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Nonlinear deformation and breakup of stretching liquid bridges

Published online by Cambridge University Press:  26 April 2006

X. Zhang
Affiliation:
Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831–6224, USA
R. S. Padgett
Affiliation:
Department of Chemical Engineering, University of Tennessee, Knoxville, TN 37996, USA
O. A. Basaran
Affiliation:
School of Chemical Engineering, Purdue University, W. Lafayette, IN 47907–1283, USA

Abstract

In this paper, the nonlinear dynamics of an axisymmetric liquid bridge held captive between two coaxial, circular, solid disks that are separated at a constant velocity are considered. As the disks are continuously pulled apart, the bridge deforms and ultimately breaks when its length attains a limiting value, producing two drops that are supported on the two disks. The evolution in time of the bridge shape and the rupture of the interface are investigated theoretically and experimentally to quantitatively probe the influence of physical and geometrical parameters on the dynamics. In the computations, a one-dimensional model that is based on the slender jet approximation is used to simulate the dynamic response of the bridge to the continuous uniaxial stretching. The governing system of nonlinear, time-dependent equations is solved numerically by a method of lines that uses the Galerkin/finite element method for discretization in space and an adaptive, implicit finite difference technique for discretization in time. In order to verify the model and computational results, extensive experiments are performed by using an ultra-high-speed video system to monitor the dynamics of liquid bridges with a time resolution of 1/12 th of a millisecond. The computational and experimental results show that as the importance of the inertial force – most easily changed in experiments by changing the stretching velocity – relative to the surface tension force increases but does not become too large and the importance of the viscous force – most easily changed by changing liquid viscosity – relative to the surface tension force increases, the limiting length that a liquid bridge is able to attain before breaking increases. By contrast, increasing the gravitational force – most readily controlled by varying disk radius or liquid density – relative to the surface tension force reduces the limiting bridge length at breakup. Moreover, the manner in which the bridge volume is partitioned between the pendant and sessile drops that result upon breakup is strongly influenced by the magnitudes of viscous, inertial, and gravitational forces relative to surface tension ones. Attention is also paid here to the dynamics of the liquid thread that connects the two portions of the bridge liquid that are pendant from the top moving rod and sessile on the lower stationary rod because the manner in which the thread evolves in time and breaks has important implications for the closely related problem of drop formation from a capillary. Reassuringly, the computations and the experimental measurements are shown to agree well with one another.

