- #1
John Creighto
- 495
- 2
I'm trying do derive the vorticity equation
[tex]\begin{align}\frac{D\vec\omega}{Dt} &= \frac{\partial \vec \omega}{\partial t} + (\vec V \cdot \vec \nabla) \vec \omega \\
&= (\vec \omega \cdot \vec \nabla) \vec V - \vec \omega (\vec \nabla \cdot \vec V) + \frac{1}{\rho^2}\vec \nabla \rho \times \vec \nabla p + \vec \nabla \times \left( \frac{\vec \nabla \cdot \underline{\underline{\tau}}}{\rho} \right) + \vec \nabla \times \vec B
\end{align}[/tex]
based on the notes give here.
I agree with the result obtained for the LHS of the equation but I am having trouble with one term on the right hand side of the equation:
[tex]+ \frac{1}{\rho^2}\vec \nabla \rho \times \vec \nabla p[/tex]
which as far as I can understand should be the curl of:
[tex]- \frac{1}{\rho} \vec \nabla p[/tex]
Looking up useful vector identities:
[tex]\nabla \times (\psi\mathbf{A}) = \psi\nabla \times \mathbf{A} - \mathbf{A} \times \nabla\psi [/tex]
I don't see how to obtain this term.
[tex]\begin{align}\frac{D\vec\omega}{Dt} &= \frac{\partial \vec \omega}{\partial t} + (\vec V \cdot \vec \nabla) \vec \omega \\
&= (\vec \omega \cdot \vec \nabla) \vec V - \vec \omega (\vec \nabla \cdot \vec V) + \frac{1}{\rho^2}\vec \nabla \rho \times \vec \nabla p + \vec \nabla \times \left( \frac{\vec \nabla \cdot \underline{\underline{\tau}}}{\rho} \right) + \vec \nabla \times \vec B
\end{align}[/tex]
based on the notes give here.
I agree with the result obtained for the LHS of the equation but I am having trouble with one term on the right hand side of the equation:
[tex]+ \frac{1}{\rho^2}\vec \nabla \rho \times \vec \nabla p[/tex]
which as far as I can understand should be the curl of:
[tex]- \frac{1}{\rho} \vec \nabla p[/tex]
Looking up useful vector identities:
[tex]\nabla \times (\psi\mathbf{A}) = \psi\nabla \times \mathbf{A} - \mathbf{A} \times \nabla\psi [/tex]
I don't see how to obtain this term.