The stream function for a given two-dimensional flow field is = 5x²y - 5y³. Determine the velocity potential if it exists. If it doesn't, explain why not.

Consider a two-dimensional flow which varies in 9me and is defined by the velocity field, u = 1 and v = 2yt. Is the flow field incompressible at all 9mes?

The stream function o in a two-dimensional flow field is given as 9 = 4x – 3y + 7xy (a) Prove that this flow field is irrotational and that it satisfies the continuity equation. Find the potential flow function P(x, y) for this flow field with boundary condition 0 = 0 at x = 2, y = 1. (b)

Consider irrotational flow past a stationary sphere of radius R located at the origin. In the limit r→∞, the velocity field v = U2, as in Fig. 8-6 in the book. (a) Calculate the velocity field v assuming potential flow given by v = Vo(r, 0, 0), where the potential can be assumed to be independent of the azimuthal coordinate and vo= 0. Here, since ə rde Ə Ər for large r/R, look for solutions of the form = f(r) cos 0. Assume a no-penetration boundary condition at the surface of the sphere. (b) Calculate the pressure P and the drag force due to pressure. Vr = U cos 0 and Vo = -U sin 0