Biomedical Engineering Reference
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The mechanical energy available for drop rebound and entirely converted into
potential energy at the maximum bouncing height, is given by the contributions
of elastic energy (only for the polymer solution drop) and surface energy, minus
the dissipation during the retraction phase,
E
D
,
r
.However,Fig.18showsthatvis-
coelastic drops exhibit higher rebounds than Newtonian drops, so that one can write
the following inequality:
E
(
P
)
E
(
P
)
E
(
P
)
D
,
r
>E
(
W
)
E
(
W
)
+
−
+
D
,
r
.
(8)
S
E
S
Introducing Eqs (21) and (22) into this inequality yields:
E
(
W
)
E
(
W
)
D
,
r
>E
(
P
)
E
(
P
)
D
,
r
.
(9)
Thus, one can conclude that the total energy dissipation during expansion and re-
traction in the viscoelastic drop is smaller than the total dissipation in the Newtonian
drop.
+
D
,
e
+
D
,
e
Appendix. Constitutive Models for Non-Newtonian Fluids
A1.1. Power-Law Fluids
The simplest type of non-Newtonian fluid behavior occurs when the viscosity co-
efficient is not a constant, but is a monomial function of the shear velocity gradient
(power-law, or Ostwald-De Waele fluid):
μ
γ
n
−
1
,
(A1)
where the consistency coefficient
K
and the power-law index
n
are empirical con-
stants. The power-law index is indicative of the shear-thinning (
n<
1) or shear-
thickening (
n>
1) behavior of the fluid, whereas for
n
=
K
˙
=
0 the Newtonian behavior
is retrieved. The consistency coefficient describes the fluid viscosity at low shear
rates, and coincides with the Newtonian viscosity for
n
=
0. Power-law fluids are
time-independent, i.e., the shear stress does not depend on the previous deformation
history. Physically, shear-thinning is usually explained by the breakdown of struc-
ture formed by interacting particles within the fluid, while shear-thickening is often
due to flow-induced jamming [64].
The Ostwald-De Waele equation implies that viscosity will change indefinitely
for any values of the shear rate; to account for a more realistic behavior, where
viscosity varies between minimum and a maximum value respectively at very low
and very high shear rates, a number of constitutive equations have been proposed,
such as the Cross model [65]:
μ
μ
∞
μ
0
−
μ
∞
=
−
1
(A2)
+
γ)
1
−
n
˙
1
(C
and the Carreau model [66]:
μ
μ
∞
μ
0
−
−
1
μ
∞
=
n)/a
,
(A3)
γ)
a
(
1
−
[
1
+
(C
˙
]
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