The aim of the research is the derivation of the isothermal principle of virtual work for an elastic solid. Methodology. By an elastic solid we mean a mathematical model of a continuous medium, in which the specific density of free energy is a function of only temperature and derivatives of displacement vectors. In the framework of equilibrium thermodynamics, the isothermal principle of virtual work for an elastic solid is obtained. Results. The isothermal principle of virtual work for an elastic solid is derived. The conditions for mechanical equilibrium inside and on the surface of an elastic solid are obtained. Research implications. For a given function of the specific density of free energy and the known value of the temperature, it is possible to determine the dependence of the displacement functions on the coordinates that characterize the deformed state of the considered elastic solid.

The aim is to study a model of a developed turbulent flow of an incompressible fluid in a boundary layer on a plate. The study is based on the analysis of the Navier-Stokes equations describing the behaviour of Tollmien-Schlichting wave amplitudes in a single-mode approximation. Methodology. A weakly nonlinear version of the wave model of a developed turbulent boundary layer is considered. The dispersion characteristics of the waves of the least damped mode are determined, and the conditions of multiple three-wave resonance of this mode of Tollmien-Schlichting waves are analyzed. Equations for the coherent part are obtained using the many-scale method. Results. In the discrete representation of the coherent structure, it is shown that the sum of the squares of the moduli of the wave amplitudes in the state of multiple three-wave resonance, multiplied by real weight factors, is an invariant of the original dynamical system. Research implications. The invariant of the dynamic system is normalized to unity, and the Birkhoff-Khinchin theory is formulated.

The aim of this work is to study the distribution of the thermal field inhomogeneity in the melt along the crystal growth axis for different growth methods. Methodology. Within the framework of the SolidWorks Premium complex, in the Flow Simulation package for the constructed geometric models of the crystallization apparatus of the Bridgman, Czochralski and HDC (horizontal directional crystallization) methods, a curvilinear block-structured base mesh was formed. The conjugate problem was solved in the thermal unit and the water-cooled jacket to study the processes of heat-mass transfer and the distribution of the thermal field inhomogeneity in the melt along the crystal growth axis for different growth methods. Results. The results of computer simulation of the distribution of temperature profiles over the vertical and cross sections of the crystals grown by the Bridgman and Czochralski methods are presented. Pictures of the characteristic arrangement of facets are obtained. The obtained data fully agree with the known corresponding experimental results for yttrium-aluminum garnet crystals. Research implications. The analysis of the obtained results of computer simulation, carried out on the basis of consideration of the general theory of crystal growth mechanisms and morphological instability of the crystallization front for the three main methods of growth, confirmed the correctness of the obtained model pictures of the appearance of faceted growth forms.

The aim of this work is to analyze the dependence of the electrical conductivity of a thin metal film on the nature of the dependence of the specularity coefficient on the angle of incidence of electrons on the metal surface. The dependence of the specularity coefficient on the angle of incidence is considered, satisfying the Andreev condition. Methodology. A model boundary condition is considered, taking into account the dependence of the specularity coefficient on the angle of incidence of electrons on the metal surface. Results. The considered model of the boundary conditions in the limiting cases transforms into the Fuchs boundary condition and into the Soffer boundary condition. In addition, the considered boundary condition, in contrast to the Fuchs and Soffer boundary conditions, satisfies the Andreev criterion for an almost tangential incidence of electrons on the metal surface. Research implications. The results obtained can find application in the study of the interaction of electromagnetic radiation with a thin metal film, wires, in small metal particles and in describing the skin effect in a metal.

The theory of evolution of the early universe predicts the formation of black holes of different mass. Depending on the physical processes, black holes can be formed with masses up to 1027 g. Because of the Hawking effect of quantum evaporation of black holes, black holes with masses over 1015 g have survived to our time. The aim of this work is to calculate the intensity of gamma radiation from the primordial black holes, to explain the diffuse gamma spectrum of the Universe’s radiation from them in the range of 10-100 MeV, and to estimate their contribution to dark matter of the Universe. Methodology. The intensity of radiation from an ensemble of black holes is calculated by using an approximation in the form of the δ-function for the radiation of the absolutely black body. Results. It is shown that if the function of the masses of primordial black holes is N(M) = K · M-γ, the intensity of Hawking’s radiation is I(ν) ~ νγ. Since the background radiation of the Universe in the range of quantum energies 10-100 MeV has the form of I(ν) ~ ν-1,3 MeV/cm2 MeV ⋅ ster, then N(M) = K ⋅ M1,3. In this range there emit primary black holes with masses of 1015g and sizes of an electron. Research implications. It is shown that the extrapolation of the resulting distribution of masses to the values of 5·1021g allows one to explain the observed mass of dark matter in the Universe. At the same time, the concentration of these black holes with masses comparable to the masses of asteroids is such that there may be hundreds of them in the solar system.

Aim: Development of new principles of controllable optical elements - diffractive, focusing and diffractive-focusing - in a wide range of their application, including both for optical and infrared radiation. Methodology. It was shown that LC composites of 4-cyano-4-octyloxydiphenyl (8OCB) in borosiloxane (BS) matrices are promising for solving the problem. After distributing the LC (at temperatures above 55 °C - the melting temperature) spatially according to some predetermined rule, then cool and thereby fix the spatial distribution. The heating-cooling cycle is technically easy to implement. The LC composite is oriented in various ways, and in this work, it is proposed to use light orientation for this purpose. Results. A light-controlled LCD lens with adjustable parameters is proposed. Both the fundamental possibility of multiple rewriting of centrosymmetric phase plates with anisotropic orientation of molecules in a thin film, and the technique of recording elements with a cylindrically symmetric and planar symmetric distribution, which provides a smooth change in the optical axis, are considered. Research implications. When changing modes, uncontrolled elements have to be replaced, which is inconvenient. Controlled optical elements are usually expensive and volatile - they require constant monitoring of their state and field regulation. Consequently, optical elements of various (according to purpose and properties) classification are relevant, rewritable (re-adjustable) at discrete moments of time and non-volatile in the rest (possibly very long) period of operation.

The aim of this work is to obtain asymptotically strict expressions for the distribution functions of pairs of molecules inside the front of a shock wave. Methodology. The work makes use of asymptomatic methods of theoretical physics, based on the selection of small parameters. Results. An asymptotically accurate expression is obtained for the distribution of pairs of heavy-component molecules of a binary shock of a compressed mixture of gases at the beginning of its progressive relaxation. Research implications. This result is extremely significant for experimental simulation of a high-speed impact effect in shock tubes.