User:Tohline/AxisymmetricConfigurations/PGE
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Axisymmetric Configurations
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 the vector time-derivative and 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):
<math>
\nabla f = {\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \biggr] + {\hat{e}}_\varphi \biggl[ \frac{1}{\varpi} \frac{\partial f}{\partial\varphi} \biggr] + {\hat{e}}_z \biggl[ \frac{\partial f}{\partial z} \biggr] ;
</math>
<math>
\nabla \cdot \vec{F} = \frac{1}{\varpi} \frac{\partial (\varpi F_\varpi)}{\partial\varpi} + \frac{1}{\varpi} \frac{\partial F_\varphi}{\partial\varphi} + \frac{\partial F_z}{\partial z} ;
</math>
<math>
\nabla^2 f = \frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial f}{\partial\varpi} \biggr] + \frac{1}{\varpi^2} \frac{\partial^2 f}{\partial\varphi^2} + \frac{\partial^2 f}{\partial z^2} ;
</math>
<math>
\frac{d}{dt}\vec{F} = {\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>
- Setting to zero all derivatives that are taken with respect to the angular coordinate <math>\varphi</math>:
After making this simplification, our governing equations become,
Equation of Continuity
<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr] = 0 </math>
Euler Equation
<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr} </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} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math>
See Also
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