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A suboptimal cross-border problem is considered about nonlinear dynamic controlled systems under deterministic, uniformly bounded external unknown disturbances. The problem is solved by applying the methods of classical variations calculus for the case when the time interval of adaptation and optimization is not set in advance. The necessary conditions for the choice of extreme motion are determined due to the proper formation of a closed suboptimal adaptive control system. The theoretical analysis is supplemented with computer calculations using a specific model example, which showed effectiveness of the considered approach. The proposed scheme of suboptimal adaptive synthesis can be used in the calculations and design of nonlinear controlled dynamic systems.

Hydrodynamic discontinuities in an external potential and incompressible flow are investigated. Using the reaction front as an example in a 2D stream, an overdetermined system of equations is obtained that describes its motion in terms of the surface itself. Assuming that the harmonic flux approaching discontinuity is additional smooth, these equations can be used to determine the motion of this discontinuity without taking into account the influence of the flow behind the front, as well as the entire external flow. It is well known that for vanishingly low viscosity, the tangential and the integral relation on the boundary (Dirichlet, Neumann problems) connects normal component of the velocity. Knowing one of them along the boundary of the discontinuity, one can determine the entire external flow. However, assuming the external flow is smooth, this will also be the case for all derivatives of velocity with respect to coordinates and time. Then a paradox arises, knowing the position of the discontinuity and the velocity data at a point on its surface, it is possible to determine the motion of this discontinuity without taking into account the influence of the flow behind the front, as well as the entire external flow. There is no physical explanation for this mechanism. It is possible that a boundary layer is formed in front of the front, where viscosity plays a significant role and Euler equations are violated. It is argued that the classical idea of the motion of hydrodynamic discontinuities in the potential and incompressible flow in the external region should be corrected.

Within the framework of the Euler equations, the possibility of achieving extreme pressure values at the inner point of a stationary flow of a nonviscous gas is considered. The flow can be non-barotropic. The well-known (G.B.Sizykh, 2018) subsonic principle of maximum pressure (SPMP) cannot be applied in transonic and supersonic flow regions. Under the conditions of the classical principle of maximum pressure by C.Truesdell (1953), there is no restriction on the values of local Mach numbers, but it has a number of features that do not allow it to be used to verify numerical calculations in the same way as it can be done when using SPMP in subsonic regions. A previously unknown principle of maximum pressure is discovered: a function of derivative flow parameters is found, which must have a certain sign (different for minimum and for maximum pressure) at the point where the pressure reaches a strict or nonstrict local extremum. This principle of maximum pressure is called general (GPMP) because its conditions do not include barotropicity, restrictions on the values of local Mach numbers, and the assumption that the gas obeys the MendeleevClapeyron equation. One of the consequences of GPMP is the conclusion that the requirement of barotropicity can be excluded from the conditions of Truesdell's principle of maximum pressure. It is proposed to use GPMP to verify numerical calculations of the ideal gas flow behind a detached shock wave formed in a supersonic flow around bodies and to verify numerical calculations of a viscous gas flow around bodies in regions remote from sources of vorticity, where the effect of viscosity can be neglected.

In this study, using the Euler equations we investigate the stagnation streamline in the general spatial case of a stationary incompressible fluid flow around a body with a smooth convex bow. It is assumed that in some neighborhood of the stagnation point everywhere, except for the stagnation point, the fluid velocity is nonzero; and that all streamlines on the surface of the body in this neighborhood start at the stagnation point. Here we prove the following three statements. 1) If on a certain segment of the vortex line the vorticity does not turn to zero, then the value of the fluid velocity in this segment is either identically equal to zero or nonzero at all points of the segment of the vortex line (velocity alternative). 2) The vorticity at the stagnation point is equal to zero. 3) On the stagnation streamline, the vorticity is collinear to the velocity, and the ratio of the vorticity to the velocity is the same at all points of the stagnation streamline (invariant of the stagnation streamline). On the basis of the obtained results, it is concluded that if in the free stream the velocity and vorticity are not collinear, a stationary flow around the body is impossible. However, the question of vorticity at the stagnation point in plane-parallel flows remains open, because the accepted assumption that the velocity of the fluid differs from zero in some neighborhood of the stagnation point everywhere, except for the stagnation point itself, excludes plane-parallel flows from consideration.

Swirling axisymmetric stationary flows of an ideal incompressible fluid are considered within the framework of the Euler equations. A number of new exact solutions to the Euler equations are presented, where, as distinct from the known Gromeka–Beltrami solutions, vorticity is noncollinear with velocity. One of the obtained solutions corresponds to the flow inside a closed volume, with the nonpermeability condition fulfilled at its boundary, the vector lines of vorticity being coiled on revolution surfaces homeomorphic to a torus.

