Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma
A stationary weakly ionized highly nonisothermal plasma is considered in the hydrodynamic approximation. Taking into account the effects of ionization, recharging, and a self-consistent field, the effect of ion viscosity on the distribution of plasma discharge parameters in the sheath was investigat...
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| Zitieren: | Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma / Ya.F. Leleko // Problems of Atomic Science and Technology. — 2023. — № 1. — С. 32-37. — Бібліогр.: 12 назв. — англ. |
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Leleko, Ya.F. 2023-12-08T14:46:00Z 2023-12-08T14:46:00Z 2023 Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma / Ya.F. Leleko // Problems of Atomic Science and Technology. — 2023. — № 1. — С. 32-37. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 52.40.Kh, 52.27.Cm, 52.25.Ya DOI: https://doi.org/10.46813/2023-143-032 https://nasplib.isofts.kiev.ua/handle/123456789/195963 A stationary weakly ionized highly nonisothermal plasma is considered in the hydrodynamic approximation. Taking into account the effects of ionization, recharging, and a self-consistent field, the effect of ion viscosity on the distribution of plasma discharge parameters in the sheath was investigated. Distributions of hydrodynamicion velocity and ion density, electron density, and self-consistent field potential in the sheath were obtained. Розглянуто стаціонарну слабко іонізовану сильно неізотермічну плазму в гідродинамічному наближенні. З урахуванням ефектів іонізації, перезарядки самоузгодженого поля досліджено вплив в’язкості іонів на розподіл параметрів плазмового розряду в перехідному шарі. Отримано розподіли гідродинамічних швидкості та густини іонів, густини електронів і потенціалу самоузгодженого поля у цьому шарі. In conclusion, the author expresses deep gratitude to Doctor of Physical and Mathematical Sciences Dmytro Leonidovich Grekov for constructive discussions and valuable advice during the work. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Basic plasma physics Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma Вплив в’язкості іонів на розподіл параметрів у перехідному шарі на межі стаціонарної слабко іонізованої сильно неізотермічної плазми Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma |
| spellingShingle |
Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma Leleko, Ya.F. Basic plasma physics |
| title_short |
Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma |
| title_full |
Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma |
| title_fullStr |
Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma |
| title_full_unstemmed |
Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma |
| title_sort |
influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma |
| author |
Leleko, Ya.F. |
| author_facet |
Leleko, Ya.F. |
| topic |
Basic plasma physics |
| topic_facet |
Basic plasma physics |
| publishDate |
2023 |
| language |
English |
| container_title |
Problems of Atomic Science and Technology |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Вплив в’язкості іонів на розподіл параметрів у перехідному шарі на межі стаціонарної слабко іонізованої сильно неізотермічної плазми |
| description |
A stationary weakly ionized highly nonisothermal plasma is considered in the hydrodynamic approximation. Taking into account the effects of ionization, recharging, and a self-consistent field, the effect of ion viscosity on the distribution of plasma discharge parameters in the sheath was investigated. Distributions of hydrodynamicion velocity and ion density, electron density, and self-consistent field potential in the sheath were obtained.
Розглянуто стаціонарну слабко іонізовану сильно неізотермічну плазму в гідродинамічному наближенні. З урахуванням ефектів іонізації, перезарядки самоузгодженого поля досліджено вплив в’язкості іонів на розподіл параметрів плазмового розряду в перехідному шарі. Отримано розподіли гідродинамічних швидкості та густини іонів, густини електронів і потенціалу самоузгодженого поля у цьому шарі.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/195963 |
| citation_txt |
Influence of ion viscosity on the distribution of parameters in the sheath at the boundary of a stationary weakly ionized strongly nonisothermal plasma / Ya.F. Leleko // Problems of Atomic Science and Technology. — 2023. — № 1. — С. 32-37. — Бібліогр.: 12 назв. — англ. |
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AT lelekoyaf influenceofionviscosityonthedistributionofparametersinthesheathattheboundaryofastationaryweaklyionizedstronglynonisothermalplasma AT lelekoyaf vplivvâzkostííonívnarozpodílparametrívuperehídnomušarínamežístacíonarnoíslabkoíonízovanoísilʹnoneízotermíčnoíplazmi |
| first_indexed |
2025-11-26T17:01:01Z |
| last_indexed |
2025-11-26T17:01:01Z |
| _version_ |
1850763802002325504 |
| fulltext |
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143).
