Singularities of the Two-Fluid Plasma Equations and Their Relation to Boundary Conditions

Abstract

The one-dimensional fluid equations are used to describe a steady state, slightly ionized plasma confined by cold walls. The singularities of the complete two-fluid and approximate one-fluid equations are investigated and compared. It is found that the singularity in the one-fluid equations, indicating a transition from plasma to sheath regions, disappears in the two-fluid equations and therefore a smooth transition is predicted. Another singularity which is present in the two-fluid but not in the one-fluid equations is shown to be compatible if a constraint relation is satisfied at the singular point. In order that the two-fluid problem be well posed—(1) the boundary conditions are modified to ensure that the constraint relation is satisfied, and (2) a kinetic model is derived which uniquely specifies the wall potential for a given plasma configuration. The occurrence of a compatible singularity is shown to be analogous to the phenomenon of transonic flow in a convergent-divergent nozzle.

Presheath and Space Charge Sheath Characteristics in Front of a Floating Wall

“…At the wall there existsdue to the microscopic character of the problemthe unique macroscopic condition [14] given by the potential of the wall. In the case of in- with normalized parameters…”

Numerical Modeling of the Non‐Isothermal Positive Column of an Ar+‐Laser

“…For a given parameter neO' however, the ion density on the axis can no longer be chosen freely, but is fixed by the postulation of a removable singularity at the ion sonic velocity (FRIEDMAN, LEVI [10)). …”

“…Thus, FRIEDMAN and LEVI [10], INGOLD [11], VALENTINI [12,13], and METZE, ERNIE and OSKAM [14] abandoned the assumption of quasi-neutrality and developed a two-fluid model, in which no derivative shows a singularity at the Bohm velocity. However, two new singularities arise at the isothermal ion and electron sonic velocities (2) While the electrons do not reach the sonic velocity within the plasma, the ion velocity of sound is reached within the positive column.…”

“…However, two new singularities arise at the isothermal ion and electron sonic velocities (2) While the electrons do not reach the sonic velocity within the plasma, the ion velocity of sound is reached within the positive column. Since the approximate quasi-neutral solution provides continuous curves for density and drift velocity at those points, FRIEDMAN and LEVI [10], INGOLD [11], FRANKLIN [15] and VALENTINI [12,13] postulated that the singularity in the two-fluid model must be removable. FRIEDMAN and LEVI [10] therefore assumed that the ratio of ion and electron densities on the axis cannot be chosen arbitrarily but is fixed by an internal condition at the ion sonic point.…”

“…Since the approximate quasi-neutral solution provides continuous curves for density and drift velocity at those points, FRIEDMAN and LEVI [10], INGOLD [11], FRANKLIN [15] and VALENTINI [12,13] postulated that the singularity in the two-fluid model must be removable. FRIEDMAN and LEVI [10] therefore assumed that the ratio of ion and electron densities on the axis cannot be chosen arbitrarily but is fixed by an internal condition at the ion sonic point. Their internal condition was n i = n. which, however, leads to physically unreasonable results.…”