Analytical solution of the second Stokes problem on behavior of gas over a oscillation surface. Part I: eigenvalues and eigensolutions

TL;DR

This paper provides an analytical solution to the second Stokes problem for rarefied gas over an oscillating surface, focusing on eigenvalues, eigensolutions, and the spectral properties of the kinetic equation.

Contribution

It derives the eigenvalues and eigensolutions for the kinetic equation with diffusive boundary conditions, analyzing the spectrum's discrete and continuous parts.

Findings

Eigenvalues correspond to zeros of the dispersion function.

The number of zeros doubles with the index of the problem coefficient.

A comprehensive spectral expansion of the solution is developed.

Abstract

The second Stokes problem about behaviour of the rarefied gas filling half-space is formulated. A plane, limiting half-space, makes harmonious oscillations in the plane. The kinetic equation with modelling integral of collisions in the form tau-model is used. The case of diffusive reflection of molecules of gas from a wall is considered. There are eigen solutions (continuous modes) the initial kinetic equation, corresponding to the continuous spectrum. Properties of dispersion function are studied. The discrete spectrum of this problem consisting of zeroes of dispersion function in complex plane is investigated. It is shown, that number of zero of dispersion function to equally doubled index of coefficient of the problem. The problem coefficient is understood as the relation of boundary values of dispersion function from above and from below on the real axis. Further there are eigen solutions (discrete modes) the initial kinetic equation, corresponding to the discrete spectrum. The common solution of the kinetic equation in the form of expansion by eigensolutions with the unknown coefficients corresponding to discrete and continuous spectra is worked out.