Vol 87, No 2 (2023)

The regular quaternion equations of the orbital motion of a spacecraft (SC) proposed by us earlier in four-dimensional Kustaanheimo–Stiefel variables (KS-variables) are considered. These equations use as a new independent variable a variable related to real time by a differential relation (Sundman time transformation) containing the distance to the center of gravity. Various new regular quaternion equations in these variables and equations in regular quaternion osculating elements (slowly varying variables) are also constructed, in which the half generalized eccentric anomaly, widely used in celestial mechanics and space flight mechanics, is used as a new independent variable. Keplerian energy and time are used as additional variables in these equations. These equations are used to construct quaternion equations and relations in variations of KS-variables and their first derivatives and in variations of Keplerian energy and real time; the isochronous derivatives of the KS-variables and of their first derivatives and the matrix of isochronous derivatives for the elliptical Keplerian motion of the spacecraft are found, which are necessary for solving the problems of predicting and correcting its orbital motion. The results of a comparative study of the accuracy of the numerical integration of the Newtonian equations of the spatial restricted three-body problem (Earth, Moon, and spacecraft) in Cartesian coordinates and the regular quaternion equations of this problem in KS-variables are presented, which show that the accuracy of the numerical integration of regular quaternion equations is much higher (by several orders) of the accuracy of numerical integration of equations in Cartesian coordinates. This substantiates the expediency of using regular quaternion equations of the spacecraft orbital motion and the quaternion equations and relations in variations constructed in the article on their basis for the prediction and correction of the orbital motion of a spacecraft.

We consider one-dimensional second order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found.

The three-dimensional stationary flow of two immiscible liquids in a layer bounded by solid parallel walls is investigated. The upper wall is thermally insulated, and the lower one has a temperature field quadratic in horizontal coordinates. Velocity fields in liquids have a special form: their horizontal components are linear in the coordinates of the same name. The resulting conjugate boundary value problem in the framework of the Oberbeck–Boussinesq model is inverse and is reduced to a system of ten integro-differential equations. For small Marangoni numbers (creeping current), the problem is solved analytically. The nonlinear problem is solved by the tau method. It is shown that the solution of the nonlinear problem with a decrease in the Marangoni number is approximated by the solution of the creeping flow problem. The analysis of the influence of physical and geometric parameters, as well as the behavior of temperature on the substrate, on the structure of convection in layers is carried out.

A numerical simulation is conducted to study the flow of a three-dimensional incompressible wall jet. The study is aimed to determine the flow structure and to compare the propagation mechanisms of turbulent and laminar wall jets. The numerical solution of the Navier–Stokes equations in the turbulent case is obtained using the wall-resolved large eddy simulation. The simulation results are compared with experimental data.

Based on the thermomechanical model of plastic deformation of an elastically compressible isotropic hardening medium, the system of relations for describing plastic shock waves of finite amplitude is obtained, which satisfies the maximum entropy production principle at the front of strong discontinuity. A classification of admissible shock-wave transitions is performed within the framework of the model of isotropic hardening under the von Mises plasticity condition.

A mathematical model of non-isothermal elastic-viscous-plastic deformation of flexible shallow shells with multidirectional reinforcement structures has been developed. Wave processes and weak resistance to transverse shear in curved panels are modeled in terms of Ambartsumian’s bending theory. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The composition components are assumed to be isotropic materials, and their plasticity is described by the flow theory with a loading function depending on the strain rate and temperature. The connectedness of the thermomechanical problem under dynamic loading of composite shallow shells is taken into account. In the transverse direction of constructions, the temperature is approximated by a 7th order polynomial. The formulated two-dimensional nonlinear initial-boundary value problem is solved using an explicit numerical scheme of time steps. The thermo-elastic-visco-plastic and thermo-elastic-plastic behavior of fiberglass and metal-composite shallow shells orthogonally reinforced in two tangential directions, loaded in the transverse direction by an air blast wave, has been studied. It is shown that flexible curved fiberglass panels at certain points can additionally heat up by 14…27°C, and similar metal-composite conctructions – by 70°С or more. In this case, peak temperature values are kept at short-term intervals – on the order of fractions of 1 ms. It is shown that, unlike flexible plates, similar shallow shells (with the same reinforcement structure and the same characteristic dimensions) under dynamic loading in the transverse direction must be calculated not only taking into account the dependence of the plastic properties of the composition components on their strain rate, but also taking into account thermal response in such thin-walled constructions. A more intense inelastic deformation of curved composite panels is observed when they are loaded from the side of the convex front surface.

We build a mathematical model describing the evolution of the pulse signal in the well in the presence of a longitudinal or transverse fracture in the bottomhole section. It is assumed that the signal is sent from the wellhead with a wavelength greater than the diameter of the well and the length of the open section of the well. According to the dynamics of the “echo” of the pulse signal returning to the wellhead, it is possible to judge the quality of hydraulic fracturing. The results of numerical calculations for a bell-shaped pulse are presented. It is shown that when diagnosing fractures, water is more preferable than oil as a fluid through which the signal propagates.