Revision as of 00:05, 4 April 2010

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>

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

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>

@@ Line 5: / Line 5: @@
 =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 [[User:Tohline/PGE|principal governing equations]] can be simplified to a coupled set of one-dimensional, ordinary differential equations.  This is accomplished by expressing each of the multidimensional spatial operators &#8212; gradient (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>) &#8212; in spherical coordinates (<math>r, \theta, \varphi</math>) (see, for example, the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates]) then setting to zero all derivatives that are taken with respect to the angular coordinates <math>\theta</math> and <math>\varphi</math>.  After making this simplification, our governing equations become,
+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 [[User:Tohline/PGE|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 [http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates Wikipedia discussion of vector calculus formulae in cylindrical coordinates]):
+<div align="center">
+<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><br />
+<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><br />
+<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><br />
+<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>
+</div>
+# 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,
 <div align="center">
@@ Line 29: / Line 57: @@
 <math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math><br />
 </div>
 =See Also=

Difference between revisions of "User:Tohline/AxisymmetricConfigurations/PGE"

Revision as of 00:05, 4 April 2010

Axisymmetric Configurations

See Also

Navigation menu

Search