Entrance flow in conduits 

(a) Estimate the entrance length for laminar flow in a circular tube. Assume that the boundary-layer thickness S is given adequately by Eq. 4.4-17, with v_∞ of the flat-plate problem corresponding to v_max in the tube-flow problem. Assume further that the entrance length L_e can be taken to be the value of x at which S = R. Compare your result with the expression for L_e, cited in S2.3 namely, L_e, = 0.035D Re. 

(b) Rewrite the transition Reynolds number xv_∞/v = 3.5 x 10⁵ (for the flat plate) by inserting δ from Eq. 4.4-17 in place of x as the characteristic length. Compare the quantity δv_∞/v, thus obtained with the corresponding minimum transition Reynolds number for the flow through long smooth tubes. 

(c) Use the method of (a) to estimate the entrance length in the flat duct shown in Fig. 4C.1. Compare the result with that given in Problem 4C.1 (d).