User:Tohline/Cylindrical 3D/Linearization
Linearized Equations in Cylindrical Coordinates
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Eulerian Formulation of Nonlinear Governing Equations
From our more detailed, accompanying discussion we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.
<math>\varpi</math> Component of Euler Equation
<math>
\frac{\partial \dot\varpi}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] +
\biggl[ \dot\varphi \frac{\partial \dot\varpi}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] - \varpi {\dot\varphi}^2 =
- \frac{1}{\rho}\frac{\partial P}{\partial\varpi} - \frac{\partial \Phi}{\partial\varpi}
</math>
<math>\varphi</math> Component of Euler Equation
<math>
\frac{\partial (\varpi\dot\varphi)}{\partial t} + \biggl[ \dot\varpi \frac{\partial (\varpi\dot\varphi)}{\partial\varpi} \biggr] +
\biggl[ \dot\varphi \frac{\partial (\varpi\dot\varphi)}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial (\varpi\dot\varphi)}{\partial z} \biggr] + \dot\varpi \dot\varphi =
- \frac{1}{\varpi} \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial \varphi} + \frac{\partial \Phi}{\partial \varphi} \biggr]
</math>
<math>z</math> Component of Euler Equation
<math>
\frac{\partial \dot{z}}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr]
+ \biggl[ \dot\varphi \frac{\partial \dot{z}}{\partial\varphi} \biggr] +\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] =
- \frac{1}{\rho}\frac{\partial P}{\partial z} - \frac{\partial \Phi}{\partial z}
</math>
Equation of Continuity
<math>
\frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr]
+ \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho \varpi \dot\varphi \biggr]
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0
</math>
These match, for example, equations (3.1) - (3.4) of Papaloizou & Pringle (1984, MNRAS, 208, 721-750), hereafter, PPI.
If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become,
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<math>~\frac{\partial {\dot\varpi}^'}{\partial t} + \biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] - \varpi ( { {\dot\varphi}_0 + {\dot\varphi}^'})^2 </math> |
<math>~=</math> |
<math>~- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial\varpi} - \frac{\partial \Phi_0}{\partial\varpi}</math> |
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<math>~\Rightarrow~~~~ \frac{\partial {\dot\varpi}^'}{\partial t} + \biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] - \varpi ( {\dot\varphi}_0)^2 - 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')</math> |
<math>~=</math> |
<math>~- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi} - \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0} \biggr) - \frac{\partial \Phi_0}{\partial\varpi}</math> |
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<math>~\Rightarrow~~~~ \frac{\partial {\dot\varpi}^'}{\partial t} + \biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] - 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^') + \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial\varpi}\biggr] </math> |
<math>~=</math> |
<math>~\biggl\{ \varpi ( {\dot\varphi}_0)^2 - \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr] - \frac{\partial \Phi_0}{\partial\varpi} \biggr\} </math> |
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<math>~\Rightarrow~~~~ \frac{\partial {\dot\varpi}^'}{\partial t} + \biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] - 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^') + \biggl[ \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr] </math> |
<math>~=</math> |
<math>~0 \, . </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |