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I'm solving the equation $$ \frac{\partial^2 \psi}{\partial r^2}+\frac{\partial^2 \psi}{\partial z^2}-\frac{1}{r}\frac{\partial \psi}{\partial r} =-\omega_\phi $$ in a cylindrical pipe, where $\psi(r,z)$ is the axisymmetric stream function, and $\omega_\phi$ is the azimuthal component of the vorticity. I'm trying to figure out the best way to deal with the boundary conditions at the outflow. So far the only thing that seems to work is $u_z=0$. Is this too restrictive? I tried making it have zero tangential stress at the outflow, but that just reduces back to $u_z=0$. Is that a result of the cylindrical symmetry?

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