Two-liquid plasma model

A hydrodynamic model in which the plasma is regarded as made up of two "liquids" (electric and ionic liquids) moving through each other. The electrical resistance of the plasma is considered to be the result of the mutual friction between these liquids.

On the assumption that the electrons are acted upon solely by the electron pressure $ p _ {e} $, while the ions are acted upon solely by the ion pressure $ p _ {i} $, the set of equations of motion has the form

$$ \tag{1 } \frac{dm \mathbf V _ {e} }{dt} = - e \left ( \mathbf E - \frac{1}{c} [ \mathbf V _ {e} \times \mathbf H ] \right ) - \frac{\nabla p _ {e} }{n _ {e} } - Rn _ {i} ( \mathbf V _ {e} - \mathbf V _ {i} ) , $$

$$ \tag{2 } \frac{dM \mathbf V _ {i} }{dt} = Z e \left ( \mathbf E + \frac{1}{c} [ \mathbf V _ {i} \times \mathbf H ] \right ) - \frac{\nabla p _ {i} }{n _ {i} } - Rn _ {e} ( \mathbf V _ {i} - \mathbf V _ {e} ) . $$

The interaction between electrons and ions is allowed for by way of a friction force which is proportional to the product of the velocity difference by the concentration of the motion-retarding particles. The quantity $ R $ is called the mutual friction coefficient or the coefficient of diffusional resistance. In view of the quasi-neutrality of the plasma ( $ n _ {e} = Zn _ {i} = n $), the equation of motion of the two-liquid plasma model is reduced to the form

$$ \frac{d \mathbf V }{dt} = - \frac{1} \rho \nabla p + \frac{1}{\rho c } [ \mathbf j \times \mathbf H ] , $$

where

$$ \mathbf V = \frac{n _ {i} M \mathbf V _ {i} + n _ {e} m \mathbf V _ {e} }{M n _ {i} + mn _ {e} } $$

is the average mass velocity, $ p = p _ {i} + p _ {e} $ is the total pressure, while $ \mathbf j = e ( Z n _ {i} \mathbf V _ {i} - n _ {e} \mathbf V _ {e} ) $ is the ionic current. If $ m / M \ll 1 $, then $ \mathbf V _ {i} \approx \mathbf V $, $ \mathbf V _ {e} \approx \mathbf V - \mathbf j / ne $.

Equations (1) and (2) may be employed to obtain the generalized Ohm law, interconnecting the current density of $ \mathbf j $ with other quantities. If the terms of the form $ ( \mathbf V _ \nabla ) \mathbf V $ can be neglected (and also if $ m / M \ll 1 $), the generalized Ohm law can be written as

$$ \frac{d \mathbf j }{dt} = \frac{ne ^ {2} }{m} \left ( \mathbf E + \frac{1}{c} [ \mathbf V \times \mathbf H ] \right ) - \frac{e}{mc} [ \mathbf j \times \mathbf H ] + \frac{e}{m} \nabla p _ {e} - \frac{\mathbf j } \tau , $$

where $ \tau = 1 / \nu _ {ei} $ is the so-called pulse transmission time and $ \nu _ {ei} $ is the effective frequency of pulse transmission, defined by the expression:

$$ R = \frac{m}{n _ {i} } \nu _ {ei} = \frac{m}{n _ {e} } \nu _ {ie} . $$

References

[1]	D.A. Frank-Kamenetskii, "Lectures on plasma physics" , Moscow (1968) (In Russian)
[2]	A.G. Kulikovskii, G.A. Lyubimov, "Magnetic hydrodynamics" , Moscow (1962) (In Russian)

References

[a1]	N.G. van Kampen, B.U. Felderhof, "Theoretical methods in plasma physics" , North-Holland (1967)
[a2]	L. Spitzer, "Physics of fully ionized gases" , Interscience (1962)