State the form of the Laplace equation in axisymmetric spherical coordinates.

Verify that the following functions satisfy this equation:

\[r \cos \theta ; \quad \cos \theta / r^{2}\]

A linear combination is also a solution by superposition. Thus the following solution for \(\phi\) obtained by taking the combination represents the potential flow around a sphere of radius \(R\) :

\[\phi=v_{\infty}\left[r \cos \theta+\frac{R^{3}}{2} \frac{\cos \theta}{r^{2}}\right]\]

Verify that the impermeability condition is satisfied at \(r=R\), the radius of the sphere, by showing that \(v_{r}=\partial \phi / \partial r\) is zero at these locations.