32 . Series: Plasma Physics (29), p. 32-37.
https://doi.org/10.46813/2023-143-032
INFLUENCE OF ION VISCOSITY ON THE DISTRIBUTION
OF PARAMETERS IN THE SHEATH AT THE BOUNDARY
OF A STATIONARY WEAKLY IONIZED STRONGLY
NONISOTHERMAL PLASMA
Ya.F. Leleko
Institute of Plasma Physics, National Science Center
“Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
E-mail: yakovleleko@gmail.com
A stationary weakly ionized highly nonisothermal plasma is considered in the hydrodynamic approximation.
Taking into account the effects of ionization, recharging, and a self-consistent field, the effect of ion viscosity on the
distribution of plasma discharge parameters in the sheath was investigated. Distributions of hydrodynamicion
velocity and ion density, electron density, and self-consistent field potential in the sheath were obtained.
PACS: 52.40.Kh, 52.27.Cm, 52.25.Ya
INTRODUCTION
The interaction of plasma with surfaces that contact
this plasma is an important area of research into the
stationary state of a gas discharge. The negative
potential of a wall with respect to the plasma, which
arises due to the high mobility of electrons, attracts ions
and repels electrons. Thus, a region of a positive space
charge is formed near the wall and this space charges
creens the neutral plasma from the negatively charged
wall. Under these conditions, it is convenient to separate
the plasma into two parts. The first part is the main
plasma volume with characteristic size L (plasma size)
where the quasi-neutrality condition is satisfied almost
exactly. This part will be called below the quasi-
neutrality region. The second part is a narrow region
near the wall where ions are mainly concentrated, and
the quasi-neutrality condition is not satisfied. This part
will be called below the sheath. The characteristic width
of the she at his in the range from several to several tens
of Debye-Hückel screening radius rDe = (Te/(4πe2ne0))1/2,
where Te – electron temperature, e – the charge of the
electron, and ne0 – hydrodynamic density of electrons in
the middle of the plasma. Typically, the value of rDe is
small compared to other characteristic quantities, such
as the plasma size L or the mean free paths of ions due
to ionization, charge exchange, or collisions.
Notable early works devoted to studies of stationary
plasma include [1-4]. In [1], it was assumed for the first
time that the velocity of ions is determined by a static
self-consistent electric field created by the balance of
electric charges of electrons and ions. Based on this, an
integral equation for the distribution of the plasma-
sheath potential for different geometries, ion mean free
paths and ionization methods was obtained. The
solution of this equation in the case of a short ion mean
free path in a cylinder with an ions generation
proportional to the electron density gave the same
potential distribution as found by Schottky [2, 3] for a
positive column using the theory of ambipolar diffusion.
As Bohm showed [4], the formation of a stationary layer
of space charge was possible only under the condition
that ions entered the region of the sheath at a speed
larger than the ion-sound speed, ʋs = ʋB = (Te/mi)1/2.
This condition was obtained in the case of cold ions
(Ti = 0) and without taking the viscosity into account. At
Ti ≠ 0 Bohm's speed is equal to ʋB = ʋs(1 + τ)1/2, where
τ = Ti/Te ≪ 1. Consequently, the ions are preliminarily
accelerated by a self-consistent electric field in the
quasi-neutrality region. If the ions move towards the
plasma-confining surfaces under the action of a self-
consistent electric field, there must be a potential
maximum in the plasma. In the case of plane walls that
will be considered in the paper, the symmetry dictates
that this maximum is located in a plane in the middle of
the plasma. It is convenient to choose the origin of
coordinates x = 0 in this plane. Then, the dielectric walls
that confine the plasma are located at x = ±L (Fig. 1).
