Biomedical Engineering Reference
In-Depth Information
determine physical parameters such as elastic modulii as well as for model vali-
dation is still important, but the flow model itself is more directly derived.
The earliest attempt to model the collecting lymphatics was that of Reddy and
co-workers [
37
,
38
]. The Navier-Stokes equations can be reduced to the following
form in cylindrical polar coordinates:
or
;
2
u
x
o
u
x
ot
þ
u
x
o
u
x
ox
þ
u
r
o
u
x
or
¼
1
o
ox
þ
l
o
ox
2
þ
1
o
or
r
ou
x
½
p
þ
qgh
ð
41
Þ
q
r
This represents the flow of fluid in a cylindrical tube oriented along the x-axis; with
the possibility of including head differences (qgh term) which will be irrelevant in
laboratory in vitro studies but may be quite important for a chain of lymphangions
across a significant section of the human body. Assuming no radial variation in the
flow, i.e.
o
p
or
¼
0
u
r
¼
0
;
ð
42
Þ
and also assuming the flow to be low-Reynolds creeping flow;
u
x
o
u
x
ox
¼
0
;
ð
43
Þ
we can simplify (Eq.
41
) as follows:
or
:
o
u
x
ot
¼
1
o
ox
þ
m
r
o
or
r
ou
x
½
p
þ
qgh
ð
44
Þ
q
Integrating this over the area of the tube (
R
...2prdr) gives
s
r
¼
a
ot
¼
pa
2
o
Q
o
ox
þ
2pa
q
½
p
þ
qgh
;
ð
45
Þ
q
where s is the wall shear stress, a the tube radius and Q the volumetric flow. For
Poiseuille flow it is straightforward to show that the shear stress at the wall is
s
r
¼
a
¼
4l
pa
3
Q
;
ð
46
Þ
(see eg. [
11
]), giving
ot
¼
pa
2
o
Q
o
ox
þ
8l
½
p
þ
qgh
qa
2
Q
:
ð
47
Þ
q
Finally, an additional term can be introduced to represent the effect of the valve,
switching between finite and infinite resistance to represent the closed valve.
For a flexible tube (i.e. one whose radius can change) it is straightforward to
show that
Search WWH ::
Custom Search