#### For citation:

Davidovich M. V., Kornev I. A., Timofeev A. I. Nonlinear temperature waves: Analysis based on the nonlinear heat equation. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2019, vol. 27, iss. 6, pp. 73-90. DOI: 10.18500/0869-6632-2019-27-6-73-90

# Nonlinear temperature waves: Analysis based on the nonlinear heat equation

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.

- Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. Moscow: Nauka, 1977. 736 p. (in Russian).
- Patankar S. Numerical Methods for Solving Problems of Heat Transfer and Fluid Dynamics. Translation from English edited by V.D. Vilensky. Moscow: Energoatomizdat, 1984. 512 p. (Patankar, Suhas V. Numerical Heat Transfer and Fluid Flow. Washington: New York: Hemisphere; McGraw-Hill, cop. 1980.)
- Shashkov A.G., Bubnov V.A., Yanovsky S.Yu. Wave Phenomena of Thermal Conductivity. Moscow: URSS, 2004. 298 p. (in Russian).
- Dulnev G.N., Parfenov V.G., Sigalov A.V. Application of Computers for Solving Heat Transfer Problems. Moscow: Higher school, 1990. 207 p. (in Russian).
- Kutateladze S.S. Fundamentals of the Theory of Heat Transfer. M: Atomizdat, 1979, 416 p. (in Russian).
- Sveshnikov A.G., Bogolyubov A.N., Kravtsov V.V. Lectures on Mathematical Physics: Studies. benefit. Moscow: Moscow State University, 1993. 352 p. (in Russian).
- Ozisik M.N. Radiative Transfer and Interactions with Conduction and Convection. C. John Wiley, 1973.
- Adrianov V.N. Fundamentals of Radiation and Complex Heat Transfer. Moscow: Energia, 1972. (in Russian).
- Muchnik G.F., Rubashov I.B. Methods of Heat Transfer Theory. Part 2. Thermal Radiation. Moscow: Higher School, 1974. 270 p. (in Russian).
- Siegel R., Howell J. Thermal Radiation Heat Transfer. McGraw-Hill Book Company, New York Google Scholar, 1972. (Siegel, Robert; Howell, John R. Thermal radiation heat transfer New Jourk a. o., 1972).
- Sparrow E.M., Sess R.D. Heat transfer by radiation, Trans. from English, Leningrad, 1970. 295. (Sparrow E.M. and Cess R.D., 1978, Radiative Heat Transfer, McGraw-Hill.Google Scholar. Tien, C.-L., 1988).
- Unsold A. Physics of Stellar Atmosphere. 2ed. Springer, Berlin, 1955.
- Ambartsumyan V.A., Mustel E.R., Severniy A.B., Sobolev V.V. Theoretical Astrophysics. Ed. V.A. Ambartsumyan. Moscow: Gostekhizdat, 1952. 635 p. (in Russian).
- Chandrasekhar S. Radiative Transfer. 2ed. Dover Publications Inc. 1960.
- Ivanov V.V. Radiation Transfer and Spectra of Celestial Bodies, Moscow: Nauka, 1969. 472 p. (in Russian).
- Favorskii O.N., Kadaner J.S. Issues of Heat Transfer in Space. M.: Higher School, 1972. 280 p. (in Russian).
- Adzerikho K.S. Lectures on the Theory of Transfer of Radiant Energy, ed. M. A. Elyashevich, Minsk, BSU publishing house, 1975, 192 p. (in Russian).
- Apresyan L.A., Kravtsov Yu.A. Theory of Radiation Transfer: Statistical and Wave Aspects, Moscow: Nauka, 1983, 216 p. (in Russian).
- Nagirner D.I. Lectures on the Theory of Radiation Transfer: Studies benefit, SPb.: Publishing house SPb. Univ., 2001, 284 p.(in Russian).
- Zeldowich Ya.B., Raizer Yu.P. Physics of Shock Waves and High-temperature Hydrodynamics Phenomena. Moscow: Nauka, 1966, 688 p. (in Russian).
- Davidovich M.V., Kornev I.A., Timofeev A.I. Nonlinear dynamics of heat transfer in cylindrical and spherical structures. In: Questions of applied physics. Intercollegiate scientific collection. Saratov, 2015. pp. 93–98. (in Russian).
- Davidovich M.V., Kornev I. A., Timofeev A.I., Yavchunovsky V.Ya. One-dimensional cylindrical thermal problem without initial conditions. In: Questions of applied physics. Intercollegiate Scientific Collection. Saratov, 2015. C. 35–37. (in Russian).
- Lifshits E.M., Pitaevsky L.P. Physical Kinetics. Moscow: Nauka, 1979, 528 p. (in Russian).
- Ortega J., Poole W. An Introduction to Numerical Methods for Differential Equations. Pilman, Marshfield, 1981.
- Kahaner D., Mouler C., Nash S. Numerical Methods and Software. Prentice-Hall, 1989.
- Amosov A.A., Dubinsky Yu.A., Kopchenova N. In. Computational Methods for Engineers. Moscow: Higher school, 1994. 544 p. (in Russian).
- Zaslavsky G.M., Sagdeev R.Z. An Introduction to Nonlinear Physics: From Pendulum to Turbulence and Chaos. Moscow: Nauka, 1988, 368 p. (in Russian).
- Tikhonov A.N. On the influence of radioactive decay on the temperature of the Earth crust. Proc. of USSR Academy of Sciences, Mathematics and Nature Sciences, 1937, pp. 431–459 (in Russian).

- 1696 reads