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User:Tohline/AxisymmetricConfigurations/Equilibria - Revision history
2024-03-28T21:55:04Z
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https://www.vistrails.org//index.php?title=User:Tohline/AxisymmetricConfigurations/Equilibria&diff=18547&oldid=prev
Tohline: /* Axisymmetric Configurations (Steady-State Structures) */
2019-08-04T00:15:28Z
<p><span dir="auto"><span class="autocomment">Axisymmetric Configurations (Steady-State Structures)</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:15, 4 August 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Axisymmetric Configurations (Steady-State Structures)=</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=Axisymmetric Configurations (Steady-State Structures)=</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><!</del>-- [[<del style="font-weight: bold; text-decoration: none;">Image</del>:<del style="font-weight: bold; text-decoration: none;">LSU_Structure_still</del>.<del style="font-weight: bold; text-decoration: none;">gif</del>|<del style="font-weight: bold; text-decoration: none;">74px|left</del>]] <del style="font-weight: bold; text-decoration: none;">--></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Equilibrium, axisymmetric '''structures''' are obtained by searching for time</ins>-<ins style="font-weight: bold; text-decoration: none;">independent, steady</ins>-<ins style="font-weight: bold; text-decoration: none;">state solutions to the </ins>[[<ins style="font-weight: bold; text-decoration: none;">User</ins>:<ins style="font-weight: bold; text-decoration: none;">Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_</ins>.<ins style="font-weight: bold; text-decoration: none;">28Part_I.29</ins>|<ins style="font-weight: bold; text-decoration: none;">identified set of simplified governing equations</ins>]]<ins style="font-weight: bold; text-decoration: none;">. </ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{LSU_HBook_header}}</del></div></td><td colspan="2"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|identified set of simplified governing equations]]. </del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{LSU_HBook_header}}</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Cylindrical Coordinate Base==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Cylindrical Coordinate Base==</div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/AxisymmetricConfigurations/Equilibria&diff=18546&oldid=prev
Tohline: /* Axisymmetric Configurations (Structure — Part II) */
2019-08-04T00:14:10Z
<p><span dir="auto"><span class="autocomment">Axisymmetric Configurations (Structure — Part II)</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:14, 4 August 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2">Line 2:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=Axisymmetric Configurations (<del style="font-weight: bold; text-decoration: none;">Structure &#8212; Part II</del>)=</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=Axisymmetric Configurations (<ins style="font-weight: bold; text-decoration: none;">Steady-State Structures</ins>)=</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><!-- [[Image:LSU_Structure_still.gif|74px|left]] --></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><!-- [[Image:LSU_Structure_still.gif|74px|left]] --></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{LSU_HBook_header}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{LSU_HBook_header}}</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l248">Line 248:</td>
<td colspan="2" class="diff-lineno">Line 248:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero &#8212; initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero &#8212; initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><table align="center" border="1" cellpadding="10"><tr><td align="center"></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><table align="center" border="1" cellpadding="10"></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><tr></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <th align="center">Spherical Coordinate Base</th></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></tr></ins></div></td></tr>
<tr><td colspan="2"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><tr><td align="center"></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/AxisymmetricConfigurations/Equilibria&diff=18545&oldid=prev
Tohline: /* Spherical Coordinate Base */
2019-08-04T00:11:05Z
<p><span dir="auto"><span class="autocomment">Spherical Coordinate Base</span></span></p>
<a href="https://www.vistrails.org//index.php?title=User:Tohline/AxisymmetricConfigurations/Equilibria&diff=18545&oldid=18544">Show changes</a>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/AxisymmetricConfigurations/Equilibria&diff=18544&oldid=prev
Tohline: Created page with '__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Axisymmetric Configurations (Structure — Part II)= <!-- left --> {{LSU_HBook_hea…'
2019-08-04T00:09:42Z
