User:Tohline/AxisymmetricConfigurations/PGE

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,

1. 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>:

   3D 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>

2. Expressing all vector time-derivatives and in cylindrical coordinates:

   3D Operators 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>

   
   

3. Setting to zero all derivatives that are taken with respect to the angular coordinate <math>\varphi</math>:

   2D Operators, Assuming Axisymmetric Conditions
   <math> \nabla f </math>	=	<math> {\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \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] + \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] + \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} + \frac{\partial F_z}{\partial z} ; </math>
   <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}\cdot\nabla)\vec{F} </math>	=	<math> {\hat{e}}_\varpi \biggl[ (\vec{v}\cdot\nabla)F_\varpi - F_\varphi \dot\varphi \biggr] + {\hat{e}}_\varphi \biggl[ (\vec{v}\cdot\nabla)F_\varphi + F_\varpi \dot\varphi \biggr] + {\hat{e}}_z \biggl[ (\vec{v}\cdot\nabla)F_z \biggr] . </math>

   
   

4. Setting (who know what?)

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

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | 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