Biomedical Engineering Reference
In-Depth Information
100 µm
a
b
Flow
w/h
= 8
FIGURE 3.2
Flow.speed.proile.for.a.rectangular.channel..(a).Plot.of.
Equation 3.7
.using.
w
/
h
.=.8..
Figure. contributed. by. Nirveek. Bhattacharjee. and. Albert. Folch.. (b). Experimental. visualization. of.
Taylor.dispersion.with.a.luorescent.dye.in.a.250.μm.×.70.μm.microchannel.under.pressure-driven.
low..(From.Subhra.Datta.and.Sandip.Ghosal,.“Characterizing.dispersion.in.microluidic.channels,”.
Lab Chip
.9,.2537-2550,.2009..Reproduced.with.permission.from.The.Royal.Society.of.Chemistry.)
large area, an efect that has been termed
Taylor dispersion
(named ater Sir Geofrey I. Taylor's
observations of low in tubes in the 1950s). In the case of a non-Newtonian luid, for the same
average low, we would observe that the luid lows faster closer to the walls, advancing more like
a plug (although there would still be no-slip at the walls), and the pressure needed to drive the
low would be smaller (because of the shear-thinning behavior).
he volumetric low rate for a rectangular cross-section microchannel can be expressed as:
∞
∑
4
3
dp
dx
192
h
w
1
n w
h
π
2
3
(
3.8
)
Q
=
wh
−
1
−
tanh
η
5
5
π
n
n
=
1 3 5
,
, ...
As expected, the low rate is a sensitive function of the height of the microchannel, its small-
est dimension. Incidentally,
Equation 3.8
is also a fascinating function: it can be seen that if the
values of
h
and
w
are swapped, the value of
Q
…is preserved! his is not trivial to prove, but it
can be veriied numerically. his invariance is what we expect from a function that describes the
low rate, a quantity that does not depend on which dimension is named “height” and which one
is named “width”; in other words, the low rate should not change if the channel, or the device,
is rotated 90 degrees (the efect of gravity is negligible).
SIR GEOFFREY INGRAM TAYLOR
Sir G. I. Taylor
(1886-1975) was born in London from a fam-
ily of mathematicians (his grandfather was George Boole, the
famous inventor of Boolean logic). He became a physicist, math-
ematician, and expert on luid dynamics and wave theory. He
worked on turbulence in the atmosphere which led to the publi-
cation of “Turbulent motion in luids” while teaching at Trinity
College. His observations as a meteorologist aboard the ship
Scotia
(International Ice Patrol) in 1913 formed the basis of his later work
on a theoretical model of turbulent air mixing. When World War I
started, he was sent to apply his knowledge to aircrat design.
Ater the war, Taylor returned to Trinity and studied turbu-
lent low applied to oceanography. He also studied a moving body as it passes through a
