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An algorithm of the search for the best (in a sense) cubature formulas on a sphere that are invariant with respect to the transformations of the icosahedral group of rotations with inversion is described. This algorithm is applied to finding the parameters of all the best cubature formulas of this symmetry type up to the 79th order of accuracy. The parameters of the new cubature formulas of the 21st, 25th and 29th orders of accuracy to 16 significant digits are given.

In this paper we investigate the two-velocity stationary hydrodynamics system with a single pressure and inhomogeneous divergent and boundary conditions for the two velocities. This system is overdetermined. By replacing the unknown functions, the problem is reduced to a homogeneous one. The solution of the resulting system is reduced to the consecutive solutions of the two boundary value problems: the Stokes problem for a single velocity and pressure, and overdetermined system for the other velocity. We present the generalized statements of these problems and their discrete approximation using the finite element method. To solve the overdetermined problem we apply a version of the regularization methods.

We say that the square matrices A and B are of the same order quasi-commute if AB = σ BA for some scalar σ. Classical relations of commutation and anti-commutation are particular cases of this definition. We give a complete description of pairs of the quasi-commuting Toeplitz and Hankel matrices for σ ≠ ± 1.

It is proved that the uniform Lebesgue constant (the norm of a linear operator from C to C) of local cubic splines with equally-spaced knots, which preserve cubic polynomials, is equal to 11/9.

The distribution of air oxygen atoms in the oxidation products of rich mixtures of syngas with air in the flame and the under autoignition conditions at constant volume was investigated by numerical simulation using the tracer method. It was found that in rich mixtures, the oxidation of hydrogen and carbon oxide occurs in stages, which is clearly visible in the profiles of the rates of formation of H₂O and CO₂. The observed stagewise nature inevitably results in the heat release rate occurring in stages. The conversion pathways and the role of the oxygen atom of the CO molecule in the heat release in these flames were investigated.

This paper describes the mathematical simulation of an industrial membrane reactor for propane dehydrogenation in the thermodynamic conjugation with hydrogen combustion (oxidation). Due to the effective removal of hydrogen through a membrane and the heat release as a result of an exothermic reaction, the temperature of the reaction stream at the inlet could be reduced to 500 gr. The fact that the process is carried out on an industrial-level membrane reactor makes it possible to reach a propane conversion of 75% with a propylene selectivity of 97%, which exceeds the figures obtained per pass in existing industrial device at higher temperatures.

A method for performance optimization of a straight-scheme hybrid rocket motor is considered. The method is based on inserting an additional amount of an oxidizer into the solid propellant with a prescribed distribution of the oxidizer mass fraction along the solid propellant charge. An analytical dependence is derived for the oxidizer fraction distribution that ensures uniform combustion and high efficiency of the solid propellant charge.

Solution is obtained to the problem of determining the conditions of hydrodynamic stability in the presence of flow of gaseous combustion products blowing over the propellant gasification surface. The blowing flow has a low gradient along the direction of motion. Analysis of the obtained dispersion equation shows the development of hydrodynamic instability with fluctuations. The transport coefficients of natural turbulence are presented in the form of the sum of two terms: the first of these is responsible for the transfer in the absence of blowing, and the second takes into account the enhancement of transport processes in blowing. Their dependence on the initial temperature of solid propellant combustion predicts a reduction in the negative erosion effect in accordance with experimental data. Accounting for the relaxation time of the evaporation process has a stabilizing effect. In the limit of strong relaxation, this leads to oscillatory stability (in the absence of blowing), the perturbations do not grow and do not decay. However, arbitrarily weak blowing leads to instability.

A mathematical model of combustion of boron particles in a ram-rocket engine is developed. The boron combustion efficiency for one-stage and two-stage injection of air into the afterburning chamber is calculated. It is demonstrated that two-stage injection of air sometimes allows the time of complete combustion of boron particles to be significantly reduced (by a factor of 1.5-3); thus, the fuel combustion efficiency in the ram-rocket engine can be increased. The simulated results are consistent with available experimental data.

The combustion of a highly exothermic mixture of calcium chromate with aluminum and boron has been studied. It has been shown that these mixtures are able to burn in a wide range of ratios of reactants. The autowave chemical conversion is accompanied by the decomposition of calcium chromate, the chemical reaction of the decomposition products with aluminum and boron, the formation of a two-phase melt of the combustion products with its subsequent gravity separation, and crystallization of the layers. Results of the study may beuseful for obtaining chromium borides.