Difference between revisions of "User:Tohline/AxisymmetricConfigurations/PGE"

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=Axisymmetric Configurations (Part I)=
=Axisymmetric Configurations (Governing Equations)=
{{LSU_HBook_header}}
{{LSU_HBook_header}}
If the self-gravitating configuration that we wish to construct is axisymmetric, then the coupled set of multidimensional, partial differential equations that serve as our [[User:Tohline/PGE|principal governing equations]] can be simplified to a coupled set of two-dimensional PDEs.   
If the self-gravitating configuration that we wish to construct is axisymmetric, then the coupled set of multidimensional, partial differential equations that serve as our [[User:Tohline/PGE|principal governing equations]] can be simplified to a coupled set of two-dimensional PDEs.   
Line 99: Line 99:
</tr>
</tr>
</table>
</table>
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>
(\vec{F} \cdot \nabla )\vec{B}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\hat{e}_\varpi \biggl[ F_\varpi \frac{\partial B_\varpi}{\partial\varpi} + \cancel{\frac{F_\varphi}{\varpi} \frac{\partial B_\varpi}{\partial\varphi}} + F_z \frac{\partial B_\varpi}{\partial z} - \frac{F_\varphi B_\varphi}{\varpi}  \biggr]
+ \hat{e}_\varphi \biggl[ F_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \cancel{ \frac{F_\varphi}{\varpi} \frac{\partial B_\varphi}{\partial\varphi}} + F_z \frac{\partial B_\varphi}{\partial z} + \frac{F_\varphi B_\varpi}{\varpi}  \biggr]
+ \hat{e}_z \biggl[ F_\varpi \frac{\partial B_z}{\partial\varpi} +\cancel{ \frac{F_\varphi}{\varpi} \frac{\partial B_z}{\partial \varphi}} + F_z \frac{\partial B_z}{\partial z} \biggr] \, .
</math>
</td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 651, Eq. (1B-54)
  </td>
</tr>
</table>
<span id="CYLconvectiveOperator">From this last expression &#8212; the so-called ''convective operator'' &#8212; we conclude as well that, for axisymmetric systems,</span>
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\hat{e}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial\varpi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi}  \biggr]
+ \hat{e}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi}  + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi}  \biggr]
+ \hat{e}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial\varpi}  + v_z \frac{\partial v_z}{\partial z} \biggr] \, .
</math>
</td>
</tr>
</table>


<li>Express all vector time-derivatives in cylindrical coordinates:
<li>Express all vector time-derivatives in cylindrical coordinates:
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[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 650, Eq. (1B-46)
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 650, Eq. (1B-46)
   </td>
   </td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>
(\vec{F} \cdot \nabla )\vec{B}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\hat{e}_r \biggl[ F_r \frac{\partial B_r}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_r}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta}  \frac{\partial B_r}{\partial \varphi} } - \frac{(F_\theta B_\theta + F_\varphi B_\varphi)}{r}\biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
&nbsp;
</td>
<td align="center">
&nbsp;
</td>
<td align="left">
<math>
+ \hat{e}_\theta \biggl[ F_r  \frac{\partial B_\theta}{\partial r} + \frac{F_\theta}{r}  \frac{\partial B_\theta}{\partial \theta } + \cancel{ \frac{F_\varphi}{r\sin\theta}  \frac{\partial B_\theta}{\partial \varphi} }
+ \frac{F_\theta B_r}{r} - \frac{F_\varphi B_\varphi \cot\theta}{r} \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
&nbsp;
</td>
<td align="center">
&nbsp;
</td>
<td align="left">
<math>
+ \hat{e}_\varphi \biggl[ F_r  \frac{\partial B_\varphi}{\partial r} + \frac{F_\theta}{r}  \frac{\partial B_\varphi}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta}  \frac{\partial B_\varphi}{\partial \varphi} }
+ \frac{F_\varphi B_r}{r} + \frac{F_\varphi B_\theta \cot\theta}{r}  \biggr] \, .
</math>
</td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 651, Eq. (1B-55)
  </td>
</tr>
</table>
From this last expression &#8212; the so-called ''convective operator'' &#8212; we conclude as well that, for axisymmetric systems,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v}
</math>
</td>
<td align="center">
=
</td>
<td align="left">
<math>
\hat{e}_r \biggl[ v_r \frac{\partial v_r}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta}  - \frac{(v_\theta^2 + v_\varphi^2 )}{r}\biggr]
+ \hat{e}_\theta \biggl[ v_r  \frac{\partial v_\theta}{\partial r} + \frac{v_\theta}{r}  \frac{\partial v_\theta}{\partial \theta }
+ \frac{v_\theta v_r}{r} - \frac{v_\varphi^2 \cot\theta}{r} \biggr]
+ \hat{e}_\varphi \biggl[ v_r  \frac{\partial v_\varphi}{\partial r} + \frac{v_\theta}{r}  \frac{\partial v_\varphi}{\partial \theta}
+ \frac{v_\varphi v_r}{r} + \frac{v_\varphi v_\theta \cot\theta}{r}  \biggr] \, .
</math>
</td>
</tr>
</tr>
</table>
</table>
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</div>
</div>


