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Technical carbon (soot) is a product obtained by thermal decomposition (pyrolysis) of hydrocarbons (usually oil) in a stream of heat carrier gas. Technical carbon is widely used as a reinforcing component in the production of rubber and plastic masses. Tire production uses 70% of all carbon produced. In furnace carbon production, the liquid hydrocarbon feedstock is injected into the natural gas combustion product stream through nozzles. The raw material is atomized and vaporized with further pyrolysis. It is important for the raw material to be completely evaporated before the pyrolysis process starts, otherwise coke, that contaminates the product, will be produced. It is impossible to operate without mathematical modeling of the process itself in order to improve the carbon production technology, in particular, to provide the complete evaporation of the raw material prior to the pyrolysis process. Mathematical modelling is the most important way to obtain the most complete and detailed information about the peculiarities of reactor operation.

A three-dimensional mathematical model and calculation method for raw material atomization and evaporation in the thermal gas flow are being developed in the FlowVision software package PC. Water is selected as a raw material to work out the modeling technique. The working substances in the reactor chamber are the combustion products of natural gas. The motion of raw material droplets and evaporation in the gas stream are modeled in the framework of the Eulerian approach of interaction between dispersed and continuous media. The simulation results of raw materials atomization and evaporation in a real reactor for technical carbon production are presented. Numerical method allows to determine an important atomization characteristic: average Sauter diameter. That parameter could be defined from distribution of droplets of raw material at each time of spray forming.

In the last few years, significant progress has been observed in the field of rotating detonation engines for aircrafts. Scientific laboratories around the world conduct both fundamental researches related, for example, to the issues of effective mixing of fuel and oxidizer with the separate supply, and applied development of existing prototypes. The paper provides a brief overview of the main results of the most significant recent computational work on the study of propagation of a onedimensional pulsating gaseous detonation wave in a non-uniform medium. The general trends observed by the authors of these works are noted. In these works, it is shown that the presence of parameter perturbations in front of the wave front can lead to regularization and to resonant amplification of pulsations behind the detonation wave front. Thus, there is an appealing opportunity from a practical point of view to influence the stability of the detonation wave and control it. The aim of the present work is to create an instrument to study the gas-dynamic mechanisms of these effects.

The mathematical model is based on one-dimensional Euler equations supplemented by a one-stage model of the kinetics of chemical reactions. The defining system of equations is written in the shock-attached frame that leads to the need to add a shock-change equations. A method for integrating this equation is proposed, taking into account the change in the density of the medium in front of the wave front. So, the numerical algorithm for the simulation of detonation wave propagation in a non-uniform medium is proposed.

Using the developed algorithm, a numerical study of the propagation of stable detonation in a medium with variable density as carried out. A mode with a relatively small oscillation amplitude is investigated, in which the fluctuations of the parameters behind the detonation wave front occur with the frequency of fluctuations in the density of the medium. It is shown the relationship of the oscillation period with the passage time of the characteristics C₊ and C₀ over the region, which can be conditionally considered an induction zone. The phase shift between the oscillations of the velocity of the detonation wave and the density of the gas before the wave is estimated as the maximum time of passage of the characteristic C₊ through the induction zone.

This paper is dedicated to discussing methods of statistical modeling the outcomes of sport events and, particularly, matches with continuous time. We propose a simulation-based approach to predicting the outcome of a match, somehow medium between pure statistical methods and agent simulation of individual players. An example of retrospective prediction is given.

Numerical investigation of mixing non-isothermal streams of sodium coolant in a T-branch is carried out in the FlowVision CFD software. This study is aimed at argumentation of applicability of different approaches to prediction of oscillating behavior of the flow in the mixing zone and simulation of temperature pulsations. The following approaches are considered: URANS (Unsteady Reynolds Averaged Navier Stokers), LES (Large Eddy Simulation) and quasi-DNS (Direct Numerical Simulation). One of the main tasks of the work is detection of the advantages and drawbacks of the aforementioned approaches.

Numerical investigation of temperature pulsations, arising in the liquid and T-branch walls from the mixing of non-isothermal streams of sodium coolant was carried out within a mathematical model assuming that the flow is turbulent, the fluid density does not depend on pressure, and that heat exchange proceeds between the coolant and T-branch walls. Model LMS designed for modeling turbulent heat transfer was used in the calculations within URANS approach. The model allows calculation of the Prandtl number distribution over the computational domain.

Preliminary study was dedicated to estimation of the influence of computational grid on the development of oscillating flow and character of temperature pulsation within the aforementioned approaches. The study resulted in formulation of criteria for grid generation for each approach.

Then, calculations of three flow regimes have been carried out. The regimes differ by the ratios of the sodium mass flow rates and temperatures at the T-branch inlets. Each regime was calculated with use of the URANS, LES and quasi-DNS approaches.

