Spatial Pattern Formation of Motile Microorganisms: From Gravitactic Bioconvection to Protozoan Culture Dynamics - Porous Media: Applications in Biological Systems and Biotechnology - page 558

It may be expected that as the Rayleigh number is gradually increased
beyond a certain threshold, the convection patterns should change smoothly
from a steady to an unsteady state with a well-defined moving flow. However,
it is dicult to control the various parameters for this transition to occur.
We have instead obtained a rather cloudy pattern, as shown in Figure 14.12,
corresponding to a Rayleigh number Ra ∗ =52 . 7, that is, rather far from the
estimated critical value Ra c =40 .
From these experiments, the following remarks are in order:

1. When H is varied, the Peclet and the Rayleigh numbers also vary

proportionally. The variations of these two parameters could change
the flow regime such that with a same initial concentration, there
may establish a diffusion state in a Hele-Shaw cell with a height H 1 ;
and a convection regime in another cell with a height H 2 >H 1 .

2. The variation of the thickness b is related to the permeability K

of the porous medium by the formula K = b 2 / 12 so that when b
is increased, the Rayleigh number is increased accordingly and a
small variation of b may strongly influence the convection pattern
in a Hele-Shaw apparatus. It should be noted that b should not be
too big to violate the two-dimensional Hele-Shaw approximation,
and not too small to affect the motility of the microorganisms.

3. The number of plumes depends strongly on the length, L, of the

apparatus. For steady convection regime, we observed regular plume
patterns of 2.5-5.5 mm length and 2.0-2.6 mm width . The distance
between two plumes is 0.4-0.8 cm in concentration of 1 10 5 -3 ×

10 5 cells/cm 3 . These observations agree with those reported for
TP (Plesset et al. 1976; wavelength λ = 0.655 cm and average
culture concentration 2.7

×

10 5 cells/cm 3 ). The size and distance
between plumes also depend on the initial concentration, which may
lead to unsteady patterns beyond a certain value. In this unsteady
regime, the shape and size of the unsteady patterns are not well
defined.

×

4. Plesset et al. (1976) have underlined that a considerable amount

of time is required before a cell turns around and resumes its nor-
mal upward swimming in the case of high concentration. Another
considerable amount of time is also required to establish the con-
vection state. The total time required for these cells to turn around
and then resume its normal upward swimming and reach the con-
vection state in our experiments is around 30 minutes to 3 hours
for steady convection regime, depending on the geometry and initial
filling concentration (i.e., the Rayleigh and the Peclet numbers). In
the case of unsteady regime (Figure 14.12), the amount of time for
the cloudy patterns to appear is around 15-30 min, depending on
the Rayleigh and the Peclet numbers of the system.