Spatial Pattern Formation of Motile Microorganisms: From Gravitactic Bioconvection to Protozoan Culture Dynamics - Porous Media: Applications in Biological Systems and Biotechnology - page 541

greater than unity. Critical conditions were obtained for various values of
Peclet number, gyrotaxis number, and cell eccentricity.
Recently, the onset and development of bioconvection of gravitactic
microorganisms within a confined fluid as well as a porous medium have been
studied by Nguyen-Quang and coworkers (Bahloul et al. 2005; Nguyen-Quang
et al. 2005; Nguyen-Quang 2006). Both linear stability analysis and numerical
simulations based on the Navier-Stokes/Darcy and concentration conserva-
tion equations were performed to determine the onset and development of fluid
flow and cell concentration in terms of the governing parameters ( Rayleigh,
Peclet, and Schmidt numbers). Attention was focused on effects of the swim-
ming speed of the microorganisms and the aspect ratio of the cavity. For a
more comprehensive bibliography, we would refer to the first excellent review
of literature from 1911 to 1992 by Pedley and Kessler (1992b) and the second
review by Hill and Pedley (2005) for the following years.
New developments of different types of bioconvection (oxytaxis, gyrotaxis,
etc.) in porous media have been presented in two review chapters by Kuznetsov
(2005, 2008).

14.2 Onset and Evolution of Gravitactic Bioconvection:

Linear Stability Analysis and Numerical Simulation

14.2.1 Mathematical Formulation of Gravitactic

Bioconvection in a Porous Medium

14.2.1.1 Description and Formulation of the Problem

Fluid flows and transport phenomena in porous media are frequently encoun-
tered in nature and industries (groundwater, building materials, composite
materials, etc.) as well as in biosystems (human organs, aquifer ecosystems,
etc.). An exact description of the flow dynamics and transport properties
within the pore length scale is beyond the capabilities of both theoretical and
experimental methods. Henry Darcy (1856) who successfully established an
empirical relation between the filtration velocity and the pressure gradient in
a porous medium has accomplished the first study of flow in porous media
in 1856. The elegantly simple statement, that the filtration velocity is propor-
tional to the pressure gradient , is now celebrated as the Darcy's law.
A rational derivation and extension of Darcy's law (which thereafter will
be called Darcy equation) from the well-established Navier-Stokes equation
was the subject of continuing research during the past 50 years. The method
of volume averaging has provided a rational foundation for the derivation of
spatially averaged equations for the fluid flow and heat and mass transfer
in porous media. This approach consists essentially of a spatial smoothing