. & Ting, L. 1995 Blob formation. Phys. Fluids 7, 226–228.Google Scholar
Keller, J. B. & Miksis, M. J. 1983 Surface tension driven flows. SIAM J. Appl. Maths 43 (2), 268–277.Google Scholar
Mansour, N. & Lundgren, T. S. 1990 Satellite formation in capillary jet breakup. Phys. Fluids A 2, 1141–1144.Google Scholar
Papageorgiou, D. T. 1994 Breakup of cylindrical jets governed by the Navier-Stokes equations. In Proceedings of the Workshop on Transition, Turbulence and Combustion, June 7-July 2, 1993, ICASE/LaRC Interdisciplinary Series in Science and Engineering (ed.) M. Y. Hussaini, T. B. Gatski & T. L. Jackson, vol. 2, pp. 225–235. Kluwer.
Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7, 1529–1544.Google Scholar
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 stability of jets. Proc. Lond. Math. Soc. 10, 4–13.Google Scholar
Renardy, M. 1994 Some comments on the surface-tension driven break-up (or the lack of it) of viscoelastic jets. J. Non-Newtonian Fluid Mech. 51, 97–107.Google Scholar
Stuart, J. T. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The besic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.Google Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid, I. Water bells. Proc. R. Soc. Lond. A 253, 289–295.Google Scholar
Ting, L. & Keller, J. B. 1990 Slender jets and thin sheets with surface tension. SIAM J. Appl. Maths 50 (6), 1533–1546.Google Scholar
Tjahadi, M., Stone, H. A. & Ottino, J. M. 1992 Satellite and subsatellite formation in capillary breakup. J. Fluid Mech. 243, 297–317.Google Scholar
Tomotika, S. 1935 On the stability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. A 150, 322–337.Google Scholar
Watson, J. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and plane Couette flow. J. Fluid Mech. 9, 371–389.Google Scholar
Yuen, M.-C. 1968 Nonlinear capillary instability of a liquid jet. J. Fluid Mech. 33, 151–163.Google Scholar