Biomedical Engineering Reference
In-Depth Information
5% of this flow rate is 0.00657 mL/min; therefore,
5% Reduction
10% Increase
Q
0
:
128 mL
=
min
Q
0
:
148 mL
=
min
5
5
4 s
8 Q
4 s
8 Q
μ
L
μ
L
R
5
R
5
πΔ
P
πΔ
P
t
8
t
8
ð
0
:
128 mL
=
min
Þð
3
:
5cP
Þð
100
μ
m
Þ
ð
0
:
148 mL
=
min
Þð
3
:
5cP
Þð
100
μ
m
Þ
5
4
5
4
πð
10 mmHg
Þ
πð
10 mmHg
Þ
34
:
5
μ
m
35
:
8
μ
m
5
5
Δ
R
0
:
5
μ
m
ð
1
:
5% change
Þ
Δ
R
0
:
8
μ
m
ð
2
:
3% change
Þ
5
5
This problem illustrates that small changes in an arteriole can significantly affect the
flow rate. Very small changes in vessel diameter cause large changes in flow.
We would like to revisit the Casson model of blood flow, to show that the volumetric
flow rate associated with this more realistic approximation of blood rheology obtains a
solution that is analogous to the Hagen-Poiseuille solution. If we compute the volumetric
flow rate of the Casson velocity profile ( Equation 5.4 ), as given by
ð
R
Q
2
π
u
ð
r
Þ
rdr
5
0
we obtain
R 4
8
5 π
dp
dx F
Q
ðΨÞ
ð
5
:
7
Þ
η
The function of
Ψ
is defined as
16
7 Ψ
4
3 Ψ 2
1
21 Ψ
0
:
5
4
F
ðΨÞ 52
1
2
1
and
2
2 1
2
τ y
R
dp
dx
Ψ 5
Comparing Equations 5.6 and 5.7 , we can see that by using the Casson model of blood
flow, the volumetric flow rate becomes a function of the yield shear stress. This would be
anticipated because if the shear stress applied to the fluid was below this threshold, the
fluid would not flow or would move as a solid material. However, there are distinct similar-
ities between Equations 5.6 and 5.7 . Therefore, if we assume that our fluid does not have a
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