Biomedical Engineering Reference
In-Depth Information
The geometrical factors
b, H,
and
L
play an important role in the devel-
opment of bioconvection patterns as the height
H
of the apparatus is directly
proportional to the Peclet number and the Rayleigh number, while
b
repre-
sents the apparent permeability of the Hele-Shaw apparatus.
We should keep in mind that although there exists a certain mathematical
analogy between Hele-Shaw and porous medium flows via the permeability
term and general form following the Darcy's law, Hele-Shaw model can not
be considered as an “exact” model for porous medium. The Hele-Shaw model
can just reproduce certain behaviors of the fluid flow within two-dimensional
porous medium. The results obtained from Hele-Shaw model might reflect
qualitatively the nature of the pattern and behavior in the equivalent porous
medium, and cannot evaluate exactly bioconvection patterning in real porous
medium. The reasons are as follows:
•
The flow in Hele-Shaw model is governed by an equation similar to the
Darcy's law only for two-dimensional case.
•
Physically, the real diffusion coecients of the considered substance (herein
the gravitactic microorganisms) are not the same for both media. In the real
porous medium, microorganisms have to diffuse through different physical
phases (real solid matrix, liquid, and pores), while in the Hele-Shaw model
they disperse with a fluid medium diffusive coecient.
•
The thickness
b
in the formula (14.30) is considered as “analogous” to the
pore size in porous medium, and should be larger than cell size, allowing
microorganisms to move easily through the medium. Tortuosity, which is an
important factor influencing significantly the convection and diffusion terms
in real porous medium, does not, hence, exist in a Hele-Shaw model and
microorganisms will interact with the inner surfaces of Hele-Shaw apparatus,
not with solid matrix in a porous medium.
14.3.4 Key Results of Experimental Study
Bioconvection patterns of
T. pyriformis
are realized in various Hele-Shaw
setups and recorded from a horizontal view. Regarding the materials and
methods for experimental issues, we refer to Nguyen-Quang et al. (2009).
From these experiments, we have obtained three different regimes of pat-
tern formation depending on experimental conditions.
•
The diffusion regime
•
The stationary convection regime
•
The unsteady convection regime
14.3.4.1
The Diffusion Regime
A
diffusion state
as predicted by the mathematical model is shown in
Figure 14.11(b). Beginning at
t
= 0, this state is established after
30 min to
2h
, with cell concentration below the critical value for the onset of convection.
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