Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-05T08:10:30.368Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dynamics of viscous droplets

Published online by Cambridge University Press:  26 April 2006

E. Becker ,
W. J. Hiller  and
T. A. Kowalewski
E. Becker
Affiliation:
Max-Planck-Institut für Strömungsforschung, Bunsenstrasse 10, D-37073 Göttingen, Germany Present address: Institut für Atmosphärenphysik an der Universität Rostock e.V., Schloßstr. 4–6, D-18221 Kühlungsborn, Germany.
W. J. Hiller
Affiliation:
Max-Planck-Institut für Strömungsforschung, Bunsenstrasse 10, D-37073 Göttingen, Germany
T. A. Kowalewski
Affiliation:
Max-Planck-Institut für Strömungsforschung, Bunsenstrasse 10, D-37073 Göttingen, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear viscous droplet oscillations are analysed by solving the Navier-Stokes equation for an incompressible fluid. The method is based on mode expansions with modified solutions of the corresponding linear problem. A system of ordinary differential equations, including all nonlinear and viscous terms, is obtained by an extended application of the variational principle of Gauss to the underlying hydrodynamic equations. Results presented are in a very good agreement with experimental data up to oscillation amplitudes of 80% of the unperturbed droplet radius. Large-amplitude oscillations are also in a good agreement with the predictions of Lundgren & Mansour (boundary integral method) and Basaran (Galerkin-finite element method). The results show that viscosity has a large effect on mode coupling phenomena and that, in contradiction to the linear approach, the resonant mode interactions remain for asymptotically diminishing amplitudes of the fundamental mode.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 258 , 10 January 1994 , pp. 191 - 216
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1968 Handbook of Mathematical Functions. Dover.
Amos, D. E. 1986 A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Software 12, 265–273.Google Scholar
Basaran, O. A. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169–198.Google Scholar
Becker, E. 1991 Nichtlineare Tropfenschwingungen unter Berücksichtigung von Oberflächenspannung und Viskosität (Nonlinear oscillations of viscous droplets driven by surface tension). Mitteilungen aus dem Max-Planck-Institut für Strömungsforschung, vol. 104. E.-A. Müller MPI Göttingen.
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
Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. 43a, 697–727.Google Scholar
Brosa, U. 1986 Linear analysis of the currents in a pipe. Z. Naturforsch. 41a, 1141–1153.Google Scholar
Brosa, U. 1988 Strongly dissipative modes. Unpublished paper, Universität Marburg.
Brosa, U. & Becker, E. 1988 Das zufällige Halsreißen (Random Neck Rupture). Movie C 1694, available (in German or English) on loan from the IWF, Nonnenstieg 72, D-37075 Göttingen, Germany.
Brosa, U., Grossmann, S., Müller, A. & Becker, E. 1989 Nuclear scission. Nuclear Phys. A 502, 423c–442cGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Defay, R. & Péatréa, G. 1971 Dynamic surface tension. In Surface and Colloid Science (ed. E. Matijevi), pp. 27–81. Wiley-Interscience.
Fehlberg, E. 1970 Klassiche Runge-Kutta-Formeln vierter und niedriger Ordnung mit Schrittweitenkontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6, 61–71.Google Scholar
Haken, H. 1990 Synergetik, 3rd edn. Springer.
Hiller, W. J. & Kowalewski, T. A. 1989 Surface tension measurements by the oscillating droplet method. Phys. Chem. Hydrodyn. 11, 103–112.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lundgren, T. S. & Mansour, N. N. 1988 Oscillation of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510.Google Scholar
Moon, P. & Spencer, D. E. 1961 Field Theory Handbook. Springer.
Natarajan, R. & Brown, R. A. 1987 Third-order resonance effect and the nonlinear stability of drop oscillation. J. Fluid Mech. 183, 95–121.Google Scholar
Patzek, T. W., Brenner, 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. 1977 Viscous effects on perturbed spherical flows. Q. Appl. Maths 35, 339–352.Google Scholar
Prosperetti, A. 1980a Normal-mode analysis for the oscillations of a viscous liquid drop immersed in another liquid. J. Méc. 19, 149–182.Google Scholar
Prosperetti, A. 1980b Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100, 333–347.Google Scholar
Rayleigh, Lord 1899 On the capillary phenomena of jets. Proc. R. Soc. Lond. A 29, 71–97.Google Scholar
Stoer, J. 1972 Einführung in die Numerische Mathematik I. Springer.
Stückrad, B., Hiller, W. J. & Kowalewski, T. A. 1993 Measurement of dynamic surface tension by the oscillating droplet method. Exp. Fluids. (to appear)Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537.Google Scholar
Wilkening, V. 1992 Betrachtungen im Hyperraum-‘Relation’ verarbeitet Multi-parameter-Daten. c't 1, 70–74.Google Scholar