Chladni figures of a circular plate floating in the bounded and unbounded water basins with the cantilevered central support

In the case of circular symmetry, we conduct numerical and analytical studies of Chladni modes of an elastic plate floating on the liquid surface and cantilevered at the center to a vertical support. Using the theory of long waves in shallow water and the approximation of the Euler beam vibrations for the bounded and unbounded water basins, we obtain the dependence of the natural and quasinatural frequencies of Chladni figures on the geometric parameters of the plate and the vibration region in presence of the bottom unevenness.     

Focusing of radiation from a source of longitudinal waves,located on a free surface

Propagation of waves from a source located on a free surface inside a circular conical horn is studied within the framework of a three-dimensional axisymmetric acoustic approximation. The horn axis is assumed to be orthogonal to the free surface. The influence of the horn geometry on the efficiency of radiation focusing in an arbitrary circular cone is studied. Criteria, objective functions, and control parameters for efficiency estimations and horn optimization are proposed. A method of optimizing the radiating system consisting of the source on the free surface and the horn on the basis of the problem geometry is developed. Geometric parameters ensuring the best focusing of radiation of the source-horn system in a circular cone for an arbitrary transmission angle are determined.     

Interaction between glow discharge plasma and dust particles

The effect of dust particle concentration on gas discharge plasma parameters was studied through development of a self-consistent kinetic model which is based on solving the Boltzmann equation for the electron distribution function. It was shown that an increase in the Havnes parameter causes an increase in the average electric field and ion density, as well as a decrease in the charge of dust particles and electron density in a dust particle cloud. Self-consistent simulations for a wide range of plasma and dust particle parameters produced several scaling laws: these are laws for dust particle potential and electric field as a function of dust particle concentration and radius, and the discharge current density. The simulation results demonstrate that the process of self-consistent accommodation of parameters of dust particles and plasma in condition of particle concentration growth causes a growth in the number of high-energy electrons in plasma, but not to depletion of electron distribution function.     

Radial Distributions of Dusty Plasma Parameters in a DC Glow Discharge

A non‐stationary non‐local kinetic model for radial distributions of dusty plasma parameters based on the solution of Boltzmann equation for electron energy distribution function is presented. Electrons and ions production in ionizing collisions and their recombination on dust particle surface were taken into account. The drift‐diffusion approximation for ions was used. To obtain the self‐consistent radial distribution of electric potential the Poisson equation was used. It is shown that at high dust particle density the recombination of electrons and ions can exceed their production in ionization collisions in the region of dusty cloud. In this case the non‐monotonous radial distribution of the electric field is formed, the radial electric field becomes reversed and the radial electron and ion fluxes change their direction toward the center of the tube (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The influence of dust particle geometry on its charge and plasma potential

In this paper, a self‐consistent numerical model describing the behaviour of plasma around isolated highly charged dust particles with different shapes of rotation figures is presented. Dust particles in the form of a sphere, oblate ellipsoids (disk‐like particles), and elongated ellipsoids (rod‐like particles) are considered in the presence of an external electric field. Using the developed model, self‐consistent distributions of a space charge and plasma potential are obtained around non‐spherical dust particles. These distributions are carefully analysed by decomposing them in a series of Legendre polynomials. Decompositions of these distributions are compared with particles of different geometry. In addition, for different geometries of dust particles, the dependencies of the charge of a dust particle on geometry in the absence of an external field are investigated.     

Two fixed-point theorems

Two fixed point theorems implementing a more general principle for partially ordered sets (which is also due to this author) are proved. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 920–923, December, 1999.     

Axial-radial acoustic eigenoscillations near a thin-walled obstacle in a cylindrical channel with stepped narrowings

The dependencies are investigated of eigenfrequencies and eigenfunctions of acoustic axial-radial oscillations near a thin-walled obstacle in a channel with stepped narrowings on the geometric parameters of the oscillation region. It is discovered that, near the thin-walled cylindrical obstacles in an inhomogeneous cylindrical channel with the stepped two-sided cylindrical narrowing, the number of eigenfrequencies of the acoustic axial-radial oscillations of a gas may increase. The dependencies are obtained of eigenfrequencies on the geometric parameters of obstacles and the inhomogeneities of the channel.     

Conditional extrema of functionals in topological linear spaces

A theorem of Lyusternik on conditional extrema of a differentiable functional defined on a Banach space is generalized to the case of a topological linear space.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 375–382, Septembers 1973.     

Optimizing an oscillation absorber in a combustion chamber

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 79–88, September–October, 1993.