Charge Convection Research Articles

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

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.

Electrohydrodynamic oscillation and stability of a charged drop

Small amplitude oscillations of a charged fluid globule immersed in a viscous dielectric are studied with special regard for the processes of surface charge relaxation and convection. Perfect dielectrics and conductors behave in identical fashion insofar as oscillation frequencies and stability are concerned whenever viscous effects are small. This similarity is due to interfacial charge convection, in the case of a perfect dielectric, and charge conduction, with a perfect conductor. The damping of stable oscillations depends, however, on the rate of interfacial charge relaxation: With instantaneous relaxation the damping rates are the same as those found with uncharged droplets, while with slower relaxation rates the dissipation is concentrated in an electrohydrodynamic boundary layer outside the drop. If viscous effects predominate then the existence of charge relaxation introduces oscillatory behavior in place of the aperiodic damping characteristic of uncharged drops. Several special circumstances are examined in detail.

Dynamics of charge monolayers on insulating liquid interfaces

At most air-liquid interfaces stressed by a static perpendicular electric field, a monolayer of charge is induced that shields the field from the liquid. For relatively inviscid and highly insulating liquids, the electrohydrodynamics can dominate in determining the monolayer and field distributions associated with additional dynamic field components. Electric shear stresses lead to a convection of surface charge that distorts the field, much as though the liquid were conducting. A configuration for studying the monolayer dynamics is developed in which a uniform field is used to induce a uniform monolayer on the interface of a liquid layer having thickness c. Superimposed is a travelling wave of potential \[ {\rm Re}\,\hat{V}_0\exp i(\omega t - kz), \] imposed in a plane parallel to, and a t a distance a above, the interface. Mechanisms for charge redistribution are reflected in the frequency response of the field transmitted through the interface to a second plane bounding the liquid layer from below. A model is developed which accounts for the self-consistent electromechanics in terms of the lumped surface parameters of surface charge σ0 and surface mobility bs, and a bulk conductivity σ. According to this model, interfacial convection dominates a t low frequencies in attenuating the field induced below the interface. For layers which are thick compared with the viscous skin depth, there is a resonance in the response a t the frequency \[ \begin{array}{@{}l@{\qquad\qquad}c@{}} &\omega = \omega_i(2^{\frac{1}{3}}),\\ {\rm where} & \omega_i = \{k\sigma^2_0/\epsilon_0(\rho\eta)^{\frac{1}{2}}[\cot {\rm h}\, ka +(\epsilon/\epsilon_0)\cot {\rm h}\,kc]\}^{\frac{2}{3}} \end{array} \] and ρ, η, ε0 and ε are the liquid mass density, liquid viscosity, and permittivities of air and the liquid, respectively. Surface and bulk conduction, characterized by the relaxation time \[ \tau_b = \epsilon_0(\cot h\,ka + (\epsilon/\epsilon_0)\cot {\rm h}\,kc)/[\sigma_0b_sk + \sigma\cot {\rm h}\,kc], \] result in a broadening of the resonance which is appreciable even for ωτb ∼ 5. At frequencies high compared with both wi and i/Tb) the response is uninfluenced by the charge monolayer. Experiments substantiate the electroviscous resonance. A time-average surface force density, and associated steady convection of the liquid, is also theoretically shown to be a consequence of the phase shifts caused by the electroviscous resonance. This is qualitatively demonstrated by measurement of the steady shear stress induced on the plane bounding the liquid from below.

Electrohydrodynamic Charge Relaxation and Interfacial Perpendicular-Field Instability

The small-amplitude motions of a plane interface between two fluids stressed by an initially perpendicular electric field are investigated. The fluids are modeled as Ohmic conductors and the convection of the surface charge caused by the dynamic interplay of interfacial electric shear stresses and the viscous stresses is highlighted. The influence of viscosity on instability growth rates in the zero-shear stress limits of perfectly conducting and perfectly insulating interfaces is described and compared to cases involving electrical shear stresses. Detailed attention is given to the instability of an interface between fluids having electrical relaxation times long compared to times of interest. It is shown that, for many common liquids, even a slight amount of surface charge makes the interface unstable at a considerably lower voltage than would be expected from theories based on the dielectrophoretic limit of no interfacial free charge. Experiments, performed using high-frequency ac stresses, gradually increased dc fields, and abruptly applied dc fields, support the theoretical model. In the general case, the electric Hartmann number is identified as an index to the dominance of the electric shear stresses over the viscous shear stresses in determining the interfacial convection of free charge.

