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We present and analyze characteristics of the runaway electron flow in a high-voltage (the voltage rise rate of up to 1.5 MV/ns) air-filled electrode gap with a strongly nonuniform electric field. It is demonstrated that such a flow contains a high-energy electron component of duration not more than 10 ps. According to numerical simulations, runaway electron generation/termination is governed by impact ionization of the gas near the cathode and switching on/off a critical (sufficient for electrons to run away) electric field at the boundary of the expanding cathode plasma. The corresponding characteristic time estimated to be 2–3 ps is defined by the ionization rate at a critical field.

The nonlinear dynamics of the free surface of an ideal dielectric liquid with a large relative permittivity in a strong horizontal electric field has been considered. It has been demonstrated that the interaction between oppositely propagating solitary waves in arbitrary geometry is elastic: they conserve their energy and momentum. The interaction between waves has been numerically simulated with the use of conformal variables. It has been shown that the interaction deforms the waves; this effect is weak for waves with a relatively small amplitude: deformation for oppositely propagating waves with the identical shape is determined by the fourth power of their amplitude. At multiple collisions of strongly nonlinear waves, a tendency to the formation of singularities, i.e., points with a high energy density of the field, is observed.

The dynamics of singularity formation on the interface between two ideal fluids is studied for the Kelvin-Helmholtz instability development within the Hamiltonian formalism. It is shown that the equations of motion derived in the small interface angle approximation (gravity and capillary forces are neglected) admit exact solutions in the implicit form. The analysis of these solutions shows that, in the general case, weak root singularities are formed on the interface in a finite time for which the curvature becomes infinite, while the slope angles remain small. For Atwood numbers close to unity in absolute values, the surface curvature has a definite sign correlated with the boundary deformation directed towards the light fluid. For fluids with comparable densities, the curvature changes its sign in a singular point. In the particular case of fluids with equal densities, the obtained results are consistent with those obtained by Moore based on the Birkhoff-Rott equation analysis.

Analytical solutions for the distributions of the electric field potential and electric charge density are derived for the outer region of a steady-state unipolar corona discharge from an ideal wedge-shaped electrode under the conditions of space-charge-limited current. Two situations are considered: a corona is initiated only from the edge of the wedge and from the entire surface of the electrode. In the former case, general solutions are obtained by sewing together exact cylindrically symmetric solutions in the drift space and plane symmetric solutions in space-charge-free regions. In the latter case, the field distribution near the edge turns out to be self-similar, i.e., invariant under extensions in the cross-sectional plane of the wedge, with the center at the top of the wedge. For both models, the dependences of the saturation current per edge’s unit length on the apex angle and applied potential difference are obtained.

The nonlinear dynamics of the interface between dielectric fluids in a strong horizontal electric field is studied. It is shown that three-dimensional waves of small but finite amplitude can propagate without shape distortions either in a direction coinciding with the direction of the external field vector or in the direction opposite to this vector, along the interface of the fluids, whose density ratio is directly proportional to the ratio of their dielectric constants. For this particular case, an analytical description of the interaction of counterpropagating weakly nonlinear waves of arbitrary shape is given.

This paper considers the problem of the possible equilibrium configurations of the free surface of a perfectly conducting fluid deformed by a nonuniform magnetic field. A family of exact solutions of the problem is obtained using conformal mappings; equilibrium is achieved due to the balance of capillary and magnetic pressures. According to these solution, the surface strain amplitude increases with increasing current and the hole is transformed into a two-dimensional bubble covering the linear conductor.

Exact partial solutions are found for the electric field distribution in the outer region of a stationary unipolar corona discharge from an ideal conical needle in the space-charge-limited current mode with allowance for the electric field dependence of the ion mobility. It is assumed that only the very tip of the cone is responsible for the discharge, i.e., that the ionization zone is a point. The solutions are obtained by joining the spherically symmetric potential distribution in the drift space and the self-similar potential distribution in the space-charge-free region. Such solutions are outside the framework of the conventional Deutsch approximation, according to which the space charge insignificantly influences the shape of equipotential surfaces and electric lines of force. The dependence is derived of the corona discharge saturation current on the apex angle of the conical electrode and applied potential difference. A simple analytical model is suggested that describes drift in the point-plane electrode geometry under saturation as a superposition of two exact solutions for the field potential. In terms of this model, the angular distribution of the current density over the massive plane electrode is derived, which agrees well with Warburg’s empirical law.

