Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 77

Fig. 4.8 Variation of wall shear stress in a pipe in the developing and fully developed region

Fig. 4.9 a Schematic of fully developed laminar velocity profile with, b differential fluid element

acts on the pipe walls τ dependent on the velocity gradient at the wall surface. At
the pipe entrance there is a sharp velocity gradient, du d / because the velocity pro-
file with a uniform velocity is abruptly brought to zero at the wall surface, thereby
producing a greater shear stress. This reduces as the velocity profile develops and
the gradient becomes less sharp. In the fully developed region the gradient becomes
constant and thus the shear stress is also constant(Fig. 4.8 ).

4.6.2 Laminar and Turbulent Velocity Profiles in a Pipe

We can obtain the velocity profile in the fully developed region for a steady laminar
incompressible flow, by solving the force balance applied to the fluid. In this region,
the fluid particles move along constant streamlines and the velocity profile in the
radial direction u ( r ) does not change in the axial flow direction (Fig. 4.9a ).
Earlier we showed that a fluid element is subjected to shear, and pressure forces
that act in pairs on opposing surfaces of the fluid element. Figure 4.9b shows a small
differential fluid element with thickness dr and length dx and the applied forces.
Since the flow is steady there is no change in the net force in the axial direction and
therefore the sum of forces is zero,