4.4.5 (WP Air enters an elbow with a uniform speed of 10 m/s as shown in Fig. P4.4.5. At the exit of the elbow, the velocity profile is not uniform. In fact, there is a region of separation or reverse flow. The fixed control volume ABCD coincides with the system at time t = 0. Make a sketch to indicate (a) the system at timet = 0.01 s and (b) the fluid that has entered and exited the control volume in that time period.

For incompressible polar coordinate flow, what is the mostthe general form of a purely circulatory motion, υ θ = υ θ ( r , θ , t )and υ r = 0, that satisfies continuity?

In plane stagnation flow, an incompressible fluid occupying the space y>0 has one velocity component given by Vx-x. The flow is two-dimensional and steady, such that V₂=0 and nothing depends on z or time t. (a) Use the continuity equation to determine Vy(x,y), given that Vy(x,0) =0. (This condition for Vy corresponds to the plane y=0 being an impenetrable boundary.) (b) is arbitrary, so you may set Y=0 at any convenient location.) Determine the stream function for this flow, (x,y). (The absolute value of

P3.5 Water at 20°C flows through a 5-in-diameter smooth pipe at a high Reynolds number, for which the velocity profile is approximated by u = U.(v/R)/8, where U, is the cen- terline velocity, R is the pipe radius, and y is the distance measured from the wall toward the centerline. If the cen- terline velocity is 25 ft/s, estimate the volume flow rate in gallons per minute.

Water flows through a two-dimensional narrowing wedgeat 9.96 gal/min per meter of width into the paper(Fig. P4.72). If this inward flow is purely radial, find anexpression, in SI units, for ( a ) the stream function and( b ) the velocity potential of the flow. Assume onedimensionalflow. The included angle of the wedge is 45 ° .