Difference between revisions of "Template:Cylindrical equations"
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\phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 | \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 | ||
\phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, | \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, | ||
| − | \quad (r,\theta,z) \in \ | + | \quad (r,\theta,z) \in \Omega |
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</math> | </math> | ||
</center> | </center> | ||
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<center> | <center> | ||
| − | <math> | + | <math>\frac{\partial \phi}{\partial z} = \alpha \phi , \quad |
| − | + | z=0 | |
| − | \ | ||
</math> | </math> | ||
</center> | </center> | ||
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<center> | <center> | ||
<math> | <math> | ||
| − | + | \frac{\partial \phi}{\partial z} = 0, z=-h | |
| − | \frac{\partial}{\partial | ||
</math> | </math> | ||
</center> | </center> | ||
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Latest revision as of 07:53, 25 August 2008
The problem for the complex water velocity potential in suitable non-dimensionalised cylindrical coordinates, [math]\displaystyle{ \phi (r,\theta,z) }[/math], is given by
[math]\displaystyle{ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, \quad (r,\theta,z) \in \Omega }[/math]
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = \alpha \phi , \quad z=0 }[/math]
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, z=-h }[/math]