Making this substitution throughout the set of governing relations gives:
Making this substitution throughout the set of governing relations gives:  


<div align="center">
<div align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{d\rho}{dt} + \rho \biggl[ \frac{1}{r^2} \frac{\partial (r^2 \dot{r})}{\partial r}  
<math>~
+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \dot\theta r \sin\theta \biggr)
\frac{\partial \rho}{\partial t}  
+ \biggl[ \frac{1}{r^2} \frac{\partial (\rho r^2 \dot{r})}{\partial r}  
+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \rho \dot\theta r \sin\theta \biggr)
  \biggr]</math>
  \biggr]</math>
   </td>
   </td>
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<span id="PGE:Euler">
<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
<font color="#770000">'''Euler Equation'''</font>
</span><br />
</span><br />
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   <td align="right">
   <td align="right">
<math>
<math>
\frac{d\dot{r}}{dt} r {\dot\theta}^2 - \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr]\sin\theta
\biggl\{ \frac{\partial \dot{r}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{r}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\}
-  r {\dot\theta}^2
</math>
</math>
   </td>
   </td>
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   <td align="left">
   <td align="left">
<math>
<math>
- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr]   \, ,
- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr] + \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr]
</math>
</math>
   </td>
   </td>
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   <td align="right">
   <td align="right">
<math>
<math>
\frac{d(r\dot\theta)}{dt} + \dot{r} \dot\theta \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{\theta}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta  
</math>
</math>
   </td>
   </td>
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   <td align="left">
   <td align="left">
<math>
<math>
- \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] \, .
- \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
</math>
</math>
   </td>
   </td>
Line 814: Line 944:
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />


{{User:Tohline/Math/EQ_FirstLaw02}}
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~
\biggl\{ \frac{\partial \epsilon}{\partial t} + \biggl[ \dot{r} \frac{\partial \epsilon}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \epsilon}{\partial\theta} \biggr]  \biggr\}
+ P\biggl\{ \frac{\partial }{\partial t} \biggl( \frac{1}{\rho}\biggr)
+ \biggl[ \dot{r} \frac{\partial }{\partial r} \biggl( \frac{1}{\rho}\biggr) \biggr]
+ \biggl[ \dot\theta \frac{\partial }{\partial\theta} \biggl( \frac{1}{\rho}\biggr) \biggr]  \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
</table>





Latest revision as of 22:15, 8 August 2019

Axisymmetric Configurations (Governing Equations)

Whitworth's (1981) Isothermal Free-Energy Surface
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If the self-gravitating configuration that we wish to construct is axisymmetric, then the coupled set of multidimensional, partial differential equations that serve as our principal governing equations can be simplified to a coupled set of two-dimensional PDEs.

Cylindrical Coordinate Base

Here we choose to …

  1. Express each of the multidimensional spatial operators in cylindrical coordinates (<math>\varpi, \varphi, z</math>) (see, for example, the Wikipedia discussion of vector calculus formulae in cylindrical coordinates) and set to zero all spatial derivatives that are taken with respect to the angular coordinate <math>\varphi</math>:

    Spatial Operators in Cylindrical Coordinates

    <math> \nabla f </math>

    =

    <math> {\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \biggr] + {\hat{e}}_\varphi \cancel{\biggl[ \frac{1}{\varpi} \frac{\partial f}{\partial\varphi} \biggr]} + {\hat{e}}_z \biggl[ \frac{\partial f}{\partial z} \biggr] ; </math>

    [BT87], p. 649, Eq. (1B-37)

    <math> \nabla^2 f </math>

    =

    <math> \frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial f}{\partial\varpi} \biggr] + \cancel{\frac{1}{\varpi^2} \frac{\partial^2 f}{\partial\varphi^2}} + \frac{\partial^2 f}{\partial z^2} ; </math>

    [BT87], p. 650, Eq. (1B-50)