The object of research is the working process of the afterburner of the combustion chamber of a turbojet dual-circuit engine with flow mixing. The research was aimed at developing a comprehensive methodology for calculating the afterburner-output device of a forced turbojet engine, taking into account the unevenness of the coefficient of oxygen excess and flow turbulence. To calculate the process of mixture formation, let’s use the model of the separate flow of the gas and liquid phases, taking into account the influence of finite transfer rates between the phases. The gas phase is calculated using a numerical method based on the Eulerian-Lagrangian approach, which allows one to calculate a three-dimensional compressible unsteady flow in an afterburner and is described by Navier-Stokes equations with Reynolds averaging and a one-parameter model of turbulent viscosity. The differential equations of the liquid phase are solved by the Runge-Kutta method. Accounting for turbulent combustion is carried out using the semi-empirical theory. The main indicator of the afterburner combustion chamber working process is the coefficient of completeness of combustion, on which the engine thrust during forced operation depends. To evaluate the combustion efficiency, the fields of velocity, temperature, pressure, mass fraction of oxygen, fuel vapor and pulsation velocity are calculated. These values are determined by numerical simulation of a two-phase flow. The work uses a model of the separate flow of the gas and liquid phases, taking into account the influence of finite transfer rates between the phases. Having data of numerical calculation and a semi-empirical model, let’s determine the completeness of fuel combustion, depending on the coefficient of excess air and the length of the combustion zone. The technique used in this work allows to calculate the completeness of fuel combustion in the afterburner, and the calculation results coincide with experimental data with an error of no more than 7 %. Having data on the completeness of combustion, one can determine the thrust of the nozzle during forced operation of the engine [ABSTRACT FROM AUTHOR]

Maximum principle for subsonic flow is fair for stationary irrotational subsonic gas flows. According to this principle, if the value of the velocity is not constant everywhere, then its maximum is achieved on the boundary and only on the boundary of the considered domain. This property is used when designing form of an aircraft with a maximum critical value of the Mach number: it is believed that if the local Mach number is less than unit in the incoming flow and on the body surface, then the Mach number is less then unit in all points of flow. The known proof of maximum principle for subsonic flow is based on the assumption that in the whole considered area of the flow the pressure is a function of density. For the ideal and perfect gas (the role of diffusion is negligible, and the Mendeleev-Clapeyron law is fulfilled), the pressure is a function of density if entropy is constant in the entire considered area of the flow. Shows an example of a stationary subsonic irrotational flow, in which the entropy has different values on different stream lines, and the pressure is not a function of density. The application of the maximum principle for subsonic flow with respect to such a flow would be unreasonable. This example shows the relevance of the question about the place of the points of maximum value of the velocity, if the entropy is not a constant. To clarify the regularities of the location of these points, was performed the analysis of the complete Euler equations (without any simplifying assumptions) in 3-D case. The new proof of the maximum principle for subsonic flow was proposed. This proof does not rely on the assumption that the pressure is a function of density. Thus, it is shown that the maximum principle for subsonic flow is true for stationary subsonic irrotational flows of ideal perfect gas with variable entropy.

A construction of building of diffeomorphisms which preserve the ball volume and are identity on boundary has proposed. A behavior of hydrodynamics incompressible fluid solutions with initial condition being null on the boundary is investigated.

The problem of a prolongation of non-divergent vector field, defined in a vicinity of zero in Rn t, to a finite non-divergent vector field on Rn is considered. Explicit formulas for the elements of the simple Lie algebra of non-divergent vector from the well-known Cartan series are obtained. This construction allows to move from the Euler equations for the ideal incompressible fluid to the Euler equations on finite-dimensional Lie groups.

This article focuses on the modification of the iterative version of Kaczmarz block algorithm for solving the problem of regularization, which is a fairly effective method for large-scale problems. An important characteristic of iterative methods is the speed of convergence, which depends on the condition number of the original problem. The main drawback of many iterative methods is the large condition number, while methods based on normal equations have the condition number of the system equal to the square of the condition number of the original problem. At the present time to increase the speed of convergence of iterative methods different types of preconditioners are used reducing the condition number of the system. The disadvantages of this approach is manifested in high computational complexity and the lack of universal preconditioner, which could be applied to any iterative method. One of the most effective approaches for improving the convergence rate of the method is to use a block variant of the method used. In this regard, in this paper we propose a modification of the original block Kaczmarz method for the regularization of the problem, which will reduce the computational complexity, and thus increase the rate of convergence of the algorithm. The article provides a detailed derivation of the proposed modification of the method and the proof of the convergence of the proposed variant of the block Kaczmarz method.

Purpose. To establish the influence of centrifugal forces of the flexible body sliding over a fixed block on the friction force between them. The influence of centrifugal forces of the flexible body is usually described by the Euler's law of flexible bodies friction (Euler's formula), which is widely used in international and domestic practice, research, engineering, technology and education. However, it was obtained in times when old notions of Amontons friction predominated and its imperfection results in the disagreement between practice and theory. Euler is the founder of analytical methods used now for study of the influences of centrifugal forces of the flexible body, but they need to be developed and improved by modern knowledge about friction and energy. Findings. The article describes an alternative formula for solution of the problem of flexible body sliding over a fixed unit, taking into account the centrifugal forces. The conclusion is based on the Coulomb's concept of friction and modern methods of mechanics, the principles of conservation of energy. Originality. The accurate data about the influence of the drum angle of contact, friction coefficient, tension in different segments of the flexible body, its linear mass and movement speed on realization of traction friction. Practical value. An alternative solution overcomes the disagreement between the accumulated empirical data and solution of Euler's formula of flexible bodies. It was found that the effect of centrifugal forces of the flexible body moving on the block is very important for the implementation of the friction force. For example, for the conveyor belt at speeds of 2 and 4 meters per second the traction under the alternative solution of the Euler problem is reduced to 15 and 75% respectively compared to the maximum. The knowledge gained develops analytical methods in mechanics, scientific concepts of friction, flexible bodies, contributing to the overall progress. [ABSTRACT FROM AUTHOR]