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Direct numerical simulation of threedimensional chaotic motion of a dielectric liquid with a free surface under the action of external horizontal electric field is carried out. The numerical model takes into account the effects of surface tension, viscosity, and external isotropic random forcing acting at large scales. A transition from dispersive capillary wave turbulence to quasi-isotropic nondispersive EHD surface turbulence with increase of the external electric field strength is numerically observed for the first time. At the regime of developed EHD wave turbulence, the total electrical energy is found to be much greater than the energy of capillary waves, i.e., electrohydrodynamic effects play a dominant role. At the same time, anisotropic effects are detected that lead to the generation of capillary wave packets traveling perpendicular to the external field direction. Despite the revealed anisotropy, the calculated spectrum of EHD wave turbulence is in very good agreement with the analytical spectrum obtained on the basis of dimensional analysis of weak turbulence spectra., 7 pages,7 figures

The possibility of the appearance of microirregularities on the surface of a liquid metal under the action of a radio frequency electric field is investigated as applied to the conditions of the compact linear collider accelerating structures. These microirregularities are formed as a result of the development of parametric instability caused by the resonant interaction between the oscillating field and surface electrocapillary waves. It is shown that if the amplitude of the boundary oscillations becomes large enough, the dynamics of the system begins to be governed by nonlinear effects. In this case, it should be expected that the instability will become explosive and the boundary of the melt will cease to be smooth in a finite time: various kinds of singularities (cuspidal points, drops, and jets) will form. This will lead to a sharp increase in the emissive ability of the electrode surface and, as a consequence, to failure of the electric strength of the system.

Emission and discharge processes during the passage of high-voltage subnanosecond pulses with variable amplitude and duration along a gas-filled coaxial transmission line (CTL) are studied. Homogeneous CTL as well as CTL with special disk inserts (electric field enhancers initiating electron emission processes) on its central electrode were used. Data on the propagation and distortion of pulses were obtained and analyzed for different gases (nitrogen, hydrogen, SF6, and air) in a wide range of pressures. The voltage/field ranges and pulse durations are established where the role of runaway electrons (RAEs) appearing near the enhancer is essential in the breakdown development. It is shown that the effect of RAEs can be suppressed by shortening the voltage pulse even for hydrogen, which has a reduced static breakdown field and a high rate of ionization. It is also demonstrated that the RAE generation is abruptly replaced by the explosive electron emission from the enhancer edge with decreasing pressure from atmospheric to technical vacuum.

An investigation of runaway electron (RE) kinetics under the influence of a strongly inhomogeneous electric field was carried out by means of 2-D analytical and 2-D numerical Monte–Carlo approaches. Several shapes of cathodes generating an inhomogeneous electric field were studied. Within the developed 2-D analytical model, blade-shaped and cone-shaped cathodes were considered. The voltage values which allow electrons to scatter at large angles close to $\pi $ /2 near a blade-shaped cathode were determined. Also, it was shown that the RE scatter angles are larger for the blade (in comparison with the ones for the cone) due to the electric field defocusing effect. Within the framework of the numerical 2-D Monte–Carlo model, a cathode was assumed to have a parabolic shape. Inapplicability of the classical runaway criterion in strongly inhomogeneous electric fields was demonstrated. Also, it was shown that the cathode surface domain of electron emission and an initial direction of electron motion have a strong influence on their energy balance and, hence, on the probability for electrons to transit into the continuous accelerating regime. Energy losses of REs were estimated. A correlation between the full-width at half-maximum of the RE energy spectrum and an inelastic energy loss value was proposed.

The first results of study of the compression of a picosecond runaway electron flow in an air electrode gap by a pulsed guiding magnetic field increasing along the electron trajectory by a factor of 10–20 are reported. The main aim is to increase the density and homogeneity of the current of runaway electrons that are generated near the edge of a tubular cathode and are accelerated in an inhomogeneous electric field to a collector inside a solenoid. Restrictions on the integral current of runaway electrons caused by the reflection of particles with high transverse velocities from a region of concentration of magnetic field lines (magnetic mirror) have been analyzed. Under “optimal” conditions (magnetic field, the diameter of the cathode, and its position with respect to the solenoid are varied), a magnetized tubular runaway electron flow radially compressed by a factor of 3–4 with a current density up to 100 А/cm2 is formed.