Fig. 1. Schematic layout of the plasma layer and the
dielectric, confining the plasma, concerning
the x coordinate
Further studies of the stationary state of plasma
generated a large number of papers. Numerous
references can be found, for example, in the works of
Riemann [5, 6].
In the majority of works, the ions viscosity is
considered a small parameter and disregarded in the
ions motion equation. For example, [7] gives the
condition when effects associated with viscosity can be
neglected in the transport equations: ʋi ≪ νLʋ , where ʋi
is the hydrodynamic velocity of ions, ν is the frequency
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143) 33
of collisions, Lʋ is the characteristic scale of change in
the hydrodynamic velocity. It is questionable that
ʋi ≪ νLʋ is satisfied in the sheath even when this
condition holds true in the quasi-neutrality region.
In [8], in the case of a stationary, weakly ionized,
strongly noni so thermal plasma, estimates were
obtained on how the viscosity of ions affects solutions
of the system of hydrodynamic equations, taking into
account the effects of ionization, charge exchange, and
the self-consistent field together with the Poisson
equation in the entire plasma volume. Solutions to this
system of equations were also obtained taking into
account the viscosity of the ions but only in the quasi-
neutrality region.
The present work is a continuation of [8]. It
investigates the solution of the system of the above
equations in the sheath at the plasma boundary, taking
into account the viscosity of the ions. The temperature
of electrons and ions and the density of neutral particles
(hydrogen) are assumed to be constant. An original
numerical algorithm was developed to solve a system of
three differential equations, two of which are of the
second order and the third one is of the first order, by
the Cauchy method for the initial values problem, taking
into account the condition on the right boundary of the
solution domain. This method is an alternative to
solving the boundary value problem for eigenfunctions
and eigenvalues. The position of the wall was
determined by the condition of the equality of electron
and ion fluxes.
The work is organized as follows. Section 2
describes the formulation of the problem and derives the
basic equations. Section 3 is devoted to the description
of the quasi-neutral approximation for solving the main
system of equations. Section 4 describes the procedure
for solving the basic equations in the sheath and
presents the main results. Section 5 presents the
conclusions of this research work.
1. BASIC EQUATIONS
To solve the problem of the stationary distribution of
plasma parameters in gas discharges, we will use the
hydrodynamic approximation. This approach can be
used when the macroscopic parameters of the plasma,
such as the hydrodynamic velocity 𝑣 and density 𝑛 of
the particles, change rather slowly in space and time.
Namely, the characteristic distances at which the values
of macroscopic quantities change are much larger than
the mean free path [9]. This approach is also valid in the
case of a collision less plasma if the thermal motion of
particles can be neglected, that is, the plasma must be
sufficiently cold [10]. However, even if these conditions
are not fulfilled, the hydrodynamic approach can be
used for a qualitative analysis of plasma parameters.
To calculate the effect of viscosity on the
distribution of plasma parameters, we will use the
stationary system of the hydrodynamic equations of
continuity and motion, complemented by the Poisson
equation. We consider the one-dimensional case for
dimensionless variables ʋ = ʋi/ʋs, n = ni/ne0, Φ = eφ/Te
taking into account the effects of ionization, charge
exchange, and a self-consistent electric field, where ʋi,
ni – hydrodynamic velocity and density of ions, φ – the
self-consistent electric field potential. We assume that
the electron density is determined by the Boltzmann
formula ne = ne0exp(eφ/Te).
