Biochips – fundamentals and applications - Electrochemical Sensors, Biosensors and Their Biomedical Applications - page 357

11.5.1 Theory of microfl uidics
Microfl uidics handles and analyzes fl uids in structures of micrometer scale. At the
microscale, different forces become dominant over those experienced in everyday
life [161]. Inertia means nothing on these small sizes; the viscosity rears its head and
becomes a very important player. The random and chaotic behavior of fl ows is reduced
to much more “smooth” (laminar) fl ow in the smaller device. Typically, a fl uid can be
defi ned as a material that deforms continuously under shear stress. In other words, a
fl uid fl ows without three-dimensional structure. Three important parameters character-
izing a fl uid are its density,

. Since the pressure in
a fl uid is dependent only on the depth, pressure difference of a few

ρ , the pressure, P , and its viscosity, η

µ m to a few hun-

dred m in a microsystem can be neglected. However, any pressure difference induced

externally at the openings of a microsystem is transmitted to every point in the fl uid.
Generally, the effects that become dominant in microfl uidics include laminar fl ow, dif-
fusion, fl uidic resistance, surface area to volume ratio, and surface tension [162].
Laminar fl ow is the defi nitive characteristic of microfl uidics. Fluids fl owing in
channels with dimensions on the order of 50 mm and at readily achievable fl ow speeds
are characterized by low Reynolds number, Re , defi ned as

µ

v 2 V/

(7)
where v is the average fl ow speed of the fl uid. The characteristic linear dimension L
is the ratio of the volume V of the fl uid to the surface area S of the walls that bound it
[163]. Due to the small dimensions of microchannels, the Re is usually much less than
100, often less than 1. In this Reynolds number region, the fl ow is completely laminar.
Laminar fl ow provides a means by which molecules can be transported in a relatively pre-
dictable manner through microchannels. To defi ne the best dimension of the microchan-
nels for calculating the Reynolds number, the hydraulic diameter, D h , is introduced as

Re ρ η vS ρ vL/ η

(4 S/p wet ) (8)
where p wet is the wetted perimeter, which is all the perimeter that is in contact with the
liquid. For a rectangular microchannel this corresponds to twice the width plus twice
the height. For a circular cross-section, Eq. (8) is simplifi ed to D h

D h

d , where d is its
circular diameter. The theoretical framework to analyze fl uid fl ow is based on Navier-
Stokes equations [164]. However, the Navier-Stokes equations contain more unknown
parameters than other equations, making complete analytical solution impossible.
Typically, several boundary conditions and/or equations of state are used for solving
the equation. An important solution to the equations, which can be used in the analysis
of a microfl uidic system, is the Poiseuille fl ow, which applies when a pressure gradient,
∆

P , is used to drive a fl uid through a capillary or channel. For a capillary with a cylin-
drical cross-section, the volume fl ow, Q , is found to be

R 4 / 8

P (9)
where R is the radius of the capillary. The velocity profi le, i.e. the velocity, v ( r ), at dif-
ferent radial positions between the center ( r

Q ( ∆ V/t ) ( π η L ) ∆

0) and the wall ( r R ) are found to be

( R 2 r 2 )( v ( r ) ∆ P/ 8 η L ) (10)