Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 146

FIGURE 5.9 The relationship
between animal mass and mean systolic
pressure. As the mass of the animal
increases, there is a general increase in
systolic pressure. The relationship be-
tween these two measurements can be
correlated to many different properties
of the animal, as described in the text.

150
140
130
120
110
100
90
80
70
60
50

Horse

Large
dog

Small
dog

Duck Human

Turkey

Mouse

Rat

Canary

10 100 1000 10000 100000 1000000

Animal mass (gram)

yield stress, the Hagen-Poiseuille solution may be applicable. Also notice that if the fluid
has no yield stress ( 0), then we would obtain the exact Hagen-Poiseuille solution.

We have already learned that the pressure variation in a normal human aorta is
80 mmHg to 120 mmHg. A comparison between blood pressure and mass shows that as the
mass of the animal increases, the mean systolic blood pressure also increases ( Figure 5.9 ).
As we discussed with changes in heart rate, correlations for this variation can be made
based on the activities and evolutionary history of the animals.

τ y 5

5.6 PR ESSURE, FLOW, AND RESISTANCE: VENOUS SY STEM

The mathematical formulation that was derived in the previous section for volumetric flow
rate through the arterial system can be used to approximate the volumetric flow rate in the
venous system as well. However, it is critical here to determine whether the assumptions that
have been made in deriving Hagen-Poiseuille's Law are still valid. In most instances, veins/
venules are not circular, and therefore, this formulation will only provide an approximation
to the volumetric flow rate. Therefore, let us revisit the derivation of the volume flow rate,
using cylindrical coordinates the velocity profile within a vein would become

r 2
4

@ p

v z ð r Þ 5 c 1 ln ð r Þ 1 c 2 ð 5 : 8 Þ

z 1 μ @

by solving the cylindrical Navier-Stokes equations (if ν z ( r ) is a function of radius only) and

assuming that flow is only pressure driven. The integration constants can only be found if
the boundary conditions of the particular flow conditions are known. With a randomly
shaped object, these conditions become difficult to define because the cross-sectional area
is not uniform. For instance, in the following figure ( Figure 5.10 ), the blood velocity along
the entire wall is zero but the radial coordinates that describe the wall are not constant.
Therefore, the solution for the volumetric flow rate would become
ð

dA

r 2
4

@ p

@

c 1 ln ð r Þ 1 c 2 ð 5 : 9 Þ

z 1

μ

A