Nonlinear dynamics of the free boundary of an ideal dielectric liquid in a vertical electric field is studied in a confined axisymmetric geometry. We derive an amplitude equation which describes the behavior of a fluid surface in the weakly nonlinear approximation. It is demonstrated that, due to the effect of quadratic nonlinearities, the regime of excitation of the surface electrohydrodynamic instability is always hard. It is also shown that cubic nonlinearities suppress the instability development for liquids with relatively small permittivity. On the contrary, the nonlinearity begins to play a destabilizing role for high values of dielectric constant. As a consequence, the instability development takes an explosive character: the amplitude of boundary perturbations grows unlimitedly over a finite time. Using an analogy with the motion of an effective Newtonian particle in a certain potential, we formulate conditions for explosive buildup of perturbations. Differences in the behavior between axisymmetric perturbations of the free surface and the cases of hexagonal, square, and planar symmetries are discussed.

The method of dynamic time-domain reflectometry is used to investigate a pulsed breakdown of an air-filled coaxial line assisted by the traveling voltage wave with amplitude of about –100 kV. The TEM-wave radial electric field exposure time was ~1 ns. The field enhancer representing a protruding disk-type insert into the central electrode of the line created conditions for spatial breakdown localization. The data obtained have been compared with the artificial short-circuit mode for the radial line gap at the location of the enhancer. With increase in the voltage amplitude and the field distortion caused by the enhancer, the criteria for electrons to run away were satisfied. In this case, a flow of fast particles is formed which initiates the breakdown with increased voltage collapse rate and discharge current and decreased resistance.

We consider the nonlinear dynamics of the free surface of a high-permittivity dielectric fluid in a strong horizontal electric field. In the framework of the weakly nonlinear approximation where we assume that the inclination angles of the boundary are small and take only the terms quadratically nonlinear in a small parameter into account in the equations of motion, we obtain a compact model equation that describes nonlinear wave processes in the system. We use this equation to investigate the interaction of counterpropagating solitary surface waves analytically and numerically. In particular, we demonstrate that the counter-propagating waves restore their shape after the interaction and thus acquire a certain phase shift. We also show that these properties of the model originate from its integrability.

The weakly nonlinear dynamics of the free surface of a dielectric liquid in an electric field directed tangentially to the unperturbed boundary is investigated numerically. Within the framework of the strong field model, when the effects of capillarity and gravity are not taken into account, it is shown that nonlinear surface waves have a tendency to break. In result of the collapse of the surface waves, the curvature of the boundary and the gradient of the local electric field undergo infinite discontinuity on the surface of the liquid. The angles of the boundary inclination remain small. The characteristic collapse time of a surface wave traveling in a given direction is calculated versus the dielectric constant of the liquid. It is shown that the time of the singularity formation increases infinitely at small and high values of the dielectric constant. The first case corresponds to the transition of the system to the neutral stability regime (the jump in electrostatic pressure at the boundary turns into zero). At high values of the dielectric constant of the liquid, the collapse time also increases. This effect is associated with the realization of a special regime of fluid motion, in which the propagation of nonlinear surface waves of an arbitrary configuration occurs without distortions. For the liquids with relative permittivity close to five, the wave breaking time reaches a minimum, i.e., the collapse of the surface waves for such liquids occurs most intensively., Comment: 7 pages, 5 figures

The investigation of the He-induced W "fuzz" electrical properties was carried out. For the research, an automated experimental setup was designed. The setup was based on a vacuum chamber operated under high vacuum condition (~10-7 Pa). The vacuum diode under investigation comprised of a flat W "fuzz" cathode with an area of about 1 cm2 and a 2 mm radius cylindrical copper anode rounded with a 2 mm radius hemisphere. The cathode-anode distance was ~100 µm. The voltage applied was up to 10 kV. An HV power supply was controlled employing the DAC/ADC module "LCardTM E20-10". This module registered automatically currents and voltages of interest also. As the result, the data on the dielectric strength of the diode with the W "fuzz" cathode were obtained. The effect of the cathode conditioning was demonstrated. An increase in the breakdown electric field strength from about 4•105 V/cm for the very first breakdown up to 9•105 V/cm for the 20th breakdown was registered. In addition, the emissive characteristics of the W "fuzz" cathode were studied in accordance with the Fowler-Nordheim theory. The estimated value of the electric field enhancement factor β near the cathode surface was found to be ~60 before the breakdown and ~80 after the breakdown. The obtained β values were somewhat smaller than the ones obtained by other research groups for the nanostructured W cathodes. This fact should be investigated additionally.

