The stream function for steady, incompressible, two dimensional flow over a circular cylinder of radius a and freestream velocity V is = V

The stream function for steady, incompressible, two dimensional flow over a circular cylinder of radius a and freestream velocity V_∞ is ψ = V∞ sinθ(r – a²/r) for the case in which the flow field is approximated as irrotational (Fig. P10–70). Generate an expression for the velocity potential function Φ for this flow as a function of r and u, and parameters V_∞ and a.

FIGURE P10–70

V AY 0 X