User:Kupirijo/Stokes stream function

Vorticity

See also: Stream_function § Vorticity

The vorticity is defined as:

$\mathbf {\omega } =\nabla \times \mathbf {v} =\nabla \times \nabla \times \mathbf {\psi }$

Derivation of vorticity $\omega$ of Stokes stream function

Consider the vorticity as defined by $\mathbf {\omega } =\nabla \times \mathbf {v}$ From the definition of the curl in spherical coordinates: ${\begin{aligned}\omega _{r}&={1 \over r\sin \theta }\left({\partial \over \partial \theta }\left(v_{\phi }\sin \theta \right)-{\partial v_{\theta } \over \partial \phi }\right){\boldsymbol {\hat {r}}}\\\omega _{\theta }&={1 \over r}\left({1 \over \sin \theta }{\partial v_{r} \over \partial \phi }-{\partial \over \partial r}\left(rv_{\phi }\right)\right){\boldsymbol {\hat {\theta }}}\\\omega _{\phi }&={1 \over r}\left({\partial \over \partial r}\left(rv_{\theta }\right)-{\partial v_{r} \over \partial \theta }\right){\boldsymbol {\hat {\phi }}}\\\end{aligned}}$ First we notice that the $r$ and $\theta$ components are equal to 0. Secondly we substitute $v_{r}$ and $v_{\theta }$ into $\omega _{\phi }$ then we get: ${\begin{aligned}\omega _{r}&=0\,{\hat {r}}\\\omega _{\theta }&=0\,{\hat {\theta }}\\\omega _{\phi }&={1 \over r}\left({\partial \over \partial r}\left(r\left(-{\frac {1}{r\sin \theta }}{\frac {\partial \psi }{\partial r}}\right)\right)-{\partial \over \partial \theta }\left({\frac {1}{r^{2}\sin \theta }}{\frac {\partial \psi }{\partial \theta }}\right)\right){\boldsymbol {\hat {\phi }}}\\\end{aligned}}$ If we do the algebra: ${\begin{aligned}\omega _{\phi }&={1 \over r}\left(-{\frac {1}{\sin \theta }}\left({\partial \over \partial r}\left({\frac {\partial \psi }{\partial r}}\right)\right)-{\frac {1}{r^{2}}}{\partial \over \partial \theta }\left({\frac {1}{\sin \theta }}{\frac {\partial \psi }{\partial \theta }}\right)\right){\boldsymbol {\hat {\phi }}}\\&={1 \over r}\left(-{\frac {1}{\sin \theta }}\left({\frac {\partial ^{2}\psi }{\partial r^{2}}}\right)-{\frac {\sin \theta }{r^{2}\sin \theta }}{\partial \over \partial \theta }\left({\frac {1}{\sin \theta }}{\frac {\partial \psi }{\partial \theta }}\right)\right){\boldsymbol {\hat {\phi }}}\\&=-{\frac {1}{r\sin \theta }}\left({\frac {\partial ^{2}\psi }{\partial r^{2}}}+{\frac {\sin \theta }{r^{2}}}{\partial \over \partial \theta }\left({\frac {1}{\sin \theta }}{\frac {\partial \psi }{\partial \theta }}\right)\right){\boldsymbol {\hat {\phi }}}\\\end{aligned}}$

If we do the calculation we find that the vorticity vector is equal to:

$\mathbf {\omega } =(0,0,-{\frac {1}{r\sin \theta }}\left({\frac {\partial ^{2}\psi }{\partial r^{2}}}+{\frac {\sin \theta }{r^{2}}}{\partial \over \partial \theta }\left({\frac {1}{\sin \theta }}{\frac {\partial \psi }{\partial \theta }}\right)\right))$

If we define the new operator $E={\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\sin \theta }{r^{2}}}{\partial \over \partial \theta }\left({\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\right)$ then we have

$\mathbf {\omega } =(0,0,-{\frac {E\psi }{r\sin \theta }})$