Abstract. In this work, methods of numerical modelling of the coronary vessels system of the human heart have been studied. This investigation includes transient flow of the liquid -blood and dynamics of zones of shear stress at vessels. The main goal of the research is obtaining of hemodynamic and shear stress for creating the digital twin of coronary heart vessels. The results were obtained for low Reynolds numbers about 20 of three-dimensional laminar flow. With this Reynolds number the turbulent flow of the blood is modelled by Realizable k-ε model, and SST models to the narrowing, expansions, and blocks inside the vessels. Loads caused by the additional energy consumption because of the turbulent flow of the blood (increase in arterial blood pressure) have been analyzed. A twodimensional model of a separated vessel with fixed blood back-flow prevention is developed. Presence of a turbulent flow core is discovered. By the means of stress-strain properties of the model, visual representation of the wearing process of the blood back-flow preventer, and heart diseases progression is obtained.

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Reasonable development and creation of any device in which there is an interaction between the fluid flow and the elements of the flow parts (for example, heat exchangers, transport and power machines, main pipelines), is impossible without detailed information about the characteristics of the flow, about the forces on the surfaces that are around, about vibroacoustic phenomena, etc. Among the various methods of obtaining information about the characteristics of the flow, about the forces on surfaces that are flown around, about vibroacoustic phenomena, an important role is played by theoretical methods that rely on the equation of hydrodynamics and numerous ways to solve them. In this case, the main efforts are aimed at solving the system of Navier-Stokes equations. In this paper, a general method is described for the numerical solution of the problem of unsteady flow of a viscous incompressible fluid in flat channels of an arbitrary shape of heat exchangers. An effective solution to the problem is achieved by using adaptive networks. The mathematical model of the flow is based on the two-dimensional Navier-Stokes equations in the variables "flow function - vortex" and the Poissonequation for pressure, which are solved on the basis of the finite-difference method. A numerical simulation of the fluid flow in a flat curvilinear elbow is carried out at the Reynolds number Re = 1000. This form reflects the most characteristic features of the flow paths of various hydraulic machines, heat exchangers, hydraulic and pipeline systems. The presentation of the numerical results was carried out on the basis of the VISSIM graphic processing package. One of the main problems (difficulties) in the numerical solution of problems of mathematical physics is the representation of boundary conditions for regions of arbitrary shape. The implementation of various artificial methods that are now used in the approximation of both the curvilinear boundaries themselves and the boundary conditions on them can lead to significant losses in the accuracy of the solution. This is especially evident in problems in which solutions in the boundary region have maximum gradients. An effective method for solving this problem is the use of adapted grids for the computational domain. The essence of this method lies in the fact that such a coordinate system, not necessarily orthogonal, is found in which the boundary lines (surfaces) of the region coincide with the coordinate lines (surfaces). In the flat case, the computational domain is transformed into a rectangular one, and the limit curve is displayed on the sides of the rectangle. In practice, the problem of constructing an adapted mesh is reduced to finding functions that describe the mappings of the canonical (rectangular) region onto the region for which the problem was originally formulated, that is, for the two-dimensional case, the functions x (ξ, η), y (ξ, η) are determined.

The process of restoration of museum exhibits is associated with the forming of various structural media, for example, drying of paints layers, varnishes and hardening of gypsum and concrete solutions during the restoration of statues, elements of room decor. The mechanics of such media is characterized by spectra of viscous, plastic and elastic properties in their various combinations. The forming of these substances is influenced by the microclimatic conditions of the premises, in which these processes take place. In technological operations, as a rule, large shear deformations take place, which greatly exceed the elastic limit. Thus, the most significant for technological calculations are viscoplastic properties, which reflect the relationship between the existing shear stresses and the rate of irreversible shear deformation. For a qualitative description of this relationship, it is customary to use consistency curves that relate shear stresses to the rate of irreversible shear deformation or the dependence of the effective viscosity on the rate of irreversible shear deformation. Structure formation is one of the main processes in the technology of processing solidifying paste-like and liquid-like media when their characteristics approach those inherent in the corresponding capillary-porous bodies. The properties of such media depend on the type and nature of the structure. The study and control of the process of structure formation to obtain a medium with desired properties is an important technological task, which can be solved, in particular, by microclimate systems. The change in the structural state of the solidifying medium can be estimated from the corresponding changes in plastic and conventional mechanical strength, degree of hydration, heat release, elastic modulus, etc. In this work, structure formation is understood as a process, as a result of which the characteristics of the material approach the characteristics of the corresponding capillary-porous body (CPB). The main parameter that determines this approximation is the degree of completeness of structure formation. Analysis of the hardening process of the considered pasty and liquid-like media on the basis of the moisture state diagram makes it possible to fix the structural state of the capillary-porous structure of these media and to evaluate the interaction of chemical reactions and the process of structure formation. And the resulting changes in the rate of structure formation allows to identify and evaluate the destructive processes that occur during solidification of the substance.

Fundamental analysis of the thermal conductivity and viscosity of quasi-solid capillary-porous bodies (CPBs), which are museum exhibits’ materials, is presented. The air environment parameters change leads to a temperature gradient in the CPBs. Non-uniform heating of the solid medium, in particular, quasi-solid CPB, is not accompanied by convection, and heat transfer is carried out only due to the mechanism of thermal conductivity. In order to create a mathematical model of this process in CPB, a system of partial differential equations in time and space coordinates is obtained. The resulting system adequately describes the thermal conductivity process in quasi-solid CPBs. The anisotropy of CPB’s thermal parameters, especially, its coefficients of thermal expansion and thermal conductivity, is also taken into account. Theoretically, the deformation process during motion in quasi-solid CPB is taken as reversible. In real conditions, the process is thermodynamically reversible only when it occurs at an infinitesimal speed. Then at each point in time, the CPB is able to establish a thermodynamic equilibrium state. Real motion occurs at a finite velocity, the CPB is not in an equilibrium state at any given moment, so there are endogenous processes that try to get it into a balanced condition. The occurrence of these processes causes the irreversibility of motion, which acts, in particular, through the dissipation of mechanical energy, which eventually turns into heat. The energy dissipation is caused by irreversible processes of thermal conductivity and processes of internal friction or viscosity. The dissipative function for isotropic and anisotropic cases was determined in order to analyze the viscosity of quasi-solid CPBs. The viscosity in the equations of motion can be considered by replacing the stress tensor with a tensor, which additionally takes into account the "dissipative" stress tensor.