Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 143

Solution

Descending Aorta Inferior Vena Cava

5 Δ

P
Q 5

20 mmHg
4

5 Δ

P
Q 5

3 mmHg
4

R R

: = : =

5L min 5L min

5 dyne

s

3 dyne

s

R 355 : R 53 :

5 5

cm 5 cm 5

This problem illustrates that to maintain the same flow rate at a higher pressure head,
the resistance to flow must be larger. This is clearly an oversimplification of the biology,
but can provide some insight into the flow through the cardiovascular system.
A more useful approximation for volumetric flow rate ( Q ) is the Hagen-Poiseuille's
Law. The general Poiseuille solution is obtained by solving for the volumetric flow rate for
laminar fluid flow within a cylindrical tube. Recall from Chapter 3 that the general veloc-
ity profile for pressure driven cylindrical flow is given by

R 2

R 2
4

@ p r

v z ð r Þ 5 1

2

μ @ z

The volumetric flow can be calculated by integrating the velocity profile over the cross-
sectional area of the vessel, as in

2
4

3
5 dr

! " # dr

ð ð ð

R R

0
@

1
A

R

0
@

1
A

2 2

R 2
4

R 2
4

@ p @ p

@

r
R

r
R

Q 2 π rv z ð r Þ dr 2 π r 1 2 π r 1

5 5 2 5 2

μ @ z μ z

0 0 0

0
@

1
A dr

2
4

3
5

2
4

3
5

R
0 5 π

ð

R

R 2
2

r 3
R 2 2

R 2
2

r 4
4 R 2 2

r 2
2

R 2
2

R 4
4 R 2 2

R 2
2 2

5 π

@ p

5 π

@ p @ p

@

r 0 1 0

μ @ z μ @ z μ z

0

R 4
8

52 π

@ p

μ @ z

As the partial derivative in space approaches the tube length (which is what can be
measured), the spatial partial derivative of pressure becomes

: @

p

z -2 Δ

P
L

As @ z L

-

@

where:

Δ P P 1 P 2

5 2