Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 76

Fig. 4.7 Fluid velocity profile development in a pipe for a laminar flow. The developing region
shows a change in the velocity profile shape but in the fully developed region, the shape becomes
constant. The parabolic profile shape is characteristic of a laminar flow. For turbulent flows the
profile would be a blunt , flatter shape

by the zero velocity at the wall. This influence continues to consecutive adjacent
fluid particles but with a diminishing value. The fluid continues downstream caused
by the pressure force pushing the fluid through, while the fluid viscosity tries to
hold the fluid together.
These conditions give rise to a velocity profile having a parabolic shape from
the wall to the centre of the pipe. The velocity profile development finally reaches a
fully developed stage where the profile remains constant as it travels further down-
stream. The length it takes to become fully developed is referred to as the entrance
length. Kays and Crawford (1993) reported the entrance lengths for laminar flow
in pipes as

(4.7)

L ≅ 0.05Re D

e lam ,

For turbulent flows in pipes Bhatti and Shah (1987) reported the entrance lengths as

1/ 4

L ≅ 1.359Re D (4.8)

e turb ,

Entrance regions are important if its distance is comparable to the total pipe length
distance. This is because the pressure drop is higher in the entrance regions of a pipe
and this influences the overall average friction factor for the entire pipe. However
if the entire pipe length is several times the length of the entrance length, then the
developing entrance length can be ignored and the entire pipe can be assumed to be
fully developed.
Earlier we saw that a shear stress arises when there is a velocity gradient in a
viscous fluid. In pipe flows where the velocity profile is not uniform, a shear stress