This paper describes an overdetermined system of equations that describe three-dimensional layered unsteady flows of a. viscous incompressible fluid at a. constant pressure. Studying the compatibility of this system makes it possible to reduce it to coupled quasilinear parabolic equations for velocity components. The reduced equations allow constructing several classes of exact solutions. In particular, polynomial and spatially localized self-similar solutions of the equations of motion are obtained. The transition to the ideal fluid limit is investigated.

We analyze nonlinear dynamics of the Kelvin–Helmholtz quantum instability of the He-II free surface, which evolves during counterpropagation of the normal and superfluid components of liquid helium. It is shown that in the vicinity of the linear stability threshold, the evolution of the boundary is described by the |ϕ|4 Klein–Gordon equation for the complex amplitude of the excited wave with cubic nonlinearity. It is important that for any ratio of the densities of the helium component, the nonlinearity plays a destabilizing role, accelerating the linear instability evolution of the boundary. The conditions for explosive growth of perturbations of the free surface are formulated using the integral inequality approach. Analogy between the Kelvin–Helmholtz quantum instability and electrohydrodynamic instability of the free surface of liquid helium charged by electrons is considered.

Unsteady plane potential flows of an ideal incompressible fluid with a free surface in the absence of external forces and capillarity are studied. An algorithm to construct exact solutions for such flows has been proposed on the basis of the analysis of compatibility conditions of the equations of motion and an auxiliary complex transport equation. This algorithm makes it possible to significantly expand the list of known exact nontrivial solutions of the considered classical problem. This list has recently consisted of only a few Dirichlet solutions: flows for which the surface of the fluid is a parabola, ellipse, or hyperbola. This algorithm allows reproducing a recently found class of solutions specified by the Hopf equation for the complex velocity and finding a fundamentally new large class of solutions for which flows are described by the Hopf equation for the inverse complex velocity.

We have theoretically studied the molten metal jets formation during an operation of vacuum arc discharge for a wide range of crater sizes for near-threshold electric currents. At the initial stage of crater formation, a liquid-metal rim forms on the cathode surface under the action of the cathode spot plasma pressure. This process has been numerically simulated in the framework of the 2-D axisymmetric heat and mass transfer problem in the approximation of viscous incompressible liquid. At the next stage, jets form due to the loss of axial symmetry of the rim (the problem becomes essentially 3-D). The development of the Rayleigh–Plateau (RP) and Rayleigh–Taylor instabilities of the rim are analyzed in terms of dispersion relations for surface waves. It is shown that maximum increments correspond to the capillary instability of the RP type. The calculated time of rim formation and that of its breakup into jets have been demonstrated to be comparable. The analysis of rim perturbation spectrum has allowed to estimate probable number of liquid-metal jets.

The formation of jets is studied that accompanies the extrusion of a molten metal under the action of the plasma pressure from microcraters formed in the cathode spot during an operation of vacuum arc discharge. We have proposed an analytic model of the melt splashing that demonstrates the threshold behavior: the extrusion rate has to exceed some threshold that is caused by capillary forces. These forces, on the one hand, lead to the reverse motion of the extruded liquid with the Taylor–Culick speed and, on the other hand, to the azimuthal instability development of the liquid-metal rim on the crater contour. The threshold values of electric current flowing through the crater and plasma pressure, at which the formation of liquid-metal jets starts, are estimated.

Conditions for electron runaway in a gas diode with a blade cathode providing a strongly inhomogeneous distribution of the electric field in the interelectrode gap have been studied theoretically. It has been demonstrated that the character of electron runaway differs qualitatively for cathodes with a different rounding radius of the edges. In the case of a relatively large edge radius (tens of microns or more), the conditions for the transition of electrons to the runaway mode are local in nature: they are determined by the field distribution in the immediate vicinity of the cathode where the electrons originate from. Here, the relative contribution of the braking force acting on electrons in a dense gas reaches a maximum. This behavior is generally similar to the behavior of electrons in a uniform field. For a cathode with a highly sharpened edge, the relative contribution of the braking force is maximum in the near-anode region. As a consequence, the runaway condition acquires a nonlocal character: it is determined by the electron dynamics in the entire interelectrode gap.

