Stokes creeping flow around a sphere, derivation of boundary conditions ,especially the velocity components?

I understand intuitively that the no-slip boundary conditions for creeping flow around a sphere, using polar coordinates (r, theta, psi) mean that Vr and Vtheta are zero at the sphere boundary. But I don't see how the velocity components


Vr = 1/r^2*sin(theta) * pdPsi/pdTheta


Vtheta = -1/r*sin(theta) * pdPsi/pdTheta


are derived. I know they can be set to zero at the sphere boundary, and that at very large R, the Vr becomes Ucos(theta), where U is the flow velocity.


Every text and reference I've checked just seems to pull those formulas out of thin air!


But I'm just an engineer, not a mathematician, and it's been a long time since college. Is it something really easy I'm missing? Or something so hard and involved nobody bothers deriving it and just takes it for granted?


HELP!


"pd" above means partial diff.

Stokes creeping flow around a sphere, derivation of boundary conditions ,especially the velocity components?
The stream function phi is chosen to satisfy the continuity equation. Here take the del dot velocity vector.





0=pd(r^2 * Vr)/(r^2 * pd(r)) + pd (Vtheta * sin(theta)/(r sin (theta) * pd (theta)





the phi function is chosen interms of Vr and Vtheta to satisfy the above equation.





This is normally kinda thrown at you in the text books but is somewhat intuitative once you've thought about stream functions long enough and had the graduate level engineering math while it is still fresh on your mind.





It may help to review how the stream function was developed for cartesion coordinates.





Hope this helps.