Problem 4 (30 pts) Considerable insight into a flow can be gained solely from the continuity equation. Consider laminar flow over a flat plate, as shown below. Fluid approaches the plate with a uniform velocity U. When the fluid is in contact with the surface, flow in the x-direction decelerates within a thin region of thickness 8. This boundary layer grows with distance along the plate (i.e., 8 = f(x)). Using the no-slip condition along the plate (y = 0, vx = 0, and vy = 0) and Vx = U (a constant outside the boundary layer), prove the following statements, using the conservation of mass for an incompressible fluid: (i) vy(x,y)is everywhere positive for 0 sy s8. (ii) vyhas a parabolic shape with respect to y very close to the plate. (iii) Vy reaches a positive maximum at y = 8. For (ii), you may assume that y, can be represented in terms of a series (e.g., Vx = a (x)y+ az(x)y2 + ...). >U Uxly) ( 1