𝑣𝑣′ = −𝛷′ − (𝜈 + 𝛼
𝑒𝛷
𝑛
) 𝑣 − 𝜏
𝑛′
𝑛
+
4
3
𝜂𝑣′′, (1)
𝑛′𝑣 + 𝑛𝑣′ = 𝛼𝑒𝛷 , (2)
𝛷′′ = 𝑒𝛷 − 𝑛, (3)
where α = αe/ωpi, ν = νex/ωpi, 𝜂 = 4𝜂
𝑖
/(3mini), α, αe, ν,
νex, 𝜂 , and 𝜂
𝑖
– dimensionless and dimensional
frequencies of electron impact ionization and charge
exchange of ions on hydrogen atoms and kinematic
viscosity coefficients of ions. The prime denotes the
derivative with respect to the dimensionless coordinate
x/rDe. A detailed derivation of the system of equations
(1)-(3) and a description of the hydrodynamic
coefficients are given in the work [8].
System of differential equations (1)-(3) considering
the viscosity of the ions is a fifth-order nonlinear system
for the unknown functions. It must be supplemented by
the boundary conditions. For reasons of symmetry, at
the center of the plasma, the following is true
𝑣(0) = 𝑣′′(0) = 𝑛′(0) = 𝛷′(0) = 0. (4)
Also, we can assume that Φ(0) = 0 because the
potential is defined up to a constant. One more
boundary condition is the condition at the plasma-wall
boundary: the hydrodynamic ions flux is equal to the
electrons flux in the direction of the coordinate x
growth. Additionally, it is assumed that the electrons are
distributed according to the Maxwell-Boltzmann
distribution, and there are no effects of reflection from
the wall and electron emission on the wall [11]
𝛤(𝐿) = 𝑛(𝐿)𝑣(𝐿) = √𝑚𝑖 (2𝜋𝑚𝑒)⁄ exp(𝛷(𝐿)). (5)
Thus, we have a system of fifth-order equations with
five boundary conditions, i.e., the problem of
determining the eigenvalues. For example, if L, Te, Ti,
and nn are given, then stationary gas discharge is
possible at a certain valuene0 that is determined by the
solution of the system.
Finding eigenfunctions and eigenvalues of a
nonlinear system (1)-(3) is a rather difficult problem.
Therefore, an alternative approach is used. We take
boundary conditions (4) as initial conditions,
supplement them with an arbitrary initial value for ne0,
and integrate system (1)-(3) in the direction of positive
𝑥, that is, we solve the Cauchy problem (see, for
example, [6], [12], where the case of cold ions was
considered).
2. QUASI-NEUTRAL APPROXIMATION
We use the smallness of the second derivative of the
potential in equation (3): Φ'' ≪ n, eΦ. Let's call this
approach the “quasi-neutral approximation”. In the 1st
iteration (Φ'' = 0), from equation (3) we get
𝑛1 = 𝑒𝛷1 , 𝛷1 = ln 𝑛1 , 𝛷1
′ =
𝑛1
′
𝑛1
, (6)
𝛷1
′′ =
𝑛1
′′
𝑛1
− (
𝑛1
′
𝑛1
)
2
. (7)
Expressions (6) are then substituted in equations (1)
and (2), which take the form
𝑣1𝑣1
′ = −(1 + 𝜏)
𝑛1
′
𝑛1
− (𝜈 + 𝛼)𝑣1 +
4
3
𝜂𝑣1
′′, (8)
𝑛1
′ 𝑣1 + 𝑛1𝑣1
′ = 𝛼𝑛1. (9)
34 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143)
In the 1st iteration, we consider Φ'' = 0 and by
solving equations (6), (8) and (9) we find the solutions
ʋ1, n1 and Φ1. In the 2nd iteration, Φ'' = Φ1'' is
determined by expression (7) and equations (6), (8) and
(9) take the form
𝑛2 = 𝑒𝛷2 − 𝛷1
′′, 𝛷2 = ln(𝑛2 + 𝛷1
′′),
𝛷2
′ =
𝑛2
′ +𝛷1
′′′
𝑛2+𝛷1
′′ , (10)
𝑣2𝑣2
′ = −
𝑛2
′ + 𝛷1
′′′
𝑛2 + 𝛷1
′′ − (𝜈 + 𝛼
𝑛2 + 𝛷1
