We pose the problem on free harmonic bending vibrations of an infinite rotating tubular elastic rod with an internal fluid flow, prestressed by a torque and a longitudinal force. We show that these vibrations can only be realized as traveling circular helical waves. It is shown that, for any wavelength, there exist four waves, two having the form of a left-handed helix and the other two having the form of a right-handed helix. Each of these waves propagates in the positive and negative directions of the longitudinal rod axis at different velocities. These phenomena can manifest themselves in deep-hole drill columns.

This paper deals with investigatson of the process of drill bit whirling on the rough plane of the well bottom. Nonholonomic kinematic models of the mechanical interaction of contacting bodies with defects at the initial stage of system self-excitation are applied. On the basis of the results of experimental studies, it is believed that one of the main factors influencing on the whirlings vibrations is the geometry of the bit. The bit is considered to be an absolutely rigid ellipsoidal body, the well bottom surface is supposed to be a plane. The resulting oscillations are associated with spontaneous bending deformations of the drill string, which are accompanied by continuous contact of the bit and the rock. The equations of motion of the bit in the linear approximation are obtained. The analysis of the solution of the linearized equations is carried out, and the frequencies of the arising periodic motions are found. The forms of oscillations under different geometrical parameters of an ellipsoidal bit are analyzed. It is shown that the bit can rotate in backward direction, moving at angular velocities that exceed the angular frequency of the drill string. The research results can be used in the development of new types of drilling rigs.

A two-dimensional periodic Stokes flow of viscous incompressible fluid in a rectangular cavity with constant velocities applied to the top and bottom walls is considered. The study of advection regimes is reduced to the sequential solving of two problems. To solve the first problem, the analytical method of superposition, which allows one to obtain any desired accuracy of the velocity field, is used. To solve the second problem, which is associated with obtaining the trajectories of individual fluid particles, numerical calculations of the Cauchy problem were performed. An analysis of the boundary conditions accuracy was performed based on the control of local integration. The advection of the selected volume of fluid in the rectangular cavity under the periodic motion of walls for a finite period time based on piecewise spline cubic interpolation is modeled. The obtained numerical results agree with well-known experimental data.

The article explores the fundamental principles and technologies of cross-platform mobile application development. It highlights the key aspects of this approach and analyzes its advantages and limitations that affect the quality and efficiency of software product creation. Special attention is given to the stages of the application lifecycle – from design to maintenance – which ensure the stability and competitiveness of solutions. The paper provides an overview of popular frameworks – React Native, Flutter, and Xamarin – with a comparative analysis of their capabilities. Practical examples of these technologies used in well-known mobile products (Instagram, Google Ads, Alaska Airlines) demonstrate the effectiveness and potential of the cross-platform approach. The author’s experience in developing a mobile application using Flutter confirms the practical relevance of the chosen technology stack within an educational project.