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(Spherical Coordinate Base)
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==Spherical Coordinate Base==
==Spherical Coordinate Base==
 +
We begin with an [[User:Tohline/AxisymmetricConfigurations/PGE#Governing_Equations_.28SPH..29|Eulerian formulation of the principle governing equations written in spherical coordinates for an axisymmetric configuration]], namely,
 +
 +
<div align="center">
 +
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<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>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~0</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
 +
<span id="PGE:Euler">The Two Relevant Components of the<br />
 +
<font color="#770000">'''Euler Equation'''</font>
 +
</span><br />
 +
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
 +
  <td align="right">
 +
<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>
 +
  </td>
 +
  <td align="center">
 +
=
 +
  </td>
 +
  <td align="left">
 +
<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>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
 +
  <td align="right">
 +
<math>
 +
r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{\theta} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta
 +
</math>
 +
  </td>
 +
  <td align="center">
 +
=
 +
  </td>
 +
  <td align="left">
 +
<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>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
 +
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
 +
 +
<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>
 +
 +
 +
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<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>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~4\pi G\rho</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</div>
 +
 +
where the pair of [[User:Tohline/AxisymmetricConfigurations/PGE#RelevantSphericalComponents|"relevant" components of the Euler equation]] have been written in terms of the specific angular momentum,
 +
<div align="center">
 +
<math>~j(r,\theta) \equiv (r\sin\theta)^2 \dot\varphi</math>,
 +
</div>
 +
which is a conserved quantity in axisymmetric systems. 
 +
 +
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:
 +
 +
<table align="center" border="1" cellpadding="10"><tr><td align="center">
 +
 +
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<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>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~4\pi G\rho</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<span id="PGE:Euler">The Two Relevant Components of the<br />
 +
<font color="#770000">'''Euler Equation'''</font>
 +
</span><br />
 +
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
 +
  <td align="right">
 +
<math>
 +
~0
 +
</math>
 +
  </td>
 +
  <td align="center">
 +
=
 +
  </td>
 +
  <td align="left">
 +
<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>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
 +
  <td align="right">
 +
<math>
 +
~0
 +
</math>
 +
  </td>
 +
  <td align="center">
 +
=
 +
  </td>
 +
  <td align="left">
 +
<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>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</td></tr></table>
=See Also=
=See Also=

Revision as of 17:11, 3 August 2019


Contents

Axisymmetric Configurations (Structure — Part II)

Whitworth's (1981) Isothermal Free-Energy Surface
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Equilibrium, axisymmetric structures are obtained by searching for time-independent, steady-state solutions to the identified set of simplified governing equations.

Cylindrical Coordinate Base

We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:

Equation of Continuity

\cancelto{0}{\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


The Two Relevant Components of the
Euler Equation

~
\cancelto{0}{\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]

~=

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

~
\cancelto{0}{\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]

~=

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

Adiabatic Form of the
First Law of Thermodynamics

~
\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\} +
P \biggl\{\cancel{\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


Poisson Equation


\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 .


The steady-state flow field that will be adopted to satisfy both an axisymmetric geometry and the time-independent constraint is, ~\vec{v} = \hat{e}_\varphi (\varpi \dot\varphi). That is, ~\dot\varpi = \dot{z} = 0 but, in general, ~\dot\varphi is not zero and can be an arbitrary function of ~\varpi and ~z, that is, ~\dot\varphi = \dot\varphi(\varpi,z). We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity ~\dot\varphi(\varpi,z), or of the specific angular momentum, ~j(\varpi,z) = \varpi^2 \dot\varphi(\varpi,z).


After setting the radial and vertical velocities to zero, we see that the 1st (continuity) and 4th (first law of thermodynamics) equations are trivially satisfied while the 2nd & 3rd (Euler) and 5th (Poisson) give, respectively,

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

~=

~0

~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]

~=

~0

~
\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 \, .

As has been outlined in our discussion of 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 ~P and ~\rho.

Spherical Coordinate Base

We begin with an Eulerian formulation of the principle governing equations written in spherical coordinates for an axisymmetric configuration, namely,

Equation of Continuity

~
\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]

~=

~0


The Two Relevant Components of the
Euler Equation

~{\hat{e}}_r:    


\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

=


- \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]

~{\hat{e}}_\theta:    


r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{\theta} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta

=


- \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

Adiabatic Form of the
First Law of Thermodynamics

~
\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\}

~=

~0


Poisson Equation

~
\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)

~=

~4\pi G\rho

where the pair of "relevant" components of the Euler equation have been written in terms of the specific angular momentum,

~j(r,\theta) \equiv (r\sin\theta)^2 \dot\varphi,

which is a conserved quantity in axisymmetric systems.

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, \dot{r} and ~\dot{\theta}, to zero — 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:

Poisson Equation

~
\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)

~=

~4\pi G\rho

The Two Relevant Components of the
Euler Equation

~{\hat{e}}_r:    


~0

=


- \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]

~{\hat{e}}_\theta:    


~0

=


- \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

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2020 by Joel E. Tohline
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