To the charging of sphere in a EHD gas flow
The interaction between conducting sphere and ions has been investigated in a presence of external electric field and electrohydrodynamic (EHD) gas flow. Gas is considered as weakly ionized under atmospheric pressure. Diffusion is assumed to be weak and the trajectory assumption is taken into consid...
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| Cite this: | НазвTo the charging of sphere in a EHD gas flowание / A.E. Sorokin // Вопросы атомной науки и техники. — 2007. — № 1. — С. 176-178. — Бібліогр.: 3 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1105022025-02-09T14:16:33Z To the charging of sphere in a EHD gas flow Щодо зарядки сфери в ЕГД потоці газа К зарядке сферы в ЭГД потоке газа Sorokin, A.E. Low temperature plasma and plasma technologies The interaction between conducting sphere and ions has been investigated in a presence of external electric field and electrohydrodynamic (EHD) gas flow. Gas is considered as weakly ionized under atmospheric pressure. Diffusion is assumed to be weak and the trajectory assumption is taken into consideration. Ion currents are obtained analytically and investigated numerically for the collinear electric field and gas flow vectors. It is shown, that sphere charging regimes depend on the key parameter ξ± - the relation of ion drift velocity far away from sphere to gas velocity. So, the cases |ξ±| < 1 and |ξ±| > 1 yield to different charging regimes. For the potential flow, the ion current has been found analytically in continuous ξ± -parameter space. Досліджено взаємодію провідної сфери і електрогідродинамічного (ЕГД) газового потоку в присутності зовнішнього електричного поля. Газ передбачається слабко іонізованим. Дифузія іонів нехтується, враховується тільки їхній дрейф і перенос потоком газу. Аналітично і чисельно отримані іонні струми на сферу для випадку потенційного обтікання для будь-яких значень керуючого параметра ξ± - відношення швидкості дрейфу іонів в однорідному електричному полі до швидкості газу удалині від поверхні сфери. Исследовано взаимодействие проводящей сферы и электрогидродинамического (ЭГД) газового потока в присутствии внешнего электрического поля. Газ предполагается слабо ионизованным. Диффузией ионов пренебрегается, учитывается только их дрейф и перенос потоком газа. Аналитически и численно получены ионные токи на сферу для случая потенциального обтекания для любых значений управляющего параметра ξ± - отношение скорости дрейфа ионов в однородном электрическом поле к скорости газа вдали от поверхности сферы. 2007 Article НазвTo the charging of sphere in a EHD gas flowание / A.E. Sorokin // Вопросы атомной науки и техники. — 2007. — № 1. — С. 176-178. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 92.60.Pw, 92.60.Mt, 52.40.Kh https://nasplib.isofts.kiev.ua/handle/123456789/110502 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Low temperature plasma and plasma technologies Low temperature plasma and plasma technologies Sorokin, A.E. To the charging of sphere in a EHD gas flow Вопросы атомной науки и техники |
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The interaction between conducting sphere and ions has been investigated in a presence of external electric field and electrohydrodynamic (EHD) gas flow. Gas is considered as weakly ionized under atmospheric pressure. Diffusion is assumed to be weak and the trajectory assumption is taken into consideration. Ion currents are obtained analytically and investigated numerically for the collinear electric field and gas flow vectors. It is shown, that sphere charging regimes depend on the key parameter ξ± - the relation of ion drift velocity far away from sphere to gas velocity. So, the cases |ξ±| < 1 and |ξ±| > 1 yield to different charging regimes. For the potential flow, the ion current has been found analytically in continuous ξ± -parameter space. |
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Sorokin, A.E. |
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Sorokin, A.E. |
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Sorokin, A.E. |
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To the charging of sphere in a EHD gas flow |
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To the charging of sphere in a EHD gas flow |
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To the charging of sphere in a EHD gas flow |
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To the charging of sphere in a EHD gas flow |
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To the charging of sphere in a EHD gas flow |
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to the charging of sphere in a ehd gas flow |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2007 |
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Low temperature plasma and plasma technologies |
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https://nasplib.isofts.kiev.ua/handle/123456789/110502 |
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НазвTo the charging of sphere in a EHD gas flowание / A.E. Sorokin // Вопросы атомной науки и техники. — 2007. — № 1. — С. 176-178. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT sorokinae tothechargingofsphereinaehdgasflow AT sorokinae ŝodozarâdkisferivegdpotocígaza AT sorokinae kzarâdkesferyvégdpotokegaza |
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2025-11-26T17:38:18Z |
| last_indexed |
2025-11-26T17:38:18Z |
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1849875447292100608 |
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176 Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 176-178
TO THE CHARGING OF SPHERE IN A EHD GAS FLOW
A.E. Sorokin
Institute of Applied Physics, RAS, 603950, Nizhny Novgorod, Russia,
e-mail: and@appl.sci-nnov.ru
The interaction between conducting sphere and ions has been investigated in a presence of external electric field and
electrohydrodynamic (EHD) gas flow. Gas is considered as weakly ionized under atmospheric pressure. Diffusion is
assumed to be weak and the trajectory assumption is taken into consideration. Ion currents are obtained analytically and
investigated numerically for the collinear electric field and gas flow vectors. It is shown, that sphere charging regimes
depend on the key parameter ± - the relation of ion drift velocity far away from sphere to gas velocity. So, the cases
±| < 1 and ±| > 1 yield to different charging regimes. For the potential flow, the ion current has been found
analytically in continuous ± -parameter space.