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).

It has been shown that waves of arbitrary configuration in 3D geometry may propagate without distortion along the surface of a dielectric liquid in the direction of a horizontal electric field. This situation occurs for the high-permittivity liquids in the case of a sufficiently high external field when the effect of electrostatic forces dominates. A general solution of the equations of motion that describes the interaction of counterpropagating waves of a small but finite amplitude has been obtained.

The dynamics of the development of an instability of a charged surface of a liquid-helium layer with a finite depth is investigated. The equations describing the evolution of the free surface are derived with the use of conformal variables for the case in which the charge completely screens the electric field above the liquid. A model of the evolution of a spatially localized perturbation of a liquid-helium surface is proposed for the strong-field limit where the dynamics of the liquid is predominantly determined by the effect of electrostatic forces. This model describes the development of an instability of the initially planar boundary to the point of the formation of cuspidal dimples. The limit of an infinitely deep liquid is considered. The stability of the previously revealed liquid flow regime described by the Laplacian growth equations is proved without significant constraints on the surface geometry.

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 influence of the space charge of ions evaporated from the surface of a conical spike that develops on a liquid-metal anode on the spike shape has been studied theoretically. A possibility of the self-similar reduction of the equations that govern the space distributions of the electric field, ion velocity field, and space charge density to a system of ordinary differential equations has been revealed. Its solution, which is compatible with the balance condition for the capillary and electrostatic forces at a free liquid surface, has allowed us to obtain the current-voltage characteristic for a liquid-metal ion source and the current dependence of the cone angle. An increase in emission current leads to a monotonic decrease of the cone angle, which agrees with the available experimental data.

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 hydrodynamic evolution of the surface of a liquid metal in the presence of an electric field is investigated using both analytical and numerical techniques. It is established that a free liquid surface with axial symmetry evolves with time into a cone-like shape, with the cone angle equal to Taylor's static value of 98.6°. The mechanism behind such interesting flow behaviour is that the system of electrohydrodynamic (EHD) equations has a self-similar asymptotic solution that generalizes Taylor's static result. The asymptotic solutions are found and the time-dependent behaviours of basic physical quantities (electric field strength, fluid velocity and surface curvature) near the singularity are established. The results and the analytical and numerical techniques used are thought to be useful in the development of time-dependent models of operating liquid metal ion sources and EHD sprayers.

Possible equilibrium configurations of the free surface of a conducting liquid in an external nonuniform electric field are investigated. The liquid is at the tip of a wedge-shaped electrode, and the counter-elecrode is a thin straight conducting filament parallel to the edge of the wedge. The conformal mapping method is used to obtain a two-parametric family of exact solutions for the shape of the surface. The solutions describe the deformation of the surface right to separation of a drop from the electrode.

Exact self-similar solutions for the problem of electric field and charge density distribution in the external region of a unipolar corona discharge from a thin point emitter have been found. The obtained solutions are valid outside the framework of the traditional Deutsch approximation based on the small influence of a volume charge on the shape of equipotential surfaces. The value of a saturated discharge current is estimated assuming that the corona discharge is generated by a certain part of the point.

The development of instability on the interface between dielectric liquids in vertical electric and gravity fields has been studied. The possibility of a special regime of motion is established, in which the velocity potential is linearly related to the electric field potential. An integro-differential equation is derived for this regime, which describes a weakly nonlinear evolution of the interface. This equation admits the existence of broad classes of exact solutions that determine the dynamics of both periodic and localized perturbations of the interface.

