UDC

533.72

Authors

Popov Vasily Nikolaevich
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)
е-mail: v.popov@narfu.ru
Gulakova Svetlana Viktorovna
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)
е-mail: s.gulakova@narfu.ru

Abstract

In the framework of the kinetic approach analytic solution (in the form of Neumann series) of the problem of isothermal slip of a dilute gas along a hard, flat surface is set up. The linearized Williams equation as the basic equation and the Maxwell mirror-diffusive reflection model as the boundary condition on a streamlined surface are used. We used the collision integral model due to the fact that the assumption of independence of the gas molecular collisions frequency on their velocity is a fairly strong simplification. More realistic, in our opinion, is the assumption about the constancy of the length of free path of gas molecules, at least for those molecules, the interaction of which can be approximated by the model of hard spheres. This assumption is equivalent to the fact that the molecular collisions frequency must be proportional to the absolute value (module) of their thermal velocity. This circumstance was taken into account when the model Williams equations was constructing. The choice of model boundary condition is due to the fact that the diffusion factor for real surfaces may be significantly different from unity. The general solution of the assumed integro-differential equation is constructed in the space of generalized functions. Substitution of the constructed general solution in the boundary conditions leads to a singular integral equation with Cauchy-type kernel, which is reduced to the Riemann boundary value problem by methods of complex variable theory. Unknown parameters of the general solution are found from the condition of resolvability of the built boundary value problems. Based on the statistical meaning of the distribution function for different values of the diffusion factor the mass flux of gas profile in half-space above the wall is constructed and the isothermal slip velocity of gas is calculated. The numerical analysis of the obtained expressions and comparison of obtained results with the similar results, published in the public media, confirms the dependence of the values of the slip coefficients on choice of a collision integral model.

Keywords

Boltzmann kinetic equation, model kinetic equations, exact analytic solution

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