Biomedical Engineering Reference
In-Depth Information
θ
r
FIGURE 5.10 Schematic of a vein with a non-standard geometry, shown by varying radial's length at various
angles,
. The method of choosing laminas of known average velocities still apply to these vessels; however, one
would need to take account for the different area of each lamina.
θ
which may be approximated for laminas of a particular thickness, if the velocity can
be assumed to be uniform within that lamina and the velocity changes with theta are
negligible. This would use a weighted average formulation. The key here is that we
must be able to neglect changes in velocity in the theta direction to apply this type of
approximation.
Example
Calculate the average volumetric flow rate for the following ellipse, which has known velocity
and geometric values for laminae of fluid ( r L is the radius for the long axis and r S is the radius
for the short axis, Figure 5.11).
Solution
First, we will calculate the area associated with each fluid lamina:
29 mm 2
A 1 5 πð
5mm
Þð
2mm
Þ 2 πð
4
:
6mm
Þð
1
:
6mm
Þ 5
8
:
29 mm 2
A 2 5 πð
4
:
6mm
Þð
1
:
6mm
Þ 2 πð
4
:
2mm
Þð
1
:
2mm
Þ 5
7
:
28 mm 2
A 3 5 πð
4
:
2mm
Þð
1
:
2mm
Þ 2 πð
3
:
8mm
Þð
0
:
8mm
Þ 5
6
:
A 4 5 πð
3
:
8mm
Þð
0
:
8mm
Þ 2 πð
3
:
4mm
Þð
0
:
4mm
Þ 5
5
:
28 mm 2
27 mm 2
A 5 5 πð
3
:
4mm
Þð
0
:
4mm
Þ 5
4
:
Using the average velocity for each lamina, the volumetric flow rate can be found:
Q
Q 1 1
Q 2 1
Q 3 1
Q 4 1
Q 5 5
v 1 A 1 1
v 2 A 2 1
v 3 A 3 1
v 4 A 4 1
v 5 A 5
5
29 mm 2
29 mm 2
28 mm 2
Q
8
:
ð
10 cm
=
s
Þ 1
7
:
ð
35 cm
=
s
Þ 1
6
:
ð
50 cm
=
s
Þ
5
28 mm 2
27 mm 2
5
:
ð
80 cm
=
s
Þ 1
4
:
ð
105 cm
=
s
Þ 5
913
:
7mL
=
min
1
Note that this type of approach likely oversimplifies the flow conditions, but would be
necessary if the assumptions made when deriving the Hagen-Poiseuille formulation can-
not be met.
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