Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-23T05:42:36.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Behaviour of a conducting drop in a highly viscous fluid subject to an electric field

Published online by Cambridge University Press:  22 May 2007

N. DUBASH  and
A. J. MESTEL
N. DUBASH
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2BZ, UK
A. J. MESTEL
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2BZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the slow deformation of a relatively inviscid conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field. The general behaviour is to deform and elongate in the direction of the field. Detailed numerical computations, based on a boundary integral formulation, are presented. For fields below a critical value, we obtain the evolution of the drop to an equilibrium shape; above the critical value, we calculate the drop evolution up to breakup. At breakup it appears that smaller droplets are emitted from the ends of the drop with a charge greater than the Rayleigh limit. As the electric field strength is increased the ejected droplet size decreases. A further increase in field strength results in the mode of breakup changing to a thin jet-like structure being ejected from the end. The shape of all drops is very close to spheroidal up to aspect ratios of about 5. Also, for fields just above the critical value there is a period of slow deformation which increases in duration as the critical field strength is approached from above. Slender-body theory is also used to model the drop behaviour. A similarity solution for the slender drop is obtained and a finite-time singularity is observed. In addition, the general solution for the slender-body equations is presented and the solution behaviour is examined. The slender-body results agree only qualitatively with the full numerical computations. Finally, a spheroidal model is briefly presented and compared with the other models.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 581 , 25 June 2007 , pp. 469 - 493
Copyright
Copyright © Cambridge University Press 2007

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

REFERENCES

Abbi, S. S. & Chandra, R. 1956 On the equilibrium of a small conducting liquid drop in a uniform external electric field. Proc. Nat. Sci. India A. 22, 363–368.Google Scholar
Baygents, J. C., Rivette, N. J. & Stone, H. A. 1998 Electrohydrodynamic deformation and interaction of drop pairs. J. Fluid Mech. 368, 359–375.CrossRefGoogle Scholar
Brazier-Smith, P. R. 1971 Stability and shape of isolated and pairs of water drops in an electric field. Phys. Fluids 14 (1), 1–6.CrossRefGoogle Scholar
Buckmaster, J. D. 1972 Pointed bubbles in slow viscous flow. J. Fluid Mech. 55 (3), 385–400.CrossRefGoogle Scholar
Buckmaster, J. D. 1973 The bursting of pointed drops in slow viscous flow. J. Appl. Mech. 40 (1), 18–24.CrossRefGoogle Scholar
Dubash, N. 2006 Behaviour of a conducting drop in a viscous fluid subject to an electric field. PhD thesis, Imperial College London.CrossRefGoogle Scholar
Dubash, N. & Mestel, A. J. 2006 Behaviour near critical for a conducting drop in an electric field. Phys. Fluids (submitted).CrossRefGoogle Scholar
Dubash, N. & Mestel, A. J. 2007 Breakup behavior of a conducting drop suspended in a viscous fluid subject to an electric field. Phys. Fluids (in press).CrossRefGoogle Scholar
Eow, J. S. & Ghadiri, M. 2003 Motion, deformation and break-up of aqueous drops in oils under high electric field strengths. Chem. Engng Process. 42, 259–272.CrossRefGoogle Scholar
Eow, J. S., Ghadiri, M. & Sharif, A. 2001 Deformation and break-up of aqueous drops in dielectric liquids in high electric fields. J. Electrostat. 51, 463–469.CrossRefGoogle Scholar
Feng, J. Q. & Scott, T. C. 1996 A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J. Fluid Mech. 311, 289–326.CrossRefGoogle Scholar
Garton, C. G. & Krasucki, Z. 1964 Bubbles in insulating liquids: stability in an electric field. Proc. R. Soc. Lond. A. 280, 211–226.Google Scholar
Grigor'ev, A. I., Sharov, A. N. & Shiryaeva, S. O. 1999 Stability of a bubble in a dielectric liquid in an external electrostatic field. Tech. Phys. 44 (8), 908–912.CrossRefGoogle Scholar
Ha, J.-W. & Yang, S.-M. 2000 Deformation and breakup of Newtonian and non-Newtonian conducting drops in an electric field. J. Fluid Mech. 405, 131–156.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics, with Special Applications to Particulate Media. Prentice Hall.Google Scholar
Hirata, T., Kikuchi, T., Tsukada, T. & Hozawa, M. 2000 Finite element analysis of electrohydrodynamic time-dependent deformation of dielectric drop under uniform DC electric field. J. Chem. Engng Japan 33 (1), 160–167.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge Univeristy Press.Google Scholar
Li, H., Halsey, T. C. & Lobkovsky, A. 1994 Singular shape of a fluid drop in an electric or magnetic field. Europhys. Lett. 27 (8), 575–580.CrossRefGoogle Scholar
Mestel, A. J. 2002 Maximal accelerations for charged drops in an electric field. Phys. Fluids 14 (4), 1396–1402.CrossRefGoogle Scholar
Miksis, M. J. 1981 Shape of a drop in an electric field. Phys. Fluids 24 (11), 1967–1972.CrossRefGoogle Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.Google Scholar
O'Konski, C. T. & Thacher, H. C. 1953 The distortion of aerosol droplets by an electric field. J. Phys. Chem. 57, 955–958.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 1997 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Ramos, A. & Castellanos, A. 1994 Conical points in liquid-liquid interfances subjected to electric fields. Phys. Lett. A. 184, 268–272.CrossRefGoogle Scholar
Rayleigh, Lord 1882 On the equilibrium of liquid conducting masses charged with electricity. Philos. Mag. 14, 184–186.CrossRefGoogle Scholar
Sample, S. B., Raghupathy, B. & Hendricks, C. D. 1970 Quiescent distortion and resonant oscillations of a liquid drop in an electric field. Intl J. Engng Sci. 8, 97–109.CrossRefGoogle Scholar
Sampson, R. A. 1891 On Stokes's current function. Phil. Trans. R. Soc. Lond. A. 182, 449–518.Google Scholar
Sherwood, J. D. 1988 Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, 133–146.CrossRefGoogle Scholar
Sherwood, J. D. 1991 The deformation of a fluid drop in a electric field: a slender-body analysis. J. Phys. A: Math. Gen. 24, 4047–4053.CrossRefGoogle Scholar
Stone, H. A., Lister, J. R. & Brenner, M. P. 1999 Drops with conical ends in electric and magnetic fields. Proc. R. Soc. Lond. A. 455, 329–347.CrossRefGoogle Scholar
Stratton, J. A. 1941 Eletromagnetic Theory. McGraw-Hill.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A. 146, 501–523.Google Scholar
Taylor, G. I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A. 280, 383–397.Google Scholar
Taylor, G. I. 1966 a The force exerted by an electric field on a long cylindrical conductor. Proc. R. Soc. Lond. A. 291, 145–158.Google Scholar
Taylor, G. I. 1966 b Studies in electrohydrodynamics i. the circulation produced in a drop by electric field. Proc. R. Soc. Lond. A. 291, 159–166.Google Scholar
Tillett, J. P. K. 1970 Axial and transverse Stokes flow past slender axisymmetric bodies. J. Fluid Mech. 44 (3), 401–417.CrossRefGoogle Scholar
Wilson, C. T. R. & Taylor, G. I. 1925 The bursting of soap bubbles in a uniform electric field. Proc. Camb. Phil. Soc. 22, 728–730.CrossRefGoogle Scholar