PACS: 92.60.Pw, 92.60.Mt, 52.40.Kh
TASK FORMULATION
A conducting sphere moves through the weakly
ionized gas with a fixed terminal velocity Up. External
electric field is collinear over sphere motion vector. All
ions which attach sphere surface recombinate on it. The
selectivity properties of the sphere surface are discarded.
The equation for ion trajectories in a spherical coordinates
may be written as
(1)
where ± = 1±cos + 0 assuming = 0 for direct flow
or = for contrary one,
(2)
and
(3)
where r is the radial spherical coordinate normalized on
particle radius R, is the azimuth, Bi± = ±|Bi±| are ions
mobilities, E is the magnitude of the external uniform
electric field and is assumed positive. U is the magnitude
of the neutral gas velocity far away from the sphere. The
value p reflects particle polarization, for high static
relative permittivities, ? 1, pε 1, as for ideal
conducting sphere, for "transparent" particle we have
pε = 1.
ION TRAJECTORIES AND SPECIAL POINTS
Steady states for ions are defined from the equations:
Vr± = V ± = 0. For the neutral particle, only ions which are
being resisted by the external electric field have stationary
points with V = 0. In direct flow, = 0, that is negative
ions. The limiting trajectories form a separatrix envelope -
sphere with a radius RS for | < 1, which is determined
from the equation Vr = 0:
(4)
The same envelope forms and for positive ions in contrary
flow, = . We need to substitute + except in (4) to
determine RS. Both saddle points, back (r = RS2) and front
(r = RS1) lie on the sphere r = RS, but have different
azimuths: = is for front saddle point, = 0 is for back
one. When sphere acquires the charge then the separating
sphere r = RS disintegrates on two independent separatrix
surfaces, each is connected with a corresponded saddle
point.
Fig. 1. Negative ions tracks (potential flow), 1.0145,
= 1
One important exclusion concerns the case of potential
flow with ±| 1. Let us consider negative ions first.
Then RS2⋅RS1 and RS2 > RS, RS1 < RS, where RS is the
spherical separatrix radius with the same but with
= 0. And in the range ±| > 1, no any stationary points
exist for the Stokes flow. But for the potential flow the
saddle point RS1 moves to the limit value RS1 = 3
when 1 while RS2 tends to infinity. When turns
through the point = 1, the saddle point RS2 returns
from infinity but from the opposite direction, = , see
Fig.1.
This state with two saddles with the same azimuth
m = contracts to one saddle ring under = m, = m for
positive particle charge ( m < 0):
(5)
At this special point, r = Rm = 3(2 m 1)/2 m we have
Vr± = Vr± r = 0, where “+” is for = 0 and “-“ is for
= . The bifurcation in this point, ( m, m), gives rise the
saddle ring, see Fig. 2, and in the bifurcation point we
have: RS1 = RS2 = RS±.
mailto:and@appl.sci-nnov.ru
177
Fig. 2. Negative ions tracks (potential flow), = 1.05,
= 1
Coordinates of the saddle rings (RS±, S±), for the
negative ions and direct flow, = 0, and for the positive
ions in opposite flow, = , are defined from the system
Vr± = V ± = 0:
(6)
When electric field became large, ±| ? 1 and the sphere
charge is fixed then the saddle ring (RS , S ) turns to
sphere equator (1, /2) asymptotically. The transition of
the point (RS , s ) through equator occurs when particle
charge has become positive (for negative ions). Positive
ions have the same tacks structure in the contrary flow,
= and the bifurcation point ( m, m) is determined by
expression (5) with substituting m against m. As it will
be showed below, the dislocation of saddle ring has an
influence on charging ion flux.
ION CURRENT, ± < 1
The solution of the equation (1), with boundary
conditions at infinity,
r sin | , r h be as following:
(7)
It is supposed that ±| < 1, otherwise the limit
will be valid only for positive ions, for negative one it
might be 0. To find the cross-section, we have to
choose those ions from the flow, which are collected by
the sphere: ±(1) = (2pε + 1) ±.. With this, we obtain an
expression connecting the impact angle i± with the shoot
parameter h±:
(8)
where sign “+” is for the positive ions flow while “ ” for
the negative ions, q% = q/qU , where qU is the maximum
charge acquired by the isolated ideal conducting sphere in
the unipolar ions environment in the electric field,
qU = (2pε + 1)E R2, see [1]. Let us consider, first,
positive ions. With varying h+ from 0 to ht+ the impact
angle i+ varies from to a+ = arccos( q% ) .
