# Mechanics

## One-dimensional equations of motion of a viscous incompressible fluid in flexible tubes

This paper describes a new variant of the averaging of the Navier–Stokes equations for axisymmetric flow of a viscous incompressible fluid with a minimum number of simplifying hypotheses. The complete system is spatially one-dimensional differential equations describing the dynamics of blood flow in the large arteries.

## Parametrical synthesis of stabilization systems

Method of feedback parameters selection for gas jet stabilization systems with elastic roads, based on minimizing the mean square deviation of the real frequency response of the designed system with respect to the real desired frequency response, was implemented. The results of analysis of transient errors stabilization functions, taking into account the effect of time delay in gas jet executive stabilization systems are given.

## Chaotic motion of top with displaced mass center

The motion of solid body with a small displacement mass center from the axis of dynamic symmetry has been studied. Analytical conditions for the existence of a hyperbolic singular point in the phase portrait of the system and the analytical solution for the separatrices have been obtained. Body makes a chaotic motion near separatrices under the influence of small perturbations caused by the asymmetry of the body.

## Asymptotic integration of dynamic elasticity theory equations in the case of multilayered thin shell

Asymptotic integration of elasticity theory 3D equations is fulfilled for the case of multilayered arbitrary-shaped thin-walled shells. The tangential and the transverse long-wave low-frequency approximations are constructed. The governing 2D equations are derived.

## Percolation of spheres in continuum

The model of the continuum percolation of hard spheres with permeable shells, which describes phase transition sol-gel, has been investigate. Spheres have hard parts in radii r, which can't be blocked with each other, and permeable shells in width d, which can be blocked. Such spheres of the equal size have been randomly packing in the cub with linear size L. The probability of joining the spheres in a cluster is proportional to the volume of overlapping of permeable shells.

## Non-stationary vibration of growth circular cylindrical shell

Small forced vibrations of growing cylindrical shell fixed on circular boundaries is studied in the framework of Kirchhoff–Love shell theory. The process of the accretion are characterized by the continuous adherence of material particles to its facial surface. Since the shell bends during the accretion, its stressed-strained state depends not only on loading, but also on the history of the process of accretion, i.e. the schedule of accretion.

## Rotational Invariance of Non-Linear Lagrangians of Type-II Micropolar Thermoelastic Continuum

The paper contains new results related to extension of the field theoretical approach and its formalism to non-linear coupled micropolar thermoelastic media. A mathematical model of micropolar (MP) type-II (GNII) thermoelastic (TE) continuum is considered. A formulation of the least thermoelastic action principle is discussed. Partial differential equations subsequent to the least action principle are derived. The translational symmetries of non-linear Lagrangians are adopted. Those include an additional symmetry: translations of the thermal displacement.

## Numerical analysis of renal artery pathologies

Mathematical modeling based on experimental data (ultrasonic imaging, angiography, 3D reconstruction via spiral computed tomography) was performed. Anatomically precise model of renal artery was created. Basic principles of blood flow dynamics with stressstrain state of artery walls were studied for normal, pathologic renal arteries and arteries with hemostasis of intraorganic branches.

## Graph approach for finite-element based model of an elastic body under conditions of axisymmetric deformation

A numerical method for analysis of the stress – strain state of elastic media based on a discrete model in form of directed graph is suggested. To analyze a deformable body using the graph approach, we partitione a solid body on elements and replace each element by its model in the form of an elementary cell. The matrices, presenting several structure elements of the graph, and the equations, describing the elementary cells, contribute to deriving the constitutive equations of the intact body. Numerical examples are presented.

## Local effects of the weak thermogravitational convective flows

The features of the natural low intensity thermo-gravitational convection occurring in microacceleration condition have been investigated numerically. The effect of thermal boundary conditions on the local characteristics of temperature field has also been studied. It was shown that the value of maximum temperature stratification depends monotonically on the intensity of heat transfer at the system boundaries. The correction algorithm has been proposed for the vorticity boundary conditions on the solid impermeable walls.