In this paper, dynamics of runaway electrons was investigated in nitrogen of 1 atm pressure for the case of a strongly non-uniform electric field. 2D Monte-Carlo technique and 1D Boltzmann's equation were employed. The discharge system under investigation comprised of two confocal parabolic electrodes: a cathode and an anode. For such a discharge system, analytical expressions of potential distribution within the discharge gap are known. A curvature radius of the parabolic cathode was varied in a wide range. The distance between apexes of the electrode parabolas was constant. As a result, the dependences of the critical electric field strength $E_{cr}$ near the cathode and of the critical voltage $U_{cr}$ applied to the discharge gap from the cathode curvature radius were obtained. A comparison of the results obtained with analytical ones was carried out. It was found, that, for the Monte-Carlo and kinetic models, both $E_{cr}$ with an increase in the cathode curvature radius and $U_{cr}$ with a decrease in the cathode curvature radius tended to some constant values, what was in qualitative agreement with analytical data. It was clearly shown that a sharp-edged cathode required much more field strength near the cathode edge compared with the classical criterion to generate runaway electrons.

We analyze features of the burst of fast electrons arriving to the anode of elongated, air-filled gap with a sharply inhomogeneous electric field with a delay relative to the primary picosecond runaway electron flow. Sufficiently high energy of the secondary particles qualifies them also as runaway electrons. The secondary flow appearance suggests impact ionization of the gas in the main part of the gap by the primary runaway electrons, as well as the presence of a residual electric field and its dynamic transformation due to propagation of the fast ionization wave toward the anode. Considering this concept, one can explain the sensitivity of the delay between two electron flows to the cathode-anode distance and to the regime of the ionization wave cut-off by an additional, floating-potential foil electrode partially transparent for primary electrons. The use of an ionization agent in the form of bremsstrahlung from a similar, tantalum foil electrode opaque to primary electrons leads to an earlier arrival of the secondary fast electrons to the anode.

Properties of the runaway electron flow (REF) generated in an air-filled high-voltage interelectrode gap with a drastically inhomogeneous electric field distribution are presented. We have shown that REF initiated near the tip of a conical cathode involves high-energy electron fraction with ultrashort length of about 10 ps. Analytical and numerical considerations show that the REF generation and cessation is determined by a combination of ionization processes in the gas in the vicinity of the cathode, resulting in multiplication of free electrons and expansion of the cathode plasma layer, and dynamic processes that switch on/off a threshold electric field at the anode side plasma border which is sufficiently high for electrons to become runaway. The characteristic time of REF generation and termination can be estimated from the rate of ionization at a near-threshold field as ~2 ps. This elementary time determines the pulse length for the runaway electron current that was calculated to be ~6 ps, which is close to its experimental value.

We consider the nonlinear dynamics of the free surface of a conducting fluid (liquid metal for applications) in an applied vertical electric field for the case of confined axisymmetric geometry: the region occupied by the fluid is limited laterally. Amplitude equation describing the evolution of a fluid boundary is derived in the weakly nonlinear approximation. In the framework of this equation, it is shown that quadratic nonlinearities accelerate the development of electrohydrodynamic instability. Also, it is demonstrated both numerically and analytically that high-order nonlinearities play a destabilizing role. Consequently, the instability takes an explosive character: the surface perturbation amplitude grows faster than exponentially.

A problem of an unsteady plane flow of an ideal incompressible fluid with a free boundary is considered. It is shown that the solution can be found by using a complex transport equation. In this case, the problem is linearized by means of the hodograph transform (the velocity components are chosen as independent variables). Examples of exact solutions are obtained. Various scenarios of formation of singularities on the free boundary within a finite time are considered.

Nonlinear dynamics of the interface of dielectric liquids under the conditions of suppression of the Kelvin–Helmholz instability by a tangential electric field has been investigated. Two broad classes of exact analytical solutions to the equations of motion describing the evolution of spatially localized and periodic interface perturbations have been found. Both classes of solutions tend to the formation of strong singularities: interface discontinuities with formally infinite amplitudes. The discontinuity sign is determined by the sign of liquid velocity jump at the interface.

Conical singularities (spikes) appearing on the surface of a liquid with ionic conduction in the electric field have been studied with allowance for the influence of bulk and surface charges. It is established that the problem of field distribution admits an exact analytical solution that corresponds to the regime of charge-limited current. Dependences of the saturation current passing via a stationary conical surface singularity on the cone angle, dielectric permittivity of liquid, and mobility of positive and negative ions are determined.

