Solution of equation by separation of variables

Now we need a functional form for ψ to solve the PDE.  Look at vrpsi above and realize that we need it to have a cos dependence.  Thus C[r] Sin[] looks like a good choice.  

Things are going well, the theta dependence divides out

This can be solved readily with a solution of the form A[r] = rα because it is an Euler equation.  

Here is the far away boundary condition, we match the r velocity to the free stream , @ r-->∞

The boundary condition on the sphere must be [r=R] = 0

Now solve for the constants

Now replace the constants in the answer to get the expression for the streamlines.