At the final stage of the work analytical comparison of numerical and experimental data was performed. Advantages and drawbacks of each approach to simulation of mixing non-isothermal streams of sodium coolant in the T-branch are revealed and formulated.

It is shown that the URANS approach predicts the mean temperature distribution with a reasonable accuracy. It requires essentially less computational and time resources compared to the LES and DNS approaches. The drawback of this approach is that it does not reproduce pulsations of velocity, pressure and temperature.

The LES and DNS approaches also predict the mean temperature with a reasonable accuracy. They provide oscillating solutions. The obtained amplitudes of the temperature pulsations exceed the experimental ones. The spectral power densities in the check points inside the sodium flow agree well with the experimental data. However, the expenses of the computational and time resources essentially exceed those for the URANS approach in the performed numerical experiments: 350 times for LES and 1500 times for ·DNS.

Theoretical and experimental investigations of water vapor interaction with porous materials are carried out both at the macro level and at the micro level. At the macro level, the influence of the arrangement structure of individual pores on the processes of water vapor interaction with porous material as a continuous medium is studied. At the micro level, it is very interesting to investigate the dependence of the characteristics of the water vapor interaction with porous media on the geometry and dimensions of the individual pore.

In this paper, a study was carried out by means of mathematical modelling of the processes of water vapor interaction with suffering pore of the cylindrical type. The calculations were performed using a model of a hybrid type combining a molecular-dynamic and a macro-diffusion approach for describing water vapor interaction with an individual pore. The processes of evolution to the state of thermodynamic equilibrium of macroscopic characteristics of the system such as temperature, density, and pressure, depending on external conditions with respect to pore, were explored. The dependence of the evolution parameters on the distribution of the diffusion coefficient in the pore, obtained as a result of molecular dynamics modelling, is examined. The relevance of these studies is due to the fact that all methods and programs used for the modelling of the moisture and heat conductivity are based on the use of transport equations in a porous material as a continuous medium with known values of the transport coefficients, which are usually obtained experimentally.

The paper proposes a new mathematical model to study the interaction dynamics of the longitudinal wall of a narrow channel with its end seal. The end seal was considered as the edge wall on a spring, i.e. spring-mass system. These walls interaction occurs via a viscous liquid filling the narrow channel; thus required the formulation and solution of the hydroelasticity problem. However, this problem has not been previously studied. The problem consists of the Navier–Stokes equations, the continuity equation, the edge wall dynamics equation, and the corresponding boundary conditions. Two cases of fluid motion in a narrow channel with parallel walls were studied. In the first case, we assumed the liquid motion as the creeping one, and in the second case as the laminar, taking into account the motion inertia. The hydroelasticty problem solution made it possible to determine the distribution laws of velocities and pressure in the liquid layer, as well as the motion law of the edge wall. It is shown that during creeping flow, the liquid physical properties and the channel geometric dimensions completely determine the damping in the considered oscillatory system. Both the end wall velocity and the longitudinal wall velocity affect the damping properties of the liquid layer. If the fluid motion inertia forces were taken into account, their influence on the edge wall vibrations was revealed, which manifested itself in the form of two added masses in the equation of its motion. The added masses and damping coefficients of the liquid layer due to the joint consideration of the liquid layer inertia and its viscosity were determined. The frequency and phase responses of the edge wall were constructed for the regime of steady-state harmonic oscillations. The simulation showed that taking into account the fluid layer inertia and its damping properties leads to a shift in the resonant frequencies to the low-frequency region and an increase in the oscillation amplitudes of the edge wall.

The aim of the work is to study a monotone finite-difference scheme of the second order of accuracy, created on the basis of a generalization of the one-dimensional Z-scheme. The study was carried out for model equations of the transfer of an incompressible medium. The paper describes a two-dimensional generalization of the Z-scheme with nonlinear correction, using instead of streams oblique differences containing values from different time layers. The monotonicity of the obtained nonlinear scheme is verified numerically for the limit functions of two types, both for smooth solutions and for nonsmooth solutions, and numerical estimates of the order of accuracy of the constructed scheme are obtained.

The constructed scheme is absolutely stable, but it loses the property of monotony when the Courant step is exceeded. A distinctive feature of the proposed finite-difference scheme is the minimality of its template. The constructed numerical scheme is intended for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems in the solution of which such equations arise is the numerical simulation of highly nonstationary medium-scale processes in the earth’s ionosphere under conditions of the appearance of the Rayleigh – Taylor instability and plasma structures with smaller scales, the generation mechanisms of which are instabilities of other types, which leads to the phenomenon F-scattering. Due to the fact that the transfer processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma incompressibility condition is fulfilled in the direction transverse to the magnetic field.

