Biomedical Engineering Reference
In-Depth Information
In the case of a tubular duct
u
max
2
U
=
Instead of using the mean velocity
U
to integrate the NS equations, we could
have used the pressure difference between inlet and outlet, and we would find a
relation between
U
and the pressure drop
D =
P P
-
P
in
out
8
U L
µ
D =
P
(2.43)
2
R
where
L
is the length of the tube. Equation (2.43) is sometimes called
Washburn's
law
[25].
2.2.5.2
ParallelPlates
The same reasoning may be done for a laminar flow limited by two parallel plates.
If the distance between the plates is
D
and the mean velocity is
U
, the velocity field
is
2
é
ù
3
y
æ
ö
u y
=
U
1
-
(2.44)
(
)
ê
ú
ç
÷
2
D
/ 2
è
ø
ê
ú
ë
û
where
y
is the transverse direction. Again, the profile does not depend on its loca-
tion and is parabolic, with a maximum velocity of
3
2
u
=
U
max
and the pressure difference between inlet and outlet is given by
12
U L
µ
D =
P
(2.45)
2
D
RectangularDucts
Generally, in microtechnologies, capillaries of a circular cross section are used to
link a fluid reservoir to the microsystem. Due to the microtechniques of etching in
silicon, glass, or plastic, capillaries in bioMEMS are often rectangular [1]. An ap-
proximated, closed form solution exists for laminar flows in rectangular channels.
The real flow profile is given by a series expansion [26], which is not always practi-
cal for applications. An approximation to this expansion was given by Purday [27].
The flow velocity in the
z
-direction inside a rectangular channel of dimensions 2
a
and 2
b
is approximated by
2.2.5.3
é
s
ù é
r
ù
x
y
æ
ö
æ
ö
(2.46)
u x y
,
=
u
ê
1
-
ú ê
1
-
ú
(
)
ç
÷
ç
÷
max
a
b
è
ø
è
ø
ê
ú ê
ú
ë
û ë
û
