Abstract. The article is dedicated to the development asymptotic theory of gas flowing at speed next to sound velocity, particularly of gas transonic flows, i.e. the flows, containing both, subsonic and supersonic areas. The main issue, when styding such flows, are nonlinearity and combined type of equations, describing the transonic flow. Based on asymptotic nonlinear equation obtained in the article, the gas transonic flows is studied, considering transverse disturbance with respect to the main flow. The asymptotic conditions at shock-wave front and conditions on the streamlined surface are found. Moreover, the equation of sound surface and asymptotic formula defining the pressure are recorded. Several exact particular solutions of such equation are given, and their application to solve several tasks of transonic aerodynamics is indicated. Specifically, the polynomial form solution describing gas axisymmetric flows in Laval nozzles with constant acceleration in direction of the nozzle's axis and flow swirling is obtained. The solutions describing the unsteady flow along the channels between spinning surfaces are presented. The asymptotic equation is obtained, describing the flow, appearing during non-separated and separated flow past, closely approximated to cylindrical one. Specific solutions are given, based on which the examples of steady flow are formed.

Nonlinear mathematical models are proposed that describe the dynamics of a pipeline with a fluid flowing in it: a) the model of bending-torsional vibrations with two degrees of freedom; b) the model describing flexural-torsional vibrations taking into account the nonlinearity of the bending moment and centrifugal force; c) the model that takes into account joint longitudinal, bending (transverse) and torsional vibrations. All proposed models are described by nonlinear partial differential equations for unknown strain functions. To describe the dynamics of a pipeline, the nonlinear theory of a rigid deformable body is used, which takes into account the transverse, tangential and longitudinal deformations of the pipeline. The dynamic stability of bending-torsional and longitudinal-flexural-torsional vibrations of the pipeline is investigated. The definitions of the stability of a deformable body adopted in this work correspond to the Lyapunov concept of stability of dynamical systems. The problem of studying dynamic stability, namely, stability according to initial data, is formulated as follows: at what values of the parameters characterizing the gas-body system, small deviations of the body from the equilibrium position at the initial moment of time will correspond to small deviations and at any moment of time. For the proposed models, positive definite functionals of the Lyapunov type are constructed, on the basis of which the dynamic stability of the pipeline is investigated. Sufficient stability conditions are obtained that impose restrictions on the parameters of a mechanical system.

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The article discusses the initial-boundary value problems for systems of differential equations, which are mathematical models of the mechanical system "pipeline - pressure sensor that is designed to measure pressure in gas-liquid media. On the basis of the proposed models, the joint dynamics of the pressure sensor sensitive element and of the working medium in the pipeline connecting the sensor to the combustion chamber of the engine is investigated. To describe the movement of the working medium, linear models of the mechanics of liquid and gas are used; to describe the dynamics of the sensitive element, both linear and nonlinear models of the mechanics of a solid deformable body are used. The solutions of stated initial-boundary value problems are carried out on the basis of the Galerkin method and the finite-difference method.

Initial-boundary value problems for systems of differential equations are considered, which are mathematical models of the mechanical system "pipeline - pressure sensor". In such a system, to mitigate the effects of vibration accelerations and high temperatures, the sensor is located at a certain distance from the engine and is connected to it via a pipeline. The "pipeline - pressure sensor" system is designed to measure pressure in gas-liquid media, for example, to control the pressure of the working medium in the combustion chambers of engines. On the basis of the proposed models, the joint dynamics of the sensitive element of the pressure sensor and the working medium in the pipeline is studied. To describe the motion of the working medium, linear models of fluid and gas mechanics are used, to describe the dynamics of a sensitive element, linear models of the mechanics of a deformable solid are applied. Analytical and numerical methods for solving initial-boundary value problems under study are presented. The numerical study of the initial-boundary value problem was carried out on the basis of the Galerkin method. In analytical study using the introduction of averaged characteristics, the solution of the original two-dimensional problem is reduced to the study of a one-dimensional model, whose further study made it possible to reduce the solution of the problem to the study of a differential equation with a deviating argument. Also, a numerical experiment is carried out and an example of calculating the deflection of the sensor’s moving element is presented.