The problem of possible equilibrium configurations of the free surface of a conducting liquid in an external spatially periodic electric field is considered. It is shown that, under identical conditions, the problem admits solutions with different wavelengths, including the wavelength corresponding to the field period and the half shorter wavelength corresponding to the period of the electrostatic pressure oscillations. In the former case, an exact partial solution is found that describes a surface profile with allowance for the capillary forces.

The problem of possible equilibrium configurations of the free surface of a conducting liquid in the electric field of a charged horizontal straight filament is considered. Using the method of conformal mapping, a one-parameter family of exact solutions of the problem is obtained, which take into account the effect of capillary forces. Conditions are found for which the region occupied by the liquid ceases to be simply connected and a charged drop is formed on the liquid surface

A broad class of exact solutions is obtained for the problem of the equilibrium configuration of the charged surface of a conducting liquid allowing for capillary forces. An analysis of the solutions showed that when the amplitudes of the perturbations reached certain critical values, the region occupied by the liquid ceases to be singly connected, which corresponds to the formation of liquid metal droplets. It is shown that a steady-state liquid metal profile may exist for which appreciable local amplification of the electric field can be achieved.

The process of electron emission from the boundary of regions with infinite and zero emissivity on a flat cathode surface of a vacuum diode in the space-charge-limited current regime has been theoretically studied. The space charge density, electric field, and velocity field distributions in the interelectrode gap are determined using self-similar solutions of the equations of motion of charged particles. It is established that a singularity, where the emission current density tends to infinity according to a power law, appears at the boundary. From this special point, electrons are emitted at an angle of 103° relative to the electrode surface.

The process of electron emission in a vacuum diode with curved cathode in the space-charge-limited current regime has been theoretically studied. The space charge density, electric field, and velocity field distributions in the interelectrode gap are determined by means of expansion with respect to a small parameter representing the ratio of the characteristic transverse and longitudinal scales. Based on these results, the current density distribution over the cathode surface is analytically described. The analytical results are compared to the data of numerical calculations available in the literature.

The propagation of waves with arbitrary amplitudes on the free surface of the ideal dielectric liquid in a strong tangential electric field is considered. A dispersion relation for the evolution of small perturbations of a stationary wave profile of the surface of a liquid with a large permittivity is obtained using conformal variables. It is established that, irrespective of the wave geometry, the amplitudes of small-scale perturbations do not increase with the time, which is evidence for the stability of wave solutions.

A particular solution to the problem of an equilibrium configuration of the surface of a conducting liquid in an external electric field is constructed based on an analysis of the balance of gravitational and electrostatic forces. The solution corresponds to a cone-shaped crater with a cone angle of 136.26°. The possibility of generalizing the obtained exact solution to the case of a dielectric liquid is considered.

The process of electron emission from a wedge-shaped cathode in the space-charge-limited current regime has been theoretically studied. The field distribution in the interelectrode gap is described using a simple analytical model, which is based on the matching of self-similar solutions for electron flows at the vertex region and at the periphery. The dependence of the perveance on the wedge angle in the wedge cathode-flat anode system is determined.

It is shown that the problem of determining the profile of an electrocapillary wave on the surface of the ideal dielectric liquid in a tangential electric field admits exact solution in the limiting case of a large permittivity of the liquid. An exact expression for the nonlinear dispersion relation between the frequency, wavenumber, and amplitude of the wave is obtained and analyzed.

The evolution of waves with finite amplitudes on the free surface of the ideal dielectric liquid in a strong tangential electric field is considered. It is shown that, in the case of a large permittivity of the liquid, nonlinear waves of arbitrary configuration can propagate in this system without distortions.

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 nonlinear dynamics of the interface between two deep dielectric fluids in the presence of a vertical electric field is studied. We consider the limit of a strong external electric field where electrostatic forces dominate over gravitational and capillary forces. The nonlinear integrodifferential equations for the interface motion are derived under the assumption of small interfacial slopes. It is shown in the framework of these equations that, in the generic case, the instability development leads to the formation of root singularities at the interface in a finite time. The interfacial curvature becomes infinite at singular points, while the slope angles remain relatively small. The curvature is negative in the vicinity of singularities if the ratio of the permittivities of the fluids exceeds the inverse ratio of their densities, and it is positive in the opposite case (we consider that the lower fluid is heavier than the upper one). In the intermediate case, the interface evolution equations describe the formation and sharpening of dimples at the interface. The results obtained are applicable for the description of the instability of the interface between two magnetic fluids in a vertical magnetic field.

