Geodesic
On certain applications of the hyperbolic heat transfer equation and methods for its solution
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 155 (2018), pp. 89-97

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When creating new technological processes based on the use of high-intensity energy flows, it is necessary to take into account the finite speed of heat transfer. This can be done by using the hyperbolic heat transfer equation obtained by A. V. Lykov within the framework of nonequilibrium phenomenological thermodynamics as a consequence of a generalization of the Fourier law for flows and the equation of thermal balance. In previous works, V. N. Khankhasaev modeled the process of switching off an electric arc in a gas flow using this equation. In this paper, we present a mathematical model of this process including the period of stable burning of an electric arc until to the shutdown moment, which consists of the replacement of the strongly hyperbolic heat transfer equation by a hyperbolic-parabolic equation. For the mixed heat transfer equation obtained, we state certain boundary-value problems, solve them by numerical algorithms, and obtain temperature fields that are well consistent with the available experimental data.
Keywords: hyperbolic-parabolic equation, hyperbolic heat transfer equation, Navier–Stokes equation, heat balance.
Mots-clés : scheme of variable directions 
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     author = {V. N. Khankhasaev and E. V. Darmakheev},
     title = {On certain applications of the hyperbolic heat transfer equation and methods for its solution},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {89--97},
     publisher = {mathdoc},
     volume = {155},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_155_a4/}
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V. N. Khankhasaev; E. V. Darmakheev. On certain applications of the hyperbolic heat transfer equation and methods for its solution. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 155 (2018), pp. 89-97. http://geodesic.mathdoc.fr/item/INTO_2018_155_a4/