    <math> (\vec{v}\cdot\nabla)f </math>

    =

    <math> \biggl[ v_\varpi \frac{\partial f}{\partial\varpi} \biggr] + \cancel{\biggl[ \frac{v_\varphi}{\varpi} \frac{\partial f}{\partial\varphi} \biggr]} + \biggl[ v_z \frac{\partial f}{\partial z} \biggr] ; </math>

    <math> \nabla \cdot \vec{F} </math>

    =

    <math> \frac{1}{\varpi} \frac{\partial (\varpi F_\varpi)}{\partial\varpi} + \cancel{\frac{1}{\varpi} \frac{\partial F_\varphi}{\partial\varphi}} + \frac{\partial F_z}{\partial z} ; </math>

    [BT87], p. 650, Eq. (1B-45)

    <math> (\vec{F} \cdot \nabla )\vec{B} </math>

    =

    <math> \hat{e}_\varpi \biggl[ F_\varpi \frac{\partial B_\varpi}{\partial\varpi} + \cancel{\frac{F_\varphi}{\varpi} \frac{\partial B_\varpi}{\partial\varphi}} + F_z \frac{\partial B_\varpi}{\partial z} - \frac{F_\varphi B_\varphi}{\varpi} \biggr] + \hat{e}_\varphi \biggl[ F_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \cancel{ \frac{F_\varphi}{\varpi} \frac{\partial B_\varphi}{\partial\varphi}} + F_z \frac{\partial B_\varphi}{\partial z} + \frac{F_\varphi B_\varpi}{\varpi} \biggr] + \hat{e}_z \biggl[ F_\varpi \frac{\partial B_z}{\partial\varpi} +\cancel{ \frac{F_\varphi}{\varpi} \frac{\partial B_z}{\partial \varphi}} + F_z \frac{\partial B_z}{\partial z} \biggr] \, . </math>

    [BT87], p. 651, Eq. (1B-54)

    From this last expression — the so-called convective operator — we conclude as well that, for axisymmetric systems,

    <math> (\vec{v} \cdot \nabla )\vec{v} </math>

    =

    <math> \hat{e}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial\varpi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi} \biggr] + \hat{e}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi} + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi} \biggr] + \hat{e}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial\varpi} + v_z \frac{\partial v_z}{\partial z} \biggr] \, . </math>


  2. Express all vector time-derivatives in cylindrical coordinates:

    Vector Time-Derivatives in Cylindrical Coordinates

    <math> \frac{d}{dt}\vec{F} </math>

    =

    <math> {\hat{e}}_\varpi \frac{dF_\varpi}{dt} + F_\varpi \frac{d{\hat{e}}_\varpi}{dt} + {\hat{e}}_\varphi \frac{dF_\varphi}{dt} + F_\varphi \frac{d{\hat{e}}_\varphi}{dt} + {\hat{e}}_z \frac{dF_z}{dt} + F_z \frac{d{\hat{e}}_z}{dt} </math>

     

    =

    <math> {\hat{e}}_\varpi \biggl[ \frac{dF_\varpi}{dt} - F_\varphi \dot\varphi \biggr] + {\hat{e}}_\varphi \biggl[ \frac{dF_\varphi}{dt} + F_\varpi \dot\varphi \biggr] + {\hat{e}}_z \frac{dF_z}{dt} ; </math>

    <math> \vec{v} = \frac{d\vec{x}}{dt} = \frac{d}{dt}\biggl[ \hat{e}_\varpi \varpi + \hat{e}_z z \biggr] </math>

    =

    <math> {\hat{e}}_\varpi \biggl[ \dot\varpi \biggr] + {\hat{e}}_\varphi \biggl[ \varpi \dot\varphi \biggr] + {\hat{e}}_z \biggl[ \dot{z} \biggr] . </math>

    [BT87], p. 647, Eq. (1B-23)


Governing Equations (CYL.)

Introducing the above expressions into the principal governing equations gives,

Equation of Continuity

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


Euler Equation

<math> {\hat{e}}_\varpi \biggl[ \frac{d \dot\varpi}{dt} - \varpi {\dot\varphi}^2 \biggr] + {\hat{e}}_\varphi \biggl[ \frac{d(\varpi\dot\varphi)}{dt} + \dot\varpi \dot\varphi \biggr] + {\hat{e}}_z \biggl[ \frac{d \dot{z}}{dt} \biggr] = - {\hat{e}}_\varpi \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - {\hat{e}}_z \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math>


Adiabatic Form of the
First Law of Thermodynamics

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math>


Poisson Equation

<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>

Conservation of Specific Angular Momentum (CYL.)

