MSC2010 35F30

Mathematical modeling of transport processes in a cylindrical channel

O. V. Germider1, V. N. Popov2

AnnotationIn the framework of the kinetic approach, a solution of heat and mass transfer problems in a long cylindrical channel is found using a mirror-diffuse model of the Maxwell boundary condition. The Williams equation is used as the main equation describing the kinetics of the process, assuming that a constant longitudinal temperature gradient is maintained in the channel. The Williams equation is written in the Cartesian coordinate system. The solution of the linearized problem of nonisothermal flow of the rarefied gas through the channel is obtained using the method of characteristics. It is shown that the type of the boundary condition becomes decisive in the construction of this solution. In a wide range of the Knudsen numbers, the reduced heat and gas mass flows through the cross-section of the channel are calculated depending on the accommodation coefficient of the tangential pulse. Limiting expressions of these flows for the free molecular and hydrodynamic flow regimes are obtained. The comparison with similar results presented in the open press is carried out. The obtained results can be used in the development of new nanotechnology.
Keywordskinetic Boltzmann equation, Williams equation, mirror-diffuse reflection, mirror-diffuse model, Maxwell model, analytic solution, Knudsen number.

1Oksana V. Germider, Postgraduate Student, Northern (Arctic) Federal University named after M. V. Lomonosov (17 Severnaya Dvina Emb., Arkhangelsk 163002, Russia), ORCID:,

2Vasily N. Popov, Head of Mathematics Chair, Northern (Arctic) Federal University named after M. V. Lomonosov, (17 Severnaya Dvina Emb., Arkhangelsk 163002, Russia), Ph.D. (Physics and Mathematics), ORCID:,

Citation: O. V. Germider, V. N. Popov, "[Mathematical modeling of transport processes in a cylindrical channel]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:1 (2018) 64–77 (In Russian)

DOI 10.15507/2079-6900.20.201801.64-77