Analytical solution of the second Stokes problem with variable amplitude on behaviour of gas over oscillation surface. Part I: eigenvalues and eigensolutions

TL;DR

This paper analytically solves the second Stokes problem for a rarefied gas over an oscillating surface with variable amplitude, focusing on eigenvalues, eigenfunctions, and the spectrum of the kinetic equation.

Contribution

It provides an analytical eigenfunction expansion solution for the second Stokes problem with variable amplitude, analyzing both continuous and discrete spectra of the kinetic equation.

Findings

Eigenvalues and eigenfunctions for the problem are derived.

Properties of the dispersion function are analyzed.

The general solution includes both discrete and continuous spectral modes.

Abstract

In the present work the second Stokes problem about behaviour of the rarefied gas filling half-space is formulated. A plane limiting half-space makes harmonious fluctuations with variable amplitude in the plane. The amplitude changes on the exponential law. The kinetic equation with model integral of collisions in the form $\tau$-model is used. The case of diffusion reflexions of gas molecules from a wall is considered. Eigen solutions (continuous modes) of the initial kinetic equation corresponding to the continuous spectrum are searched. Properties of dispersion function are studied. It is investigated the discrete spectrum of the problem consisting of zero of the dispersion functions in the complex plane. It is shown, that number of zero of dispersion function to equally doubled index of problem coefficient. The problem coefficient is understood as the relation of boundary values of dispersion function from above and from below on the real axis. Further are eigen solutions (discrete modes) of the initial kinetic equation corresponding to the discrete spectrum are searched. In the end of work the general solution of the kinetic equation in the form expansion under eigen solutions with unknown coefficients corresponding to discrete and continuous spectra is constructed.