Type
Research Article
Copyright
© 1996 Cambridge University Press

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References

Anilkumar, A. V., Grugel, R. N., Shen, X. F., Lee, C. P. & Wang, T. G. 1993 Control of thermocapillary convection in a liquid bridge by vibration. J. Appl. Phys. 73, 41654170.Google Scholar
Basaran, O. A. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169198.Google Scholar
Basaran, O. A. & DePaoli, D. W. 1994 Nonlinear oscillations of pendant liquid drops. Phys. Fluids 6, 29232943.Google Scholar
Borkar, A. & Tsamopoulos, J. 1991 Boundary-layer analysis of the dynamics of axisymmetric capillary bridges. Phys. Fluids A 3, 28662874.Google Scholar
Brown, R. A. 1988 Theory of transport processes in single crystal growth from the melt. AIChE J. 34, 881911.Google Scholar
Brown, R. A. & Scriven, L. E. 1980 The shapes and stability of captive rotating drops. Phil. Trans. R. Soc. Lond. A 297, 5179.Google Scholar
Chen, T.-Y. & Tsamopoulos, J. 1993 Nonlinear dynamics of capillary bridges: theory. J. Fluid Mech. 255, 373409.Google Scholar
Chen, T.-Y., Tsamopoulos, J. A. & Good, R. J. 1992 Capillary bridges between parallel and non-parallel surfaces and their stability. J. Colloid Interface Sci. 151, 4969.Google Scholar
Coriell, S. R., Hardy, S. C. & Cordes, M. R. 1977 Stability of liquid zones. J. Colloid Interface Sci. 60, 126136.Google Scholar
Denn, M. M. 1980 Drawing of liquids to form fibers. Ann. Rev. Fluid Mech. 12, 365387.Google Scholar
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier—Stokes equation. J. Fluid Mech. 262, 205221.Google Scholar
Ennis, B. J., Li, J., Tardos, G. I. & Pfeffer, R. 1990 The influence of viscosity on the strength of an axially strained pendular liquid bridge. Chem. Engng Sci. 45, 30713088.Google Scholar
Fowle, A. A., Wang, C. A. & Strong, P. F. 1979 Experiments on the stability of conical and cylindrical liquid columns at low Bond numbers. In Proc. 3rd European Symp. Mat. Sci. Space, pp. 317325.
Gaudet, S., McKinley, G. H. & Stone, H. A. 1994 Extensional deformation of Newtonian and non-Newtonian liquid bridges in microgravity. AIAA 940696, pp. 19.Google Scholar
Gillette, R. D. & Dyson, D. C. 1971 Stability of fluid interfaces of revolution between equal solid circular plates. Chem. Engng J. 2, 4454.Google Scholar
Gonzalez, H., McCluskey, F. M. J., Castellanos, A. & Barrero, A. 1989 Stabilization of dielectric liquid bridges by electric fields in the absence of gravity. J. Fluid Mech. 206, 545561.Google Scholar
Green, A. E. 1976 On the non-linear behaviour of fluid jets. Intl J. Engng Sci. 14, 4963.Google Scholar
Gresho, P. M., Lee, R. L. & Sani, R. C. 1979 On the time-dependent solution of the incompressible Navier—Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in Fluids (ed. C. Taylor & K. Morgan), vol. 1, pp. 2779. Pineridge, Swansea.
Harris, M. T. & Byers, C. H. 1989 An advanced technique for interfacial tension measurement in liquid-liquid system. Oak Ridge National Laboratory/TM-10734.
Haynes, J. M. 1970 Stability of a fluid cylinder. J. Colloid Interface Sci. 32, 652654.Google Scholar
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, 141156.Google Scholar
Kheshgi, H. S. 1989 Profile equations for film flows at moderate Reynolds numbers. AIChE J. 35, 17191727.Google Scholar
Kheshgi, H. S. & Scriven, L. E. 1983 Penalty finite element analysis of unsteady free surface flows. In Finite Elements in Fluids (ed. R. H. Gallagher, J. T. Oden, O. C. Zienkiewicz, T. Kawai & M. Kawahara), vol. 5, pp. 393434. Wiley.
Kistler, S. F. & Scriven, L. E. 1983 Coating flows. In Computational Analysis of Polymer Processing (ed. J. R. A. Pearson & S. M. Richardson), pp. 243299. Applied Science Publishers.
Kroger, R., Berg, S., Delgado, A. & Rath, H. J. 1992 Stretching behavior of large polymeric and Newtonian liquid bridges in plateau simulation. J. Non-Newtonian Fluid Mech. 45, 385400.Google Scholar
Kroger, R. & Rath, H. J. 1995 Velocity and elongation rate distributions in stretched polymeric and Newtonian liquid bridges. J. Non-Newtonian Fluid Mech. 57, 137153.Google Scholar
Lapidus, L. & Pinder, G. F. 1982 Numerical Solution of Partial Differential Equations in Science and Engineering. John Wiley.
Lee, H. C. 1974 Drop formation in a liquid jet. IBM J. Res. Dev. 18, 364369.Google Scholar
Luskin, M. & Rannacher, R. 1982 On the smoothing property of the Crank-Nicholson scheme. Applicable Anal. 14, 117135.Google Scholar
