View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945601.png" />, while the ions are acted upon solely by the ion pressure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945602.png" />, the set of equations of motion has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945605.png" /> is called the mutual friction coefficient or the coefficient of diffusional resistance. In view of the quasi-neutrality of the plasma (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945606.png" />), the equation of motion of the two-liquid plasma model is reduced to the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945607.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945608.png" /></td> </tr></table>

is the average mass velocity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t0945609.png" /> is the total pressure, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456010.png" /> is the ionic current. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456013.png" />.

Equations (1) and (2) may be employed to obtain the generalized Ohm law, interconnecting the current density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456014.png" /> with other quantities. If the terms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456015.png" /> can be neglected (and also if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456016.png" />), the generalized Ohm law can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456018.png" /> is the so-called pulse transmission time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456019.png" /> is the effective frequency of pulse transmission, defined by the expression:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094560/t09456020.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.A. Frank-Kamenetskii,   "Lectures on plasma physics" , Moscow  (1968)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kulikovskii,   G.A. Lyubimov,   "Magnetic hydrodynamics" , Moscow  (1962)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.G. van Kampen,   B.U. Felderhof,   "Theoretical methods in plasma physics" , North-Holland  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Spitzer,   "Physics of fully ionized gases" , Interscience  (1962)</TD></TR></table>