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Stability of a charged drop near a conductor wall

Regular Article

Abstract.

The effect of conductor boundaries on the deformation and stability of a charged drop is presented. The motivation for such a study is the occurrence of a charged conductor drop near a conductor wall in experiments (Millikan-like set-up in studies on Rayleigh break-up) and applications (such as electrospraying, ink-jet printing and ion mass spectroscopy). In the present work, analytical (linear stability analysis (LSA)) and numerical methods (boundary element method (BEM)) are used to understand the instability. Two kinds of boundaries are studied: a spherical, conducting, grounded enclosure (similar to a spherical capacitor) and a planar conducting wall. The LSA of a charged drop placed at the center of a spherical cavity shows that the Rayleigh critical charge (corresponding to the most unstable l = 2 Legendre mode) is reduced as the non-dimensional distance \( \hat{d}\) = \( {\frac{{b-a}}{{a}}}\) decreases, where a and b are the radii of the drop and spherical cavity, respectively. The critical charge is independent of the assumptions of constant charge or constant potential conditions. The trans-critical bifurcation diagram, constructed using BEM, shows that the prolate shapes are subcritically unstable over a much wider range of charge as \( \hat{d}\) decreases. The study is then extended to the stability of a charged conductor drop near a flat conductor wall. Analytical theory for this case is difficult and the stability as well as the bifurcation diagram are constructed using BEM. Moreover, the induced charges in the conductor wall lead to attraction of the drop to the wall, thereby making it difficult to conduct a systematic analysis. The drop is therefore assumed to be held at its position by an external force such as the electric field. The case when the applied field is much smaller than the field due to inherent charge on the drop ( \( {\frac{{a^3\rho g}}{{3\epsilon_0\psi^2}}}\) ≪ 1 is considered. The wall breaks the fore-aft symmetry in the problem, and equilibrium, predominantly prolate shapes corresponding to the legendre mode, l = 2 , are observed. The deformation increases with increasing charge on the drop. The breakup of the prolate equilibrium shapes is independent of the legendre modes of the initial perturbations. The prolate perturbations are subcritically unstable. Since the equilibrium prolate shapes cannot continuously exchange instability with equilibrium oblate shapes, an imperfect transcritical bifurcation is observed. A variety of highly deformed equilibrium oblate shapes are predicted by the BEM calculations.

Keywords

Flowing matter: Nonlinear Physics 

References

  1. 1.
    N. Ashgriz, Handbook of Atomization and Sprays (Springer, New York, 2011)Google Scholar
  2. 2.
    A. Kazutoshi, J. Electrost. 18, 63 (1986)CrossRefGoogle Scholar
  3. 3.
    J.W.S. Rayleigh, Philos. Mag. 14, 184 (1882)Google Scholar
  4. 4.
    D. Duft, T. Achtzehn, R. Muller, B.A. Huber, T. Leisner, Nature 421, 128 (2003)ADSCrossRefGoogle Scholar
  5. 5.
    E. Giglio, B. Gervais, J. Rangama, B. Manil, B.A. Huber, D. Duft, R. Muller, T. Leisner, C. Guet, Phys. Rev. E. 77, 036319 (2008)ADSCrossRefGoogle Scholar
  6. 6.
    A. Doyle, D.R. Moffett, B. Vonnegut, J. Colloid Sci. 19, 136 (1964)CrossRefGoogle Scholar
  7. 7.
    C.B. Richardson, H. Pigg, R.L. Hightower, Proc. R. Soc. London, Ser. A 422, 319 (1988)ADSGoogle Scholar
  8. 8.
    J.A. Tsamopoulos, T.R. Akylas, R.A. Brown, Proc. R. Soc. London, Ser. A 401, 67 (1985)ADSCrossRefGoogle Scholar
  9. 9.
    A. Gomez, K. Tang, Phys. Fluids 6, 404 (1994)ADSCrossRefGoogle Scholar
  10. 10.
    K. Tang, A. Gomez, Phys. Fluids 6, 2317 (1994)ADSCrossRefGoogle Scholar
  11. 11.
    W. Deng, A. Gomez, Int. J. Heat Mass Transfer 54, 2270 (2011)CrossRefGoogle Scholar
  12. 12.
    S. Chandra, C. Avedisian, Proc. R. Soc. London, Ser. A 432, 13 (1991)ADSCrossRefGoogle Scholar
  13. 13.
    M. Pasandideh-Fard, Y.M. Qiao, S. Chandra, J. Mostaghimi, Phys. Fluids 8, 650 (1996)ADSCrossRefGoogle Scholar
  14. 14.
    C. Tropea, I.V. Roisman, Atomiz. Sprays 10, 387 (2000)Google Scholar
  15. 15.
    J. Fukai, Y. Shiba, T. Yamamoto, O. Miyatake, D. Poulikakos, M. Megaridis, Z. Zhao, Phys. Fluids 7, 236 (1995)ADSCrossRefGoogle Scholar
  16. 16.
    M.R. Davidson, Chem. Eng. Sci. 55, 1159 (2000)CrossRefGoogle Scholar
  17. 17.
    C. Pozrikidis, J. Fluid Mech. 215, 331 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    S.I. Betelu, M.A. Fontelos, Physica D 209, 28 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    S.U. Ryu, S.Y. Lee, Int. J. Multiphase Flow 35, 1 (2009)CrossRefGoogle Scholar
  20. 20.
    T. Erneux, P. Mandel, SIAM J. Appl. Math. 46, 1 (1986)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    O.A. Basaran, L.E. Scriven, Phys. Fluids A 1, 799 (1989)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    T. Achtzehn, R. Muller, D. Duft, T. Leisner, Eur. Phys. J. D 34, 311 (2005)ADSCrossRefGoogle Scholar
  23. 23.
    R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena (John Wiley and Sons, New York, 2002)Google Scholar
  24. 24.
    H.A. Stone, L.G. Leal, J. Fluid Mech. 220, 161 (1990)ADSMATHCrossRefGoogle Scholar
  25. 25.
    C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge University Press, Cambridge, 1992)Google Scholar
  26. 26.
    W.J. Cody, Math. Comput. 19, 105 (1965)MathSciNetMATHGoogle Scholar
  27. 27.
    C. Hastings, Approximations for Digital Computers (Princeton University Press, New Jersey, 1965)Google Scholar
  28. 28.
    R.M. Thaokar, S.D. Deshmukh, Phys. Fluids 22, 034107 (2010)ADSCrossRefGoogle Scholar
  29. 29.
    J. Blake, Proc. Cambridge Philos. Soc. 70, 303 (1971)ADSMATHCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • S. E. Mhatre
    • 1
  • S. D. Deshmukh
    • 1
  • R. M. Thaokar
    • 1
  1. 1.Department of Chemical EngineeringIndian Institute of Technology BombayMumbaiIndia

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