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basically, i am wondering how to find the velocity profile of slow flow around a sphere in terms of a stream function ##\psi = f(r,\theta)## where we are in spherical coordinates and ##\theta## is the angle with the ##z##-axis. (i think this is a classical problem).

i understand the situation like this: if we are trying to satisfy continuity then we have ##\nabla \cdot \vec{V} = 0##. thus, since we can describe the flow in two parameters ##r, \theta##, we say ##\vec{V} = \nabla \times \psi \hat{\phi}## where ##\hat{\phi}## is the phi unit vector. doing this in spherical coordinates gives me a different solution than the proposed ##\vec{V} = -\frac{1}{r^2 sin \theta} \frac{\partial \psi}{\partial \theta}\hat{r} + \frac{1}{r sin \theta} \frac{\partial \psi}{\partial r} \hat{\theta}##

can someone shed some light on this? thanks so much!