Analytical solution of the second Stokes problem with variable amplitude on behaviour of gas over oscillation surface. Part I: eigenvalues and eigensolutions
Abstract
In the present work the second Stokes problem about behaviour of the rarefied gas filling halfspace is formulated. A plane limiting halfspace 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2016
 arXiv:
 arXiv:1603.04750
 Bibcode:
 2016arXiv160304750L
 Keywords:

 Physics  Fluid Dynamics;
 82C40;
 82B40;
 80A20;
 80A99
 EPrint:
 32 pages, in Russian, 8 figures