# On some applications of the hyperbolic heat equation and the methods for solving it

### Abstract

Creation of new technological processes based on the use of high-intensity energy fluxes makes it necessary to take into account the final rate of heat propagation when determining the temperature state. This account can be realized with the help of the hyperbolic heat equation obtained by A. V. Lykov in the framework of nonequilibrium phenomenological thermodynamics as a consequence of the generalization of the Fourier law for flows and the heat balance equation. In the previous works by V. N. Khankhasaev, the process of switching off the electric arc in a spiral gas flow was simulated using this equation. In this paper, a mathematical model of this process is developed with the addition of a period of steady burning of the arc until the moment of disconnection and replacement of the strictly hyperbolic heat conduction equation by a hyperbolic-parabolic equation. For the resulting mixed heat conduction equation, a number of boundary value problems in the Fortran and Matcad software environments are correctly posed and numerically solved, obtaining temperature fields that are in good agreement with the available experimental data.

### References

[1]

Poltev A. I., Construction and Calculation of High-Voltage SF6 Apparatuses [in Russian], Energia, Leningrad (1979).

[2]

Buyantuev S. L., Besprozvannykh M. N., Batorov S. S., and Borodyansky G. Ya., “Method of quenching a high-current arc of a high-voltage switch,” USSR Authors’ Cert. No. 1634042, Gos. Reg. 08.10.1990.

[3]

Buyantuev S. L. and Khankhasaev V. N., “Numerical investigation of nonstationary processes at the current zero [in Russian],” in: Sb. Dokl. Vsesoyuz. Semin. "Nonstationary Arc and Near-Electrode Processes in Electrical Apparatuses and Plasma Torches", pp. 29–36, Inst. Mat. Mekh. Akad. Nauk Kazakh. SSR, Alma-Ata (1991).

[4]

Buyantuev S. L. and Khankhasaev V. N., “On a generalization of the Navier–Stokes equations in mathematical models of an electric arc in a spiral gas flow at the moment of current interruption [in Russian],” Elektrichestvo, No. 11, 17–23 (1996).

[5]

Khankhasaev V. N. and Buyantuev S. L., “Numerical calculation of a mathematical model of an electric arc in a gas flow [in Russian],” in: Sb. Trudov Mezhdunar. Nauchn.-Prakt. Konf. "Energy Saving and Environmental Technologies at Baikal", pp. 168–172, Ulan-Ude (2001).

[6]

Formalev V. F., Selin I. A., and Kuznetsova E. L., “The occurrence and propagation of thermal waves in a nonlinear anisotropic space [in Russian],” Izv. Ross. Akad. Nauk., Ser. Energetika, No. 3, 136–141 (2010).

[7]

Shashkov A. G., Bubnov V. A., and Yanovsky S. Yu., Wave Phenomena of Thermal Conductivity [in Russian], URSS, Moscow (2004).

[8]

Cattaneo G., “Sur une forme de l’´equation de la chaleur eliminant le paradoxe d’une propagation instantan´ee,” C. R. Acad. Sci., Paris, 247, No. 4, 431–433 (1958).

[9]

Vernotte P., “Les paradoxes de la th´eorie continue de l’´equation de la chauleur,” C. R. Acad. Sci., Paris, 246, No. 22, 3154–3155 (1958).

[10]

Lykov A. V., Heat and Mass Transfer [in Russian], Energia, Moscow (1972).

[11]

Bubnov V. A., “Molecular-kinetic substantiation of the heat transfer equation [in Russian],” Inzh.-Fiz. Zh., 28, No. 4, 670-676 (1975).

[12]

Shablovsky O. N., “Propagation of a plane shock heat wave in a nonlinear medium [in Russian],” Inzh.-Fiz. Zh., 49, No. 3, 436–443 (1985).

[13]

Kuvyrkin G. N., “Thermodynamic derivation of the hyperbolic heat equation [in Russian],” Termofiz. Vysokih Temperatur 25, No. 1, 78–82 (1987).

[14]

Borodyansky G. Ya., “Relaxation model of electric arc dynamics [in Russian],” in: Tez. Dokl. 7th Vsesoyuz. Sessii Nauchn. Soveta "Low-Temperature Plasma Physics", pp. 19, Ulan-Ude (1988).

[15]

Danilenko V. A., Kudinov V. M., and Makarenko A. S., “Influence of memory effects on the formation of dissipative structures during fast processes [in Russian],” Preprint Akad. Nauk Ukrain. SSR, Inst. Elektrosvarki, No. 83-1, Kiev (1983), 58 p.

[16]

Khankhasaev V. N. and Mizhidon G. A., “Algorithm for numerical calculation of the mixed heat equation in the one-dimensional case,” in: Proc. 6th Int. Conf. "Mathematics, its Applications and Mathematical Education", pp. 357–359, Ulan-Ude (2017).

[17]

Khankhasaev V. N. and Mestnikova N. N., “A scheme of alternating directions for the numerical solution of a hyperbolic-parabolic equation [in Russian],” in: Sb. Trudov Mezhdunar. Konf. "Cubature Formulas, Monte Carlo Methods and Their Applications", pp. 117–120, Inst. Kosmich. Informats. Tekhnol. Sib. Feder. Univ., Krasnoyarsk (2011).

[18]

Khankhasaev V. N., Mestnikova N. N., and Khankhasaeva Ya. V., “Numerical solution of a hyperbolic-parabolic equation by a scheme of alternating directions [in Russian],” in: Sb. Trudov Mezhdunar. Nauchn.-Prakt. Konf. "Innovative Technologies in Science and Education", pp. 79–82, Buryatsk. Gos. Univ., Ulan-Ude (2011).

[19]

Ladyzhenskaya O. A., Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).

[20]

Romanova N. A., On the convergence of difference schemes of a boundary-value problem for equations of mixed type. Diss. kand. fiz.-mat. nauk (01.01.02), Yakutsk. Gos. Univ., Yakutsk, 1994, 109 p.

[21]

Dulnev G. N., The Use of Computers for Solving Heat Transfer Problems [in Russian], Vyssh. Shkola, Moscow (1990).

[22]

Khankhasaev V. N. and Darmakheev E. V., “A scheme of alternating directions for the numerical solution of the mixed heat equation [in Russian],” in: Mat. Semin. "Aktual’nye Voprosy Veshchestvennogo i Funktsional’nogo Analiza", pp. 122–127, Ulan-Ude (2015).

[23]

Ragaller K., Egli W., and Brand K., “Dielectric recovery of an axially blown SF6-arc after current zero,” IEEE Trans. Plazma Sci., PS-10, No. 3, 154–161 (1982).

[24]

Ferziger J. and Kaper G., Mathematical Theory of Transport Processes in Gases [in Russian], Mir, Moscow (1976).

*Mathematical notes of NEFU*, 25(1), pp. 98-111. doi: https://doi.org/10.25587/SVFU.2018.1.12772.

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