′′
𝑛2
) 𝑣2 − 𝜏
𝑛2
′
𝑛2
+
+
4
3
𝜂𝑣2
′′, (11)
𝑛2
′ 𝑣2 + 𝑛2𝑣2
′ = 𝛼(𝑛2 + 𝛷1
′′). (12)
It is convenient to find the initial values of the
plasma parameters at the origin of coordinates using
expansions of the plasma parameters near this point
𝛷 = (𝑎0 + 𝑎1�̅�2+. . )�̅�2, 𝑣 = (𝑏0 + 𝑏1�̅�2+. . )�̅�,
𝑛 = 𝑐0 + 𝑐1�̅�2+. ., (13)
where �̅� = x/rDe. We find coefficients of these
expansions with the required accuracy by substituting
(13) into (1)-(3) and equating terms with equal powers
while considering the sufficient number of expansion
terms. In [6], a similar expansion was used to overcome
the problem of the singularity in the middle of the
plasma in the region near x = 0 for the case of cold ions
(Ti = 0).
Using the quasi-neutral approximation for the
system of equations (1)-(3) without the viscosity was
necessary to overcome the singularity at the point
ʋ = ʋTi. When the viscosity was included in
consideration, the order of equation (1) increased, and
this singularity was eliminated. However, the necessity
of using the quasi-neutral approximation remained.
The following values of dimensional and
dimensionless quantities were used in the calculations:
ne0 = 1010 cm-3, Te = 2 eV, Ti = 0.1 eV, nn = 1014 cm-3,
ʋTe = 5.93∙107 cm/c, ʋTi = 3.09∙105 cm/с,
ʋs = 1.38∙106 cm/c, rDe = 1.05∙10-2 cm, ωpi=1.32∙108 c-1,
αe = 1.39∙103 c-1, νex = 2.74∙105 c-1, νCi = 1.49∙105 c-1,
𝜂
𝑖
= 2.25∙105 cm2/c, α = 1.05∙10-5, ν = 2.08∙10-3,
νC = 1.13∙10-3, 𝜂 = 15.5. Solutions of the systems of
equations (8), (9) using (6) and (11), (12) using (7) and
(10) taking into account the viscosity were obtained in
the work [8]. Viscosity influences the variation of
plasma parameters in the quasi-neutrality region very
little and produces only a slight increase of the size of
this region. Fig. 2 shows the dependences of the
dimensionless hydrodynamic ions velocity ʋ2, the
densities of ions n2 and electrons ne2/ne0, the potential of
the self-consistent electric field – Φ2, the flow of ions
n2ʋ2 in the second iteration of the quasi-neutral
approximation and the hydrodynamic density of ions n1
in the first iteration of the quasi-neutral approximation
on the coordinate x at the plasma boundary in the case
when the viscosity is taken into account. Dependencies
in Fig. 2 are limited when the condition (5) on the wall
(Lq ≈ 105.857 cm) is satisfied. It should be noted that
deviation of the quasi-neutral solution n2from the exact
solution n of the system of equations (1)-(3) increases
with an increase of the space charge (n2 – ne2/ne0). As is
known, the beginning of the sheath and the boundary of
the region of applicability of the quasi-neutral
approximation are defined by the point in space where
the fraction of the space charge becomes noticeable:
(n2 – ne2/ne0)/n2 ≈ 0.01. In Fig. 2, this condition
corresponds to the point x ≈ 105.5 cm. Therefore, when
x ≿ 105.5 cm, the dependencies in this figure are
qualitative only and are different from the exact
solutions of the system of equations (1)-(3).
