Spatial Pattern Formation of Motile Microorganisms: From Gravitactic Bioconvection to Protozoan Culture Dynamics - Porous Media: Applications in Biological Systems and Biotechnology - page 550

TABLE 14.1
Critical Values of Rayleigh Number for a Porous Cavity with F = L cr
Length Corresponding to Swimming Speed of Microorganisms

Swimming Speed or Critical Values of

Peclet Number Pe Rayleigh Number

By Linear Stability By Simulation

0.1 12.188 12.22

1. 14.823 14.83

5. 51.028 51.437

10. 103.374 100

(a) (c)

(b) (d)

FIGURE 14.6
Streamlines ψ and isolines of concentration for the case of F =4, Ra = 100,
n cell = 1 (a) Pe = 0.001, (b) Pe = 0.1, (c) Pe = 1, and (d) F = L cr ,
n cell =1, Pe =5 ,Ra= 52 , close to Ra cr = 51.

the initial conditions. This is also predicted by the linear stability analysis.
Nevertheless, the growth rate and the convergence time for simulations are
strongly affected by the initial conditions.
For Pe smaller than 1, the supercritical convection patterns at steady
state are very similar to the ones obtained from thermoconvection (Rayleigh-
Benard convection) for the case of heating from below by a constant heat
flux.
For F>L cr and Pe 2, the morphology of steady state bioconvective

cells depends strongly not only on the value of Rayleigh, but also on the initial
condition and the aspect ratio of the cavity. When the Rayleigh number is close
to the threshold, the steady state morphology predicted by linear stability

≥