Self-similar processes of heat transfer in a solid with axisymmetric heat source

Аuthors

Attetkov A. V.^*, Volkov I. K.

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: fn2@bmstu.ru

Abstract

Self-similar (self-congruent) heat transfer processes occupy an important place in mathematical theory of solids thermal conductivity. The results of self-similar heat transfer processes in the isotropic solid under the impact of the instant flat or point (spherically symmetric) heat source on the object under study are well known. The specific features of the self-similar heat transfer process in a solid with spherical focus of various physical properties, i.e. spherical cavity filled with high-temperature gas with its fixed or moving according to the specified law boundary, and spherical inclusion or an inclusion in the form of a spherical layer absorbing penetrating radiation are known as well.

Despite the results achieved in the self-similar heat transfer processes in the analyzed system studying, a number of questions is still unanswered. It refers, in particular, to the issue of the possibility of self-similar process existence in the isotropic body with a heat source wielding axial symmetry.

The article formulates a problem of the temperature field determining of an isotropic solid with an axisymmetric heat source in the form of a circular cylinder. The mathematical model being analyzed of the heat transfer process in the system under study is based on the hypothesis that the heat source is thermally thin, i.e. on the realization of the «concentrated capacity» idea. It represents a mixed problem for a parabolic type second-order partial differential equation with a specific boundary condition actually accounting for the heat source presence in the system.

Using a self-similar substitution proposed in the article, a self-similar problem of heat transfer in an isotropic solid with a thermally thin heat source wielding axial symmetry was formulated and solved. A sufficient condition setting the possibility of a self-similar heat transfer process implementing in the system being analyzed has been identified. The obtained results were employed in a qualitative analysis of physical properties of the self-similar heat transfer process under study.

Keywords:

isotropic solid body, axisymmetric heat source, temperature field, self-similar solution

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