Chapter 7 – Complementary Experimental Tools 293
Thus,
I
m
r
z
d
r
z d r
AU
Au
Au
AU
Au
AU
=
=
⋅
=
⋅
(
)
=
×
×
−
/
/
/
4
(10
10
m
9
π
ρ
δ
π
ρ
π
ρ
δ
2
2
2
2
4
4
)
((5 10
m)/(0.5
50
10
m))
1.6
10
m
16 cm
2
6
2
1
×
×
×
×
=
×
=
−
−
−
7.6.2 MICROFLUIDICS
Microfluidics (for a good overview, see Whitesides, 2006) deals with systems that control
the flow of small volumes of liquid, anything from microliters (i.e., volumes of 10−9 m3) down
to femtoliters (10−18 m3), involving equivalent pipes or fluid channels of cross-sectional
diameters of ~1 μm up to a few hundred microns. Pipes with smaller effective diameters
down to ~100 nm can also be used, whose systems are often referred to as nanofluidics,
which deal with smaller volumes still down to ~10−21 m3, but our discussion here is relevant
to both techniques.
Under normal operation conditions, the flow through a microfluidics channel will be lam
inar. Laminar flow implies a Reynolds number (Re) ca. < 2100 compared to turbulent flow
that has an Re ca. > 2100 (see Equation 6.8). Most microfluidics channels have a diameter in
the range of ~10–100 μm and a wide range of mean flow speeds from ~0.1 up to ~10 m s−1.
This indicates a range of Re of ~10−2 to 103 (see Worked Case Example 7.2).
The fluid for biological applications is normally water-based and thus can be approximated
as incompressible and Newtonian. A Newtonian fluid is one in which viscous flow stresses
are linearly proportional to the strain rate at all points in the fluid. In other words, its vis
cosity is independent of the rate of deformation of the fluid. Under these conditions, flow
in a microfluidics channel can be approximated as Hagen–Poiseuille flow, also known as
Poiseuille flow (for non-French speakers, Poiseuille is pronounced, roughly, “pwar-zay”),
which was discussed briefly in Chapter 6. A channel of circular cross-section implies a para
bolic flow profile, such that
(7.13)
v
z
p
x
a
z
x ( ) = −∂
∂
−
(
)
1
4
2
2
η
where
η is the dynamic (or absolute) viscosity
p is the fluid pressure along an axial length of channel x
a is the channel radius
vx(z) is the speed of flow of a streamline of fluid at a distance z perpendicular to x from
the central channel axis
For a fully developed flow (i.e., far away from exit and entry points of the channel), the
pressure gradient drop is constant, and so equals Δp/l where Δp is the total pressure drop
across the channel of length l. It is easy to demonstrate a dependence between Δp and the
volume flow rate Q given by Poiseuille’s law:
(7.14)
∆p
l
a Q
R Q
H
=
=
8
4
η
π
RH is known as the hydraulic resistance, and the relation Δp = RHQ applies generally to
noncircular cross-sectional channels. In the case of noncircular cross-sections, a reasonable