The <math>\hat{e}_\varphi</math> component of the Euler equation leads to a statement of conservation of specific angular momentum, <math>j</math>, as follows.

<math> \frac{d(\varpi\dot\varphi)}{dt} + \dot\varpi \dot\varphi = \frac{1}{\varpi}\biggl[ \varpi \frac{d(\varpi\dot\varphi)}{dt} + \varpi \dot\varpi \dot\varphi \biggr] =0 </math>

<math> \Rightarrow ~~~~~ \frac{d(\varpi^2 \dot\varphi)}{dt} = 0 </math>

<math> \Rightarrow ~~~~~ j(\varpi,z) \equiv \varpi^2 \dot\varphi = \mathrm{constant} ~(\mathrm{i.e.,}~\mathrm{independent~of~time}) </math>


So, for axisymmetric configurations, the <math>\hat{e}_\varpi</math> and <math>\hat{e}_z</math> components of the Euler equation become, respectively,

<math>~{\hat{e}}_\varpi</math>:    

<math> \frac{d \dot\varpi}{dt} - \frac{j^2}{\varpi^3} </math>

=

<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] </math>

<math>~{\hat{e}}_z</math>:    

<math> \frac{d \dot{z}}{dt} </math>

=

<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math>

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>,


<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>

Making this substitution throughout the set of governing relations gives:

Equation of Continuity

<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>


The Two Relevant Components of the
Euler Equation

<math>~{\hat{e}}_\varpi</math>:    

<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>

<math>~=</math>

<math>~ - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3} </math>

<math>~{\hat{e}}_z</math>:    

<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>

<math>~=</math>

<math>~ - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math>

Adiabatic Form of the
First Law of Thermodynamics

<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>


Poisson Equation

<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>

Spherical Coordinate Base

Here we choose to …

  1. Express each of the multidimensional spatial operators in spherical coordinates (<math>r, \theta, \varphi</math>) (see, for example, the Wikipedia discussion of vector calculus formulae in spherical coordinates) and set to zero all spatial derivatives that are taken with respect to the angular coordinate <math>\varphi</math>:

    Spatial Operators in Spherical Coordinates

    <math> \nabla f </math>

    =

    <math> {\hat{e}}_r \biggl[ \frac{\partial f}{\partial r} \biggr] + {\hat{e}}_\theta \biggl[ \frac{1}{r} \frac{\partial f}{\partial\theta} \biggr] + {\hat{e}}_\varphi \cancel{\biggl[\frac{1}{r\sin\theta}~ \frac{\partial f}{\partial \varphi} \biggr]} ; </math>

    [BT87], p. 649, Eq. (1B-38)

    <math> \nabla^2 f </math>

    =

    <math> \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial f}{\partial r} \biggr] + \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta \frac{\partial f}{\partial\theta}\biggr) + \cancel{ \biggl[\frac{1}{r^2 \sin^2\theta} \frac{\partial^2 f}{\partial \varphi^2} \biggr]} ; </math>

    [BT87], p. 650, Eq. (1B-51)

    <math> (\vec{v}\cdot\nabla)f </math>

    =

    <math> \biggl[ v_r \frac{\partial f}{\partial r} \biggr] + \biggl[ \frac{v_\theta}{r} \frac{\partial f}{\partial\theta} \biggr] + \cancel{\biggl[\frac{v_\varphi}{r\sin\theta}~ \frac{\partial f}{\partial \varphi} \biggr]} ; </math>

    <math> \nabla \cdot \vec{F} </math>

    =

    <math> \frac{1}{r^2} \frac{\partial (r^2 F_r)}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( F_\theta \sin\theta \biggr) + \cancel{ \biggl[ \frac{1}{r\sin\theta}~\frac{\partial F_\varphi}{\partial \varphi} \biggr]} ; </math>

    [BT87], p. 650, Eq. (1B-46)

    <math> (\vec{F} \cdot \nabla )\vec{B} </math>

    =

    <math> \hat{e}_r \biggl[ F_r \frac{\partial B_r}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_r}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} } - \frac{(F_\theta B_\theta + F_\varphi B_\varphi)}{r}\biggr] </math>

     

     

    <math> + \hat{e}_\theta \biggl[ F_r \frac{\partial B_\theta}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\theta}{\partial \theta } + \cancel{ \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi} } + \frac{F_\theta B_r}{r} - \frac{F_\varphi B_\varphi \cot\theta}{r} \biggr] </math>

