Geodesic
Nonlinear temperature waves: Analysis based on the nonlinear heat equation
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 27 (2019) no. 6, pp. 73-90.

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The aim of the work is to introduce a nonlinear equation of thermal conductivity, which takes into account the radiation according to the Stefan–Boltzmann law inside the structure from each virtual surface (which assumes the introduction of body-blackness coefficient variation), and on its basis to consider temperature waves. Studied model. A nonlinear wave in a plane one-dimensional well-transparent layer in a Cartesian coordinate system with a large temperature gradient and thermostats at the boundaries is studied. It is believed that the variation of the blackness coefficient does not depend on the coordinate and temperature. The model of cooling of a cylindrical volume with water in a heat-insulating shell is also considered. Results. Nonlinear equation of thermal conductivity based on the energy balance is obtained, which is applied to a region transparent to radiation with a temperature gradient. Numerical study of temperature waves, showing a strong nonlinear properties: steepness increase of the front without possibility of overturning, the increase in wave velocity with increasing temperature gradient. It is also shown that accounting for radiation is important for cooling dynamics even at low temperatures, and in the considered problem leads to an increase in the calculated cooling rate by several tens of percent. Discussion. Limits of applicability of the equation and models are given and discussed. In terms of methodology, the proposed material may be of interest to engineers, students and postgraduates engaged in thermophysics. Results can be applied to calculation of thermal processes in transparent atmospheres of celestial bodies, as well as to analysis of temperature fields in micro-and nanostructures, for example, during heating of auto-emission structures.
Keywords: equation of radiant heat transfer, Stefan–Boltzmann law, Kirchhoff law, blackness coefficient, nonlinear temperature waves. 
@article{IVP_2019_27_6_a5,
     author = {M. V. Davidovich and I. A. Kornev and A. B. Timofeev},
     title = {Nonlinear temperature waves: {Analysis} based on the nonlinear heat equation},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {73--90},
     publisher = {mathdoc},
     volume = {27},
     number = {6},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2019_27_6_a5/}
}
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M. V. Davidovich; I. A. Kornev; A. B. Timofeev. Nonlinear temperature waves: Analysis based on the nonlinear heat equation. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 27 (2019) no. 6, pp. 73-90. http://geodesic.mathdoc.fr/item/IVP_2019_27_6_a5/