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Resonances of a conducting drop in an alternating electric field

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

Resonance phenomena of a conducting drop forced by an alternating electric field are studied by perturbation analysis. Although the motions are assumed to be irrotational, weak viscous effects are included in the boundary condition of the normal stress balance. Without an external field, the first-order expansion of the domain perturbations yields the same result as that obtained by Lamb (1932) for the viscous decay of the free oscillation modes. A primary resonance occurs in the first-order forced oscillation problem. Under strong excitation, superharmonic, sub-harmonic, and coupled resonances are revealed in the second-order solutions. Hence, large-amplitude drop oscillations may occur even if the excitation frequencies are away from the characteristic drop frequencies and the spatial forms of the excitation modes do not directly match the drop shape modes. In order to obtain comparable response amplitudes, however, the magnitudes of external forcing required to excite secondary resonances are shown to be about an order greater than that for the primary resonances.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 417 - 435
Copyright
© 1991 Cambridge University Press

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References

Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Beard, K. V.: 1984 Oscillation models for predicting raindrop axis and backscatter ratios. Radio Sci. 19, 67–74.Google Scholar
Beard, K. V., Johnson, D. B. & Jameson, A. R., 1983 Collisional forcing of raindrop oscillations. J. Atmos. Sci. 40, 455–462.Google Scholar
Beard, K. V., Ochsr, H. & Kubesh, R. J., 1989 Natural oscillations of small raindrops. Nature 342, 408–410.Google Scholar
Blanchard, D. C.: 1950 The behavior of water drops at terminal velocity in air. Trans. Am. Geophys. Union 31, 836–842.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. In Proc. Intl Colloq. Drops Bubbles (ed. D. J. Collins, M. S. Plesset & M. M. Saffren). Pasadena: Jet Propulsion Laboratory.
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
Gunn, R.: 1949 Mechanical resonance in freely falling raindrops. J. Geophys. Res. 54, 383–385.Google Scholar
Jones, D. M. A.: 1959 The shape of raindrops. J. Met. 16, 504–510.Google Scholar
Jeoseph, D. D.: 1973 Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Rat. Mech. Anal. 51, 295–303.Google Scholar
Kangr, I. S. & Leal, L. G., 1987 The drag coefficient for a spherical bubble in a uniform streaming flow. Phys. Fluids 31, 233–237.Google Scholar
Kang, I. S. & Leal, L. G., 1988 Small-amplitude perturbations of shape for a nearly spherical bubble in an inviscid straining flow (steady shapes and oscillatory motion). J. Fluid Mech. 187, 231–266.Google Scholar
Lamb, H.: 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Landau, L. D. & Lifshitz, E. M., 1959 Electrodynamics of Continuous Media. Addison-Wesley.
Lundgren, T. S. & Mansour, N. N., 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510.Google Scholar
Marston, P. L.: 1980 Shape oscillation and static deformation of drops and bubbles driven by modulated radiation stress: theory. J. Acoust. Soc. Am. 67, 15–26.Google Scholar
Miller, C. A. & Scriven, L. E., 1968 The oscillations of a fluid 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
Nayfeh, A. H.: 1983 The response of single degree of freedom systems with quadratic and cubic non-linearities to a subharmonic excitation. J. Sound Vib. 89, 457–470.Google Scholar
Nayfeh, A. H. & Mook, D. T., 1979 Nonlinear Oscillations. Wiley-Interscience.
Prosperetti, A.: 1980 Free oscillations of drops and bubbles: the initial-value problems. J. Fluid Mech. 100, 333–347.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
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. & 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