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Nonlinear oscillations of viscous liquid drops

Published online by Cambridge University Press:  26 April 2006

Osman A. Basaran
Osman A. Basaran
Affiliation:
Chemical Technology Division. Oak Ridge National Laboratory, Oak Ridge. TN 37831. USA
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Abstract

A fundamental understanding of nonlinear oscillations of a viscous liquid drop is needed in diverse areas of science and technology. In this paper, the moderate- to large-amplitude axisymmetric oscillations of a viscous liquid drop, which is immersed in dynamically inactive surroundings, are analysed by solving the free boundary problem comprised of the Navier–Stokes system and appropriate interfacial conditions at the drop–ambient fluid interface. The means are the Galerkin/finite-element technique, an implicit predictor-corrector method, and Newton's method for solving the resulting system of nonlinear algebraic equations. Attention is focused here on oscillations of drops that are released from an initial static deformation. Two dimensionless groups govern such nonlinear oscillations: a Reynolds number, Re, and some measure of the initial drop deformation. Accuracy is attested by demonstrating that (i) the drop volume remains virtually constant, (ii) dynamic response to small-and moderate-amplitude disturbances agrees with linear and perturbation theories, and (iii) large-amplitude oscillations compare well with the few published predictions made with the marker-and-cell method and experiments. The new results show that viscous drops that are released from an initially two-lobed configuration spend less time in prolate form than inviscid drops, in agreement with experiments. Moreover, the frequency of oscillation of viscous drops released from such initially two-lobed configurations decreases with the square of the initial amplitude of deformation as Re gets large for moderate-amplitude oscillations, but the change becomes less dramatic as Re falls and/or the initial amplitude of deformation rises. The rate at which these oscillations are damped during the first period rises as initial drop deformation increases; thereafter the damping rate is lower but remains virtually time-independent regardless of Re or the initial amplitude of deformation. The new results also show that finite viscosity has a much bigger effect on mode coupling phenomena and, in particular, on resonant mode interactions than might be anticipated based on results of computations incorporating only an infinitesimal amount of viscosity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 169 - 198
Copyright
© 1992 Cambridge University Press

