Biomedical Engineering Reference
In-Depth Information
steady state pattern for a given Rayleigh number and for a given size medium
seems to obey the principle of minimum total potential energy.
In contrast to the continuum model, Hopkins and Fauci presented a
“discrete” mathematical model and a numerical method of solution for
the fluid/microorganisms/nutrient system (Hopkins and Fauci 2002). Their
approach was based on the method of Childress and Peyret (1976) in using
a simplified representation of microbes as point particles, performing calcula-
tions with a large enough number of particles to model realistic cell concen-
trations and macroscopic fluid effects.
The aforementioned works were devoted exclusively to bioconvection in a
fluid medium, while it has been noted that “
the fluid may be dynamically con-
strained so that convection is heavily damped. This constraint may be imposed
by a porous medium..., which is located throughout the fluid or only near
the top surface. The damping of fluid convection essentially eliminates down-
ward re-mixing of the cells
” (Kessler 1986b). In fact, although bioconvection
in porous media is frequently encountered in nature as well as in industrial
applications, very few studies have been done to explicitly address bioconvec-
tive patterns in porous media.
According to Hill and Pedley (2005), a rational extension of the model of
bioconvection from a fluid to a porous medium should take into account the
fact that the local flow (through the pores) may cause the cells to tumble
and strongly affect their ability to reorient if the pores are too small. So far,
bioconvection in porous media has been described by the
continuum model
of
Pedley et al. (1988), with the fluid flow being determined by Darcy equation,
instead of the Navier-Stokes equation.
In 2001, Kuznetsov and Jiang presented a study of
gravitatic
bioconvection
in a porous medium. The Darcy equation for the fluid flow and the diffusion-
convection equation for the concentration of microorganisms were numerically
solved to obtain the flow and concentration fields in a square cavity in term
of the permeability of the porous medium. They concluded that there exists
a critical value of permeability for the onset of convection and “
after numer-
ous computations, it was determined that the critical value of permeability is
approximately 4 × 10
−
7
m
2
. If permeability is smaller than this value, then no
convection develops. This, in turn, causes the cells to accumulate in the top
layer and stay there
.” (Kuznetsov and Jiang 2001).
Later, Kuznetsov and Avramenko (2002) obtained a criterion for stabil-
ityofa
gyrotactic
suspension in a porous medium using linear perturbation
theory. This criterion gives the critical permeability in terms of the bacterial
cell eccentricity, average swimming velocity, and other parameters. The onset
of bioconvection has also been examined under the effect of cell deposition
and declogging by Kuznetsov and Jiang (2003).
Nield et al. (2004) studied the onset of
gyrotactic
bioconvection in a hor-
izontal saturated porous medium. The Darcy flow model was employed, and
it was assumed that the
Peclet
number (dimensionless cell velocity) is not
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