Exact solutions of the equations of motion of liquid helium with a charged free surface

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 than 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 describe 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 change in the surface topology as a result of formation of charged bubbles.     

Exact partial solutions for the surface dynamics of a dielectric liquid with a charged surface in the gravitational field

We study the evolution of instability in the boundary of a perfect dielectric liquid with a free surface charge in an external electric field. Conformal variables are used to find exact partial solutions to the equations of motion for the case when the charge completely shields the field above the liquid, the electrostatic and gravitational forces being taken into account. The solutions describe the development of instability of the initially planar boundary until sharp dimples are formed on it.     

Exact solution of a static charged sphere in general relativity

An exact solution of the field equations of general relativity is obtained for a static, spherically symmetric distribution of charge and mass. Their physical properties are studied in some detail. Our solution includes as special cases the results given previously by Cooperstock and De La Cruz, Mehra and Bohra, Santos, and Shi-Chang.     

带电绳子在电荷体系下的平衡位形

推导了均匀带电绳子在固定电荷体系下受力平衡的微分方程,数值求解这个方程,计算发现不同参数下的绳子位形呈现凸曲线、花生状和多叶扭结曲线.     

Long-wavelength structure on a charged liquid surface

The problem of the equilibrium form of a charged surface of a dielectric liquid in a strong electric field, such that a flat surface becomes unstable, is studied. A periodic long-wavelength structure with a small amplitude can arise when the gap between the surface and a charged electrode is small compared with the capillary length and the charge completely screens the electric field. The equilibrium form of the surface is calculated assuming that the resulting wave is one-dimensional. The effect of the boundary conditions at the vessel walls on the dependence of the amplitude of the standing wave on the applied voltage is estimated. It is shown that this dependence is very sensitive to the conditions of contact between the vessel walls and the liquid. The possibility is discussed of using the theory developed in this paper to explain the experimental results obtained with a charged liquid-hydrogen surface. Zh. éksp. Teor. Fiz. 115, 43–49 (January 1999)     

Classical charged fluids at equilibrium near an interface: Exact analytical density profiles and surface tension

The structure of equilibrium density profiles in an electrolyte in the vicinity of an interface with an insulating or conductive medium is of crucial importance in chemical physics and colloidal science. The Coulomb interaction is responsible for screening effects, and in dilute solutions the latter effects give rise to universal leading corrections to nonideality, which distinguish electrolyte from nonelectrolyte solutions. An example is provided by the excess surface tension for an air-water interface, which is determined by the excess particle density, and which was first calculated by Onsager and Samaras. Because of the discrepancy between the dielectric constants on both sides of the interface, every charge in the electrolyte interacts with an electrostatic image, and the Boltzmann factor associated with the corresponding self-energy has an essential singularity over the length scalel from the wall. Besides Coulomb interactions, short-range repulsions must be taken into account in order to prevent the collapse between charges with opposite signs or between each charge and its image when the solvent dielectric constant is lower than that of the continuous medium on the other side of the interface. For a dilute and weaklycoupled electrolyte,l is negligible with respect to the bulk Debye screening length ξ_D. In the framework of the grand-canonical ensemble, systematic partial resummations in Mayer diagrammatics allow one to exhibit that, in this regime, the exact density profiles at leading order are the same as if they were calculated in a partially-linearized mean-field theory, where the screened pair interaction obeys an inhomogeneous Debye equation. In the latter equation the effective screening length depends on the distancex from the interface: it varies very fast over the lengthl and tends to its bulk value over a few ξDs. The equation can be solved iteratively at any distancex, and the exact density profiles are calculated analytically up to first order in the coupling parameter l/ξ_D. They show the interplay between three effects: (1) the geometric repulsion from the interface associated with the deformation of screening clouds, (2) the polarization effects described by the images on the other side of the interface, (3) the interaction between each charge and the potential drop created by the electric layer which appears as soon as the fluid has not a charge-symmetric composition. Moreover, the expressions allow us to go beyond Onsager-Samaras theory: the surface tension is calculated for charge-asymmetric electrolytes and for any value of the ratio between the dielectric constants on both sides of the interface. Similar diagrammatic techniques also allow one to investigate the charge renormalization in the dipolar effective pair interaction along the interface with an insulating medium.     

Stability of the equilibrium states of a charged bubble in a dielectric liquid

The stability of equilibrium states of a charged spherical bubble in dielectric fluid with respect to centrally symmetric variations of its volume is studied by analyzing a nonlinear equation describing radial oscillations of the bubble in the neighborhoods of its singularities. It is shown that only one of the two possible equilibrium states of the bubble is stable. The boundaries separating the domains in the physical parameter space are found that correspond to stable and unstable states. It is found that when the bubble carries an electric charge, the domains of physical parameters corresponding to equilibrium states of the bubble expand.     

