Biomedical Engineering Reference
In-Depth Information
We now take the derivative (the cosine becomes a negative sine), which evaluated at
y
= −
h
is
±1, depending on the value of
n
, so we obtain:
n z
h
π
cosh
n
−
1
∞
∑
n
−
1
2
2
16
h
dp
dx
π
(
−
1
)
2
F
=
−
(
−
1
)
1
−
2
(
3.21
)
x
3
2
ηπ
n w
h
π
2
hn
cosh
n
=
1 3
,
,
5...
2
Rearranging,
n z
h
n w
h
π
cosh
∞
∑
8
h
dp
dx
2
1
2
F
x
=
−
1
−
(
3.22
)
2
n
ηπ
π
cosh
n
=
1 3 5
,
, ...
2
his formula cannot be further reduced, even for the special case of the high-aspect ratio
microchannel (
h
≫
w
or
w
≫
h
), because the hyperbolic cosine is a function that grows extremely
fast and any approximation must be valid for all
n
(but
n
is inside the cosh). However, the large
numbers and the 1/
n
2
term make the summation converge very fast, so a graph can be built
with a spreadsheet in a few minutes. he force acting on the cells as a function of position
across the channel is plotted in
Figure 3.3
for two diferent aspect ratios commonly encoun-
tered in BioMEMS,
h
/
w
= 10 and
h
/
w
= 1. To be fair, we need to specify in which conditions we
are performing the comparison: it is not the same to compare a 1:1 =
h
/
w
microchannel with a
10:1 =
h
/
w
microchannel at equal low rates than comparing them at equal (average) low speeds.
Indeed, note that the forces at equal average low speeds are very similar in both cases (blue and
red curves in
Figure 3.3
), although the force in the center of the channel is approximately 4.5%
larger for the
h
/
w
= 10 case. On the other hand, close to the walls, at about a distance correspond-
ing to 5% of the width of the microchannel, the forces are 12%
weaker
for the
h
/
w
= 10 case (at
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-w
h/w
= 10
y
(same u
avg
as h/w = 1)
x
z
h/w
= 1
u
x
F
x
F
x
2
h
h/w
= 10
(same Q as h/w
= 1)
-w/2
0
Z
w/2
w
FIGURE 3.3
Force.acting.on.the.bottom.surface.of.a.rectangular.microchannel..For.computation.
purposes,.the.expression.μ8
h
(-
dp
/
dx
)/(
ηπ
2
).in.
Equation 3.22
.is.forced.to.take.the.value.of.1,.and.
as.a.result.
F
x
is.in.nonconventional.units.that.depend.on.the.values.of.
h
.and.of.
dp
/
dx
..(Figure.con-
tributed.by.the.author.)