     

     

    <math> + \hat{e}_\varphi \biggl[ F_r \frac{\partial B_\varphi}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi} } + \frac{F_\varphi B_r}{r} + \frac{F_\varphi B_\theta \cot\theta}{r} \biggr] \, . </math>

    [BT87], p. 651, Eq. (1B-55)

    From this last expression — the so-called convective operator — we conclude as well that, for axisymmetric systems,

    <math> (\vec{v} \cdot \nabla )\vec{v} </math>

    =

    <math> \hat{e}_r \biggl[ v_r \frac{\partial v_r}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta} - \frac{(v_\theta^2 + v_\varphi^2 )}{r}\biggr] + \hat{e}_\theta \biggl[ v_r \frac{\partial v_\theta}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_\theta}{\partial \theta } + \frac{v_\theta v_r}{r} - \frac{v_\varphi^2 \cot\theta}{r} \biggr] + \hat{e}_\varphi \biggl[ v_r \frac{\partial v_\varphi}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_\varphi}{\partial \theta} + \frac{v_\varphi v_r}{r} + \frac{v_\varphi v_\theta \cot\theta}{r} \biggr] \, . </math>

  2. Express all vector time-derivatives in spherical coordinates:

    Vector Time-Derivatives in Spherical Coordinates

    <math> \frac{d}{dt}\vec{F} </math>

    =

    <math> {\hat{e}}_r \frac{dF_r}{dt} + F_r \frac{d{\hat{e}}_r}{dt} + {\hat{e}}_\theta \frac{dF_\theta}{dt} + F_\theta \frac{d{\hat{e}}_\theta}{dt} + {\hat{e}}_\varphi \frac{dF_\varphi}{dt} + F_\varphi \frac{d{\hat{e}}_\varphi}{dt} </math>

     

    =

    <math> {\hat{e}}_r \frac{dF_r}{dt} + F_r \biggl[ {\hat{e}}_\theta \dot\theta + {\hat{e}}_\varphi \dot\varphi \sin\theta \biggr] + {\hat{e}}_\theta \frac{dF_\theta}{dt} + F_\theta \biggl[ - {\hat{e}}_r \dot\theta + {\hat{e}}_\varphi \dot\varphi \cos\theta \biggr] + {\hat{e}}_\varphi \frac{dF_\varphi}{dt} + F_\varphi \biggl[ - {\hat{e}}_r \dot\varphi \sin\theta - {\hat{e}}_\theta \dot\varphi \cos\theta \biggr] </math>

     

    =

    <math> {\hat{e}}_r \biggl[ \frac{dF_r}{dt} - F_\theta \dot\theta - F_\varphi \dot\varphi \sin\theta \biggr] + {\hat{e}}_\theta \biggl[ \frac{dF_\theta}{dt} + F_r \dot\theta - F_\varphi \dot\varphi \cos\theta \biggr] + {\hat{e}}_\varphi \biggl[ \frac{dF_\varphi}{dt} + F_r \dot\varphi \sin\theta + F_\theta \dot\varphi \cos\theta \biggr] ; </math>

    <math> \vec{v} = \frac{d\vec{x}}{dt} </math>

    =

    <math> \frac{d}{dt}\biggl[ \hat{e}_r r \biggr] = {\hat{e}}_r \dot{r} + {\hat{e}}_\theta~ r \dot\theta + {\hat{e}}_\varphi ~r \sin\theta ~ \dot\varphi . </math>

    [BT87], p. 648, Eq. (1B-30)

Governing Equations (SPH.)

Introducing the above expressions into the principal governing equations gives,

Equation of Continuity

<math>~\frac{d\rho}{dt} + \rho \biggl[ \frac{1}{r^2} \frac{\partial (r^2 \dot{r})}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \dot\theta r \sin\theta \biggr)

\biggr]</math>

<math>~=</math>

<math>~0</math>


Euler Equation

<math>~ {\hat{e}}_r \biggl[ \frac{d\dot{r}}{dt} - r {\dot\theta}^2 - r {\dot\varphi}^2 \sin^2\theta \biggr] + {\hat{e}}_\theta \biggl[ \frac{d(r\dot\theta)}{dt} + \dot{r} \dot\theta - r { \dot\varphi }^2 \sin\theta \cos\theta \biggr] + {\hat{e}}_\varphi \biggl[ \frac{d(r \sin\theta \dot\varphi)}{dt} + \dot{r} \dot\varphi \sin\theta + r \dot\theta \dot\varphi \cos\theta \biggr] </math>

<math>~=</math>

<math>~- {\hat{e}}_r \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr] - {\hat{e}}_\theta \biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr] </math>


Adiabatic Form of the
First Law of Thermodynamics

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math>


Poisson Equation

<math>~ \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] + \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr) </math>

<math>~=</math>

<math>~4\pi G\rho</math>

Conservation of Specific Angular Momentum (SPH.)