Flow of runaway electrons (RAEs) propagating in a radial, air-filled gap of coaxial line (CL) changes the dynamics of breakdown in the field of traveling voltage pulse. However, despite the effect of RAEs, breakdown does not occur if subnanosecond pulse is less in duration and amplitude than some values. In this work, we study the influence of an external axial magnetic field (B z) on the breakdown development. We demonstrate the transformation of the voltage pulse reflection from the ionized (breakdown) zone with changing B z. Due to gyration of fast electrons in an applied magnetic field, the gas region ionized by RAEs does not reach the anode. The ionized bridge between the cathode and anode is gradually replaced by a near-cathode plasma layer representing a discrete, reflecting/absorbing inhomogeneity in the CL.

The dynamics of runaway electrons in a gas diode in a sharply nonuniform electric field determined by the geometry of electrodes is considered. The analytical solution of the equation of motion of electrons for an edge cathode shows that their runaway at the periphery in the region of weak field is possible only if the applied potential difference exceeds a certain threshold determined by the interelectrode distance and the parameters of the gas. This condition supplements a classical runaway condition that the field strength at the emission edge of the cathode should be higher than a threshold value depending only on the parameters of the gas. According to our estimates, this new condition imposes higher requirements than the classical condition on the field strength in the limit of the strongly sharp edge of the cathode.

We report on direct numerical simulation of quasi-one-dimensional bidirectional capillary-wave turbulence. Although nontrivial three-wave and four-wave resonant interactions are absent in this peculiar geometry, we show that an energy transfer between scales still occurs concentrated around the linear dispersion relation that is broadened by nonlinearity. The wave spectrum displays a clear wave number power-law scaling that is found to be in good agreement with the dimensionally prediction for capillary-wave turbulence involving four-wave interactions. The carried out high-order correlation analysis (bicoherence and tricoherence) confirms quantitatively the dominant role of four-wave quasi-resonant interactions. The Kolmogorov-Zakharov spectrum constant is also estimated numerically. We interpret our results as the first numerical observation of anisotropic capillary-wave turbulence in which four-wave interactions play a dominant role, Comment: 8 pages, 5 figures

We consider the behavior of the free surface of an ideal dielectric liquid in an applied normal electric field for the case of confined axisymmetric geometry of the system. The quadratic nonlinear amplitude equation which describes the evolution of the boundary is derived in the framework of the Hamiltonian formalism. According to this equation, the hard regime of excitation of electrohydrodynamic instability is always realized. Also, it is shown that the part of the potential energy functional which is responsible for the higher-order nonlinearities is negatively defined if the dielectric constant of the liquid is sufficiently large, e > 2.78. Under this condition, the growth of surface perturbations has an explosive character.

The nonlinear dynamics of the interface between ideal dielectric fluids in the presence of tangential discontinuity of the velocity at the interface and the stabilizing action of the horizontal electric field is examined. It is shown that the regime of motion of the interface where liquids move along the field lines occurs in the state of neutral equilibrium where electrostatic forces suppress Kelvin–Helmholtz instability. The equations of motion of the interface describing this regime can be reduced to an arbitrary number of ordinary differential equations describing the propagation and interaction of structurally stable solitary waves, viz. rational solitons. It is shown that weakly interacting solitary waves recover their shape and velocity after collision, whereas strongly interacting solitary waves can form a wave packet (breather).

We have theoretically studied the dynamics of molten metal during crater formation in the cathode spot of vacuum arc discharge. At the initial stage, a liquid-metal ridge is formed around the crater. This process has been numerically simulated in the framework of the two-dimensional axisymmetric heat and mass transfer problem in the approximation of viscous incompressible liquid. At a more developed stage, the motion of liquid metal loses axial symmetry, which corresponds to a tendency toward jet formation. The development of azimuthal instabilities of the ridge is analyzed in terms of dispersion relations for surface waves. It is shown that maximum increments correspond to instability of the Rayleigh–Plateau type. Estimations of the time of formation of liquid metal jets and their probable number are obtained.

