Aim. Reduction of temperature unevenness in a shell-type turbine blade under conditions of heat supply and removal.

Methodology. The mathematical formulation of the problem of reducing the temperature unevenness of the shell using a system of curved heat sink channels is carried out. The mathematical model was constructed using the condition of continuity of the cooling flow in the channel, data on the source of thermal stress, boundary conditions at the entrance and exit to the cooling channel, restrictions on the height of the channels using interpolation polynomials.

Results. The variable height and trajectories of the heat sink channels are calculated for a given uneven temperature field, allowing to intensify cooling in the most thermally loaded area of the shell.

Research implications. The theoretical and practical significance lies in the possibility of using the proposed model to develop a deflector shape with curved channels for blades of a gas turbine engine with an internal cooling system.

Aim. To consider the viscoelastic characteristics of a lyotropic liquid crystal, namely, a solution of the synthetic polypeptide poly-β-benzyl-aspartate (PBA) in m-cresol, which were obtained at different values of the strain amplitude.

Methodology. The experimental data of dynamic measurements were approximated by the equations of a structural rheological model at separate intervals of the cyclic frequency of shear vibrations.

Results. The possibility of using the equations of the structural model to describe the frequency dependences of dynamic modules in conditions of nonlinear viscoelasticity is shown. It is shown that the coefficients of the rheological equations depend on the amplitude of the deformation in accordance with the provisions of the structural model.

Research implications. It is shown that the equations of the structural rheological model are capable of approximating the experimental data of dynamic measurements in the case of a lyotropic liquid crystal. Rheological equations retain their form at different set values of the strain amplitude, which are in the region of nonlinear viscoelasticity.

Aim. Development of a microscopic model of thermophoresis of a single particle that goes beyond the linear approximation and takes into account significant nonlinear effects that occur under conditions of strong temperature gradients.

Methodology. The methods of stochastic thermodynamics were applied and the modified Langevin equation with temperature-dependent parameters was used, which made it possible to analytically derive the expression for the thermophoretic velocity taking into account quadratic corrections for the temperature gradient.

Results. The results obtained demonstrate qualitatively new features of thermophoretic drift: the possibility of inverting the direction of particle motion when critical values of the temperature gradient are reached, significant deviations from the predictions of linear theory in the field of strong temperature field inhomogeneities, as well as a pronounced dependence of the observed effects on the parameters of the medium. The analysis of fluctuation-dissipative ratios established a connection between the microscopic characteristics of the system and the macroscopic manifestations of thermophoresis.

Research implications. lie in a significant expansion of the fundamental concepts of thermophoretic transfer mechanisms, which for the first time systematically takes into account second-order nonlinear effects. From a practical point of view, the developed model creates the basis for new methods for controlling particle motion in microfluidic devices and nanotechnology applications, and also allows us to explain a number of experimentally observed anomalies in the behavior of colloidal systems and biological objects in inhomogeneous temperature fields.

Aim. Determination of the catalytic activity of a single molecule of horseradish peroxidase (HRP) in the oxidation reaction of the substrate 2,2ʹ-azino-bis-[3-ethylbenzthiazoline-6- sulfonate] (ABTS) with hydrogen peroxide.

Methodology. To determine (monitor) the catalytic activity of HRP, pore technology has been used; it has allowed us to analyze the activity of a single HRP molecule without introducing additional components into the system to enhance the signal. For this purpose, a solid-state pore of about 5 nm in size, formed by electron-beam drilling in a silicon nitride plate of ~40 nm thickness, has been used. A HRP molecule has been embedded in this pore, after which the catalytic activity of the molecule embedded in the pore in the presence of ABTS and H₂O₂ has been analyzed by measuring the ion current through the pore with the HRP molecule embedded in it.

Results. A pore detector has been proposed to study the catalytic activity of HRP in the reaction of ABTS oxidation. It has been found that this detector made it possible to monitor the activity of this enzyme by registering of ion current through the pore.

Research implications. It has been shown that the manufactured pore can be used to monitor HRP activity. The results obtained are important for the development of work in the field of enzyme research at the level of single molecules.

Aim: to analytically investigate the existence or absence of the effect of overlapping values of distribution functions for relative velocities of molecules in a shock-compressed binary mixture of gases.

Methodology. Asymptotic and approximation methods of mathematical physics were used.

Results. It is analytically shown that in the modified bimodal model of a shock-compressed binary mixture of gases, there are effects of overlapping distribution functions with respect to the relative velocities of molecules. These effects, consisting in an excess of the amounts of energetically active molecules overcoming the threshold of barrier processes inside the wave front compared with a similar amount beyond it, can occur in a wide range of molecular weights and concentrations in a shock-compressed binary mixture of gases.

Research implications. The obtained analytical results are essential for clarifying the issue of the need to take into account translational disequilibrium when determining the velocity coefficients of energetically activated inelastic collisions inside shock wave fronts.

Aim. The classical lag theorem from the course of operational calculus has shown unsatisfactory results on a variety of specific examples made up of elementary functions. The article presents a new formula for the delay theorem, which gives correct results.

Methodology. The method consists in determining the images of functions with a delay by direct calculation from the Laplace integral, or using a linear combination of tabular images. The solutions obtained are compared with the images obtained using the classical delay theorem. The comparison of the results obtained by the two methods turned out to be unsatisfactory for all the examples.

Results. A new, correct delay theorem is formulated and the corresponding formula is presented. The results of applying the new formula gave correct results. An error has been identified that occurred during the derivation of the classical delay formula. It consists in the fact that in the process of deducing the formula, one integral term was unlawfully deleted.

Research implications. Operational calculus is used in automatic control theory and in electrical circuit calculations. The corrected delay theorem allows one to obtain correct results in the named systems, where signals with delay are present.