In Silico Biofluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 353

From the above example, which obtained the dimensionless group that related the drag
force to the flow properties, we are able to end up with a mathematical statement that
states

F
v 2 d 2 ρ 5 f ρ vd μ

Therefore, if we were considering a flow where drag forces were important, it would be a
necessary requirement that the Reynolds number be matched between the actual scenario
and the model. By satisfying this criterion, the forces acting on the spheres would be scal-
able between the model and the realistic case. This same analysis can be conducted for
any of the dimensionless parameters (or as many as necessary) in order to satisfy similar-
ity between the flow that is being modeled and the model itself.

Example
Imagine that we are making a model of blood flow through an artery; however, we could
only use water as our fluid flowing through the model artery. Calculate what the angular fre-
quency of the pulsatile waveform and the initial inlet velocity should be if the characteristic
length (diameter) of the blood vessel is 10 cm, the heartbeat is 72 beats/min (angular frequency
is 5.24 rad/sec), and the inlet velocity is 50 cm/s.

Solution
Using the Reynolds number, we can calculate the initial inlet velocity of the model:

Re blood 5Re model
ρ b v b d b

μ b 5 ρ m v m d m μ m

v m 5 ρ b v b d b μ m

=

m 3

ð = Þð Þð Þ

1050 kg 50 cm s 10 cm 1cP

μ b ρ m d m 5

5 15 cm = s

1000 kg = m 3 ð 10 cm Þð 3 : 5cP Þ

Using the Womersley number, we can calculate the angular frequency of the model:

α blood 5α model

1 = 2 1 = 2

d b ω b
ν b

5d m ω m
ν m

1 = 2 1 = 2

d b ρ b ω b
μ b

5d m ρ m ω m
μ m

2
4

3
5

2
4

3
5

0
@

1
A

0
@

1
A

0
@

1
A

2 2

1 = 2 1 = 2

ω m 5 μ m
ρ m

d b
d m

ρ b ω b
μ b

1cP
1000 kg

10 cm
10 cm

ð 1050 kg = m 3 Þð 72 beats = min Þ

5

=

m 3

:

3 5cP

5 21 : 6 beats = min