Possible equilibrium configurations of the free surface of a jet of an ideally conducting liquid placed in a nonuniform magnetic field are considered. The magnetic field is generated by two thin wires that are parallel to the jet and bear oppositely directed currents. Equilibrium is due to a balance between capillary and magnetic forces. For the plane symmetric case, when the jet deforms only in the plane of its cross section, two one-parameter families of exact solutions to the problem are derived using the method of conformal mapping. According to these solutions, a jet with an initially circular cross section deforms up to splitting into two separate jets. A criterion for jet splitting is derived by analyzing approximate two-parameter solutions.

Exact solutions are obtained for the problem of an equilibrium configuration of an uncharged cylindrical jet of a conducting liquid in a transverse electric field. The transverse cross section of the jet moving between two planar electrodes is deformed under the action of electrostatic forces (capillary forces play a stabilizing role). According to the solutions obtained, the initially circular cross section of the jet may be significantly (formally, unboundedly) stretched along the lines of forces of the field, and the boundaries of the jet asymptotically approach the electrodes.

The formation of microjets is studied during the extrusion of a melted metal by the plasma pressure from craters formed on a cathode in a burning vacuum arc. An analytic model of liquid-metal splashing that includes two stages is proposed. At the first stage, the liquid motion has the axial symmetry and a liquid-metal wall surrounding the crater is formed. At the second stage, the axial symmetry is broken due to the development of the Plateau–Rayleigh instability in the upper part of the wall. The wall breakup process is shown to have a threshold. The minimal plasma pressure and the minimal electric current flowing through the crater required for obtaining the liquid-metal splashing regime are found. The basic spatial and temporal characteristics of the jet formation process are found using the analytic model.

High-voltage picosecond breakdown of an atmospheric-pressure air-filled coaxial line in the radial electric field of a propagating transverse electromagnetic wave has been studied both experimentally and theoretically. On the one hand, we demonstrate that gas preionization by runaway electrons (RAEs) plays a decisive role in the breakdown development process: the breakdown delay time drastically increases in the absence of RAEs. On the other hand, it is established that, for sufficiently short pulses, the radial gap switching process does not have enough time to develop even in a situation where the RAE flow is effectively generated. Fundamental limitations on the breakdown strength of gas coaxial feeders imposed by this effect are discussed.

We investigate the formation of a secondary flow of runaway electrons with energies on the order of 100 keV in an air gap after the passage of a primary picosecond flow of similar fast particles, arising in the cathode region where the distribution of the electric field is very inhomogeneous. The dependence of the secondary flow onset delay on the gap length and the field strength is demonstrated. The reasons for the discrimination of the two flows in time are considered analytically. The occurrence of fast secondary electrons in an anode-located part of the gap is demonstrated under conditions when the ionization wave generated in the cathode region by the primary runaway electrons is cutoff. For this, an intermediate aluminum foil electrode under a floating potential has been used. This third electrode, which divides the gap into two sections, is partially transparent for the runaway electrons. It has been also shown that runaway electrons demonstrate the earliest arrival to the anode if they arise due to the photoionization of the gas by bremsstrahlung of the primary particles bombarding an intermediate electrode made of tantalum foil.

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 behavior of the free surface of a perfectly conducting liquid in an external uniform electric field is considered in the framework of the Hamiltonian formalism for the case of bounded axisymmetric geometry of the system (the fluid is bounded by a cylindrical rigid wall). Taking into account the influence of quadratic nonlinearities, we derive an amplitude equation which describes the evolution of the boundary. Using this equation, we find the condition for the hard excitation of boundary instability that leads to an explosive growth of surface perturbations. The differences in the description of the dynamics of axisymmetric perturbations of the boundary from the cases of plane, square, and hexagonal symmetries of the problem are discussed.

The formation of jets is studied during the extrusion of a melted metal by the plasma pressure from microcraters formed in the cathode spot of a burning vacuum arc. An analytic model of liquid-metal splashing is proposed. At the initial stage, a axially symmetric liquid-metal wall is formed around the crater. Its radius is determined by comparing the rate of liquid extrusion from a forming crater and the Taylor-Culick velocity of the reverse motion of the toroidal rim of a liquid-metal wall. At the developed stage, the motion of liquid metal loses axial symmetry, which corresponds to a tendency to jet formation. According to the proposed model, this is due to the development of the Plateau-Rayleigh instability of the rim. The basic spatial and temporal parameters of the jet formation process are calculated.

