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Electron Fluid Dynamical
Equations for Anti-Force
Current Bearing Waves
Carerabas Clark
Mostafa Hemmati, Ph.D.
Department of Physical Science
Arkansas Tech University
Russellville , AR, 72801
Objectives
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Introduction of breakdown waves
Introduction of the set of Electron Fluid Dynamical
Equations for Breakdown Waves
Dimensionless Equations for Pro-Force and Anti-Force
Waves
Inclusion of the current condition behind the wave front
Derivation of the set of Electron Fluid Dynamical
Equations for current bearing Anti-Force Waves
Background
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Charles Wheatstone -Ionizing potential waves was first observed
in low pressure discharge tube subjected to high potential difference
by Sir Charles Wheatstone. In his study, he was able to report
luminous front velocities in excess of 108 m/s, but due to lack of
effective time resolution equipment, he was unable to verify his
suspicions.
J.J Thomson - In 1893 conducted a series of experiments with
cathode ray tubes in which he reported observing fast moving
luminous pulses. He was able to improve the apparatus (discharge
tube), but was unable to solve the problem of synchronization of
pulse initiation with observation of front passage. He found that the
velocity of these luminous pulses was independent of size, shape
and material of the electrode.
Background

J.W. Beams - Offered a qualitative analysis for the phenomena observed by J.J
Thomson. Beams concluded that the luminosity always moves from the high
voltage electrode toward the electrode maintained at ground potential
regardless of the polarity. Beams theorized that the electrons are the main
element in the wave propagation, and near the pulsed electrode the field is very
high and intense ionization takes place.

Paxton and Fowler (1962) - Proposed a fluid model and theory for
breakdown wave propagation. They were able to write down equations of
conservation of the flux of mass, momentum and energy.
Shelton and Fowler (1968) - Were able to advance Paxton’s set of
equations and concluded a zero current condition.

Electron Fluid Dynamical Equations

Fowler et al. (1984) – modified the electron fluid
dynamical equations for breakdown waves propagating
into a non-ionized medium. The most significant
correction terms were inclusion of the heat conduction
term in the equation of conservation of energy.
Electron Fluid Dynamical Equations

R.G. Fowler and M. Hemmati – Completed
Shelton's set of equations. The set of equations
represent a one-dimensional, steady-state, electron
fluid-dynamical wave for which the electric field
force on electrons is in the direction of wave
propagation and the waves are propagating into a
neutral medium at constant velocity. These EFD
equations are the equations of conservation of mass,
momentum, and energy plus the Poisson equation.
Types of Breakdown Waves
Pro-Force waves are defined as waves in
which the electric field force on electrons is in
the same direction as the propagation of the
wave
 Anti-Force waves are waves for which the
electric field force on electrons is in the
opposite direction of the propagation of the
wave

Electron Fluid Dynamical Equations
The basic equations for analyzing breakdown
waves are the conservation equations of mass,
momentum, and energy coupled with Poisson’s
equation
Equations of Conservation of Mass,
Momentum, Energy and Poisson’s Equations
d ( nv )
 n
dx
(1)
(Mass)
d
mnvv  V   nkTe  enE  Kmnv  V 
dx
(Momentum)
(2)
d 
5nk 2Te dTe 
2
mnvv  V   nkTe5v  2V   2env 
 =
dx 
mK dx 
m
m
2
 3 nkKTe    Kmnv  V  (Energy) (3)
M 
M 
dE e
 N i  n 
dx  0
(Poisson’s Equation) (4)
Variables in Conservation
Equations
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E
= Electric field
x = position
 = Ionization frequency.
K = Elastic collision frequency.
V = Wave velocity.
M = Neutral particle mass
E0 = Electric field at the wave front.
= Ionization potential.
e = Electron charge.
Te = Electron Temperature.
n = Electron Number Density
v = Electron Velocity.
m = Electron Mass.
k = Boltzmann’s constant

Dimensionless Variables

In order to handle these equations more
effectively, dimensionless variables are
substituted in place of the non-dimensionless
variables seen previously.

