Study of linear parabolic and linear hyperbolic thermal conduction operators
https://doi.org/10.25587/2411-9326-2024-1-88-101
Abstract
We study the numerical solution to the nonlinear heat conduction problem for a plate with a nonlinear heat source (thermal conductivity coefficient and internal heat source are exponential functions of temperature). In particular, for the nonlinear problem, the phenomena of self-similarity, inertia, and heat localization were found, which also manifest themselves in solutions of linear hyperbolic heat equations. With a self-similar change in temperature in some ranges of spatial and temporal variables, similarity (self-similarity) of temperature curves is observed. When heat is localized in a certain range of spatial variable, the temperature does not change over time. The inertia of heat is revealed in the finite speed of its propagation, despite the solution of the parabolic heat equation. The listed phenomena are also observed in solutions of linear hyperbolic heat equations, the derivation of which takes into account the time dependence of the heat flow in the formula of Fourier’s law, leading to a finite rate of heat propagation. In nonlinear problems, a similar effect manifests itself due to the dependence of the physical properties and heat source on temperature, leading to a similar delay in heat flow.
About the Authors
V. V. Zhukov
RKTs Progress
Russian Federation
Russian Federation
Vitaliy V. Zhukov
Yu. A. Kryukov
RKTs Progress; Physics Department, Samara State Technical University
Russian Federation
Russian Federation
Yuri A. Kryukov
244 Molodogvardeyskaya Street, Samara 443100
K. V. Trubitsyn
Department of Theoretical Foundations of Heat Engineering and Hydromechanics, Faculty of Thermal Power Engineering, Samara State Technical University
Russian Federation
Russian Federation
Konstantin V. Trubitsyn
244 Molodogvardeyskaya Street, Samara 443100
V. A. Kudinov
Department of Theoretical Foundations of Heat Engineering and Hydromechanics, Samara State Technical University
Russian Federation
Russian Federation
Vasily A. Kudinov
244 Molodogvardeyskaya Street, Samara 443100
E. V. Kotova
Department of Theoretical Foundations of Heat Engineering and Hydromechanics, Samara State Technical University
Russian Federation
Russian Federation
Evgenia V. Kotova
244 Molodogvardeyskaya Street, Samara 443100
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Review
For citations:
Zhukov V.V., Kryukov Yu.A., Trubitsyn K.V., Kudinov V.A., Kotova E.V. Study of linear parabolic and linear hyperbolic thermal conduction operators. Mathematical notes of NEFU. 2024;31(1):88-101. (In Russ.) https://doi.org/10.25587/2411-9326-2024-1-88-101
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