Analytical Solution of Second Stokes Problem on Behavior of Gas over Oscillation Surface. Part III: Solving of Problem and Applications

TL;DR

This paper derives an analytical solution to the second Stokes problem for a rarefied gas over an oscillating surface, analyzing eigen solutions, velocity profiles, and energy dissipation.

Contribution

It provides a novel analytical solution to the second Stokes problem using eigenfunction expansion and studies the properties of the dispersion function.

Findings

Velocity of gas in half-space calculated

Gas velocity at the oscillating boundary determined

Friction force and energy dissipation quantified

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 model 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 general solution of the kinetic equation in the form of expansion by eigen solutions with the unknown coefficients corresponding to discrete and continuous spectra is worked out. In the present work the analytical solution of second Stokes problem is constructed. On the basis of the analytical solution velocity of gas in half-space is calculated. Also velocity of gas directly at oscillating boundary is calculated precisely. The force of a friction operating from gas on the oscillating plate is found, and also dissipation of energy of a plate is obtained.