Fundamentals on Bionanotechnologies - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 47

l

U

U

l

z

x

h 2 w

w

y

Fig. 1.47 Microchannels of various shapes

The fluid inside microchannels is considered a continuous media, and the
velocity flow is described by the Navier-Stokes equation

Œ.@ v =@t/ C v r v Dr¢ C f Dr p C v C f ; (1.44)

where v is the fluid velocity, f denotes the external forces (gravitational, electric, or
magnetic) per unit volume exerted on the fluid, is the fluid density, O¢ is the fluid

stress tensor (viscosity contribution), is the shear viscosity, and p is the pressure.
If the inertial force, which is nonlinear, is small (situation encountered in almost
all microfluidic devices), the Navier-Stokes equation becomes the Stoke equation

.@ v =@t/ Dr¢ C f Dr p C v C f : (1.45)

In both equations, the mass conservation requirement imposes that

@=@t Cr . v / D 0 (1.46)

holds. To decide if the inertial forces are important in the steady state condition

@ v =@t D 0; (1.47)

we must calculate the Reynolds number, defined as the ratio between the inertial
force and the viscous force

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D UL = D UL =;

. v r v /
v

Re D (1.48)

where U is a characteristic flow velocity scale, L is a typical length scale, and
D = is the viscous coefficient.
Relation ( 1.48 ) tells us that the Reynolds number decreases as the dimensions of
the systems decrease. If the fluid velocity is in the range of cm s 1 and L D 10m,
we obtain Re Š 10 1 , so that the convective (inertial) force is negligible compared
to the viscous force. No convection means that there is no turbulence in microfluidic
systems. In microfluidic devices, the flow is linear or laminar, is described by the