Temperature field of three-phase material with inclusions in the form of a ball layer absorbing penetrating radiation


Аuthors

Attetkov A. V.*, Volkov I. K., Gaydaenko K. A.**, Kotovich A. V.

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

*e-mail: fn2@bmstu.ru
**e-mail: kseniyagaydaenko@gmail.com

Abstract

In studies on the problem of laser initiation of explosive decomposition in energy materials, an important place is occupied by the «microbeam model» of the heat transfer process in a two-phase material with spherical inclusions absorbing penetrating radiation. A theoretical and significant practical interest in studying heat transfer processes in a radiation-transparent material is absorbing inclusions in the form of a ball layer. Difficulties are known in finding an analytical solution to the corresponding problem for a system of partial second-order parabolic derivatives even in the simplest situation of thermal isolation of the outer boundary of the ball layer. It is noted that the analytical solution of the problem under consideration using the Laplace’s integral transformation on a time variable in the space of its images is not associated with overcoming significant difficulties, but the subsequent transition to the space of originals of the used integral transformation is very problematic. To overcome the difficulties encountered in the work, another version of the analytical solution to the problem under consideration was proposed. Using the general theory of integral transformations, a finite integral transformation over a spatial variable for a three-layer region was developed, its core, a spectrum of eigenvalues and a weight function were identified. The obtained results are used to find analytically closed solution of the problem presented by the studied mathematical model of heat transfer in a material transparent for radiation with absorbing inclusions in the form of a ball layer. The presented results show that during the parametric analysis of the studied temperature field, significant technical difficulties can arise due to the complex nature of the dependence of the nucleus and the spectrum of eigenvalues of the developed integral transformation on the parameters of the original model. In this regard, it is advisable, using the original mathematical model as the basic one, to develop a hierarchy of its simplified analogues and then determine the range of possible application of each of them.

Keywords:

three-phase material, laser radiation, absorbing inclusions in the form of ball layer, temperature field, integral transformation

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