Convective effects in plasmas with beams

A short review is presented of those plasma instabilities the development of which requires the presence of both a directed motion of the particles and a transverse inhomogeneity of their distribution function. The mechanism of the instabilities considered here arises from the effect of charge convection across the perturbing electric field and the main magnetic field. Since the wave vector of the unstable oscillations is directed almost perpendicularly to the magnetic field, ion motion plays an important part. As a result of interaction with these oscillations, the ions are able to acquire from the beam a large fraction of its energy. This possibly explains the experimentally discovered effect of strong heating of the ions when an electron beam with finite boundaries passes through a plasma.

Effect of Space Charge Convection on the Magnetic Field Transition towards the Plasma Régime

Effect of Space Charge Convection on the Magnetic Field Transition towards the Plasma Régime

Measurements of current density in the fair weather atmosphere

Simultaneous measurements of potential gradient and conductivity have been made from an airplane over an altitude range of from 15 m to 6 km. Within the exchange layer over the oceans, the conduction current density decreases with altitude, requiring a convection current which is probably caused by the upward diffusion of positive space charge. The conduction current is a maximum at 15 m above the earth's surface, where it is found to be on the average some 40 per cent, and on some soundings 200 per cent, higher than that above the exchange layer. Overland, space charge convection produces a conduction current within the exchange layer which is randomly variable, and on the average some 20 per cent higher than that above the layer. A convection current of some 50 per cent of the total current is sometimes observed at the top of the exchange layer. Above it, the current density is constant with altitude to within 10 per cent, and is found to vary from 1.1×10−12 amp/m2 over Maryland to 4.1×10−12 amp/m2 over Greenland, approximately inverse to the variation in columnar resistance. The average value of current density over the oceans is 2.7×10−12 amp/m2, giving an estimated world-wide conduction current of 1400 amp. Over Greenland, the total potential and average current density aloft on individual soundings both vary together with time and are in phase with the universal variation in thunderstorm activity. A comparison of simultaneous measurements at the surface and aloft in Greenland and near Key West, Florida, show that surface measurements of current density at a single station cannot be used to provide an index of world-wide effects on a day-to-day basis.

Experiments in magneto-fluid dynamics

Magneto-fluid dynamics deals with the motion of electrically conducting gases and liquids in the presence of magnetic fields. The macroscopic description of such motions involves the coupling of the equations of hydrodynamics and Maxwell's equations. For velocities small compared to that of light, and if one neglects displacement and charge convection currents, the coupling in the equations of motion occurs through a Lorentz body force term—a term involving the current density j and the magnetic induction, B, viz., j×B/c. The requirement of Lorentz invariance effects the introduction of the coupling into Maxwell's equations, so that one calculates j from the generalized Ohm's law j = σ(E+V×B/c), where σ is the fluid conductivity, E the electric field intensity and V the fluid velocity. Both j×B and V×B are basically nonlinear terms, whence the already uncomfortable situation with nonlinearities in hydrodynamics is quite considerably aggravated. Linearization on the one hand and “simple” experiments on the other must be invoked to convey a physical feeling for the phenomena to be expected from this coupling.

Airborne measurements of atmospheric electricity

An experimental investigation of the variations of atmospheric electric quantities in space and time was conducted. A review of past measurements reveals several inaccuracies and inconsistencies which stimulated the investigation. The simultaneous measurement of both polar conductivities, of potential gradient, and of several meteorological quantities from a multi‐engine airplane, and the derivation of other atmospheric electric variables, are described. Altitude ‘profiles’ of conductivity and potential gradient up to 6 km obtained on a representative sounding over the Chesapeake Bay show a secondary exchange layer above the more common surface exchange layer. The space‐charge density profile exhibits positive charge concentrations at the tops of these layers, near the bases of the associated temperature inversions. Upward convection of positive space charge causes the computed conduction‐current density within the exchange layers to be some 40 pct higher than its relatively constant value above the layers. The product of columnar resistance and current density, employed by previous workers to obtain atmospheric potential, provides erroneous values whether the conduction current near the Earth's surface or the average value over the entire altitude range is used. On the basis of a 6‐km atmospheric potential of 327 kv obtained from the integrated potential gradient, the true extrapolated total potential of the high atmosphere is 341 kv at a time near the diurnal maximum.