Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 168

5.5. The pressure, flow, and resistance in the arterial system are related by

5 Δ

P
R

Q

The volumetric flow rate can also be approximated using the Hagen-Poiseuille solution:

5 πΔ

Pr 4

Q

8 μ L

which is valid for laminar flow within a cylindrical tube.
5.6. The formulas derived for the relationship between pressure, flow, and resistance in the
arterial system can also be used in the venous system. However, it is unlikely for the veins
to be perfectly cylindrical. In this case, the Navier-Stokes solution would need to be solved
for the randomly shaped vein, which can find a relationship between pressure, flow, and
resistance. The general Navier-Stokes for cylindrical components under these conditions
would be

r 2
4

@ p

v z ð r Þ 5 c 1 ln ð r Þ 1 c 2 :

z 1 μ @

where the integration constants would need to be obtained for the particular conditions. It
is possible to do a weighted average of laminae of fluid with known average velocities.
5.7. Wave propagation within the arterial circulation can occur in two ways. The first is the
pressure pulse that displaces the arterial wall, where a differential area would need to be
considered within the properties. The mechanical properties of vessel wall would also need
to be considered in these formulations. Making assumptions about the homogeneity of the
arterial wall mechanical properties, the wave speed can be computed from

s

Eh
4 pr i

c

5

The second wave that can be propagated throughout the arterial system is the fluid pres-
sure (or velocity) at bifurcations. The geometry of the bifurcation would determine the rela-
tive division of the pressure between the branches. However, one must consider that some
of the energy is reflected back along the main feed branch. In this case, the relative pressure
within the daughter branches and the main branch can be solved by

Z 2 1
p

Z 2 1
d 1

Z 2 1
d 2

2 ð

Þ

1

p r
p p 5 Z 2 1

p

1 ð

Z 2 1
d 1

Z 2 1
d 2

Þ

1

2 Z 2 1
p

p d 1
p p 5

p d 2
p p 5 Z 2 1

d 1

Z 2 1
d 2

Z 2 1
p

1 ð

Þ

1