Consider the flow within a round pipe. At the entrance, the flow velocity is uniform at U.. Near the inlet to the pipe, the flow velocity will be a function of of r and z. Downstream of the entrance region, the flow becomes fully-developed and the velocity profile is not a function of 2 and 8. Apply the continuity and the Navier-Stokes equations in cylindrical coordinate (Eq. 9-62, p. 475) to determine the fully-developed velocity profile within the round pipe.
a. List all the assumptions
b. Find all velocity components of the fully-developed flow in the pipe (V, V, V:)
c. Determine the expression for the fully-developed velocity profile in terms of the pres- sure gradient, 3
d. Determine the expression for the fully-developed velocity profile in terms of the max- imum velocity in the fully-developed profile, Umax.
e. Determine the expression for the fully-developed velocity profile in terms of the inlet velocity, U..
f. What is the ratio of Umax to U. for fully-developed pipe flow?