Molten metal jets formation during operation of vacuum arc discharge for a wide range of cathode spot cell sizes and electric currents has been studied. In our model, this process has been divided into two stages. At the initial one, the melt is extruded from the microcrater on the cathode surface under the action of plasma pressure. As a result, a liquid-metal toroidal rim is formed around the crater. This process has been numerically simulated in the framework of the two-dimensional axisymmetric heat and mass transfer problem in the approximation of viscous incompressible liquid. At the next stage, the geometry of the problem loses axial symmetry due to azimuthal instabilities development (the liquid metal rim breaks into jets). The development of the instabilities of the rim is analyzed in terms of dispersion relations for surface waves. It is shown that maximum increments correspond to the capillary instability of the Plateau-Rayleigh type. The time of formation of liquid metal jets and their probable number are estimated.

Possible equilibrium configurations of the surface of a drop of a conducting liquid, moving relative to a dielectric medium at a certain angle to the external electric field, are considered in the 2D formulation. A two-parametric family of exact particular solutions to this problem is obtained using the conformal mapping method. These solutions are characterized by a considerable strain of the 2D drop surface (the highest possible ratio of its longitudinal and transverse sizes is 11/2).

The nonlinear dynamics of the free surface of an ideal dielectric liquid that is exposed to an external oblique electric field has been studied theoretically. In the framework of the Hamiltonian formalism, a system of nonlinear integro-differential equations has been derived that describes the dynamics of nonlinear waves in the small-angle approximation. It is established that for a liquid with high dielectric permittivity, these equations have a solution in the form of plane waves of arbitrary shape that propagate without distortion in the direction of the horizontal component of the external field.

Processes of propagation and interaction of nonlinear gravity-capillary waves on the free surface of a deep non-conducting liquid with high dielectric constant under the action of a tangential electric field are numerically simulated. The computational method is based on the time-dependent conformal transformation of the region occupied by the fluid into a half-plane. In the limit of a strong electric field, when the gravitational and capillary forces are negligibly small, there exists an exact analytical solution of the electro-hydrodynamic equations describing propagation without distortions of nonlinear surface waves along (or against) the electric field direction. In the situation where gravity and capillarity are taken into account, the results of numerical simulations indeed show that, for large external field, the waves traveling in a given direction tend to preserve their shape. In the limit of a strong electric field, the interaction of counter-propagating waves leads to the formation of regions, where the electrostatic and dynamic pressures undergo a discontinuity, and the curvature of the surface increases infinitely. The Fourier spectrum of the surface perturbations tends to the power-law distribution (k^-2). In the case of a finite electric field, the wave interaction results in a radiation of massive cascade of small-scale capillary waves that causes the chaotic behavior of the system. The investigated mechanism of interaction between oppositely-traveling waves can enhance development of the capillary turbulence of the fluid surface., Submitted

The dynamics of a bubble in a dielectric liquid under the influence of a uniform external electric field is considered. It is shown that in the situation where the boundary motion is determined only by electrostatic forces, the special regime of fluid motion can be realized for which the velocity and electric field potentials are linearly related. In the two-dimensional case, the corresponding equations are reduced to an equation similar in structure to the well-known Laplacian growth equation, which, in turn, can be reduced to a finite number of ordinary differential equations. This allows us to obtain exact solutions for asymmetric bubble deformations resulting in the formation of a finite-time singularity (cusp).

Nonlinear dynamics of the free surface of finite depth non-conducting fluid with high dielectric constant subjected to a strong horizontal electric field is considered. Using the conformal transformation of the region occupied by the fluid into a strip, the process of counter-propagating waves interaction is numerically simulated. The nonlinear solitary waves on the surface can separately propagate along or against the direction of electric field without distortion. At the same time, the shape of the oppositely traveling waves can be distorted as the result of their interaction. In the problem under study, the nonlinearity leads to increasing the waves amplitudes and the duration of their interaction. This effect is inversely proportional to the fluid depth. In the shallow water limit, the tendency to the formation of a vertical liquid jet is observed.

A relative motion of the normal and superfluid components of Helium II results in Kelvin-Helmholtz instability (KHI) at their common free surface. We found the exact solutions for the nonlinear stage of the development of that instability. Contrary to the usual KHI of the interface between two fluids, the dynamics of Helium II free surface allows decoupling of the governing equations with their reduction to the Laplace growth equation which has the infinite number of exact solutions including the formation of sharp cusps at free surface in a finite time., 5 pages, 1 figure