Editing AxisymmetricConfigurations/PGE (section)

===Eulerian Formulation (CYL.)===

Each of the above simplified governing equations has been written in terms of Lagrangian time derivatives.  An Eulerian formulation of each equation can be obtained by replacing each Lagrangian time derivative by its Eulerian counterpart.  Specifically, for any scalar function, <math>f</math>, 


<div align="center">
<math>
\frac{df}{dt} \rightarrow \frac{\partial f}{\partial t} + (\vec{v}\cdot \nabla)f =
\frac{\partial f}{\partial t} + \biggl[ \dot\varpi \frac{\partial f}{\partial\varpi} \biggr] + 
\biggl[ \dot{z} \frac{\partial f}{\partial z} \biggr]   .
</math>
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Making this substitution throughout the set of governing relations gives:

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<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span>

<math>\frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] 
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math><br />


<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
</span>
<table border="0" cellpadding="5">
<tr>
  <td align="right"><math>~{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>~
\frac{\partial \dot\varpi}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + 
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr]  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}  
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>~{\hat{e}}_z</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>~
\frac{\partial \dot{z}}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] + 
\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>
</table>

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

<math>~
\biggl\{\frac{\partial \epsilon}{\partial t} + \biggl[ \dot\varpi \frac{\partial \epsilon}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \epsilon}{\partial z} \biggr]\biggr\} +
P \biggl\{\frac{\partial }{\partial t}\biggl(\frac{1}{\rho}\biggr) + 
\biggl[ \dot\varpi \frac{\partial }{\partial\varpi}\biggl(\frac{1}{\rho}\biggr) \biggr] + 
\biggl[ \dot{z} \frac{\partial }{\partial z}\biggl(\frac{1}{\rho}\biggr) \biggr] \biggr\} = 0
</math>


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho . 
</math><br />
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