Biomedical Engineering Reference
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and
light
on bioconvection patterns in
Chlamydomonas reinhardtii
and con-
cluded that those factors had a significant effect on pattern formation. A
study of
galvanotaxis
was published by Itoh and Toida (2001) who tried to
control bioconvection of
Tetrahymena thermophila
by applying an electrical
field. Other authors have also analyzed the spatiotemporal characteristics of
TP
bioconvection patterns under altered gravity acceleration (Mogami et al.
2004). Two-dimensional bioconvective plumes of
Tetrahymena pyriformis
have
been recently reported from experiments by Nguyen-Quang et al. (2009) who
noted that, two-dimensional systems are more amenable to experiment and
simulation than three-dimensional systems as the reduction of dimension sig-
nificantly reduces the amount of data required to specify the flow. Hence,
experiments can straightforwardly determine an entire two-dimensional scalar
field, and two-dimensional calculations can be performed at Rayleigh numbers
much higher than those in three dimension (DeLuca et al. 1990). Nguyen-
Quang et al. (2009) therefore considered a very thin cavity that constrains the
fluid motion to a two-dimension vertical plane, making it possible to intro-
duce a number of simplifications in the Navier-Stokes equation that lead to
the so-called Hele-Shaw flow. The physical concept of this apparatus is well
described by Hele-Shaw (1898) and Bear (1972).
14.3.2 Hele-Shaw Apparatus and Darcy's Law
A Hele-Shaw apparatus consists of two rigid parallel transparent plates (glass
or Plexiglass), separated by a very thin spacer to create a two-dimensional
fluid medium. The setup of Hele-Shaw apparatus is shown in Figure 14.10 with
dimensions
H
L
, separated by a thin spacer
b
. The Hele-Shaw apparatus is
a well-known model of porous media where the law of Darcy is expressed as
follows:
×
K
µ
∇
V
=
−
P
(14.29)
The Hele-Shaw cell is equivalent to a “porous medium” having a perme-
ability
K
=
b
2
12
(14.30)
which is well known under the name of “
Permeability of the Hele-Shaw cell.
”
More details on the Hele-Shaw cell approach to gravitactic bioconvection can
be found in Nguyen-Quang (2006).
14.3.3 Geometrical and Physicobiological Parameters
It is instructive to distinguish two groups of parameters that determine
the development of bioconvection (Nguyen-Quang et al. 2009): one group
is related to the geometry of the Hele-Shaw cell and the other one to the
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