Двовимірна спектральна задача з конвективною складовою для двошарової прямокутної області
Journal Title: Математичне моделювання - Year 2018, Vol 1, Issue 2
Abstract
TWO-DIMENSIONAL SPECTRAL PROBLEM WITH A CONVECTIVE COMPONENT FOR A TWO-LAYER RECTANGULAR AREA Pyshnograev Y.N., Shtanko A.I. Abstract Among the analytical methods used in solving problems of diffusion type, the method of finite integral transformations is quite powerful. The catolicityof this method allows to find solutions to a wide range of heat and mass transfer problems. The success of the method of finite integral transformations depends on the possibility of solving the corresponding spectral problem, to which the initial equations in partial derivatives and boundary conditions are reduced. In this connection, the complexity of obtaining a solution ofa spectral problem is directly related to the geometry and structure of the body in question. In engineering practice, the study of the processes occurring in multilayer structures, whose geometry is modeled by a two-dimensional region, is of great interest. For this reason, consideration of the corresponding spectral problem and the construction of an algorithm for its solution is an relevant scientific study, the results of which will make it possible to obtain solutions of important engineering problems. One of the key stages of the application of the method of finite integral transformations is the solution of the corresponding spectral problem. The process of its solution enables to build a complete system of eigenfunctions, which are further used as kernels of finite integral transformations. This makes it possible to obtain analytical solutions for a large number of heat and mass transfer problems using standard algorithms. In this context, the purpose of the study is to solve a two-dimensional spectral problem with a convective component for a two-dimensionaldomain, expressly to study the constructing peculiarities of orthogonal system of functions and the corresponding set of eigenvalues. The undertaken studies have shown that in the construction of a solution to a spectral problem, the main feature is the need to consider three intervals of the eigenvalues location. This is due to the presence of the convective component, as well as the fact that the domain under consideration is two-dimensional and two-layer. The discovered eigenfunctions can be further used as kernels of integral transformations for the complete solution of the corresponding heat and mass transfer problems. The analysis of the performed computationsshows that promising are the directions connected with the solution of spectral problems for three-dimensional regions and domains with more than two layers. References [1] Kartashov E.M. Analiticheskie metody v teorii teploprovodnosti tverdykh tel [Analytical methods in the theory of thermal conductivity of solids]. Moskow, 1985. 480 p. [2] Plyatt S.N. Raschety temperaturnykh poley betonnykh gidrosooruzheniy [Calculations of temperature fields of concrete hydro structures]. Moskow, 1974. 407 p. [3] Pyshnograev Y.N. The problem of the propagation of heat in an orthotropic two-layer plate when heated by point sources of heat. Trudy 1 VK "Tekhnologicheskie problemy prochnosti nesushchikh konstruktsii" Zaporozh'e,1991, vol.1, ch.1, pp. 155–160 (in Russian). [4] Pyshnograev Y.N., Pyshnograev E.Y. Construction of a system of eigenfunctions for the convective diffusion equation with piecewise constant coefficients. Zbirnyk prats In-tu matematyki NAN Ukrainy, 2012, vol. 9, no.1, pp. 7–12 (in Russian). [5] Pyshnograev Y.N., Shtanko A.I., Pyshnograev E.Y. Analytical solution of the problem of convective heat exchange in two-layer medium. Vіsnik Zaporіz'kogo natsіonal'nogo unіversitetu. Fіziko-matematichnі nauki, 2017, no. 2, pp. 236–242(in Russian).
Authors and Affiliations
Ю. М. Пишнограєв, Г. І. Штанько
Keywords
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- EP ID EP444530
- DOI 10.31319/2519-8106.2(39)2018.154206
- Views 107
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