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We consider conical structures appearing on the free surface of a liquid with ionic conductivity in an applied electric field. An exact self-similar solution for the electric field distribution near the cone apex is found. It corresponds to the space-charge-limited current mode: the electric current through the liquid cone is limited by both bulk and surface charges of ions with different signs. Dependences of the saturation current and of the potential drops inside and outside the conical formation on the cone half-angle, dielectric constant of liquid, and mobility of anions and cations are determined and analyzed.

The elastic interaction of spherical particles is studied. The distribution of the stress, normal to the contact plane, is determined by the rod model suggested recently, which is applicable in the more wide range of deformations as compared with the classical Hertz law. In the rod model context an inner part of compressed particles is regarded as an elastic cylindrical rod, which radius is equal to the contact radius. The rod reaction is added to the normal particle interaction corresponding with the Hertz solution. The resulting normal force passes into the Hertz solution for infinitesimal deformations and gives stronger particle repulsion for finite deformations. Here we solve the Mindlin problem for the rod model, i.e., derive the tangential interaction of initially compressed particles when a relative displacement takes place. The analytical expressions, which determine the total displacement of the sphere’s centers and the corresponding tangential force, are derived. So, the generalization of the classical Mindlin law is obtained for the rod model.

The problem of possible equilibrium configurations of the free surface of a conducting liquid deformed by a non-uniform external electric field is considered. The liquid rests on a flat electrode; the opposite electrode is a straight thin conducting filament placed parallel to it. It is assumed that the problem has plane symmetry: the surface is invariant under shift along the filament. Using the method of conformal mapping, a family of exact solutions of the problem is obtained; equilibrium is achieved as a result of the balance between capillary and electrostatic forces. According to the solutions, the two-dimensional drop considerably deforms with an increase in the strength of the external electric field (this increase can be related to changes in the linear charge density of the filament or in the electrode spacing). It turns out that the drop aspect ratio varies from unity (the charge density equals zero) to 11/2 (the filament recedes to infinity).

Solutions for the problem on the equilibrium configurations of uncharged conducting liquid jets in a transverse electric field are obtained. These solutions correspond to finite-amplitude non-axisymmetric azimuthal deformations of the surface of a round jet: the jet is stretched along the field in its cross-section. The range of electric fields is determined for which solutions of the problem exist. If the electric field strength is over some critical value, the electrostatic equations have no solution, and the jet splits. The obtained solutions are qualitatively examined for stability under small azimuthal perturbations.

The nonlinear dynamics of the free surface of an ideal dielectric liquid in a strong electric field is studied. The equation for the evolution of surface electrohydrodynamic waves is derived in the approximation of small surface-slope angles. It is established that the equation can be solved for liquids with sufficiently high values of the permittivity. This makes it possible to describe the interaction of the counter-propagating waves., Comment: 10 pages. accepted to Phys. Lett. A

The evolution of large-scale convective instabilities in a plasma-like medium in the field of its own electric current is considered in the framework of the magneto-hydrodynamics of incompressible viscous resistive liquids. The simplest low-mode model of the nonlinear interaction of perturbations is proposed and analytically investigated. According to this model, if the amplitude of the current exceeds its critical value, the effective resistance of the medium can break into oscillations. The model has enabled us to estimate with reasonable exactness the threshold commutation current and the frequency of current oscillation for experiments on repeated interruption and restoration of electric current through the InGa eutectic.

The initial stages of the stratification of a current-carrying metal conductor are considered in the framework of the magnetohydrodynamic equations. We find that if an electric current in the conductor exceeds its threshold value, then it is necessary to take into consideration an additional characteristic longitudinal size corresponding to the large-scale hydrodynamic and current fluctuations with the maximum increment. This size is shown to agree well with the stratum size observed in experiments.