Dimensionless Variables for
Breakdown
Waves
E
E0
 2e 
n
  
2 
  0 E0 
mVK

eE 0
v

V
KTe

2e


K
 
eE0 x
mV 2
2e

mV 2
2m

M
Description of Dimensionless
Variables

 = Electron number density.
 







= Electron velocity.
 = Electron Gas Temperature.
 = Relates Electric Field to Wave Speed.
 = Net Electric field.
 = Ionization Rate.
 = Position inside the wave.
ω = Ratio of Electron mass to Neutral mass
α = Wave Velocity.
Dimensionless Variables for
Pro-Force Waves

Substituting these dimensionless variables into
Eqs.[1-4] yields the set of Dimensionless EFD
equations for Pro-Force Waves
Dimensionless Electron Fluid Dynamical
Equations for Pro-Force Waves
d
   
d
(5)
d
   1          1
d
d 
5 2 d 
2
2
   1   5  2     
d 
 d 

  3    1
d 
   1
d 
2

(6)

(7)
(8)
Electron Fluid Dynamical Equations
for Anti-Force Waves


For Anti-Force Waves, the electron gas pressure is
assumed to be large enough to provide the driving
force, which implies that the electron temperature
must be large enough to sustain the motion.
In the Fluid model the wave is considered to be a
plane wave propagating in the positive direction.
The heavy particles are considered to be at rest
relative to the laboratory frame and the wave
extends from x = 0 to − ∞. The set of EFD equations
will be different from the set listed above and have
been provided by Hemmati (1999).
Modifications to the Dimensionless
Variables

In the wave, the heavy particles will be moving in
the negative x direction with a speed V. Therefore,
V < 0, E > 0, and K1 > 0. This leads to both ξ and
0
κ being negative.
Dimensionless Variables for
Anti-Force Waves
E

E0
 2e 
n
  
2 
  0 E0 
mVK
 
eE0
v

V
KTe

2e


K
 
eE0 x
mV 2
2e

mV 2
2m

M
Dimensionless Variables for
Anti-Force Waves

Substituting these dimensionless variables into
Eqs.[1-4] yields the set of Dimensionless EFD
Equations for Anti-Force Waves
Electron Fluid Dynamical Equations
for Anti-Force Waves
d
   
d
(9)
d
   1          1
d
d
d
(10)

5 2 d 
2
   1   5  2   


d





2   1   3    1
d

    1
d

2

(11)
(12)
Current Behind the Wave Front

Considering the Ion Number Density and
Velocity behind the wave to be Ni and Vi, the
net current behind the wave is
I = eNiVi - env
1

Solving this equation for Ni results in
Current Behind the Wave Front
Substituting Ni into the previous Poisson’s
equation[4] results in
dE
e I1
nV

(

 n)
dx
 0 eV
V
(17)
Current Behind the Wave Front
Now substituting the dimensionless variables
for Anti-Force Waves into Eq.[17] reduces it to
d
I1
v

 (  1)
d  0 KE0 
(18)
Current Behind the Wave Front
If you substitute ι for
in the above
equation it reduces the Poisson’s equation to
I1
 0 KE0
d
v
   (  1)
d

(19)
Current Behind the Wave Front

Solving for v(ψ -1) from the previous equation
and substituting it into the equation for
conservation of energy for Anti-Force waves
gives the final form of the equation with a
large current.
Electron Fluid Dynamical Equations
for Current-Bearing Anti-Force Waves
Therefore, the final form of the set of electron
fluid-dynamical equations describing Anti-Force
current bearing waves will be:
Electron Fluid Dynamical Equations
for Current-Bearing Anti-Force Waves
d
    ,
d
(20)
d
   1          1,
d
d
d
(21)

5 2 d
2
2
2
  
   1   5  2     
 d




 2   3    1 ,
d

     1
d

2
(22)
(23)