Biomedical Engineering Reference
In-Depth Information
η
η
i
η
m
η
i
η
η
=0.005Pas=
= 0.022 Pa s
= 0.005 Pa s =
=0.022Pas
o
o
m
80
40
η
i
η
==0
η
m
=0
35
all viscosities
60
30
25
40
20
average angle
15
η
η
20
10
i
m
=0
==0
η
amplitude
all viscosities
Tran-Son-Tay et al. (1984)
Fischer (2007)
5
0
0
0
50
100
150
200
0
50
100
150
200
γ
γ
-1
-1
(s )
(s )
Fig. 10.10.
Tumbling and tank-treading frequency (left) of a RBC in shear flow and swinging
average angle and amplitude (right) for different cases: 1) η
o
=
5
×
10
−
3
Pa
·
s
, η
i
=
η
m
=
0 (circles);
2) η
o
=
η
i
=
5
×
10
−
3
Pa
·
s
, η
m
=
0 (squares); 3) η
o
=
η
i
=
5
×
10
−
3
Pa
·
s
, η
m
=
22
×
10
−
3
Pa
·
s
(triangles) (from [51])
haviour is related to existence of a RBC minimum energy state shown in the exper-
iments by Fischer [15], where a RBC relaxed to its original state marked by several
attached microbeads after some time of tank-treading motion. Hence, the RBC has
to exceed a certain energy barrier in order to transit into a tank-treading motion in
shear flow.
Theoretical predictions [14, 33] attempt to capture RBC dynamics in shear flow
depending on the shear rate and the viscosity contrast defined as
λ
=(
η
i
+
η
m
)
/
η
o
.
According to the theories, for a small
3 a RBC tumbles at low shear rates and
tank-treads at high shear rates. Near the tumbling-to-tank-treading transition there
exists a narrow intermittent region where theories predict an instability such that
RBC tumbling can be followed by tank-treading and vise versa. However, in case of
a large viscosity contrast (
λ
<
3) the theories predict a well-defined tumbling regime
followed by an intermittent region, while stable tank-treading may not be present. In
addition, the tank-treading state is also characterized by RBC swinging around the
tank-treading axes with certain frequency and amplitude.
A simulated RBC is suspended into a solvent placed between two parallel walls
moving with constant velocities in opposite directions. Fig. 10.10 (left) shows tum-
bling and tank-treading frequencies with respect to shear rates in comparison with
experiments [63, 64]. Comparison of the simulated dynamics with experiments
showed that a purely elastic RBC with or without inner solvent (circles and squares)
results in an overprediction of the tank-treading frequencies, because the membrane
assumes no viscous dissipation. Addition of the membrane viscosity (triangles) re-
duces the values of the tank-treading frequencies and provides a good agreement
with experiments for the membrane viscosity
λ
>
10
−
3
Pa
=
×
·
s
. Note that for
all cases a finite intermittent region is observed and it becomes wider for a non-
zero membrane viscosity. This result is consistent with the experiments, but it dis-
agrees with the theoretical predictions. Similar results for the intermittent region
were reported in simulations of viscoelastic vesicles [65]. Moreover, an increase in
the internal fluid or membrane viscosities results in a shift of the tumbling-to-tank-
η
22
m
