"A viscous flow moves over a flat plate of length L. As the flow continues, the boundary layer grows quickly. At the transition point, the boundary layer thickness is δ. What is the value of the velocity potential function ɸ for this flow?"
In viscous flow over a flat plate, a boundary layer forms and grows along the plate. Inside this layer, the flow has friction and rotation. Since the velocity potential only exists in frictionless, smooth (irrotational) flow, it is not valid in this case.
So, the velocity potential function does not exist for this viscous flow.