Biomedical Engineering Reference
In-Depth Information
FIGURE 14.1
Tetrahymena pyriformis
protozoan population. (Picture taken by T. Nguyen-
Quang at the University of Minho, Braga, Portugal.)
then derived the equations for the onset of bioconvection in a fluid medium.
The basis of their analysis is the equilibrium solution which is the balance
between the upward swimming and downward diffusion in quiescent water.
They obtained analytically the preferred wavenumber and growth rates of
bioconvection, which qualitatively agree with experiments on both
TP
and
algae. As noted by Pedley and Kessler, the model of Childress et al. was the
first self-consistent
continuum model
of bioconvection (Pedley and Kessler
1992b). Subsequently, many researchers have extended the Childress et al.
theory to suspensions of algae with gyrotactic behavior in fluid media (Hill
et al. 1989; Bees and Hill 1998; Ghorai and Hill 2000).
One of the first simulations was performed by Childress and Peyret who,
instead of using the continuum cell conservation equation, treated the cells as
individual moving points each of which exerts a force on the fluid and move
relative to the fluid by a superposition of upswimming and random walk. In
this approach, each microorganism acts as a point mass in the Navier-Stokes
equation. Inversely, the fluid flow affects the swimming of these organisms as
each one is convected by the fluid (Childress and Peyret 1976).
In 1986, Fujita and Watanabe presented a numerical study of bioconvec-
tion based on the equations derived by Childress et al. (1975). They found
a transition from steady flow to periodic and finally to nonperiodic motions
through a sequence of period doubling bifurcations as the Rayleigh number
increased. This route to chaos is quite similar to that of Benard convection
(Fujita and Watanabe 1986). A subsequent study by Harashima, Watanabe
and Fujishiro was focused on the evolution of convection patterns in gravitac-
tic suspensions (Harashima et al. 1988). Following the Childress et al. theory,
they numerically solved the Navier-Stokes and cell conservation equations for
upswimmers such as
TP
. They found that the evolution of the system pro-
ceeds in the direction to intensify the downward advection of microorganisms
and to reduce the total potential energy of the system. In other words, the
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