МОДЕЛЮВАННЯ КОНВЕКТИВНОЇ ТЕПЛОПРОВІДНОСТІ ДВОШАРОВОГО СЕРЕДОВИЩА ПРИ НЕОДНОРІДНИХ ГРАНИЧНИХ УМОВАХ
Journal Title: Математичне моделювання - Year 2018, Vol 1, Issue 1
Abstract
PROBLEM OF CONVECTIVE HEAT CONDUCTIVITY OF A DOUBLE ENVIRONMENT UNDER HEAVY BORDER CONDITIONS Pyshnograev Y.N., Shtanko A.I. Abstract In this paper, we consider the special cases of the application of the method of finite integral transformations in the problem of convective heat conductivity of a two-layer medium. In this case inhomogeneous boundary conditions of the first kind in any form are given at external boundaries. It is shown that after carrying out the integral transformation, the formal solution is written in the form of functional series with nonuniform convergence. This leads to difficulties in the numerical solution of the problem. Especially large errors occur when calculating the temperature function near the outer boundaries and common interfaces of the layers. The main idea of this research is obtaining a solution in the form of rapidly converging series. The physical definition of the problem can be found below. We consider a two-layer medium. Heat is distributed according to the laws of heat conduction and convection. On the outer boundaries, non-homogeneous boundary conditions of the first kind in common form are given. At the common interface of the layers - the conditions for an ideal thermal contact. At the initial moment of time, the temperature of the layers is represented as an arbitrary function of the spatial variable. The temperature function is determined, which depends on the spatial variable and time. The mathematical model consists of a one-dimensional nonstationary heat conduction equation with a convective component, boundary conditions and initial conditions. It is shown that it is necessary to represent the target temperature function in the form of a sum of two terms: nonstationary and quasi-stationary. An algorithm for solving the problem with respect to the quasi-stationary component is given. Its final representation is written in the form of a linear combination of linear and exponential functions. The analysis of the obtained solution is carried out. It is concluded that taking into account the quasistationary term makes possible to improve the convergence of the functional series representing the formal solution of the initial problem for inhomogeneous boundary conditions. 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] Farlow, S. (1985) Uravneniya s chastnymi proizvodnymi dlya nauchnykh rabotnikov i inzhenerov [Partial differential equations for scientists and engineers]. Moskow, 1985, 384 p. [5] 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). [6] 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 EP294836
- DOI 10.31319/2519-8106.1(38)2018.128939
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