Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T07:35:26.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Electro-convection in a dielectric liquid layer subjected to unipolar injection

Published online by Cambridge University Press:  29 March 2006

J. C. Lacroix ,
P. Atten  and
E. J. Hopfinger
J. C. Lacroix
Affiliation:
Laboratoire d'Electrostatique, CNRS, Grenoble, France
P. Atten
Affiliation:
Laboratoire d'Electrostatique, CNRS, Grenoble, France
E. J. Hopfinger
Affiliation:
Laboratoire de Mécanique des Fluides (Laboratoire associéau CNRS), Université de Grenoble, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of electric charge convection in a dielectric liquid layer of high ionic purity, when subjected to unipolar injection, is in many ways analogous to that of thermal convection in a horizontal fluid layer heated from below, although no formal analogy can be established. The problem treated is intrinsically more nonlinear than the thermal problem. We consider two asymptotic states of convection: one where the whole motion is dominated by viscosity, and one where inertial effects dominate. In each state, two or three spatial regions are distinguished. From the approximate equations that hold in the different regions, information about the variation of the different quantities with distance from the injector is obtained, and further approximations permit us to establish the dependence of the current density ratio I/I0 (called the electric Nusselt number) on the stability parameter T = M2R = εϕ0/Kρν, and on 1/R = ν/Kϕ0, which is an equivalent Prandtl number (ε is the permittivity, ρ the fluid density, K the mobility, ν the kinematic viscosity, and ϕ0 the applied voltage). In the viscous state, the analysis gives I/I0 ∞ T½; in the inertial state the law I/I0 ∞ (T/R)1/4 = M½ is obtained. Since M is independent of the applied voltage, the latter law shows the saturation in the electric Nusselt number observed in earlier experiments. The transition in the states is associated with a transition number (MR)T [gap ] 30, which is an electric Reynolds number, related to an ordinary Reynolds number of about 10.

The experimental results, obtained in liquids of very different viscosities and dielectric constants, verify these theoretical predictions; further, they yield more precise numerical coefficients. As for the transition criteria, the experiments confirm that the viscous and inertial effects are of the same order when Re [gap ] 10. It was also possible to determine roughly the limits of the viscous and inertial states. The viscous analysis remains valid up to a Reynolds number of about 1; the inertial state can be considered valid down to a Reynolds number of 60. Schlieren observations show that the motion has the structure of very stable hexagonal cells at applied voltages just above the critical voltage, which are transformed into unstable filaments when the voltage is increased further. At even higher voltages, the motion finally breaks down into turbulence. It may be of interest to point out that, when M < 3, the electric Nusselt number approaches 1, which is equivalent to the situation in thermal convection at low Prandtl numbers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 69 , Issue 3 , 10 June 1975 , pp. 539 - 563
Copyright
© 1975 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

Atten, P. & Gosse, J. P. 1969 Transient of one-carrier injections in polar liquids J. Chem. Phys. 51, 2804–2811.Google Scholar
Atten, P. & Lacroix, J. C. 1974 Stabilité hydrodynamique non linéaire d'un liquide isolant soumis à une injection unipolaire forte. C. R. Acad. Sci. B 278, 385–387.Google Scholar
Atten, P. & Moreau, R. 1972 Stabilité électrohydrodynamique des liquides isolants soumis à une injection unipolaire J. Mécan. 11, 471–520.Google Scholar
Avsec, D. & Luntz, M. 1937 Tourbillons électroconvectifs dans une nappe Liquide. C. R. Acad. Sci. 204, 420–422.Google Scholar
Budde, W. 1962 Appl. Opt. 1, 201–205.
Chu, T. Y. & Goldstein, R. J 1973 Turbulent convection in a horizontal layer of water J. Fluid Mech. 60, 141–159.Google Scholar
Deardorff, J. W. & Willis, G. E. 1967 Investigation of turbulent thermal convection between horizontal plates J. Fluid Mech. 28, 675–704.Google Scholar
FéLICI, N. 1969 Phénomènes hydro- et aérodynamiques dans la conduction des diélectriques fluides Rev. Gén. Electr. 78, 717–734.Google Scholar
FéLICI, N. & Sauviat, M. 1972 Electrodialytic varnishes as ion injectors. Proc. Int. Conf. Conduction and Breakdown in Dielectric Liquids, Dublin, pp. 43–46.Google Scholar
Hopfinger, E. J. & Gosse, J. P. 1971 Charge transport by self-generated turbulence in insulating liquids submitted to unipolar injection Phys. Fluids, 14, 1671–1682.Google Scholar
Hopfinger, E. J. & Lacroix, J. C. 1972 Hydrodynamic mechanisms of charge transport near the electrodes. Proc. Int. Conf. Conduction and Breakdown in Dielectric Liquids, Dublin, pp. 97–100.Google Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number Phys. Fluids, 5, 1374–1389.Google Scholar
Lacroix, J. C. & TOBAZéON, R. 1972 Experimental study of charge transfer phenomena in viscous fluids with single carrier injection. Proc. Int. Conf. Conduction and Breakdown in Dielectric Liquids, Dublin, pp. 93–96.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. Roy. Soc. A 225, 196–212.Google Scholar
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh—Bénard convection J. Fluid Mech. 58, 289–312.Google Scholar
Ostroumov, G. A. 1954 Zh. Tekh. Fiz. 24, 1915–1919.
Schmidt, R. J. & Milverton, S. W. 1935 On the instability of a fluid when heated from below. Proc. Roy. Soc A 152, 586–594.Google Scholar
Schneider, J. M. & Watson, P. K. 1970 Electrohydrodynamic stability of space-charge-limited currents in dielectric liquids. I. Theoretical study Phys. Fluids, 19, 1948–1954.Google Scholar
Spiegel, E. 1962 On the Malkus theory of turbulence. Mécanique de la Turbulence, Colloque International du CNRS, no. 108.Google Scholar
Stuetzer, O. M. 1962 Magnetohydrodynamics and electrohydrodynamics Phys. Fluids, 5, 534–544.Google Scholar
Veronis, G. 1966 Large-amplitude Bénard convection J. Fluid Mech. 26, 49–68.Google Scholar