Unsteady mono-dimensional model for Hall-effect thrusters

Unsteady mono-dimensional model for Hall-effect thrusters 

Hall-effect thrusters exhibit a geometric configuration facilitating the use of a quasi-one-dimensional model, somewhat like the nozzles in chemical propiilsion (often, we can even adopt the one-dimensional model with a constant straight section on this subject, see Appendix A2). 

However, in view of the interactions at the walls and the presence of the magnetic field inducing an azimuthal motion of the electrons, we need to integrate these interactions into the balance equations as a fimction of time t and abscissa value x. This is what we usually do in a one-dimensional flow with heat-exchange at the walls 

However, there are a few differences here with regard to the interaction between the plasma and the wall. Indeed, in Chapter 7, the metal/plasma interface and the sheath of thickness Ld exhibit a prevailing direction perpendicular to the axis, whereas here, the surfaces of electromagnetic interaction are generally parallel to it. 

A number of authors have established balance equations for these flows, using simplifying hypotheses. For examples, see Barral et al. [BAR 03] and Gascon, Dudeck and Barral [GAS 03] in the steady-state case, or Boeuf and Garrigues [BOE 98] for an initial approach to the low-frequency oscillations in plasma thrusters and, more recently, Barral and Miedzik [BAR 11] for a more elaborate model of these phenomena in the context of a closed-loop study of Hall-effect accelerators. Also see Dabiri et al. [DAB 13]. 

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