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.

The problem of determining equilibrium configurations of the free surface of a conducting liquid is considered with allowance for a finite interelectrode distance. The analogy is established between this electrostatic problem and that of finding the profile of a progressive capillary wave on the free surface of a liquid layer of a finite depth, which was solved by Kinnersley. This analogy allowed exact solutions to be obtained for the geometry of liquid electrodes, which expand the existing notions about the possible stationary states of the system., Comment: 5 pages, 1 figure

A region of physical parameters is found where the equilibrium configurations of charged cylindrical jets of a conducting liquid can exist; these configurations correspond to the azimuthal mode n=2 of the surface deformations of a round jet. A critical value of the jet linear charge is determined for which the jet splits into two. It is shown that such instability is excited in a soft regime.

Stability of a liquid crater wall formed under the action of an ytterbium-fiber laser in the course of the Nd3+:Y2O3 nanopowder production is studied theoretically. It has been shown that hydrodynamic instability can develop on the melt-vapor interface as a result of the tangential discontinuity of the velocity between the vapor stream and molten crater wall. The characteristic spatial and temporal scales are estimated in the framework of the proposed qualitative model, they are found to be 20-90 μm and 20-50 μs, respectively, that is in good agreement with experimental data. Thus, the droplet formation time (during which the amplitude of the boundary perturbation reaches the wavelength order) is much smaller than a pulse duration of the ytterbium-fiber laser (1360 μs). This means that a significant amount of material can be removed from the crater due to formation of microscale droplets during the irradiation. This mechanism can explain the much greater crater depth for the fiber laser than for CO2 laser (a pulse duration for which is 370 μs).

The influence of the space charge of ions emitted from the surface of a conical spike on its shape has been studied. The problem of the calculation of the spatial distributions of the electric field, ion velocity field, and the space charge density near the cone tip has been reduced to the analysis of a system of ordinary differential equations. As a result of numerical solution of these equations, the criterion of the balance of the capillary and electrostatic forces on the conic surface of a liquid-metal anode has been determined. It has allowed us to relate the electrical current flowing through the system, the applied potential difference and the cone angle. We have compared the results of our calculations with available experimental data concerning emission from the surface of pure liquid gallium (Ga), indium (In), tin (Sn), and some liquid alloys, such as Au+Si, Co+Ge, and Au+Ge. On the basis of the proposed model, explanations have been given for a number of specific features of the emissive behavior of different systems., 11 pages, 8 figures

The stability of a current-carrying rectangular liquid metal conductor with respect to small longitudinal surface perturbations was theoretically studied. It is shown that the instability increment tends to zero in both short-and longwave limits and, hence, passes through a maximum at a finite wavenumber. The characteristic times of instability development are estimated.

It is shown that the problem of describing a profile of a progressive electrocapillary wave on the boundary of a dielectric liquid, with a free surface charge completely screening the electric field above the liquid, admits an analytical solution. An exact expression for the nonlinear dispersion relationship between the frequency, wavenumber, and amplitude of such a wave is derived and analyzed.

The integral estimation method is used to study the wave collapse dynamics for the Klein-Gordon equation with a nonlinearity of the general form in the case of periodic boundary conditions. Sufficient integral criteria (generalizing the known ones) for the wave collapse are formulated.

We will prove that, within the framework of planar geometry, no equilibrium configuration of a free charged surface of a conducting liquid can exist ensuring unlimited local amplification of the applied electric field.

