Electron Fluid Dynamical Equations for Anti-Force Current Bearing Waves

Carerabas ClarkMostafa Hemmati, Ph.D.

Department of Physical Science Arkansas Tech University

Russellville , AR, 72801

Objectives

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 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

ndx

nvd

)(

VvKmnenEnkTeVvmnvdx

d

dx

dTe

mK

TenkenvVvnkTeVvmnv

dx

d 22 5

225

23 VvKmnM

mnkKTe

M

m

nNe

dx

dEi

0

=

(1)

(2)

(3)

(4)

(Mass)

(Momentum)

(Energy)

(Poisson’s Equation)

Variables in Conservation Equations

= Electric field = position = Ionization frequency. = Elastic collision frequency. = Wave velocity. = Neutral particle mass = Electric field at the wave front. = Ionization potential. = Electron charge. = Electron Temperature. = Electron Number Density = Electron Velocity. = Electron Mass. = Boltzmann’s constant

ExKVM

0Ee

eTnvmk

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

M

m2

0eE

mVK

K

2

2

mV

e

V

v

nE

e

2

00

2

e

KTe

2

0E

E

20

mV

xeE

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. = Ratio of Electron mass to Neutral mass α = Wave Velocity.

ω

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

(5)

(6)

(7)

(8)

d

d

11 d

d

d

d

d

d 222 5

251

1

d

d

213

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, E0 > 0, and K1 > 0. This leads to both ξ and

κ being negative.

Dimensionless Variables for Anti-Force Waves

M

m2

0eE

mVK

K

2

2

mV

e

V

v

nE

e

2

00

2

e

KTe

2

0E

E

20

mV

xeE

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

(9)

(10)

(11)

(12)

d

d

11 d

d

d

d

d

d 22 5

251

1

d

d

21312

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

Solving this equation for Ni results in

I1 = eNiVi - env

Current Behind the Wave Front

)( 1

0

nV

nV

eV

Ie

dx

dE

(17)

Substituting Ni into the previous Poisson’s equation[4] results in

Current Behind the Wave Front

)1(00

1

v

KE

I

d

d

Now substituting the dimensionless variables for Anti-Force Waves into Eq.[17] reduces it to

(18)

Current Behind the Wave Front

)1(

v

d

d

00

1

KE

I

(19)

If you substitute ι for in the above equation it reduces the Poisson’s equation to

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

,11 d

d

2

222 5

251

d

d

d

d

,132 2

1

d

d

(21)

(20)

(22)

(23)