Biomedical Engineering Reference
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t
= 32 min
t
= 44 min
t
= 34 min
t
= 46 min
t
= 36 min
t
= 48 min
t
= 38 min
t
= 50 min
t
= 40 min
t
= 52 min
t
= 42 min
t
= 54 min
10 mm
FIGURE 14.12
Evolution of unsteady convection patterns. (Nguyen-Quang, T., Nguyen, T.H.,
Guichard, F., Nicolau, A., Smatzari, G., LePalec, et al.,
Zoo. Sci
., 26, 2009,
doi:10.2108/zsj.26.54.)
initial filling concentration. Plumes illustrated in Figure 14.12 were obtained in
a Hele-Shaw cell of dimensions
H
0
.
075 cm, with a concentra-
tion of 3
.
08
×
10
5
cells/cm
3
after centrifugation, corresponding to a Rayleigh
number of approximately
Ra
∗
=52
.
7.
×
L
×
b
=7
.
6
×
8
×
14.3.4.4
Critical Threshold for the Transition
In the same Hele-Shaw apparatus of dimensions
H
0
.
075 cm,
we have observed the unsteady flow for a range of concentrations from
2
.
29
×
L
×
b
=7
.
6
×
8
×
10
5
cells/cm
3
,
corresponding to values of the Rayleigh
number
Ra*
varying from
39.2 to 41.9
, with a dimensionless swimming speed
of 228 and a diffusion coecient
D
c
=1
.
5
10
5
to 2
.
45
×
×
10
−
3
cm
2
/
sec. We estimate that
the critical Rayleigh number for the transition from steady state to unsteady
state is around 40
,
that is, four times higher than the first critical Rayleigh
number
Ra
* = 10.2 predicted by the linear stability analysis.
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