Biomedical Engineering Reference
In-Depth Information
ρ
, and the fluid viscosity
m
. Hence,
n
= 4. The units intervening in the problem
are to the number of 3: kilogram, meter, and second. The Buckingham theorem
then states that there is only 4 − 3 = 1 dimensionless parameter characterizing the
problem. This parameter is the well-known Reynolds number (1.9).
1.4
ScalingNumbersandCharacteristicScales
1.4.1
Micro-toNanoscales
The Knudsen number defines the transition between micro- and nanoscales [2]. This
transition is extremely important; it defines the lower limit where the continuum
hypothesis can be used. The Knudsen number is defined as
Kn L
(1.4)
where
L
is a representative physical length scale and
l
the mean free path.
Kn
is
small at the microscale, and is larger than 1 at the nanoscale. For a gas at normal
conditions,
l
is of the order of 1
m
m, whereas it is smaller for a liquid (5-10 nm).
=
λ
1.4.2
HydrodynamicCharacteristicTimes
In microhydrodynamics, four characteristic times are usually defined. These times
will be used in the following sections to establish some dimensionless groups:
1. The convective (or viscous) time scales the time for a perturbation to propa-
gate in the liquid
τ
C
R V
(1.5)
where
R
is a dimension and
V
is the velocity of the liquid. Depending on the
flow configuration (shear or elongational), the convective time may also be
written as
τ
=
�
1
γ ε
or
1
�
C
where
γ
� �
,
are, respectively, a shear rate and an elongation rate.
2. The diffusional time is the time taken by a perturbation to diffuse in the
liquid
2
τ
D
R
=
ν
(1.6)
where
ν
=
m
/
ρ
is the kinematic viscosity (units m
2
/s).
Figure1.1
Schematic of microscale (
Kn
<<) and nanoscale (
Kn
>) chambers.
