Biomedical Engineering Reference
In-Depth Information
A number of common shapes (and not so common ones), including the circular and the rect-
angular cross-section, can be solved analytically. he isosceles triangle is also included in the
next section because it is a geometry commonly used in more traditional silicon-based micro-
luidics (resulting from the wet etch of Si(100), see Section 1.4.1 and 1.4.2).
3.2.4.1 Circular Cross-Section
For a channel of circular cross-section and radius
r
0
, the reader can verify by substitution that
the following is a solution of
Equation 3.3
:
dp
dx
(
)
2
2
u r
x
( )
= −
r
−
r
(
3.5
)
0
4
η
he low proile is parabolic:
u
x
is maximum at the center of the channel (where
r
= 0) and
gradually decreases toward the walls, where it is zero [
u
x
(
r
=
r
0
) = 0].
Integrating over the whole area
A
, one obtains the volumetric low rate
Q
:
dp
dx
4
r
π
r
dp
dx
(
)
∫∫
∫
2
2
0
Q
=
u r dA
( )
= −
r
−
r
2
π
r dr
=
−
(
3.6
)
x
0
4
η
8
η
A
0
BLOOD: A NON-NEWTONIAN FLUID
Blood is a very important fluid
in BioMEMS, one that eludes the simple math-
ematical treatment of these pages because it is non-Newtonian. Stated plainly, blood con-
tains a large concentration of red blood cells which tend to elongate under high shear, so
its viscosity is a function of the shear rate (unlike a Newtonian luid, in which it is a con-
stant). In other words, there is a critical shear rate value past which the viscosity decreases
with increasing shear rate. his property of blood is critical to life, as it saves the heart
enormous amounts of energy when it comes to pumping blood through small capillaries.
If red blood cells did not elongate (a property called “shear thinning”), our hearts would
have to spend much more energy to make blood reach all the corners of our body.
3.2.4.2 Rectangular Cross-Section
Here, the solutions are more complicated but the method is the same:
n z
h
n w
h
π
n y
h
π
2
3
cosh
cos
∞
∑
2
16
h
dp
dx
2
)
(
1 2
)
/
u y z
( , )
=
−
(
−
1
n
−
1
−
(
3.7
)
x
3
ηπ
π
n
cosh
n
=
1 3 5
,
, ...
2
where 2
w
is the width and 2
h
is the height of the microchannel (the origin is (0,0), -
w ≤ z ≤
+w
, -
h ≤ y ≤ +h
). his function is plotted in
Figure 3.2a
. Interestingly, the low proile is also
approximately parabolic, but the proile stays constant as one moves away from the sidewalls and
becomes smaller near the sidewalls; next to the sidewalls, the no-slip condition prevails because
the luid, because of its own viscosity, “feels” the friction exerted by the walls. he low proile
can be readily visualized by introducing a plug of luorescent dye (
Figure 3.2b
). An immediate
efect of the existence of a nonuniform velocity ield across the channel's cross-section is that
the dye—
even in the hypothetical absence of molecular difusion
—will rapidly disperse over a
