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Three-dimensional oscillation characteristics of electrostatically deformed drops

Published online by Cambridge University Press:  26 April 2006

James Q. Feng  and
Kenneth V. Beard
James Q. Feng
Affiliation:
Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Kenneth V. Beard
Affiliation:
Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
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Abstract

A three-dimensional asymptotic analysis of the oscillations of electrically charged drops in an external electric field is carried out by means of the multiple-parameter perturbation method. The mathematical framework allows separate treatments of the quiescent deformation due to the electric field and the oscillatory motions caused by other physical factors. Without oscillations, the solution for the quiescent drop shape exhibits a prolate deformation with a slight asymmetry about the drop's equatorial plane. This axisymmetric quiescent deformation of the equilibrium drop shape is shown to modify the oscillation characteristics of axisymmetric as well as asymmetric modes. The expression of the characteristic frequency modification is derived for the oscillation modes, manifesting fine structure in the frequency spectrum so the degeneracy of Rayleigh's normal modes for charged drops is removed in the presence of an electric field. Physical reasoning indicates that the degeneracy of the oscillation modes is associated with the spherical symmetry of the system, so the removal of the degeneracy may be regarded as a consequence of the symmetry breaking caused by the electric field. In addition, the small-amplitude oscillation mode shapes are also modified as a result of the coupling between the oscillatory motions and the electric field as well as the quiescent deformation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 227 , June 1991 , pp. 429 - 447
Copyright
© 1991 Cambridge University Press

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References

Adornato, P. M. & Brown, R. A. 1983 Shape and stability of electrostatically levitated drops.. Proc. R. Soc. Lond. A 389, 101–117.Google Scholar
Beard, K. V., Feng, J. Q. & Chuang, C. C. 1989 A simple perturbation model for the electrostatic shape of falling drops. J. Atmos. Sci. 46, 2404–2418.Google Scholar
Brazier-Smith, P. R., Brook, M., Latham, J., Saunders, C. P. & Smith, M. H. 1971 The vibration of electrified water drops.. Proc. R. Soc. Lond. A 322, 523–534.Google Scholar
Carruthers, J. R. 1974 The application of drops and bubbles to the science of space processing of materials, Proc. Intl. Colloq. Drops Bubbles (ed. D. J. Collins, M. S. Plesset & M. M. Saffren). Pasadena: Jet Propulsion Laboratory.
Davis, E. I. & Ray, A. K. 1980 Single aerosol particle size and mass measurements using an electrodynamic balance. J. Colloid Interface Sci. 75, 566–576.Google Scholar
Feng, J. Q. 1990 A method of multiple-parameter perturbations with an application to drop oscillations in an electric field. Q. Appl. Maths 48, 555–567.Google Scholar
Feng, J. Q. & Beard, K. V. 1990 Small-amplitude oscillations of electrostatically levitated drops.. Proc. R. Soc. Lond. A 430, 133–150.Google Scholar
Joseph, D. D. 1967 Parameter and domain dependence of eigenvalues of elliptic partial differential equations. Arch. Rat. Mech. Anal. 24, 324–351.Google Scholar
Joseph, D. D. 1973 Domain perturbations: The higher order theory of infinitesimal water waves. Arch. Rat. Mech. Anal. 51, 294–303.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Electrodynamics of Continuous Media. Addison-Wesley.
Natarajan, R. & Brown, R. A. 1987a The role of three-dimensional shapes in the break-up of charged drops.. Proc. R. Soc. Lond. A 410, 209–227.Google Scholar
Natarajan, R. & Brown, R. A. 1987b Third-order resonance effects and the nonlinear stability of drop oscillations. J. Fluid Mech. 183, 95–121.Google Scholar
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley-Interscience.
O'Konski, C. T. & Thacher, H. C. 1953 The distortion of aerosol droplets by an electric field. J. Phys. Chem. 57, 955–958.Google Scholar
Pruppacher, H. R. & Klett, J. D. 1978 Microphysics of Clouds and Precipitation. Reidel.
Rayleigh, Lord 1882 On the equilibrium of liquid conducting masses charged with electricity. Phil. Mag. 14, 184–186.Google Scholar
Rhim, W. K., Chung, S. K., Hyson, M. H., Trinh, E. H. & Elleman, D. D. 1987 Large charged drop levitation against gravity. IEEE Trans. Indust. Applics. IA-23, 975–979.Google Scholar
Rogers, R. R. 1984 A review of multiparameter radar observations of precipitation. Radio Sci. 19, 23–26.Google Scholar
Rosenkilde, C. E. 1969 A dielectric fluid drop in an electric field.. Proc. R. Soc. Lond. A 312, 473–494.Google Scholar
Taylor, G. I. 1964 Disintegration of water drops in an electric field.. Proc. R. Soc. Lond. A 280, 383–397.Google Scholar
Trinh, E. & Wang, T. G. 1982 Large-amplitude free and driven drop-shape oscillation: experimental observations. J. Fluid Mech. 122, 315–338.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1984 Resonant oscillations of inviscid charged drops. J. Fluid Mech. 147, 373–395.Google Scholar
Tsamopoulos, J. A., Aklylas, T. R. & Brown, R. A. 1985 Dynamics of charged drop break-up.. Proc. R. Soc. Lond. A 401, 67–88.Google Scholar
Weatherburn, C. E. 1927 Differential Geometry of Three Dimensions. Cambridge University Press.