Vol 25, No 3 (2023)

In a Banach space, using branching theory methods, a periodic solution of a linear inhomogeneous differential equation with a small perturbation at the derivative (perturbed equation) is constructed. Under the condition of presence of a complete generalized Jordan set, the uniqueness of this periodic solution is proven. It is shown that when a small parameter is equal to zero and certain conditions are met, the periodic solution of the perturbed equation transforms into the family of periodic solutions of the unperturbed equation. The result is obtained by representing the perturbed equation as an operator equation in Banach space and applying the theory of generalized Jordan sets and modified Lyapunov-Schmidt method. As is known, the latter method reduces the original problem to study of the Lyapunov-Schmidt resolving system in the root subspace. In this case, the resolving system splits into two inhomogeneous systems of linear algebraic equations, that have unique solutions at ε ≠ 0, and 2n parameter families of real solutions at ε = 0 respectively.

This paper studies mathematical models for a joint calculation of electric and thermal fields in electrochemical systems (in electrolytes). As is known, passing of electric current through an electrolyte is accompanied by release of Joule heat. The consequence of temperature redistribution in an electrochemical system is a change in the main physical and chemical parameters: viscosity and density of the conductive medium, specific electrical and thermal conductivity, heat transfer coefficient, etc. Taking into account in mathematical models the mutual influence of thermal and electric fields is of particular importance in the technologies of electrolysis of non-ferrous metals (primarily in the industrial production of aluminum) that is accompanied by high-temperature conditions and intensive electrical heat and mass transfer. The processes of electrolytic-plasma removal of coatings, polishing of parts and plasma-electrolytic oxidation attract special attention from the machine-building industry due to the possibility of qualitative improvement of surface properties. In the article correctness of statements of nonlinear models are investigated. The question of unique solvability of the system is explored and a priori estimates of the generalized solution in the Sobolev norms are established. Also, the system of nonlinear partial derivative equations considered in the paper can describe a mathematical model for the joint calculation of electric and thermal fields in solid conductors of electricity and heat.

The article presents a study of methane mixtures' laser conversion under various conditions of the computational experiment. Such flows are characterized by sharp local changes in the gas dynamic characteristics and the mixture components' concentrations. Their dynamics and mutual transformations are described by a stiff system of Navier-Stokes equations and chemical kinetics, which imposes serious restrictions on the choice of a computational algorithm. Numerical experiments were carried out using previously developed 2D code for modeling subsonic axisymmetric flows of a multicomponent medium, supplemented by modules that take into account laser radiation and solve equations of methane conversion's chemical kinetics. Verification of the results was made by calculating the conversion of methane under the influence of walls' external heating. Comparison of the substances' concentrations at the pipe outlet with the direct solution of the chemical kinetics system at different reaction temperatures showed a good agreement of the results. Computational experiments on the laser radiation's effect on the flow of a chemically active absorbing medium have been completed. It is shown that the laser radiation introduced into the mixture and absorbed by ethylene, changes the flow pattern and significantly increases the temperature of the gas mixture. An increase in temperature contributes to an increase in the yield of target products (ethylene, acetylene, hydrogen) at the reactor shorter length, while in the absence of radiation, the maximum concentrations of products appear at the reactor's outlet. The effect of the initial composition of the gas mixture on the methane's conversion is investigated, and it is concluded that the presence of ethylene significantly increases the formation of target products at moderate reactor walls' temperatures in the presence of laser radiation.

While modelling dispersions in containers or tubes it may be necessary to find distortions brought by the suspended particles in the temperature distribution in a vessel. Essential step of such a calculation is to determine of temperature field emerging when the particles are placed near the plane wall of the vessel. For simplicity one may suppose additionally that the carrying medium is stationary and that the particles are spherical. Solving this problem, the authors replace the plane by a fictitious particle that is mirror-positioned with respect to a given one. This allows to use multipole expansion for representation of the temperature that is a harmonic function in the case discussed. The obtained solution is used to find effective heat conduction coefficient of particles’ stratum placed in a half-space bounded by a plane with constant temperature. To do this, the authors average the temperature in the medium by the particles’ positions and compare the result with the solution of reference problem about temperature distribution in a half-space with a uniform stratum of other thermal conductance. The calculation is provided under the assumption that suspended spheres are placed rarely and therefore interact only with the plane but not with each other. A correction term is obtained that must be included in the expression for heat conduction coefficient if the total medium longitude in the direction orthogonal to the plane is finite.