Biomedical Engineering Reference
In-Depth Information
3. The Rayleigh time is a time scale of the perturbation of an interface under
the action of inertia and surface tension [3-5]
=
3
τ
ρ γ
R
R
(1.7)
4. The capillary time—sometimes called the Tomotika time [3-5]—is the time
taken by a perturbed interface to regain its shape against the action of viscosity
η
R
η
0
0
τ
=
=
T
γ γ
R
(1.8)
1.4.3
NewtonianFluids
The terms of Newtonian and non-Newtonian fluids will be defined in Chapter 2.
Let us mention here that usual liquids like water belong to the Newtonian category
and polymeric fluids belong to the non-Newtonian category.
The Reynolds number characterizes the relative importance of inertial and vis-
cous forces. It is usually written under the form
Re
V R
(1.9)
where
V
is the average fluid velocity and
R
is a length characteristic of the geometry.
Written under the form
=
ν
2
ρ
µ
V
V R
Re
=
/
the Reynolds number is the ratio of a dynamic pressure—linked to inertia—to a
shearing stress—linked to viscous forces. In microfluidics, the Reynolds number is
generally small, corresponding to a laminar flow regime. Very few microfluidic sys-
tems use turbulent flows. It is recalled here that, in enclosures, a number of Reyn-
olds of 2,000 is required to reach the transition to turbulence [6]. A more subtle
subclassification can be made for the laminar regime. A very low Reynolds number
(less than 0.5) indicates a creeping flow, for which the Stokes approximation is valid
(see Chapter 2); this is the general case. However, recently, with the development
of microsystems for cells handling, larger velocities are often used corresponding to
Reynolds numbers in the range from 1 to 20. In these two cases, the physical phe-
nomena for the convective transport are different, as will be shown in Chapter 2.
Figure 1.2 schematizes the definition of the Reynolds number for some geometrical
configurations.
The capillary number is very important in two-phase microfluidics. It compares
viscous/elongational forces to surface tension forces. The capillary number can take
different forms, depending on the physics of the problem. It can be built using the
average velocity, the shear rate, or the elongational rate:
Ca
=
η γ
V
�
Ca
=
ηγ γ
R
(1.10)
�
Ca
=
ηε γ
R
