Question: Continuity equation for two-dimensional flow in polar coordinates

(a) Consider a two-dimensional flow field expressed in terms of the cylindrical coordinate system (r, ?, z), where all flow variables are independent of the azimuthal angle ?-for example, the flow over a circular cylinder. If the velocity components (u, v) correspond to the coordinate directions (r, ?), respectively, show that the continuity equation is given by

(b) Show that the continuity equation can be automatically satisfied by a stream function ψ of a form such that