Exact solution for a static charged gas sphere in general relativity

A general solution of the Einstein-Maxwell field equations has been obtained for a static charged gas sphere having maximum matter density at the centre. The density decreases along the radius and finally becomes zero at the surface of the sphere.     

Exact hierarchical solution of the damped oscillator problem

As a alternative to other exact approaches we present a procedure, in which the eigenvectors of a coupled oscillatory system are calculated in a hierarchically exact (N) manner. In the system a single oscillator is coupled to a dense sequence of background oscillators, which among themselves are not mutually coupled (Oscillatory Fano problem). The eigenvectors beeing known, all decay laws become computable. Two of them are given. It is strongly emphasized thatquite different decay laws evolve fordifferent initial conditions.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Exact solution of the standard transfer problem in a stellar atmosphere

We come back to the analytical solution of the standard transfer problem in a stellar atmosphere. It consists in solving the radiative transfer equation in a homogeneous and isothermal plane-parallel atmosphere, with light scattering taken as isotropic and monochromatic. The literature on the subject is reviewed and the existing solution in a finite slab is improved thanks to the introduction of non classical auxiliary functions. Eleven-figure tables of the solution are given for typical values of the input parameters currently met in stellar atmospheres.     

Exact solution for the gravitational field of a charged,magnetized, spinning mass

A family of solutions of the Einstein-Maxwell field equations is presented, corresponding to the exterior of stationaryaxisymmetric sources with charge, mass, angular momentum, and magnetic dipole moment. The Riemann tensor vanishes asymptotically for each member of the family; some solutions are asymptotically flat and some have NUT-like behavior asymptotically. For the asymptotically flat solutions, the gyromagnetic ratio may vary from zero to one. The corresponding value for the Kerr-Newman solution is one. A method for generating infinite chains of families of solutions of the Einstein-Maxwell equations is described.     

Exact solution to the averaging problem in cosmology

The exact solution of a two-scale Buchert average of the Einstein equations is derived for an inhomogeneous universe that represents a close approximation to the observed universe. The two scales represent voids, and the bubble walls surrounding them within which clusters of galaxies are located. As described elsewhere [New J. Phys. 9, 377 (2007)10.1088/1367-2630/9/10/377], apparent cosmic acceleration can be recognized as a consequence of quasilocal gravitational energy gradients between observers in bound systems and the volume-average position in freely expanding space. With this interpretation, the new solution presented here replaces the Friedmann solutions, in representing the average evolution of a matter-dominated universe without exotic dark energy, while being observationally viable.     

Exact solution of the Landau-Lifshitz equations for a radiating charged particle in the Coulomb potential

We solve exactly the classical non-relativistic Landau-Lifshitz equations of motion for a charged particle moving in a Coulomb potential, including radiation damping. The general solution involves the Painlevè transcendent of type II. It confirms our physical intuition that a negatively charged classical particle will spiral into the nucleus, supporting the validity of the Landau-Lifshitz equation.     

On the stability of a weakly charged surface of liquid helium

The traditional approach to the development of instability of a weakly charged helium surface needs correction. It is shown that the well-known electrostatically “equipotential” Frenkel-Tonks scenario should be transformed to a more general sequence of events that would remain reasonable when the 2D charge density tends to zero. Under these conditions, the priorities change and the instability development through nucleation (with the formation of separate multicharged dimples) becomes preferable. The experiment qualitatively confirms the predictions of the theory.     

Shear flow of a nematic liquid crystal near a charged surface

The maximum alignment angle θ_eff and the effective rotational viscosity coefficient γ _i ^eff of polar liquid crystals, such as 4-n-octyloxy-4′-cyanobiphenyl (8OCB), are investigated in the vicinity of charged bounding surfaces. The quantities θ_eff and γ ₁ ^eff are calculated in the framework of the Ericksen-Leslie theory. The results of calculations demonstrate that, for a homeotropic alignment of molecules on charged indium tin oxide surfaces, the effective rotational viscosity coefficient γ _i ^eff can increase by 7.8% as compared to the bulk rotational viscosity coefficient γ₁.     

Particular solution of self-consistent-field equations of charged liquid

The system of Maxwell equations and Euler equations describing the interaction of a charged liquid with its intrinsic electromagnetic field reduces to nonlinear equations of a vector field. A class of accurate particular solutions is obtained for equations of motion in which, for media with a constant invariant density, the field equations transform to equations of Prock type.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 55–58, June, 1978.It remains to thank Yu. S. Vladimirov, V. N. Mel'nikov, N. V. Mitskevich, and S. I. Syrovatskii for discussions of the problem and for useful remarks, and also I. K. Fetisov for supporting the work.