Spatial Pattern Formation of Motile Microorganisms: From Gravitactic Bioconvection to Protozoan Culture Dynamics - Porous Media: Applications in Biological Systems and Biotechnology - page 554

Y

b

H

Z

g

X

L

FIGURE 14.10
Conceptual model of the Hele-Shaw apparatus.

physicobiological characteristics of the system. The Rayleigh number is a com-
bination of these two groups and is the parameter governing the development
of the bioconvection patterns. From the linear stability analysis results, we
arrive at two following parameters:

1. The geometrical parameter

C 1 = b 2 H 2 when Pe

≤ 1

C 1 = b 2 H when Pe> 1

(14.31)

2. The physicobiological parameter

C 2 = 144 D c v
gnβV c

when Pe ≤ 1

C 2 = 122 . 4 D c v
gnβ

when Pe> 1 (14.32)

The condition for the onset of bioconvection may then expressed as

C 1 ≥ C 2 (14.33)

With a given species of microorganisms, C 2 can be a fixed constant. The
geometry of the Hele-Shaw apparatus is then determined according to the
criterion (14.33). On the other hand, it follows from (14.33) that a minimum
concentration, n , is required for convection to develop in a fixed-geometry
Hele-Shaw cell. This minimum concentration is called critical concentration
for the onset of convection.