The <math>\hat{e}_\varphi</math> component of the Euler equation leads to a statement of conservation of specific angular momentum, <math>~j</math>, as follows.

<math>~0</math>

<math>~=</math>

<math>~ \frac{d(r \sin\theta \dot\varphi)}{dt} + \dot{r} \dot\varphi \sin\theta + r \dot\theta \dot\varphi \cos\theta </math>

 

<math>~=</math>

<math>~ \frac{1}{r\sin\theta} \biggl[ r\sin\theta\frac{d(r \sin\theta \dot\varphi)}{dt} + r\sin\theta \dot\varphi ( \dot{r}\sin\theta + r\dot\theta \cos\theta) \biggr] </math>

 

<math>~=</math>

<math>~ \frac{1}{r\sin\theta} \biggl[\frac{d(r^2 \sin^2\theta \dot\varphi )}{dt} \biggr] \, . </math>

<math> \Rightarrow ~~~~~ j(r,\theta) \equiv (r\sin\theta)^2 \dot\varphi = \mathrm{constant} ~(\mathrm{i.e.,}~\mathrm{independent~of~time}) </math>


So, for axisymmetric configurations, the <math>\hat{e}_r</math> and <math>\hat{e}_\theta</math> components of the Euler equation become, respectively,

<math>~{\hat{e}}_r</math>:    

<math>

\frac{d\dot{r}}{dt} -  r {\dot\theta}^2 - \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr]\sin\theta

</math>

=

<math> - \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr] \, , </math>

<math>~{\hat{e}}_\theta</math>:    

<math> \frac{d(r\dot\theta)}{dt} + \dot{r} \dot\theta - \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta </math>

=

<math> - \biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr] \, . </math>

Eulerian Formulation (SPH.)

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>,


<math> \frac{df}{dt} \rightarrow \frac{\partial f}{\partial t} + (\vec{v}\cdot \nabla)f = \frac{\partial f}{\partial t} + \biggl[ v_r \frac{\partial f}{\partial r} \biggr] + \biggl[ \frac{v_\theta}{r} \frac{\partial f}{\partial\theta} \biggr] = \frac{\partial f}{\partial t} + \biggl[ \dot{r} \frac{\partial f}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial f}{\partial\theta} \biggr] \, . </math>

Making this substitution throughout the set of governing relations gives:

Equation of Continuity

<math>~ \frac{\partial \rho}{\partial t} + \biggl[ \frac{1}{r^2} \frac{\partial (\rho r^2 \dot{r})}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \rho \dot\theta r \sin\theta \biggr)

\biggr]</math>

<math>~=</math>

<math>~0</math>


The Two Relevant Components of the
Euler Equation

<math>~{\hat{e}}_r</math>:    

<math> \biggl\{ \frac{\partial \dot{r}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{r}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\} - r {\dot\theta}^2 </math>

=

<math> - \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+ \frac{\partial \Phi }{\partial r} \biggr] + \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr] </math>

<math>~{\hat{e}}_\theta</math>:    

<math> r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{\theta}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta </math>

=

<math> - \biggl[ \frac{1}{\rho r} \frac{\partial P}{\partial\theta} + \frac{1}{r} \frac{\partial \Phi}{\partial\theta} \biggr] + \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta </math>

Adiabatic Form of the
First Law of Thermodynamics

<math>~ \biggl\{ \frac{\partial \epsilon}{\partial t} + \biggl[ \dot{r} \frac{\partial \epsilon}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \epsilon}{\partial\theta} \biggr] \biggr\} + P\biggl\{ \frac{\partial }{\partial t} \biggl( \frac{1}{\rho}\biggr) + \biggl[ \dot{r} \frac{\partial }{\partial r} \biggl( \frac{1}{\rho}\biggr) \biggr] + \biggl[ \dot\theta \frac{\partial }{\partial\theta} \biggl( \frac{1}{\rho}\biggr) \biggr] \biggr\} </math>

<math>~=</math>

<math>~0</math>


Poisson Equation

<math>~ \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] + \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr) </math>

<math>~=</math>

<math>~4\pi G\rho</math>

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation