Biomedical Engineering Reference
In-Depth Information
dp
dx
−
L
∆
P
Q
R
=
=
(
3.11
)
Q
For a rectangular microchannel, we substitute
Equation 3.8
into
Equation 3.11
to obtain:
−
1
∞
3
4
L
wh
η
192
h
w
∑
1
n w
h
π
3
4
η
L
wh
1
1 0 63
(
3.12
)
R
=
1
−
tanh
≈
3
5
5
2
3
−
.
h w
/
π
n
n
,
, ...
=
1 3 5
where the approximation simply states that that the second term of the sum in
Equation 3.12
(
n
= 3) is approximately 3
5
= 243 times smaller than the irst term (for “standard” microchan-
nels where
h
<
w
), so all the terms
n
> 1 can be neglected. If the microchannel has a high aspect
ratio, that is,
h
≪
w
, then the expression in brackets is approximately 1 and the resistance can
be approximated as:
3
4
L
wh
η
(
3.13
)
R
≈
(
h w
)
3
Strictly speaking,
Equation 3.13
is valid both for
h
≪
w
and
w
≪
h
because the microchannel
conserves the same resistance ater a 90 degree rotation.
For a circular cross-section microchannel (i.e., a glass or a blood capillary), it is straightfor-
ward to see (by substitution of
Equation 3.6
into
Equation 3.11
) that the resistance is:
η
π
L
=
8
R
(
3.14
)
r
4
0
he expression for the pressure drop in a circular pipe (or microchannel) is known as the
Hagen-Poiseuille equation
(Poiseuille derived it experimentally in 1838):
= × =
8
η
π
LQ
r
∆
P R Q
(
3.15
)
4
0
It is important to keep in mind that the Hagen-Poiseuille equation only applies to Newtonian
luids.
3.2.6 Shear Stress
Shear stress
is mathematically more complex to deine because it is a tensor, which means that it
is the manifestation of a force that can act in many directions. Like all tensors, it can be manipu-
lated as a matrix of numbers using matrix calculus. he advantage of using matrix calculus is
that one can visualize how the
x
,
y
, and
z
components afect each other in a straightforward way.
he shear stress tensor (a 3 × 3 matrix) is usually denoted without a subindex, τ, and each one of
its nine components is expressed with a subindex: τ
xx
, τ
xy
, τ
xz
, etc.:
τ
τ
τ
xx
xy
xz
τ
=
τ
τ
τ
(
3.16
)
yx
yy
yz
τ
τ
τ
zx
zy
zz
