A sphere of radius \(R\) is rotating in an infinite fluid at an angular velocity of \(\Omega\). Derive the following expression for the velocity field:

\[v_{\phi}=r \Omega \sin \theta \frac{R^{3}}{r^{3}}\]

Find an expression for the torque exerted by the sphere on the fluid. 

Assume a solution of the form \(v_{\phi}=f(r) \sin \theta\), where \(f(r)\) is a function just of \(r\) to be determined from the \(\phi\)-momentum balance.