Abstract
A number of mathematical problems of the theory of ionization in relation to processes in stationary plasma thrusters (SPT) are solved in this work. Two main one-dimensional mathematical models of ionization are considered – hydrodynamic and kinetic. The main question is the existence of ionization oscillations (breathing modes). On the basis of a hydrodynamic model, a boundary value problem for stationary ionization equations is solved. Its unique solvability and the absence of breathing modes are proved. In the case when the ion velocity in the flow region has a single zero with a positive derivative, it is proved that the stationary boundary value problem has a countable number of solutions, and a necessary and sufficient condition for the existence of breathing modes is formulated. Finally, an analytical solution of the ionization equations is given in the case of constant velocities of atoms and ions, and the formulas obtained are applied to the solution of the Cauchy problem, boundary value and mixed problems in the simplest regions. In the case of the kinetic model of ionization, the existence of breathing modes is numerically shown and the results obtained are compared with the hydrodynamic case.
Highlights
В работе решён ряд математических задач теории ионизации применительно к процессам в стационарных плазменных двигателях (СПД)
The main question is the existence of ionization oscillations
In the case when the ion velocity in the flow region has a single zero with a positive derivative, it is proved that the stationary boundary value problem has a countable number of solutions, and a necessary and sufficient condition for the existence of breathing modes is formulated
Summary
V z i a a0 где константы C и D однозначно ищутся из граничных условий:. (z) C 1 , k vi0 z где константы C и D однозначно находятся из граничных условий. Va 1 DexpU (z) где константы C и D однозначно определяются из граничных условий:. В частности, na (z) – монотонно убывающая функция, а ni (z) – периодичная функция (1 a cos z) 1 , промодулированная по амплитуде монотонно возрастающей по z функцией DC expU(z) / (1 DexpU(z)) , сходящейся при z к значению C. Тогда система (1) относительно безразмерных значений всех величин перепишется в виде: na / t (nava ) / z kI nani , ni / t (nivi ) / z kI nani ,. Где vi,k vi (zk ) , vi (z ) и счёт по второму уравнению в (14) ведётся для k z k z. Начальное условие t1 – точка-тире, t2 – тире, t3 – точка, установившиеся решения – сплошные линии
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