On the fundamental solution of the heat transfer problem in one-dimensional harmonic crystals
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O R I G I NA L A RT I C L E
O. S. Loboda · E. A. Podolskaya A. M. Krivtsov
· D. V. Tsvetkov ·
On the fundamental solution of the heat transfer problem in one-dimensional harmonic crystals
Received: 28 March 2020 / Accepted: 21 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The work is devoted to the description of unsteady thermal processes in low-dimensional structures. To obtain the relationship between the microscopic and macroscopic descriptions of solids, it is necessary to understand the heat transfer mechanism at the micro-level. At the latter, in contrast to the macro-level, analytical, numerical, and experimental studies demonstrate significant deviations from the Fourier’s law. The paper bases on the ballistic heat transfer model, according to which the heat is carried by the thermal waves. This effect can be applied, for example, to signal transmission and heat removal problems. The influence of nonnearest neighbors on processes in discrete media, as well as processes in polyatomic lattices, is investigated. To describe the evolution of the initial thermal perturbation, the dispersion characteristics and group velocities in one-dimensional crystal are analyzed for (i) a diatomic chain with variable masses or stiffnesses and for (ii) a monatomic chain with regard for interaction with second neighbors. A fundamental solution to the heat distribution problem for the corresponding crystal models is obtained and investigated. The fundamental solution allows to obtain a description of waves traveling from a point source, and can serve as the basis for constructing all other solutions. In both cases, the solution consists of two thermal fronts moving one after another with different speeds and intensities. Quantitative estimates of the intensity of the thermal wave front are given, and the dynamics of changes in the velocities and intensities of the waves depending on the parameters of the problem is analyzed. Thus, a simple method for estimating the wavefronts intensity is proposed and tested on two models. These results can be used to identify and analyze those parts of the wave processes that are of interest in terms of interpretation of the effects observed in the experiments. Keywords Thermal processes · Kinetic temperature · One-dimensional crystal · Fundamental solution · Ballistic heat transfer · Group velocity
1 Introduction At the macroscopic level, the heat propagation in the majority of the materials is described by the Fourier’s law, which assumes a linear relationship between the heat flux and the temperature gradient with a proportionality coefficient, defined as the thermal conductivity coefficient. However, at small temporal and spatial scales, Communicated by Andreas Öchsner. This work was supported by the Government of the Russian Federation (state assignment 0784-2020-0027). O. S. Loboda (B) · E. A. Podolskaya · D. V. Tsvetkov · A. M. Krivtsov Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russian Federation E-mail: loboda_o@
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