Mason, G. 1970 An experimental determination of the stable length of cylindrical liquid bubbles. J. Colloid Interface Sci. 32, 172176.Google Scholar
Meseguer, J. 1983 The breaking of axisymmetric slender liquid bridges. J. Fluid Mech. 130, 123151.Google Scholar
Meseguer, J. & Perales, J. M. 1991 A linear analysis of g-jitter effects on viscous cylindrical liquid bridges. Phys. Fluids A 3, 23322336.Google Scholar
Meseguer, J. & Sanz, A. 1985 Numerical and experimental study of the dynamics of axisymmetric slender liquid bridges. J. Fluid Mech. 153, 83101.Google Scholar
Michael, D. H. 1981 Meniscus stability. Ann. Rev. Fluid Mech. 13, 189215.Google Scholar
Miller, C. A. & Scriven, L. E. 1968 The oscillations of a droplet immersed in another fluid. J. Fluid Mech. 32, 417435.Google Scholar
Mollot, D. J., Tsamopoulos, J., Chen, T.-Y. & Ashgriz, A. 1993 Nonlinear dynamics of capillary bridges: experiments. J. Fluid Mech. 255, 411435.Google Scholar
Papageorgiou, D. T. 1995a On the breakup of viscous liquid threads. Phys. Fluids 7, 15291544.Google Scholar
Papageorgiou, D. T. 1995b Analytical description of the breakup of liquid jets. J. Fluid Mech. 301, 109132.Google Scholar
Patzek, T. W. & Scriven, L. E. 1982 Capillary forces exerted by liquid drops caught between crossed cylinders: A 3-D meniscus problem with free contact line. In Proc. 2nd Intl Colloq Drops Bubbles (ed. D. H. Le Croissette), pp. 308314. JPL Publication 82–7, Jet Propulsion Laboratory, Pasadena, California.
Perales, J. M. & Meseguer, J. 1992 Theoretical and experimental study of the vibration of axisymmetric viscous liquid bridges. Phys. Fluids A 4, 11101130.Google Scholar
Peregrine, D. H., Shoker, G. & Symon, A. 1990 The bifurcation of liquid bridges. J. Fluid Mech. 212, 2539.Google Scholar
Plateau, J. 1863 Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity. In Annual Report of the Board of Regents of the Smithsonian Institution, pp. 270283. Washington, DC.
Rayleigh, Lord 1879 On the instability of jets. Proc. Lond. Math. Soc. 10, 413.Google Scholar
Ruschak, K. J. 1978 Flow of a falling film into a pool. AIChE J. 24, 705709.Google Scholar
Russo, M. J. & Steen, P. H. 1986 Instability of rotund capillary bridges to general disturbances: Experiment and theory. J. Colloid Interface Sci. 113, 154163.Google Scholar
Sankaran, S. & Saville, D. A. 1993 Experiments on the stability of a liquid bridge in an axial electric field. Phys. Fluids A 5, 10811083.Google Scholar
Sanz, A. 1985 The influence of the outer bath in the dynamics of axisymmetric liquid bridges. J. Fluid Mech. 156, 101140.Google Scholar
Sanz, A. & Diez, J. L. 1989 Non-axisymmetric oscillations of liquid bridges. J. Fluid Mech. 205, 503521.Google Scholar
Schulkes, R. M. S. M. 1993a Nonlinear dynamics of liquid columns: a comparative study. Phys. Fluids A 5, 21212130.Google Scholar
Schulkes, R. M. S. M. 1993b Dynamics of liquid jets revisited. J. Fluid Mech. 250, 635650.Google Scholar
Shipman, R. W. G., Denn, M. M. & Keunings, R. 1991 Mechanics of the “falling plate” extensional rheometer. J. Non-Newtonian Fluid Mech. 40, 281288.Google Scholar
Sridhar, T., Tirtaatmadja, V., Nguyen, D. A. & Gupta, R. K. 1991 Measurement of extensional viscosity of polymer solutions. J. Non-Newtonian Fluid Mech. 40, 271280.Google 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
Strang, G. & Fix, G. J. 1973 An Analysis of the Finite Element Method. Prentice-Hall.
Strani, M. & Sabetta, F. 1988 Viscous oscillations of a supported drop in an immiscible fluid. J. Fluid Mech. 189, 397421.Google Scholar
Timmermans, J. 1960 The Physico-Chemical Constants of Binary Systems in Concentrated Solutions, vol. 4. Interscience.
Tirtaatmadja, V. & Sridhar, T. 1993 A filament stretching device for measurement of extensional viscosity. J. Rheol. 37, 10811102.Google Scholar
Tsamopoulos, J., Chen, T.-Y. & Borkar, A. 1992 Viscous oscillations of capillary bridges. J. Fluid Mech. 235, 579609.Google Scholar
Ungar, L. H. & Brown, R. A. 1982 The dependence of the shape and stability of captive rotating drops on multiple parameters. Phil. Trans. R. Soc. Lond. A 306, 347370.Google Scholar
Zhang, Y. & Alexander, J. I. D. 1990 Sensitivity of liquid bridges subject to axial residual acceleration. Phys. Fluids A 2, 19661974.Google Scholar
Zhang, X. & Basaran, O. A. 1995 An experimental study of dynamics of drop formation. Phys. Fluids 7, 11841203.Google Scholar
Zhang, X. & Basaran, O. A. 1996 Dynamics of drop formation from a capillary in the presence of an electric field. J. Fluid Mech. 326, 239263.Google Scholar