Difference between revisions of "User:Tohline/Cylindrical 3D/Linearization"
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Revision as of 00:19, 11 March 2016
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.
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>
<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>
See Also
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