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. Here we accomplish this by,
- Expressing 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 setting 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>
<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>
<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>
- Expressing 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>
Governing Equations
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
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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<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> |
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<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
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>
See Also
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