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References

Alonso C. T. 1974 The dynamics of colliding and oscillating drops. In Proc. Intl Colloq. on Drops and Bubbles (ed. D. J. Collins, M. S. Plesset & M. M. Saffren), vol. I. pp. 139–157. Jet Propulsion Laboratory, Pasadena, CA.
Alonso C. T., LeBlanc, J. M. & Wilson J. R. 1982 Studies of the nuclear equation of state using numerical calculations of nuclear drop collisions. In Proc. Second Intl Colloq. on Drops and Bubbles (ed. D. H. LeCroissette), pp. 268–279. Jet Propulsion Laboratory, Pasadena, CA.
Aris R. 1962 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall.
Basaran, O. A. & Patzek T. W. 1991 Three-dimensional oscillations of liquid drops. Paper presented at the Winter Annual Meeting of the AIChE, 17–22 Nov. 1991, Los Angeles.
Basaran O. A., Scott, T. C. & Byers C. H. 1989 Drop oscillations in liquid–liquid systems. AIChE J. 35, 1263–1270.Google Scholar
Beard K. V., Ochs, H. T. & Kubesh R. J. 1989 Natural oscillations of small raindrops. Nature 342, 408–410.Google Scholar
Carruthers, J. R. & Testardi L. R. 1983 Materials processing in the reduced-gravity of space. Ann. Rev. Mater. Sci. 13, 247–278.Google Scholar
Coyle D. J. 1984 The fluid mechanics of roll coating: steady flows, stability, and rheology. Ph.D. thesis, University of Minnesota. Available from University Microfilms, Inc., AnnArbor, MI.
Foote G. B. 1973 A numerical method for studying simple drop behavior: simple oscillation. J. Comput. Phys. 11, 507–530.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. 27–79. Swansea: Pineridge.
Harlow F. H. 1963 The particle-in-cell method for numerical solution of problems in fluid dynamics. Proc. Symp. Appl. Maths 15, 269–288.Google Scholar
Harlow F. H., Amsden, A. A. & Nix, J. R. 1976 Relativistic fluid dynamics calculations with the particle-in-cell technique. J. Comput. Phys. 20, 119–129.Google Scholar
Hood P. 1976 Frontal solution program for unsymmetric matrices. Intl J. Numer. Meth. Engng 10, 379–399 (and Correction, Intl J. Numer. Meth. Engr. 11, (1977). 1055).Google Scholar
Huyakorn P. S., Taylor C. Lee, R, L. & Gresho, P. M. 1978 A comparison of various mixed interpolation finite elements in the velocity-pressure formulation of the Navier–Stokes equations. Computers Fluids 6, 25–35Google 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. 393–434. 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. 243–299. Applied Science Publishers.
Lamb H. 1932 Hydrodynamics. Dover.
Lundgren, T. S. & Mansour N. N. 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510 (referred to herein as L & M).Google Scholar
Luskin, M. & Rannacher R. 1982 On the smoothing property of the Crank–Nicholson scheme. Applicable Anal. 14, 117–135.Google Scholar
Miller, C. A. & Scriven L. E. 1968 The oscillations of a droplet immersed in another fluid. J. Fluid Mech. 32, 417–435.Google Scholar
Natarajan, R. & Brown R. A. 1986 Quadratic resonance in the three-dimensional oscillations of inviscid drops with surface tension. Phys. Fluids 29, 2788–2797.Google Scholar
Natarajan, R. & Brown R. A. 1987 Third-order resonance effects and the nonlinear stability of drop oscillations. J. Fluid Mech. 183, 95–121.Google Scholar
Nix, J. R. & Strottmann D. 1982 Fluid dynamical description of relativistic nuclear collisions. In Proc. Second Intl Colloq. on Drops and Bubbles (ed. D. H. LeCroissette), pp. 260–267. Jet Propulsion Laboratory, Pasadena, CA.
Orszag S. A. 1980 Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70–92.Google Scholar
Patzek T. W., Benner R. E., Basaran, O. A. & Scriven L. E. 1991 Nonlinear oscillations of inviscid free drops. J. Comput. Phys. 97, 489–515.Google Scholar
Prosperetti A. 1980 Normal-mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid. J. MeAc. 19, 149–182.Google Scholar
Rayleigh Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Reid W. H. 1960 The oscillations of a viscous liquid drop. Q. Appl. Maths 18, 86–89.Google Scholar
Saito, H. & Scriven L. E. 1981 Study of coating flow by the finite element method. J. Comput. Phys. 42, 53–76.Google Scholar
Scott T. C., Basaran, O. A. & Byers C. H. 1990 Characteristics of electric-field-induced oscillations of translating liquid droplets. Indust. Engng Chem. Res. 29, 901–909.Google Scholar
Silliman W. J. 1979 Viscous film flows with contact lines. Ph.D. thesis, University of Minnesota. Available from University Microfilms, Inc., AnnArbor, MI.
Stone, H. A. & Leal L. G. 1990 The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161–186.Google Scholar
Strang, G. & Fix G. J. 1973 An Analysis of the Finite Element Method. Prentice-Hall.
Trinh, E. & Wang T. G. 1982 Large-amplitude free and driven drop-shape oscillations: experimental observations J. Fluid Mech. 122, 315–338.Google Scholar
Tsamopoulos J. A. 1989 Nonlinear dynamics and break-up of charged drops. In AIP Conf. Proc. 197: Third Intl Colloq. on Drops and Bubbles (ed. T. G. Wang), pp. 169–187. American Institute of Physics.
Tsamopoulos, J. A. & Brown R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537 (referred to herein as T & B).Google Scholar
Walters R. A. 1980 The frontal method in hydrodynamics simulations. Computers Fluids 8, 265–272.Google Scholar
Weatherburn C. E. 1927 Differential Geometry of Three Dimensions. Cambridge University Press.