Fig. 2. Dependencies of the dimensionless
hydrodynamic ions velocity ʋ2 (1), the densities of ions
n2 (2) and electrons ne2/ne0 (3), the potential of the self-
consistent electric field – Φ2 (4), the flow of ions
Γ2=n2ʋ2 (5) in the second iteration of the quasi-neutral
approximation and the dimensionless hydrodynamic
density of ions n1 (6) in the first iteration of this
approximation on the coordinate x at the plasma
boundary, taking the viscosity into account
3. SOLUTION OF THE SYSTEM OF BASIC
EQUATIONS IN THE SHEATH
Without taking the viscosity into account, the system
of equations (1)-(3) has a singularity at the point ʋ = ʋTi.
Therefore, from the middle of the plasma (x = 0) to the
point ʋ = ʋs, the problem was solved by the Cauchy
method with the help of the quasi-neutral
approximation. The quasi-neutral solutions in the
second iteration n2, ʋ2 and Φ2 were obtained.
At the point ʋ = ʋs, the quasi-neutral solution has a
singularity. After passing the singularity point ʋ = ʋTi at
the point ʋTi < ʋ < ʋs (the joining point), these solutions
n2, ʋ2, Φ2, and Φ2' were used as initial conditions for
solving the system of equations (1)-(3) without the last
term in (1). In this way, the solution n of the system of
equations (1)-(3) was obtained from the joining point to
the plasma boundary without taking into account the
viscosity [8]. However, it should be understood that the
mathematical solutions of systems of equations without
viscosity n2 and n always differ by some value n2 – n.
This value n2 – n ≪ |Φ1''| is small in the quasi-neutrality
region and increases as the plasma boundary is
approached. For a smoother joining, one can take into
account the difference ∆n in the initial conditions of the
system of equations (1)-(3) at the joining point without
taking into account the viscosity. Any iteration of the
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143) 35
quasi-neutral approximation can be chosen as an
approximate solution. Then, these initial conditions can
be written as:
𝑛 = 𝑛2 + ∆𝑛 (14)
and from the system of equations (1) - (3)
𝑣 = 𝑣2 + ∆𝑣 ≈
𝑛2𝑣2
𝑛2+∆𝑛
, (15)
𝛷 = 𝛷2 + ∆𝛷 ≈ ln(𝑒𝛷2 + ∆𝑛). (16)
When obtaining (15) and (16), we neglected changes in
the derivatives of the plasma parameters when the
density changes by ∆n at the joining point.
Our calculations have shown that the system of
equations (1)-(3) is stable against a change in the value of
∆n in the initial conditions (14)-(16). When ∆n is taken
into account, the solution of the system of equations (1)-
(3) without taking into account viscosity is shifted by a
distance x(n2+∆n) – x(n2), which is much smaller than the
size of the sheath. Therefore, for calculations without
taking into account the viscosity [8] in the initial
conditions of the system of equations (1)-(3), the value
∆n = 0 was chosen at the joining point.
As already noted above, taking into account the
viscosity increases the order of the equation of motion
of ions (1). This leads to the situation when two second-
order equations appear in the system (1)-(3). The
eigenfunctions of these equations are exponential
functions. In addition, the singularity disappears from
systems (1)-(3), (8)-(9), and (11)-(12). In this case, the
system of equations (1)-(3) is unstable against a change
in the value of ∆n in the initial conditions (14)-(16).
