User:Tohline/AxisymmetricConfigurations/PGE
Axisymmetric Configurations (Part I)
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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 …
- 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)
- 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,
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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 …
- 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)
- 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.
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<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> |
|
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<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> |
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<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,
|
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:
See Also
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