Biomedical Transport Processes - Introduction to Biomedical Engineering - page 966

Since u z needs to be finite at r ¼ 0, c 1 ¼ 0. The no slip boundary condition at the pipe wall

requires that

u z ¼

0at

r ¼ R

(radius of the pipe), which yields

4m @ p

1

2

c ¼

@ z R

:

2

Thus, we have finally the following parabolic velocity profile:

4m @ p

1

2 2

u z ¼

@ z ð R

r Þ:

The maximum velocity occurs at the pipe centerline ( r ¼ 0):

:

2

u z max ¼ R

4m @ p @ z

The blood flow is thus a paraboloid with a maximum velocity at the center of the blood ves-
sel with a zero velocity at the wall, as shown in Figure 14.32. The velocity, varying in the
radial direction, can be thought of as flowing within concentric layers, each with a different
velocity at a different radial value. The zero value at the wall corresponds to the no slip con-
dition associated with Poiseuille flow. The fluid velocity is symmetric about the center of
the tube, corresponding to the axisymmetric assumption associated with Poiseuille flow.
The no slip condition is a result of the flow of a viscous fluid against a nonmoving
boundary.

(a)

(b)
FIGURE 14.32 The parabolic velocity profile associated with the solution of the Navier-Stokes equations for
Poiseuille flow. The velocity is maximal at the center of the vessel and zero at the wall. The velocity is axisymmetric,
corresponding to the associated Poiseuille assumption.