Biomedical Engineering Reference
In-Depth Information
rotating luid. He did pioneering work on the theory of dislocations in crystals during his
appointment as a Royal Society research professor. He also introduced a new approach to
turbulent low through a statistical study of velocity luctuations.
Even ater his oicial retirement, he made contributions such as developing a method
for measuring the second coeicient of viscosity and creating an incompressible liquid
with separated gas bubbles suspended in it. Shear viscosity of the liquid resulted in the
dissipation of the gas in the liquid during expansion, thus allowing the bulk viscosity to be
easily calculated. In 1947, ater viewing declassiied movies of the irst atomic bomb test on
1945 in New Mexico, he was able to correctly estimate the energy released in the explosion
using only dimensional analysis. Other late work included the longitudinal dispersion in
low in tubes (in 1953), movement through porous surfaces, and the dynamics of sheets of
liquids. In 1969, he published his last research paper at 83 years of age, to describe jets of
conducting liquid produced by electrical ields that, to this day, bear the name of
Taylor
cone
in his honor. More than a dozen other phenomena, such as the
Taylor-Couette low
(which occurs when a viscous luid is conined between two rotating cylinders) and the
Taylor vortex
(an instability occurring in Taylor-Couette low) are named ater him; else-
where in this topic, we describe the
Taylor dispersion
that is ubiquitous in microchannels.
[
Excerpt adapted from Wikipedia
.
Photograph of G. I. Taylor obtained from
http://old.lms
.ac.uk/newsletter/335/335_12.html].
3.2.4.3 Triangular (Isosceles) Cross-Section
he low velocity proile
u
x
(
y
,
z
) and total volumetric low rate
Q
for a microchannel of isosceles
triangle cross-section (with base
a
, height
b
, and angle subtended by the top vertex ϕ) are:
B
−
2
2
2
2
1
dp
dx
y
−
−
z
tan
tan
φ
z
b
u y z
x
( , )
=
−
−
1
(
3.9
)
2
η
1
φ
2
3
2
4
3
ab
dp
dx
(
B
2
2 1
−
)tan
φ
Q
=
−
tan )
(
3.10
)
2
η
(
B
+
)(
−
φ
5
2
1
where
B
≡
4
+
−
1
to simplify the previous notations.
2
tan
φ
3.2.5 Microchannel Resistance
Because of the large surface-to-volume ratio of microchannels, the walls of a microchannel exert
a lot of friction when compared with the force required to keep the luid moving (the luid's iner-
tia). When the source of energy used to pump the luid is removed, the luid stops immediately.
In considering pressure-driven lows, an important implication of
Equation 3.8
for the micro-
luidics designer is the calculation of
microchannel resistance
, which is deined (in analogy
with Ohm's law of electricity,
R
=
V
/
I
) as the ratio between the applied pressure
P
(which plays
the role of voltage
V
) and the volumetric low rate
Q
(which plays the role of current
I
).
Note that in all the formulas for
Q
above,
Q
is proportional to the pressure gradient (−
dp
/
dx
)
and does not depend on
x
, so for any microchannel segment of length
L
and constant cross-
section, we can write
