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Generation and breakup of Worthington jets after cavity collapse. Part 2. Tip breakup of stretched jets

Published online by Cambridge University Press:  15 October 2010

Área de Mecánica de Fluidos, Departamento de Ingenería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos, 41092 Sevilla, Spain
Department of Applied Physics and J.M. Burgers Centre for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands Physik Department, Technische Universität München, 85748 Garching, Germany
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The capillary breakup of the high-speed Worthington jets ejected after a cavity collapse in water occurs due to the high-Reynolds-number version of the capillary end-pinching mechanism first described, in the creeping flow limit, by Stone & Leal (J. Fluid Mech., vol. 198, 1989, p. 399). Using potential flow numerical simulations and theory, we find that the resulting drop ejection process does not depend on external noise and can be described as a function of a single dimensionless parameter, WeS = ρ R30S20/σ, which expresses the ratio of the capillary time to the inverse of the local strain rate, S0. Here, ρ and σ indicate the liquid density and the interfacial tension coefficient, respectively, and R0 is the initial radius of the jet. Our physical arguments predict the dimensionless size of the drops to scale as Ddrop/R0 ~ We−1/7S and the dimensionless time to break up as TS0 ~ We2/7S. These theoretical predictions are in good agreement with the numerical results.

Copyright © Cambridge University Press 2010

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Antkowiak, A., Bremond, N., Dizès, S. L. & Villermaux, E. 2007 a Inertial jets. Bull. Am. Phys. Soc. 52, 104.Google Scholar
Antkowiak, A., Bremond, N., Dizès, S. L. & Villermaux, E. 2007 b Short-term dynamics of a density interface following an impact. J. Fluid Mech. 577, 241250.CrossRefGoogle Scholar
Antkowiak, A., Bremond, N., Duplat, J., Dizès, S. L. & Villermaux, E. 2007 c Cavity jets. Phys. Fluids 19, 091112.CrossRefGoogle Scholar
Bolanos-Jiménez, R., Sevilla, A., Martínez-Bazán, C. & Gordillo, J. M. 2008 Axisymmetric bubble collapse in a quiescent liquid pool. Part II. Experimental study. Phys. Fluids 20, 112104.CrossRefGoogle Scholar
Boulton-Stone, J. M. & Blake, J. R. 1993 Gas bubbles bursting at a free surface. J. Fluid Mech. 254, 437466.CrossRefGoogle Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
Frankel, I. & Weihs, D. 1985 Stability of a capillary jet with linearly increasing axial velocity (with application to shaped charges). J. Fluid Mech. 155, 289307.CrossRefGoogle Scholar
Gekle, S. & Gordillo, J. M. 2010 Generation and breakup of Worthington jets after cavity collapse. Part 1. Jet formation. J. Fluid Mech. doi:10.1017/S0022112010003526.Google Scholar
Gekle, S., Gordillo, J. M., van der Meer, D. & Lohse, D. 2009 High-speed jet formation after solid object impact. Phys. Rev. Lett. 102, 034502.Google Scholar
Gordillo, J. M. & Pérez-Saborid, M. 2005 Aerodynamic effects in the break-up of liquid jets: on the first wind-induced breakup regime. J. Fluid Mech. 541, 120.CrossRefGoogle Scholar
Gordillo, J. M., Sevilla, A. & Martínez-Bazán, C. 2007 Bubbling in a co-flow at high Reynolds numbers. Phys. Fluids 19, 077102.CrossRefGoogle Scholar
Keller, J. B., Rubinow, S. I. & Tu, Y. O. 1973 Spatial instability of a jet. Phys. Fluids 16, 20522055.CrossRefGoogle Scholar
Mikami, T., Cox, R. G. & Manson, S. G. 1975 Breakup of extending liquid threads. Intl J. Multiph. Flow 2, 113138.CrossRefGoogle Scholar
Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223256.CrossRefGoogle Scholar
Plateau, J. 1873 Statique Expérimentale et Théorique des Liquides. Gauthier-Villars.Google Scholar
Rayleigh, W. S. 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 413.CrossRefGoogle Scholar
Rein, M. 1996 The transitional regime between coalescing and splashing drops. J. Fluid Mech. 306, 145165.CrossRefGoogle Scholar
Savart, F. 1833 Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi. Ann. Chim. 53, 337386.Google Scholar
Schulkes, R. M. S. M. 1996 The contraction of liquid filaments. J. Fluid Mech. 309, 277300.Google Scholar
Shield, T. W., Bogy, D. B. & Talke, F. 1986 A numerical comparison of one-dimensional fluid jet models applied to drop-on-demand printing. J. Comput. Phys. 67, 327347.CrossRefGoogle Scholar
Shield, T. W., Bogy, D. B. & Talke, F. 1987 Drop formation by DOD ink-jet nozzles: a comparison of experiment and numerical simulation. IBM J. Res. Dev. 31, 96110.CrossRefGoogle Scholar
Sterling, A. & Sleicher, C. 1975 The instability of capillary jets. J. Fluid Mech. 68, 477495.CrossRefGoogle Scholar
Stone, H. A., Bentley, B. J. & Leal, L. G. 1986 An experimental study of transient effects in the breakup of viscous drops. J. Fluid Mech. 173, 131158.Google Scholar
Stone, H. & Leal, L. 1989 Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399427.CrossRefGoogle Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. Part III. Disintegration of fluid sheets. Proc. R. Soc. Lond. A 253, 313321.Google Scholar
Thoroddsen, S. T., Takehara, K., Etoh, T. G. & Ohl, C. D. 2009 Spray and microjets produced by focusing a laser pulse into a hemispherical drop. Phys. Fluids 21, 112101.Google Scholar
Tomotika, S. 1936 Breaking up of a drop of viscous liquid immersed in another viscous fluid which is extending at a uniform rate. Proc. R. Soc. 153, 302318.Google Scholar
Yarin, A. L. 2006 Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38, 159192.CrossRefGoogle Scholar