Application of Porous Media Theories in Marine Biological Modeling - Porous Media: Applications in Biological Systems and Biotechnology - page 368

transmission electron microscopy images (Leppard et al. 2004) clearly reveal
that aggregates are rather porous organisms. Hence, implementing porous
media theories can enhance the current quantitative estimations of the nutri-
ent exchange mediated by the aggregates from one side, and provide an
improved picture of biological consequences. Certainly there exist more bio-
geochemical problems, which are treated by the means of porous media theo-
ries, however, we settle for the examples mentioned to not explode the given
framework.
This manuscript is organized as follows. First, a brief description of the
mathematical model is brought. In the next sections, some recent examples
are given with application in the field of marine microbiology. Finally, some
concluding remarks and examples of future applications of porous media in
marine microbiology and biogeochemistry have been mentioned.

10.2 Description of the Mathematical Model
For the numerical solution of the porous media equations different tech-
niques such as finite difference method, finite element method, and finite vol-
ume method have been suggested. However, lattice Boltzmann model (LBM)
has proved to be a promising technique to be applied in porous domains
(Guo et al. 2002; Jue 2003; Wu et al. 2005). Compared to other numerical
methods, LBM has the advantage of being most suitable for parallel algo-
rithms. Besides, LBM is known to have a simple structure, which makes it
most attractive for program coding. Being based on lattices, LBM has the
ability of tackling complex meshes, dealing with multiphase, multicomponent
fluids or domains (Succi 2001), which frequently occur in the field of marine
biogeochemistry. For the sake of completeness, only a brief description of the
LBM is brought here. The interested reader may refer to Succi (2001) for
further details.

10.2.1 BGK Model
The Boltzmann equation describes a fluid from a microscopic viewpoint as an
ensemble of discrete particles following the distribution f = f ( u , x ,t ), where
f is the probability of finding a particle with velocity (or momentum) in the
range ( u , u + d u ) and position in the range ( x , x + d x ) at time t . Then the
discretized Boltzmann equation in D2Q9 lattice is expressed as follows:

f i ( x + e i δ t ,t + δ t ) − f i ( x ,t )= − Ω i ( f ( x ,t )) (10.1)

where the subscript i is the direction of the velocity. Furthermore, δt is the
time increment and Ω i denotes the collision operator. The discrete velocities