The background social strain of a society can be quantitatively estimated using various statistical indicators. Mathematical models, allowing to forecast the dynamics of social strain, are successful in describing various social processes. If the number of interacting groups is small, the dynamics of the corresponding indicators can be modelled with a system of ordinary differential equations. The increase in the number of interacting components leads to the growth of complexity, which makes the analysis of such models a challenging task. A continuous social stratification model can be considered as a result of the transition from a discrete number of interacting social groups to their continuous distribution in some finite interval. In such a model, social strain naturally spreads locally between neighbouring groups, while in reality, the social elite influences the whole society via news media, and the Internet allows non-local interaction between social groups. These factors, however, can be taken into account to some extent using the term of the model, describing negative external influence on the society. In this paper, we develop a continuous social stratification model, describing the dynamics of two societies connected through migration. We assume that people migrate from the social group of donor society with the highest strain level to poorer social layers of the acceptor society, transferring the social strain at the same time. We assume that all model parameters are constants, which is a realistic assumption for small societies only. By using the finite volume method, we construct the spatial discretization for the problem, capable of reproducing finite propagation speed of social strain. We verify the discretization by comparing the results of numerical simulations with the exact solutions of the auxiliary non-linear diffusion equation. We perform the numerical analysis of the proposed model for different values of model parameters, study the impact of migration intensity on the stability of acceptor society, and find the destabilization conditions. The results, obtained in this work, can be used in further analysis of the model in the more realistic case of inhomogeneous coefficients.

In this paper, the system of equations of magnetic hydrodynamics (MHD) is considered. The exact solutions found describe fluid flows in a porous medium and are related to the development of a core simulator and are aimed at creating a domestic technology «digital deposit» and the tasks of controlling the parameters of incompressible fluid. The central problem associated with the use of computer technology is large-dimensional grid approximations and high-performance supercomputers with a large number of parallel microprocessors. Kinetic methods for solving differential equations and methods for «gluing» exact solutions on coarse grids are being developed as possible alternatives to large-dimensional grid approximations. A comparative analysis of the efficiency of computing systems allows us to conclude that it is necessary to develop the organization of calculations based on integer arithmetic in combination with universal approximate methods. A class of exact solutions of the Navier – Stokes system is proposed, describing three-dimensional flows for an incompressible fluid, as well as exact solutions of nonstationary three-dimensional magnetic hydrodynamics. These solutions are important for practical problems of controlled dynamics of mineralized fluids, as well as for creating test libraries for verification of approximate methods. A number of phenomena associated with the formation of macroscopic structures due to the high intensity of interaction of elements of spatially homogeneous systems, as well as their occurrence due to linear spatial transfer in spatially inhomogeneous systems, are highlighted. It is fundamental that the emergence of structures is a consequence of the discontinuity of operators in the norms of conservation laws. The most developed and universal is the theory of computational methods for linear problems. Therefore, from this point of view, the procedures of «immersion» of nonlinear problems into general linear classes by changing the initial dimension of the description and expanding the functional spaces are important. Identification of functional solutions with functions makes it possible to calculate integral averages of an unknown, but at the same time its nonlinear superpositions, generally speaking, are not weak limits of nonlinear superpositions of approximations of the method, i.e. there are functional solutions that are not generalized in the sense of S. L. Sobolev.

A promising method of increasing precipitation in arid climates is the method of creating a vertical high-temperature jet seeded by hygroscopic aerosol. Such an installation makes it possible to create artificial clouds with the possibility of precipitation formation in a cloudless atmosphere, unlike traditional methods of artificial precipitation enhancement, which provide for increasing the efficiency of precipitation formation only in natural clouds by seeding them with nuclei of crystallization and condensation. To increase the power of the jet, calcium chloride, carbamide, salt in the form of a coarse aerosol, as well as NaCl/TiO₂ core/shell novel nanopowder, which is capable of condensing much more water vapor than the listed types of aerosols, are added. Dispersed inclusions in the jet are also centers of crystallization and condensation in the created cloud to increase the possibility of precipitation. To simulate convective flows in the atmosphere, a mathematical model of FlowVision large-scale atmospheric flows is used, the solution of the equations of motion, energy and mass transfer is carried out in relative variables. The statement of the problem is divided into two parts: the initial jet model and the FlowVision large-scale atmospheric model. The lower region, where the initial high-speed jet flows, is calculated using a compressible formulation with the solution of the energy equation with respect to the total enthalpy. This division of the problem into two separate subdomains is necessary in order to correctly carry out the numerical calculation of the initial turbulent jet at high velocity (M > 0.3). The main mathematical dependencies of the model are given. Numerical experiments were carried out using the presented model, experimental data from field tests of the installation for creating artificial clouds were taken for the initial data. A good agreement with the experiment is obtained: in 55% of the calculations carried out, the value of the vertical velocity at a height of 400 m (more than 2 m/s) and the height of the jet rise (more than 600 m) is within an deviation of 30% of the experimental characteristics, and in 30% of the calculations it is completely consistent with the experiment. The results of numerical simulation allow evaluating the possibility of using the high-speed jet method to stimulate artificial updrafts and to create precipitation. The calculations were carried out using FlowVision CFD software on SUSU Tornado supercomputer.