Microfluidics - Introduction to BioMEMS - page 109

equal average low speeds) compared with the h / w = 1 case, and between 27% (for h / w = 10) and
31% (for h / w = 1) of the peak value at the center of the channel. his picture changes radically
when the comparison is made at equal low rates (experimentally, the value of dp / dx would need
to be adjusted accordingly by almost a factor of 10), because the high aspect ratio microchannel
has a much wider cross-sectional area, resulting in much lower low speeds, and consequently,
much lower forces (i.e., the green curve is exactly 10 times smaller than the blue curve in Figure
3.3 ). In all cases, the force is zero at the walls, as expected from the no-slip condition.

3.2.7 Capillary Flow
For a very small channel on the order of ~1 μm in its smallest dimension, the low resistance
is so large that pressure-driven low can be impractical (resulting in bursting of the channel,
for example). On the other hand, capillary illing scales well as the size of the channel is scaled
down. he capillary pressure P c of a liquid-air meniscus in a rectangular microchannel of width
w and height h is:

cos θ + cos θ cos θ + cos θ

b t l r

P = − γ +

( 3.23 )

c

h w

where γ is the surface tension of the luid and θ b , θ t , θ l , and θ r are the contact angles of the liquid
on the bottom, top, let, and right microchannel walls, respectively. his is the pressure that is
pulling the low.
Note that Equations 3.11 , 3.12 , and 3.13 are still valid here. However, Δ P is not a term pro-
duced by, say, a human operator but by the luid itself. Note that for a capillary open at only one
end, we have Δ P = P c + Δ P int , where Δ P int is the internal pressure; for channels that are open
at both ends, Δ P int = 0 and Δ P = P c . In this case, we can see that to increase low speed (which
is proportional to P c ), it is to our advantage to (a) decrease the size of the microchannel; and
(b) decrease the contact angle (the maximum value of the numerator is 2), for example, by sur-
face functionalization or by adding a surfactant.

3.2.8 Flow through Porous Media
A wealth of applications in microluidics requires the control of low through a layer of porous
material, for example, a membrane containing nanopores, a slab of gel, or paper. he equation
describing the low rate Q for a luid through a porous medium under a pressure diferential, Δ P ,
now known as Darcy's law , was irst established experimentally by Henry Darcy in 1856 and has
since been derived from the Navier-Stokes equation:

= κ

A
L

Q ∆ P ( 3.24 )

where:
μ = dynamic viscosity (in Pa.s)
L = length of the porous medium over which the pressure drop Δ P is taking place
A = cross-sectional area of the low
κ = permeability of the medium (in units of area).

3.2.9 Diffusion
he behavior of luids is not fully predicted by the Navier-Stokes equation, in that it does not
consider difusion. Difusion is the macroscopic result of the thermally driven microscopic
motion of particles (down to the size of atoms). It is inherently irreversible and random in
nature, such that the motion of a single particle is not predictable. Fortunately, the motion of