### Mathematical modeling of the process heat transfer in a long cylindrical channel

#### O. V. Germider^{1}, V. N. Popov^{2}, А. А. Yushkanov^{3}

Annotation | Within the framework of the kinetic approach the problem of the flow of rarefied gas in a cylindrical channel with longitudinal temperature gradient is solved. As the basic equation, that describes the kinetics of the process, the Williams kinetic equation, and as a boundary condition on the wall of the channel - a model of diffuse reflection are used. The deviation from the equilibrium state of the gas is assumed small. It allowed to consider the solution of the problem in the linearized form. In order to find a linear correction to the local equilibrium distribution function the problem is reduced to solving a linear homogeneous differential equation of the first order. It's solution is constructed using the method of characteristics. The obtained solution is used to construct the profile of the heat flux vector in the channel and the heat flux through the cross section of the channel. The numerical analysis of the final expressions is done. A comparison with similar results obtained by using the method of discrete ordinates, showed that the procedure proposed in the decision leads to the correct result in a wide range of channel radius. |
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Keywords | Boltzmann kinetic equation, model kinetic equations, method of characteristics, models of boundary conditions |

^{1}Post graduate student, Northern Arctic federal university named after M. V. Lomonosov, Arkhangelsk; o.germider@narfu.ru.^{2}Head of Mathematics Chair, Northern Arctic federal university named after M. V. Lomonosov, Arkhangelsk; v.popov@narfu.ru.^{3}Professor of the Department of Theoretical Physics, Moscow State Regional University, Moscow; yushkanov@inbox.ru.

**Citation**: O. V. Germider , V. N. Popov , А. А. Yushkanov , "[Mathematical modeling of the process heat transfer in a long cylindrical channel]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,17:1 (2015) 22–29 (In Russian)