Fig. 3 shows the dependences of the difference n2 - n of
the dimensionless hydrodynamic ion densities, which
were obtained in the second iteration of the quasi-
neutral approximation and when solving the system of
equations (1)-(3), on x coordinate for different values of
∆n in the initial conditions on different scales. The blue
line corresponds to the solution with ∆n ≈ 4.079∙10-6
and L1 ≈ 104.29 cm. In this case, as L is approached
(condition (5) is satisfied), the ion density sharply
increases. The green line corresponds to the solution
with ∆n ≈ 4.086∙10-6 and L2 ≈ 104.25 cm. Here, when
approaching L (condition (5) is not satisfied), the
potential and density of electrons → ∞.These two
solutions are not physical because their graphical
dependences cannot pass near the nominal boundary of
the quasi-neutrality region (x ≈ 105.5 cm).The magenta
line corresponds to a solution with ∆n ≈ 4.081∙10-6 and
L3 ≈ 105.833 cm (condition (5) is satisfied).This
solution can be a possible solution of the system of
equations (1)-(3). Unfortunately, it is impossible to find
the unique “correct” solution of the system of equations
(1)-(3) with this formulation of the problem. Boundary
condition (5) determines the potential Φ(L) and the
hydrodynamic ion flux n(L)ʋ(L) ≈ const(L) at the
plasma boundary uniquely, but there is a certain
arbitrariness in the choice of n(L) and ʋ(L). Therefore,
the system of equations (1)-(3) can have many possible
solutions in this case. The physically meaningful
solutions can be defined as follows: near the nominal
boundary of the quasi-neutrality region (x ≈ 105.5 cm),
the graphic dependences of the solutions of the system
of equations (1)-(3) on x coordinate should not differ
much from the second iteration of the quasi-neutral
solution.
By setting ∆n at the joining point between the blue
and green lines (Fig. 3), the dichotomy method can be
used to obtain a solution with any boundary values of
the hydrodynamic ions density n(L) or ions velocity
ʋ(L) and the corresponding value of L.
The joining of the solutions of the system of
equations (1)-(3) and the second iteration of the quasi-
neutral approximation was carried out at the point
x ≈ 103.52 cm. This point was chosen from the
condition that the number of decimal places in the
variable ∆n is sufficient to obtain solutions with
different values of hydrodynamic ions density n(L) or
velocity ʋ(L) by the dichotomy method. That is, if the
value of this point is taken less, then 32 decimal places
may not be enough to reach solutions with the desired
n(L) or ʋ(L). In the chosen case, x = 9850 rDe.
Fig. 3. Dependences of the difference n2 - n of the
dimensionless hydrodynamic ion densities, which were
obtained in the second iteration of the quasi-neutral
approximation and when solving the system of
equations (1)-(3), on x coordinate for different values of
∆n in different scales (a, b)
As an example, let us consider the behavior of plasma
parameters for three cases when the graphical dependences
are slightly different from the second iteration of the quasi-
neutral solution near the point x ≈ 105.5 cm.
Fig. 4 shows the dependences of the ratio of the fraction
of the dimensionless space charge – Φ'' to the
dimensionless electron density eΦ on the distance L – x to
the edge of the plasma for solving the system of equations
(1)-(3) for various plasma sizes, taking the viscosity into
36 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143)
account, L3, L4 ≈ 105.869 cm, L5 ≈ 105.920 cm and without
viscosity L0 ≈ 104.593 cm. In the absence of viscosity, the
size of the sheath is ≈ 0.335 cm. Taking the viscosity into
account, for the chosen solutions this value increases to
0.34...0.44 cm (≾15 %). In this case, the plasma size L
increases by ~1 %.
Fig. 4. Dependences of the ratio of the proportion of the
dimensionless space charge – Φ'' to the dimensionless
electrons density eΦ on the distance L - x for solutions of
the system of equations (1)-(3) for different plasma
sizes, taking the viscosity into account L3 (1), L4 (2),
L5 (3) and without viscosity for L0(4)
Fig. 5. Dependences of the dimensionless hydrodynamic
densities of ions n (a, 1–4) and electrons ne/ne0 (a, 5–8),
the ions velocity ʋ (b, 1–4) and the potential of the self-
consistent electric field – Φ (b, 5–8) on x coordinate for
various plasma sizes L3 (1.5), L4 (2.6), L5 (3.7) and the
second iteration of the quasi-neutral approximation for
Lq (4.8) taking the viscosity into account
Fig. 5 shows the dependences of the dimensionless
hydrodynamic ions velocity ʋ, ions density n, electrons
density ne/ne0, and self-consistent electric field potential
– Φ on x coordinate for various values of the plasma
size L3, L4, L5 and the second iteration (Lq) of the quasi-
neutral approximation when the viscosity is taken into
account. Solutions of the system of equations (1)-(3),
taking into account the viscosity for L3, L4, and L5 at
x ≈ 105.5 cm, do not visually differ from the quasi-
neutral solution. It should be understood that such a
condition can be satisfied by an infinite number of
solutions with different L.
