# Mathematical Modeling of Disturbed Zone Near the Plane Electrode Streamlined by a Rarefied Plasma Flow

### Аuthors

Kotel'nikov V. A.*, Kotelnikov M. V.*, Nguyen Xuan Thau M. **

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: mvk_home@mail.ru
**e-mail: nguoikinhbac.rus.@gmail.com

### Abstract

Plane electrode can be considered as an element of the surface of a satellite moving in the ionospheric plasma. It can be in front of the satellite, in which case the velocity vector is perpendicular to the plasma flux incident onto the surface thereof. If the plane electrode is disposed on the side surface of the satellite, the velocity vector of the plasma flow is parallel to its surface.
A mathematical model of the problem in both cases involves the kinetic Vlasov equations for ion and electron distribution functions and the Poisson equation for the self-consistent electric field. Maxwell distribution with appropriate values of ​​directed velocities is used as the initial and boundary conditions for the distribution functions. The electrode surface potential is considered as given, while at the outer boundary of the computational domain the potential is assumed to be zero.
If the electrode is represented as an elongated rectangle, the problem is formulated in a four-dimensional phase space. Thus, the problem is time-dependent, multi-dimensional and multi-parameter. A numerical method of successive iterations in time is used to solve the problem. Kinetic equations are solved by the method of characteristics or the Davydov method of large particles, and the Poisson equation is solved by spectral methods. The computational algorithm is optimized and adapted for the computer of Pentium type with medium processing power.
Distribution functions for ions (IDF) and electrons (EDF) for two mentioned cases were obtained as a result of numerical modeling. The self-consistent electric field isolines were obtained also. With the known IDF and EDF, their moments were calculated. The presented velocity profiles of charged particles allow qualitative assessment for the end and edge effects that influence the distributions of ion and electron current densities over the plate in a complicated and non-linear way.

### Keywords:

parietal plasma, flat electrode, distribution function, satellite, a fully ionized plasma, Poisson equation, Vlasov equation

### References

1. Langmuir I. Collected Works of Irwing Langmuir , Phys. Review, 1926, vol. 11, pp.101-119.
2. Al'pert YA. A., Gurevich A. V., Pitayevskiy A. P. Iskusstvennyye sputniki v razrezhennoy plazme (Artificial satellites in a rarefied plasma), Nauka 1975, 352 p.
3. Kotel'nikov M. V. Mekhanika i elektrodinamika pristenochnoy plazmy (Mechanic and electrodynamics of wall plasma), Doctor’s thesis, Moscow, MAI, 2008, 276 p.
4. Shan'kov A.V. Matematicheskoye modelirovaniye protsessov perenosa vblizi ploskikh pristenochnykh zondov (Mathematical modeling of transfer processes near by the flat wall probes), PhD thesis, Moscow, MAI 1985, 218 p.
5. Kotel'nikov M.V., Gidaspov V.Y., Kotel'nikov V.A. Matematicheskoye modelirovaniye obtekaniya tel potokami besstolknovitel'noy i stolknovitel'noy plazmy (Mathematical modeling of flows with objects in collisionless and collisional plasma), Moscow, Fizmatlit, 2010, 288 p.
6. Kotel'nikov V.A., Kim V.P., Kotel'nikov M.V. Vzaimodeystviye tel s potokami razrezhennoy plazmy (The interaction between objects and the flow of rarefied plasma), Moscow, MAI, 2010, 186p.
7. M.V. Kotel'nikov, V.A. Kotel'nikov, S.B. Ul'danov. Protsessy perenosa v pristenochnykh sloyakh plazmy (Transfer processes in the wall layers of plasma), Moscow, Nauka, 2004, 475p.
8. Alekseev B.V., Kotel'nikov V.A. Zondovyy metod diagnostiki plazmy (Probe method for plasma diagnostics), Moscow, Еnergoatomizdat, 1988, 240p.