Problem Consider a cylindrical pipe of length L and diameter D = 2R. The angle that the axis of the pipe forms with the vertical direction is a. Assume that when the fluid enters the pipe its velocity is uniform (i.e., it has the same value over the entire cross-section of the pipe) and equal to U in the axial direction. In the radial and angular directions, the velocity is zero. So, it is: 2 =0: v(r, 0, z) = Ue, (1.1) Here v is the fluid velocity and e, is a vector of unit magnitude parallel to the coordinate axis z; furthermore, we have assumed that the pipe inlet is located at z = 0. Near the entrance of the pipe, the velocity profile varies in the axial direction. But after a certain entrance length, the profile becomes fully developed, no longer changing with z. The evolution of the velocity profile is sketched in Fig. 1, where, for clarity, the pipe inclination is not shown. The entrance length is denoted by L. For z2 L, the fluid velocity is no longer a function of the axial coordinate. In engineering design, pressure drop calculations are based on relations that hold only for fully developed flows. However, entrance effects are always present. To judge whether these effects are negligible, one must estimate the value of the entrance length. This is our main goal. Let pand ja denote the density and viscosity of the fluid, respectively (assumed to be constants); we will restrict the analysis to steady, laminar flows in which Re = pUD/p > 1. Developing Velocity Profile Fully Developed Velocity Profile Inlet Velocity Profile 8(2) Figure 1: Evolution of the velocity profile in the pipe entrance region. L, is the entrance length, while 6(2) is the thickness of the "wall layer," i.e., the region where a,v, 0. Answer all the questions below in the reported order. a) Using scaling arguments and the mass balance equation (written in cylindrical coordinates), identify the length and velocity scales in the wal layer. b) Using scaling arguments and the equation governing the evolution of the velocity component v,, obtain the pressure scale and estimate the order of magnitude of L.. c) There is an alternative method for estimating L,. First, estimate the (mean) residence time of the fluid in the entrance region; then, use appropriutely the results of linear momentum penetration theory (that we obtained when we studied the pure diffusion of linear momentum). Use this method and verify that the result is the same as that found in part b). d) Using scaling arguments and the equation that governs the evolution of the velocity component u,, show that in the wall layer P is approximately a function of z only. e) Suggest a criterion for judging whether entrance effects may be neglected. If Re = 1500, R = 2 cm and L = 250 m, can entrance effects be safely neglected? ) The estimate you have made for L, is valid for laminar flows. For turbulent flows, do you expect it to be much larger, much shorter or about the same as for the laminar case? That is, for turbulent flows, do you expect the condition for neglecting entrance effects to be more, less or equally demanding compared to the condition holding for the laminar case? Justify your answer.