So, the threshold shoot distance ht+ is obtained from
the (8) with cos i+ = q% :
(9)
If q% > 0 then Bi+Er(1, ) > 0 for any and, as
consequence, all positive ions will be repelled from the
aerosol surface and J% + = 0, see (10). Oppositely, if
q% < 1 then Bi+Er(1, ) < 0 for any azimuths and all
ions in the vicinity of aerosol particle will be attracted by
it and J% + = q. The corresponding collision cross-
section is: 2
th + = 4 +/(1 + +). Negative ions are
resisted by the external electric field for = 0, and it
gives income to the current if only q > 0. Then negative
ions bend around the particle and impact the surface at the
range i ∈ [0, a ], where cos( a ) = 1 2 q% , as it
follows from the (8). It defines the ring at the particle
surface which is connected by the separatrix paths with
the front saddle point (rS , ). Ions from the maximum
distance ht move to the back saddle point and attach the
sphere surface with i = 0. As it follows from the
equation (8) the cross-section be as:
2
th − = 4 /(1 + ).
So, the expressions for ht± determine positive and
negative charging current. The dimensional current, J±,
may be expressed through the collision cross-section
2
th + :
(10)
or
(11)
where J0± is the maximum charging current. Summarizing
the results, the expressions for the currents be as:
(12)
After the similar arguments as in the case = 0 charging
currents for = may be expressed as:
(13)
The formulae (12), (13) are the same as obtained in [2] by
another way. Actually, the method of integration over the
sphere surface (to sum all ions tracks and deduce the
current) was used in [3] for fast ions ±| >1. Following
this assumption and integrating the value 3 cos +
over the sphere surface from = 0 till = a we have:
(14)
But, as we will find in the next paragraph, that is
correct only for ±| ? 1 - it’s not for the general case.
For | q% | > 1 integration should be performed over the all
sphere surface that gives J% + = 0 and J% = 4 q% for
q% 1; J% + = q and J% = 0 for q% 1.
ION CURRENT, ± 1
The new results are presented below for the "fast" ions.
178
Potential flow with ±| > 1 has stationary points as it is
shown in Fig. 1,2 and has closed ion paths. Let us
consider the parameter space ( ±, ±) for negative ions and
let = 0. When |∈ [1, | m|] the current is determined
by formula (12), right side. But for | > m we have to
find the attachment azimuth which is not equal to the old
value a = arccos (1 2 q% ) for q% > 0 (positive net
charge, negative ions).
Since, the coordinate for saddle ring is known, we can
connect two conjugate points (1, a ) and (1, θ% a ) with
separatrix ring state (RS , S ). Thus, all these points lie on
the same ion path, see Fig. 2. Solving (1) for closed
separatrix path and using the relation
cos a± + cosθ% ±= -2 q% gives:
(15)
where q% S = ( 1(RS ) + 0(RS ))/3 , 0, 1 are the flow
function for potential flow determined by (2). Extending
this expression for the positive ions too, q% S± is written as:
(16)
where (RS±, S±) are the coordinates of the saddle ring.
Integrating the radial electric field over the sphere and
using the relation cos S = q% / q% S the negative ion
current be as:
(17)
At the point ( m, m) we have q% S = q% and formula (17)
yields: J% = 4 q% that is the same as for all range
| < | m|. From the other side, in the limit | ? 1,
current turns to the value (1 + q% )2, since q% S 1.
Finally, for all q% range, | > 1, = 0, negative ions
current is:
,
(18)
where q% S = (1 + 2 |)/2 |RS and it is always positive.
If q% 1 the current remains a constant for all ±. For
positive ions there are no closed tracks if +| > 1, = 0.
So, J% + can be obtained by the simple integration over the
sphere with limit azimuth cos a+ = 1 2 q% . It gives the
left part of equation (17). Due to the symmetry, currents
J% ± for = can be obtained using the following
substitution in (18): J% ± J
m
% , +, q% q% ,
q% s q% S+, and setting - in formula for q% S :
(19)
where q% S+ = (1 + 2 +)/2 +RS+, and J% is determined by
the right side of (20). Near the point ±| = 1 we can find
that q% S± ≈ (1.5)2/3(δξ)1/3
= 1, where = ± 1. But if
±| ? 1 then q% s± 1 1/3 ±
2 ∼ 1. At this limit formulas
(19) and (18) became equivalent. This means that current
does not depend on angle if ±| ? 1. That is right also
for Stokes flow for any ±| > 1, then current will be
expressed by the formula (19) with q% S+ = 1.
REFERENCES
1. M.M. Pouthenier and M. Moreau-Hanot. La charge
des particules spheriques dans un champ ionise // J.
Physique. 1932, v.12, p.590–613.
2. J.A. Chalmers, F.J.W. Whipple. On Wilson’s theory
of the collection of charge by falling drops // Quart.
Journal of Royal Meteorologycal Society. 1944, v.70,
N304, p.103–120.
3. G.F. Drukarev. To charges of raindrops // Proc. USSR
Acad. Sci. 1947, v.6, N8, pp. 330–336.
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