Biomedical Engineering Reference
In-Depth Information
Viscoelastic fluids are characterized by a relaxation time. This time can be seen
as the inverse of the critical shear rate, that is, the shear threshold that starts to
change the value of the viscosity
τ
= 1
cr
(1.14)
or as a time for the polymer chains to change its configuration—from stretched to coiled,
for instance. Even if this time may seem short (0.05 second for alginate polymers), it has
important consequences for the liquid behavior as will be shown in Chapter 2.
The Weissenberg and Deborah numbers are characteristic of non-Newtonian
flows. Depending on the authors, the definition of these two dimensionless num-
bers may differ somewhat. Both numbers are the ratio of the relaxation time of the
polymeric liquid and a specific process time or a characteristic time frame. If we
follow Bird, Armstrong, and Hassager [13], the Weissenberg number is the equiva-
lent of the Reynolds number for viscoelastic fluids. It is defined as the ratio of the
relaxation time
τ
—characterizing the elasticity of the fluid—to the convective time
τ
C
, defined in (1.5) as
τ
C
=
R
/
V
Wi
=
τ
V R
(1.15)
In a shear or elongational flow, the Weissenberg number can be defined as
Wi
=
τ γ
�
(1.16)
�
Wi
=
τ ε
A large Weissenberg number indicates a strong viscoelastic behavior. Figure 1.5
shows the definition of the Weissenberg number in the different flow configurations
of a flow focusing device (FFD).
The Deborah number—which name was given by one of the founder of rheol-
ogy, Markus Reiner—is usually defined by the ratio of the relaxation time and the
Rayleigh time defined by (1.7) as
τ
3
/
=
ρ γ
R
R
Figure1.5
Weissenberg numbers in different flow configurations: (a) in a convergent, (b) in a free
falling jet [], and (c) in a low focusing device [7].