<p>Created page with '__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Axisymmetric Configurations (Structure — Part II)= <!-- <a href="/index.php/File:LSU_Structure_still.gif" title="File:LSU Structure still.gif">74px|left</a> --> {{LSU_HBook_hea…'</p>
<p><b>New page</b></p><div>__FORCETOC__<br />
<!-- __NOTOC__ will force TOC off --><br />
<br />
=Axisymmetric Configurations (Structure &#8212; Part II)=<br />
<!-- [[Image:LSU_Structure_still.gif|74px|left]] --><br />
{{LSU_HBook_header}}<br />
<br />
<br />
Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|identified set of simplified governing equations]]. <br />
<br />
==Cylindrical Coordinate Base==<br />
We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:<br />
<br />
<div align="center"><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<br />
<math>\cancelto{0}{\frac{\partial\rho}{\partial t}} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] <br />
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math><br /><br />
<br />
<br />
<span id="PGE:Euler">The Two Relevant Components of the<br /><br />
<font color="#770000">'''Euler Equation'''</font><br />
</span><br />
<table border="0" cellpadding="5"><br />
<tr><br />
<td align="right"><br />
<math>~<br />
\cancelto{0}{\frac{\partial \dot\varpi}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + <br />
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] <br />
</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~<br />
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3} <br />
</math><br />
</td><br />
</tr><br />
<tr><br />
<td align="right"><br />
<math>~<br />
\cancelto{0}{\frac{\partial \dot{z}}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] + <br />
\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr]<br />
</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~<br />
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] <br />
</math><br />
</td><br />
</tr><br />
</table><br />
<br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br /><br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br /><br />
<math>~<br />
\biggl\{\cancel{\frac{\partial \epsilon}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \epsilon}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \epsilon}{\partial z} \biggr]\biggr\} +<br />
P \biggl\{\cancel{\frac{\partial }{\partial t}\biggl(\frac{1}{\rho}\biggr)} + <br />
\biggl[ \dot\varpi \frac{\partial }{\partial\varpi}\biggl(\frac{1}{\rho}\biggr) \biggr] + <br />
\biggl[ \dot{z} \frac{\partial }{\partial z}\biggl(\frac{1}{\rho}\biggr) \biggr] \biggr\} = 0<br />
</math><br />
<br />
<br />
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br /><br />
<br />
<math><br />
\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 . <br />
</math><br /><br />
</div><br />
<br />
<br />
The steady-state flow field that will be adopted to satisfy both an axisymmetric geometry and the time-independent constraint is, <math>~\vec{v} = \hat{e}_\varphi (\varpi \dot\varphi)</math>. That is, <math>~\dot\varpi = \dot{z} = 0</math> but, in general, <math>~\dot\varphi</math> is not zero and can be an arbitrary function of <math>~\varpi</math> and <math>~z</math>, that is, <math>~\dot\varphi = \dot\varphi(\varpi,z)</math>. We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity <math>~\dot\varphi(\varpi,z)</math>, or of the specific angular momentum, <math>~j(\varpi,z) = \varpi^2 \dot\varphi(\varpi,z)</math>.<br />
<br />
<br />
<span id="2DgoverningEquations">After setting the radial and vertical velocities to zero,</span> we see that the <math>1^\mathrm{st}</math> (continuity) and <math>4^\mathrm{th}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> &amp; <math>3^\mathrm{rd}</math> (Euler) and <math>5^\mathrm{th}</math> (Poisson) give, respectively,<br />
<br />
<div align="center"><br />
<table border="0" cellpadding="5"><br />
<tr><br />
<td align="right"><br />
<math>~<br />
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3} <br />
</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~0</math><br />
</td><br />
</tr><br />
<tr><br />
<td align="right"><br />
<math>~<br />
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] <br />
</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~0</math><br />
</td><br />
</tr><br />
<tr><br />
<td align="right"><br />
<math>~<br />
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2}<br />
</math><br />
</td><br />
<td align="center"><br />
<math>~=</math><br />
</td><br />
<td align="left"><br />
<math>~4\pi G \rho \, .</math><br />
</td><br />
</tr><br />
</table><br />
<br />
</div><br />
<br />
As has been outlined in our discussion of [[User:Tohline/SR#Time-Independent_Problems|supplemental relations for time-independent problems]], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}. <br />
<br />
==Spherical Coordinate Base==<br />
<br />
=See Also=<br />
* Part I of ''Axisymmetric Configurations'': [[User:Tohline/AxisymmetricConfigurations/PGE|Simplified Governing Equations]]<br />
<br />
<br />
<br />
{{LSU_HBook_footer}}</div>
Tohline