Analytical Approaches to the Analysis of Unsteady Heat Conduction for Partially Bounded Regions
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AND MASS TRANSFER AND PHYSICAL GASDYNAMICS
Analytical Approaches to the Analysis of Unsteady Heat Conduction for Partially Bounded Regions E. M. Kartashov* Moscow Institute of Radio Engineering, Electronics, and Automation (MIREA), Russian Technological University, M.V. Lomonsov Institute of Fine Chemical Technologies, Moscow, 119454 Russia *e-mail: [email protected] Received November 22, 2019; revised December 6, 2019; accepted December 24, 2019
Abstract—A mathematical theory is developed for the construction of integral transforms for the following partially bounded regions: a space with an internal cylindrical cavity in the cylindrical coordinates (radial heat flux); a space with an internal spherical cavity in the spherical coordinates (central symmetry); and a space bounded by a planar surface in the Cartesian coordinates. Expressions are proposed for the integral transforms, Laplace operator images, and inversions for images. The formulated approach differs from the classical theory of differential equations of mathematical physics for the construction of integral transforms with a continuous spectrum of eigenvalues based on the corresponding singular Sturm–Liouville problems. The proposed method is based on the operational solutions of the initial boundary-value problems of unsteady heat conduction with an inhomogeneous initial function and homogeneous boundary conditions. The formulated approach makes it possible to develop the Green’s function method and to construct integral representations of the analytical solutions of the boundary-value problems simultaneously based on the Green’s functions and inhomogeneities in the main equation and boundary conditions of the problem. The proposed functional relations can be used in numerous particular cases of practical thermal physics. Examples of the application of the obtained results in some fields of science and technology are presented. DOI: 10.1134/S0018151X20030086
INTRODUCTION Modern structural materials (a combination of micro- or nanostructured materials) are referred to as structure-sensitive. The use of nanotechnologies for the production of these materials is an important direction in the development of modern materials science, because they have unique physico-mechanical properties, which condition their efficient application in designs subjected to high-intensity external effects [1, 2]. An important stage in the preparation and use of these materials is the construction of the corresponding mathematical models describing their behavior in a wide range of variation of the external effects. The general methodology for the construction and analysis of these models is far from completely established and requires further development. This statement primarily concerns mathematical models of some physical processes, which are based on the principle of local thermodynamic equilibrium and the continuous-medium hypothesis [2]. These models often contain a temperature component in the form of the corresponding boundary-value heat conduction problem under co
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