User:Tohline/Cylindrical 3D/Linearization

Linearized Equations in Cylindrical Coordinates

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,

<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>
<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>
<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>
<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

© 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