Vol 87, No 1 (2023)

The article is devoted to the problem on motion of a rigid body with a fixed point under the action of a force field, which is the superposition of three homogeneous force fields. Existence conditions are investigated for precession motions, which are characterized by the following property: the angular velocity of the precession of the body is two times more than the angular velocity of its proper rotation. It is established that the body has the dynamic symmetry with respect to an axis making a constant angle with a vector fixed in the immovable space. In the case of the body with the spherical mass distribution this angle equals to \({\text{arccos}}\frac{1}{4}\). Solution of the equations of motion of the body is expressed through the elliptic functions of time.

A regularized problem has been formulated to describe the flow of a molecular gas in a hypersonic macrokinetic nonequilibrium thin viscous shock layer (kinetic TVSL) near a sharp edge. It is shown that the solution of the regularized kinetic TVSL problem on the sharp edge, the description of friction and heat transfer coincides with the solution of the Navier–Stokes analogue of this TVSL problem. It is shown that in the kinetic TVSL the friction stress and the normal heat flow on the wall in the vicinity of the sharp edge are identical with the corresponding values in the Navier–Stokes TVSL. The similarity of currents near the sharp edge in the kinetic and Navier–Stokes versions of the TVSL is indicated. A method for constructing a solution to the kinetic TVSL problem for the vicinity of a sharp edge based on the Navier–Stokes TVSL solution is indicated. A formula is obtained for calculating the pressure on the surface in the kinetic TVSL flow near the sharp edge through the components of the solution (on the wall) of the Navier–Stokes TVSL flow problem.

Within the framework of integral constitutive relations for linear isotropic viscoelastic media with difference-type kernels in the case of a nonrelaxing volume, possible, complementary to the known, setup experiments for determining the kernels of Ilyushin operators \({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{g} }_{\beta }}\) are proposed. One of them is based on the use of a sample from an auxiliary viscoelastic solid, the material functions of which are related to the creep function and the volume compression module of the given material. Similar schemes of setup experiments for finding operator kernels \({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{h} }_{\gamma }}\), in a certain sense conjugated with \({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{g} }_{\beta }}\), are also proposed.

Problems of definition of thermotension of a thermoelastic core with use of model of the connected thermoelasticity are considered. In this case in the equation of heat conductivity there is the divergence of speed of the movement of material points of the rod, and the elasticity equations contains the temperature gradient. On the basis of generalized functions method the generalized solutions of non-stationary and stationary boundary value problems have been solved at action of the power and thermal sources of various type including ones described singular generalized functions, under various boundary conditions on the ends of a core. Thermoshock waves which arise in such designs at action of impact loads and heat fluxes are considered, conditions on their fronts are received. The uniqueness of the set boundary tasks, including taking into account shock waves has been proved. Regular integral representation of the generalized solutions are given, which give the analytical solution of the tasks. Numerical implementation of solutions of a number of direct, return and semi-return boundary value problems of stationary fluctuations is carried out and results of computer experiments are presented