On the inertia term in the momentum equation in the free-fall regime of discharge maintenance

Abstract

AbstractThe study, being on two-dimensional modelling of low pressure discharges, suggests an approach to the nonlinear inertia term in the momentum equation of the positive ions needed to be accounted for in the free-fall regime of the discharge maintenance. On the basis of conclusions that the inertia term acts in the wall sheath, where the ions fly perpendicularly to the walls, it is shown that (i) the parallel – to the walls – velocity component can be neglected, and (ii) the rest of the convective derivative can be determined by using the energy conservation law in the collisionless case. In a way, the inertia term acting as a retarding force is joined to the momentum loss term by introducing effective collision frequencies. The validity of the procedure is proved in a model of a low pressure argon discharge by comparison with the exact solutions for the two-dimensional spatial distribution of the discharge characteristics (ion velocity, electron density and temperature and DC electric field and its potential). The conclusion is that (i) ignoring the velocity component that is parallel to the walls does not cause deviation from the exact solution, and (ii) the approximation of using the energy conservation law in the collisionless case is good enough.