Biomedical Engineering Reference
In-Depth Information
2
Q
s
(
)
V
=
N i
-
,
i
=
0,...
N
-
1
(2.74)
i
ρ
a b
0
Step 2: Pressure at the Axial Nodes
We have already seen in relations (2.49) and (2.50) that Washburn's law for a rect-
angular capillary of cross dimensions
a
and
b
(
λ
=
b
/
a
) is
8
L V
η
(
)
D =
P
g
λ
2
a
where the function
g
(
λ
) is defined by using the Heaviside function
H
2
1
+
λ
3
æ
ö
( )
g
λ
=
H
(4.45
-
λ
)
+
H
(
λ
-
4.45)
ç
÷
λ
2
è
ø
At an intersection, there is a distortion of the laminar flow lines. This problem
is complex in a rectangular geometry and we simplify by
(
)
8
η
13
a V
( )
(2.75)
D
P
=
g
λ
int
er
2
a
and in a side branch of length
L
, the linear pressure drop is reduced to
(
)
8
η
L
-
4
a V
( )
(2.76)
D
P
=
g
λ
linear
2
a
so that the total pressure drop of a side branch is
(
)
8
η
L
+
9
a V
( )
(2.77)
D = D
P
P
+ D
P
=
g
λ
linear
int
er
2
a
Again, starting from the last node in the axial channel and applying (2.77), we
obtain the pressure at the last node
(
)
*
8
η
L
+
9
a
V
s
N
N
(
)
P
=
P
+
g
λ
N
o
N
2
a
N
where
V
*
is the velocity in the
N
th side branch and
P
o
is the pressure at the outlets
(atmospheric pressure). By replacing the flow rate by the flow velocity in the
N
th
side branch, the pressure can be cast into the form
(
)
8
η
L
+
9
a Q
s
N
o
(
)
P
P
g
=
+
λ
(2.78)
N
o
N
3
b a
N
Now, we progress towards the front end of the axial channel and deduce the
recurrence relation
(
)
8
η
L
+
9
a Q
8
η
L Q
s
N
o
(
)
o
(
0
(2.79)
P
=
P
+
g
λ
+
g
λ
(
N i N i
-
)(
- +
1)
i
o
N
3
3
0
ba
ba
N
