Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 157

distance that is required for the fluid to become fully developed again after it has become
skewed. As an approximation, if the vessel cross-sectional area remains constant after the
bifurcation, within two-five tube diameters, the flow would be fully developed again. The
significance of flow skewing at a bifurcation is that the wall shear stress is not what one
would anticipate based on a fully developed flow. Along the inner wall, the shear stress
increases and along the outer wall the shear stress decreases. This will be discussed in
detail later, but cardiovascular diseases tend to initiate in locations where the shear stress
diverges from the normal conditions (either heightened or reduced shear stress).

5.9 FL OW THROUGH TAPERING AND CURVED CHA NNELS

In most of the examples that we have discussed in this chapter, we made the assumption
that the blood vessel is circular with a constant diameter and that the vessel remains straight
(in Chapter 3, we worked some examples without these assumptions, but the rate of change
of the taper was not considered in those problems). However, this assumption does not match
the real scenarios; most blood vessels experience a small reduction in diameter and do not
remain in the same plane in three-dimensional space (although they are relatively straight
and not that twisted). In most blood vessels, the taper is relatively small until the branch point
occurs; however, it is fairly easy to accommodate the effects of taper within our calculations.
Also, the curves within vessels can easily be accounted for as we have shown in Chapter 3.
The first assumption that can be made about a tapered blood vessel is that it tapers in
discrete steps ( Figure 5.19 ). To model a taper with steps, the assumption would be made
that the velocity, pressure, shear stress, among others are all constant throughout each sec-
tion at one instant in time. The analysis of changes within these flow variables would
need to be conducted only at the discrete steps 1, 2, and 3. These calculations can also be
carried out independent of each other (only the output flow variables of each step would
be needed to calculate the flow variables within the next step). The only reason that this
step analysis technique would be conducted instead of considering the exact blood vessel
taper is because 1) the change in area cannot be represented accurately or 2) the change in
area is too complicated to compute by hand and the use of numerical methods is needed.
To be able to model a taper and compute the fluid parameters of interest, one must use
the same formulas for Conservation of Mass and Conservation of Momentum throughout
the entire blood vessel. For an easy example, a continuous taper that reduces the blood
vessel radius by half over the entire vessel length, can be modeled as

x
2 L

r ð x Þ 5 r i 1 ð 5 : 27 Þ

2

FIGURE 5.19 The modeling of a continuous taper with
evenly spaced steps. To calculate the velocity profile through a
continuous taper, one would need to determine the exact taper
function, which may not be as simple as the examples shown in
the text. However, if the taper is complicated, it can be approxi-
mated with discrete steps.

1 2 3