4.3 Fluid Dynamics
93
4.3.4 Applications to Blood Flow in Arteries
Conservation of mass for flow in an unbroken artery means that whatever blood
went into the artery minus what came out must be still inside. For a steady flow, we
would have, from point 0 to point 1 along the flow
0
v · dS =
1
v · dS .
(4.36)
If we let v i be the average fluid velocity within the cross-sectional area A i at the
location i along the artery, then
v 0 A 0 = v 1 A 1 .
(4.37)
In the case the artery divides into N smaller arteries,
v 0 A 0 =
N
k=1
v k A k .
(4.38)
This relation is very useful in determining the speed of blood in the branches after
the location of arterial bifurcation.
Even though blood flow is not uniform, under ordinary circumstances, the flow
is sufficiently smooth that Poiseuille’s law still applies approximately. The dramatic
part of Poiseuille’s Law is the very strong dependence of flow on the radius of
the artery. If a constriction, such as that produced by arteriolosclerosis or plaque
buildup, diminishes the diameter of an artery by half, the same pressure produces
only an eighth of the original flow!
The nature of blood flow determined by physical laws leads to useful observations about our circulatory system. For example, using fluid volume conservation,
we can estimate the number of capillaries in the body, a number difficult to get by
direct counting. The net flow in the aorta should match the net flow through all the
capillaries:
v a A a = N c v c A c
(4.39)
Figure 4.6 shows how the cross-sectional area of the vessels carrying blood
varies with distance from the heart, and also shows the average velocity of the
blood at these same distances. Note that variations in blood pressure will induce
small changes in the arterial cross-sectional areas along with the velocities. These
variations are not shown in the figure.
The forcing term from a pumping healthy heart produces a pressure above
atmospheric varying from systole of about 120 mmHg to diastole of 70 mmHg with
a cardiac output of about 90 cc/cycle, with about 1.2 cycles per second. From this
we can calculate by pressure times volume flow that the adult heart performs at
93
4.3.4 Applications to Blood Flow in Arteries
Conservation of mass for flow in an unbroken artery means that whatever blood
went into the artery minus what came out must be still inside. For a steady flow, we
would have, from point 0 to point 1 along the flow
0
v · dS =
1
v · dS .
(4.36)
If we let v i be the average fluid velocity within the cross-sectional area A i at the
location i along the artery, then
v 0 A 0 = v 1 A 1 .
(4.37)
In the case the artery divides into N smaller arteries,
v 0 A 0 =
N
k=1
v k A k .
(4.38)
This relation is very useful in determining the speed of blood in the branches after
the location of arterial bifurcation.
Even though blood flow is not uniform, under ordinary circumstances, the flow
is sufficiently smooth that Poiseuille’s law still applies approximately. The dramatic
part of Poiseuille’s Law is the very strong dependence of flow on the radius of
the artery. If a constriction, such as that produced by arteriolosclerosis or plaque
buildup, diminishes the diameter of an artery by half, the same pressure produces
only an eighth of the original flow!
The nature of blood flow determined by physical laws leads to useful observations about our circulatory system. For example, using fluid volume conservation,
we can estimate the number of capillaries in the body, a number difficult to get by
direct counting. The net flow in the aorta should match the net flow through all the
capillaries:
v a A a = N c v c A c
(4.39)
Figure 4.6 shows how the cross-sectional area of the vessels carrying blood
varies with distance from the heart, and also shows the average velocity of the
blood at these same distances. Note that variations in blood pressure will induce
small changes in the arterial cross-sectional areas along with the velocities. These
variations are not shown in the figure.
The forcing term from a pumping healthy heart produces a pressure above
atmospheric varying from systole of about 120 mmHg to diastole of 70 mmHg with
a cardiac output of about 90 cc/cycle, with about 1.2 cycles per second. From this
we can calculate by pressure times volume flow that the adult heart performs at