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

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 ratioI/I0(called theelectric Nusselt number) on the stability parameterT=M2R= εϕ0/Kρν, and on 1/R= ν/Kϕ0, which is an equivalent Prandtl number (ε is the permittivity, ρ the fluid density,Kthe mobility, ν the kinematic viscosity, and ϕ0the applied voltage). In the viscous state, the analysis givesI/I0∞T½; in the inertial state the lawI/I0∞ (T/R)1/4=M½is obtained. SinceMis 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 whenRe[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, whenM< 3, the electric Nusselt number approaches 1, which is equivalent to the situation in thermal convection at low Prandtl numbers.

Two-dimensional numerical analysis of electroconvection in a dielectric liquid subjected to strong unipolar injection

“…It would certainly result in a different vortical activity and for this reason a different mechanism of charge mixing which determines the charge transport and, hence, charge distribution which lead consequently to the electrical Nusselt number. This difference in mixing efficiency between 2D and 3D fully turbulent expected flow could explain the discrepancy between the level of values of electrical Nusselt number found in our 2D numerical simulations compared to the experimental data of Lacroix et al 40 Following the same underlying thought, extending the size domain to L = 10 (15 cells) instead of L = 0.614 (1 cell) should reduce the discrepancy between the electrical Nusselt number in the numerical experiments compared to experimental data.…”

“…As in the experiments, the value of T at saturation depends on M. This fact is analysed in detail by Lacroix et al, 40 where a scaling law Ne ∼ M 1/2 is predicted. Although Figure 19 shows that the value of Ne at saturation increases with M, it is difficult to decide whether or not our results follow the scaling law Ne ∼ M 1/2 , this uncertainty is probably related to the two-dimensional nature of the simulation.…”

“…Although Figure 19 shows that the value of Ne at saturation increases with M, it is difficult to decide whether or not our results follow the scaling law Ne ∼ M 1/2 , this uncertainty is probably related to the two-dimensional nature of the simulation. Lacroix et al 40 arrived theoretically at this behaviour from two previous relations. One of them is that the velocity scales as v ∼ ME.…”

“…It is interesting to compute the electrical Nusselt number from the electric current, since experimental data are available in the literature, Lacroix et al 40 The electrical Nusselt is defined as: N e = I e I 0 ,…”

Patterns at the onset of electroconvection in freely suspended smectic films

“…This mechanism has been extensively studied both theoretically (21 26) and experimentally. (24)(25)(26) However, the injection process can be treated theoretically only in a rather idealized way, and in practice liquid crystals are too impure to be regarded as insulators. An isotropic system which is more relevant to liquid crystals is an electrochemical cell (2v '28) in which the fluid is an electrolyte with a nonzero conductivity.…”