Biochips – fundamentals and applications - Electrochemical Sensors, Biosensors and Their Biomedical Applications - page 358

TABLE 11.7
Terms for common geometries

Cross-section Fluidic resistance

8η L /π R 4 Circular

12η L / wh 4 ( h : height, w : width) Rectangular (low aspect)

28.454η L / a 4 ( a : side length)

Square

184.751η L / a 4 ( a : side length)

Regular triangle

clearly describing a parabolic fl ow profi le and also satisfying the no-slip condition.
The term 8 R 4 in Eq. (9) is called the fl uidic resistance. For channels with non-

cylindrical cross-sections, the expressions are similar to those in Eq. (9), but with dif-
ferent terms for the resistance. Table 11.7 gives the terms for common geometries.
There are more important and useful relations that can be used in design and analy-
sis of fl ow in microfl uidic channel network with changing sections. Since

η L /[ πι ] π

S 1 v 1 S 2 v 2 constant (11)

the Bernoulli equation is a direct application of the law of energy conservation, relat-
ing pressure, kinetic energy, and potential energy in the following way,

v 2

P 1/2 ρ ρ gh constant (12)

A knowledge of v can give an indication of the transit time of a plug of chemical or an
ensemble of cells through a microfl uidic channel network and thus to assess whether there
is enough time for complete mixing or chemical reaction. Both Eq. (11) and Eq. (12) are
strictly only valid under idealized conditions (i.e. incompressible and non-viscous fl uids
and steady fl ow), but can still be helpful for overall estimation and assessment.
The surface tension is of great importance when dealing with bubbles and particu-
late contaminations in microchannels and determining how strong the capillary forces
are in a microchannel. For a cylindrical cross-section, the capillary force, F cap , can be
expressed quantitatively as shown in the following equation,

F cap 2 π r γ cos Θ (13)

where are the surface tension and the contact angle, respectively.

There are three types of mass transport processes within a microfl uidic system: con-
vection, diffusion, and immigration. Much more common are mixtures of three types
of mass transport. It is essential to design a well-controlled transport scheme for the
microsystem. Convection can be generated by different forces, such as capillary effect,
thermal difference, gravity, a pressurized air bladder, the centripetal forces in a spin-
ning disk, mechanical and electroosmotic pumps, in the microsystem. The mechanical
and electroosmotic pumps are often used for transport in a microfl uidic system due to
their convenience, and will be further discussed in section 11.5.2. The migration is a
direct transport of molecules in response to an electric fi eld. In most cases, the moving

γ and Θ