Nonlinear waves on sheets of dielectric liquid in the presence of an external tangential electric field are studied theoretically. It is shown that waves of arbitrary shape in three-dimensional geometry can propagate along (or against) the electric field direction without distortion, i.e., the equations of motion admit a wide class of exact traveling wave solutions. This unusual situation occurs for nonconducting ideal liquids with high dielectric constants in the case of a sufficiently strong field strength. Governing equations for evolution of plane symmetric waves on fluid sheets are derived using conformal variables. A dispersion relation for the evolution of small perturbations of the traveling wave solutions is obtained. It follows from this relation that, regardless of the wave shape, the amplitudes of small-scale perturbations do not increase with time and, hence, the traveling waves are stable. We also study the interaction of counterpropagating symmetric waves with small but finite amplitudes. The corresponding solution of the equations of motion describes the nonlinear superposition of the oppositely directed waves. The results obtained are applicable for the description of long waves on fluid sheets in a horizontal magnetic field.

Subcritical behavior of a free surface of the ideal dielectric liquid in an external electric field has been theoretically studied. A system of amplitude equations is obtained, which describe nonlinear interaction of three standing waves forming a hexagonal structure. Within the framework of a model described by these equations, a sufficient integral criteria are formulated for the rigid instability excitation on the flat surface of the liquid medium.

A conformal mapping method was used to obtain exact solutions of the problem of the equilibrium shape of a charged two-dimensional liquid-metal droplet. It was found that when the perturbation amplitudes reach certain critical values, the regions occupied by the liquid ceased to be singly connected, which corresponds to the breakup of the droplet into parts.

The nonlinear dynamics of the free surface of an ideal conducting liquid in a strong external electric field is studied. It is establish that the equations of motion for such a liquid can be solved in the approximation in which the surface deviates from a plane by small angles. This makes it possible to show that on an initially smooth surface for almost any initial conditions points with an infinite curvature corresponding to branch points of the root type can form in a finite time., 14 pages

The evolution of the interface between two ideal dielectric liquids in a strong vertical electric field is studied. It is found that a particular flow regime, for which the velocity potential and the electric field potential are linearly dependent functions, is possible if the ratio of the permittivities of liquids is inversely proportional to the ratio of their densities. The corresponding reduced equations for interface motion are derived. In the limit of small density ratio, these equations coincide with the well-known equations describing the Laplacian growth., Comment: 10 pages

A wide class of exact solutions is obtained for the problem of finding the equilibrium configurations of charged jets of a conducting liquid; these configurations correspond to the finite-amplitude azimuthal deformations of the surface of a round jet. A critical value of the linear electric charge density is determined, for which the jet surface becomes self-intersecting, and the jet splits into two. It exceeds the density value required for the excitation of the linear azimuthal instability of the round jet. Hence, there exists a range of linear charge density values, where our solutions may be stable with respect to small azimuthal perturbations., 7 pages, 5 figures, to appear in Physical Review E

The formation dynamics is studied for a singular profile of a surface of an ideal conducting fluid in an electric field. Self-similar solutions of electrohydrodynamic equations describing the fundamental process of formation of surface conic cusps with angles close to the Taylor cone angle 98.6 are obtained. The behavior of physical quantities (field strength, fluid velocity, surface curvature) near the singularity is established., LaReX, 10 pages, 1 figure

Applying the method of integral estimates to the analysis of three-wave processes we derive the sufficient criteria for the hard loss of stability of the charged plane surface of liquids with different physical properties. The influence of higher-order wave interactions on the instability dynamics is also discussed., 9 pages

The dynamics of the development of instability of the free surface of liquid helium, which is charged by electrons localized above it, is studied. It is shown that, if the charge completely screens the electric field above the surface and its magnitude is much larger then the instability threshold, the asymptotic behavior of the system can be described by the well-known 3D Laplacian growth equations. The integrability of these equations in 2D geometry makes it possible to described the evolution of the surface up to the formation of singularities, viz., cuspidal point at which the electric field strength, the velocity of the liquid, and the curvature of its surface assume infinitely large values. The exact solutions obtained for the problem of the electrocapillary wave profile at the boundary of liquid helium indicate the tendency to a charge in the surface topology as a result of formation of charged bubbles., Comment: LaTeX, 26 pages, 5 figures