On some problems of modelling the non-stationary heat conductivity process in an axisymmetric multilayer medium

Some possibilities of solving the non-stationary heat equation in an axisymmetric multilayer medium are considered. The problem is solved by combining Fourier method and the matrix method. The solutions of the first and third boundary value problems are considered. Examples of calculations by the specified method for three-layer and five-layer axisymmetric systems are given.


Introduction
Multilayer materials in the form of plates, shells, and screens are finding increasing practical application in technology. During operation, they are exposed to external thermal influences (as a result of heating by electromagnetic radiation and/or charged particles) and often operate under extreme conditions -for example, in space, in nuclear power, etc. The study of thermal regimes in a multilayer shell makes it possible to predict the behaviour of individual layers and to identify conditions conducive to possible deformation, melting, and other changes in the physical or chemical nature of individual layers. To predict the influence of external factors on materials and multilayer structures, it is necessary to have good calculation methods that allow predicting possible undesirable phenomena.
We give a short review of some recent research on this issue. In a paper [1], an example was considered when the coefficient of thermal conductivity changes exponentially depending on the coordinate. The resulting solution is expressed in terms of Bessel functions. A paper [2] includes conditions for the electromagnetic exchange of energy between layers, and the material layers, except for the extreme ones, are considered quite thin. In a paper [3], the most general formulation of the heat transfer problem in the fuel piping system is studied at a high theoretical level, including the possibility of solving it in various classes of functions.
In our research, we propose to use a combination of the matrix method and the method of generalized powers of Bers and the Fourier method in the case of a nonstationary process to solve the problem of heat and mass transfer in multilayer media.
The matrix method as applied to problems of conduction of heat in composite slabs is described in [4]. However, it was not widely used to solve heat-and mass-transfer problems in multilayer media, possibly due to the fact that the formulas of the analytical solution were extremely complex, computer algebra systems were just beginning to emerge at that time, and therefore numerical methods were preferred. At present, due to the development of symbolic computation systems, this method can be effectively used for calculations. Also, the idea of the matrix method is currently successfully applied in theoretical research [5,6].
are respectively, the coefficient of thermal conductivity, heat capacity and density of the medium. The flow is directed along the x-axis Figure 1. Scheme of an axisymmetric multilayer medium.
If the physical parameters are constant on each layer of the multilayer medium, then the heat conduction process can be described by a system of differential equations  are the thermal conductivity coefficient, heat capacity and density on the i-th layer respectively. Then the flux on each layer is determined by the formula At the boundaries between the layers of the medium, we take the conditions of ideal contact The initial temperature distribution is given gx is a function that can be specified layer by layer, i.e. may have discontinuities of the first kind at the contact points of the layers.
In this paper, we will consider the solution of the first boundary value problem and the solution of the third boundary value problem , , where (1) r and ( 1) n r + are thermal resistance coefficients at the boundaries of a multilayer medium, 1 T and 2 T are external temperatures at the boundaries of the medium.

The method of calculation
The solution of the posed problems (1 -4) and (1,2,3,5) will be sought by the combined application of the matrix method and the Fourier method.
We write a solution of equations (1) in the form And nonstationary subproblem has the form , .
We solve the stationary subproblem (6) by the matrix method. On the segment   The solution of the Cauchy problem for a segment   1 , ii xx + in the Bers formalism [7,8] has the form where generalized power of Bers takes the form for axisimmetric layers and the solution for the next ( 1) i + layer is The solution for the entire layer system is built sequentially, starting from the first layer. Solution for the first layer is (1) (1) then for the second layer the solution takes the form Performing further sequential substitution through the layers, we obtain the general result for the ith layer Then, at the end point 1 n x + of the layer system, we obtain  (9) and obtain a solution to the stationary subproblem (6).
Next, we solve non-stationary problems (7). On the i-th layer, we define the Cauchy problem then the solution to the Cauchy problem for each layer has the form Here the solution is written in the Bers formalism [7,8]. We denote  11 1 We write the ideal contact of the layers in matrix form The solution for the first layer is For the end point of the first layer, then we get (1) ( , and, considering the conditions of ideal contact, the solution for the second layer can be written (2) (2) Performing further sequential substitution by layers, we obtain a solution on the i-th layer Thus, at the end point of the layer system, we have Equation (11) connects the values of the function and the flow at the initial and end points, which makes it possible to obtain a condition for determining the eigenvalues

Results of calculations and their discussion
Some model problems were solved by this method.
In figure 2 shows the result of modelling the first boundary value problem for a three-layer axisymmetric system with the radii of the boundaries of layers 1 0,1 m, x =  == At the initial moment of time, only the middle layer is heated to 20°C, that is, the initial conditions are symmetric. The graphs are based on 25 eigenvalues. Modelling shows that the cooling of a layer with a smaller radius occurs faster than a layer with a large radius, and since the outer layers have a significantly lower thermal conductivity than the inner layer, the temperature on it practically does not change along the coordinate, which corresponds to the physical course of the process. . The temperature distribution is based on 10 eigenvalues. Modelling shows that at smaller radii of the layer system, the transition to the stationary regime occurs faster, while the temperature drop at the boundary points of the multilayer cylinder is more pronounced. Calculations also show that at large radii of the cylindrical system, the temperature distribution approaches the temperature distribution in the corresponding system of plane layers of the same thickness, which corresponds to the physical meaning of the simulated phenomenon.  . Figure 4 shows the modelling results in a five-layer axisymmetric medium with parameters typical for the "plastic-glue-aluminum-glue-plastic" system: All calculations were performed using the mathematical package Maple (Widows 10 Home, processor Intel® Core™ i5-8250U CPU@ 1.60GHz 1.80GHz, 6.00GB). Most of the computational time according to the described algorithm is spent on finding eigenvalues. The computation time noticeably increases with an increase in the number of layers or the number of eigenvalues. Particularly, for a threelayer medium, modelling on 10 eigenvalues usually does not exceed 10 minutes.
Also note that, as before, the results of modelling using the matrix method described in this work correlate well with the results of numerical analysis of the considered heat and mass transfer processes.

Conclusion
The paper describes the combined use of the analytical matrix method and the classical Fourier method for solving the problem of heat conduction in an axisymmetric multilayer medium. Some possibilities of this approach for modelling first boundary value problem and third boundary value problem are considered. It is shown that the proposed matrix method makes it possible to calculate temperature distributions in a relatively short time with an accuracy sufficient for practical use.