In Fig. 6 the dependences of the dimensionless
hydrodynamic velocities ʋ of ions, the densities of ions
n and electrons ne/ne0 and the potential of the self-
consistent electric field – Φ on the coordinate L – x in
the sheath at the edge of the plasma are shown for cases
with the viscosity for L3, L4, and L5 and without it for L0.
The size of the sheath for various cases is visible in
Fig. 4.
Fig. 6. Dependences of the dimensionless hydrodynamic
densities of ions n (a, 1–4) and electrons ne/ne0 (a, 5–8),
the ions velocity ʋ (b, 1–4) and the potential of the self-
constrained electric field – Φ (b, 5–8) on x coordinate
with the viscosity for L3 (1,5), L4 (2,6), L5 (3,7)
and without the viscosity for L0 (4,8)
Taking the viscosity into account increases the
hydrodynamic velocity of ions at the boundary of the
sheath from ʋs to ≈ 1.5ʋs and makes the change in
plasma parameters smoother.
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №1(143) 37
CONCLUSIONS
As is known, in a stationary gas discharge near a
surface with which the plasma interacts, a sheath is
formed. This is because electrons, due to their greater
mobility, charge the wall negatively relative to the rest
of the plasma volume. The negative potential of the wall
attracts the ions. As a result, a narrow region (the
sheath) is formed that screens the surface from the rest
of the plasma volume (the quasi-neutrality region). The
sheath boundary can be determined from the condition
ni – ne ≈ 0.01ne. The size of the sheath can be
≈ 30...50 rDe.
The system of hydrodynamic equations of motion
and continuity for ions together with the Poisson
equation was solved by the Cauchy method. The values
of the parameters at the center of the plasma were used
as the initial conditions. The plasma boundary was
determined from the condition of equality of the
hydrodynamic ions flux and the kinetic electrons flux.
The paper proposes and applies a method for finding
the parameters of a weakly ionized, strongly
nonisothermal stationary plasma with Ti ≠ 0, taking into
account the viscosity of ions in the sheath. Calculations
have shown that when the viscosity is taken into
account, plots of the dependences of the plasma
parameter on the spatial coordinate become smoother,
the size of the sheath increases lightly (≾15 %), and the
hydrodynamic velocity of ions at the boundary of the
sheath increases approximately by ~ 50 %.
ACKNOWLEDGEMENTS
In conclusion, the author expresses deep gratitude to
Doctor of Physical and Mathematical Sciences Dmytro
Leonidovich Grekov for constructive discussions and
valuable advice during the work.
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Article received 13.12.2022
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ВПЛИВ В’ЯЗКОСТІ ІОНІВ НА РОЗПОДІЛ ПАРАМЕТРІВ У ПЕРЕХІДНОМУ ШАРІ НА МЕЖІ
СТАЦІОНАРНОЇ СЛАБКО ІОНІЗОВАНОЇ СИЛЬНО НЕІЗОТЕРМІЧНОЇ ПЛАЗМИ
Я.Ф. Лелеко
Розглянуто стаціонарну слабко іонізовану сильно неізотермічну плазму в гідродинамічному наближенні.
З урахуванням ефектів іонізації, перезарядки самоузгодженого поля досліджено вплив в’язкості іонів на
розподіл параметрів плазмового розряду в перехідному шарі. Отримано розподіли гідродинамічних
швидкості та густини іонів, густини електронів і потенціалу самоузгодженого поля у цьому шарі.
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