$9-42$ A common flow encountered in practice is the cross$-$flow of a fluid approaching a long cylinder of radius $R$ at a free stream speed of $U_{}.$ For incompressible inviscid flow, the velocity field of the flow is given as

$u_r=U_{}(1-\frac{R^2}{r^2})cos$

$u_{}=-U_{}(1+\frac{R^2}{r^2})sin$

Show that the velocity field satisfies the continuity equation, and determine the stream function