Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 266

maximum growth rate; k d is the decay rate; and K A and K B are the half-
saturation constants for A and B. The advection and diffusion of electron
donors and acceptors is governed by the continuity and momentum conserva-
tion equations (NS equations) coupled with diffusion equations for A and B :

dA
dt

A
K A + A

B
K B + B

2 A

= D A ∇ − k s M (7.23)

and

dB
dt

A
K A + A

B
K B + B

2 B

= D B ∇ − k s M (7.24)

The NS, diffusion and biomass evolution equations were solved using
a Lagrangian particle method based on SPH. SPH uses a meshless dis-
cretization of the computational domain and an interpolation scheme,
A ( r )= i ( A i /n i ) W ( r − r i ,h ), allowing approximation of a continuous field
A ( r ) using the values of A at a set of discretization points. Here, W is the
bell-shaped SPH weighting function with compact support of scale h , r i is the
positions of discretization point i , A i = A ( r i ), n i = ρ i /m i is the particle num-
ber density, ρ i and m i are the density and the mass of the phase associated
with point i . Because each point possesses a mass and volume, it is natural to
think of discretization points as physical particles. The SPH approximation of
continuous fields allows the mass and momentum conservation equations to
be written in the form of a system of ordinary differential equations (ODEs)
(Tartakovsky et al. 2005c),

n i =
j

W ( r j − r i ,h ) (7.25)

and

m i d v i
dt

= F N−S
i

(7.26)

where

P j
n j

+ P i
n i

F N−S
i

= − ∇ i W ( r i − r j ,h )

j

+
j

4 µ i µ j
( µ i + µ j )

v j )
n i n j ( r i −

( v i −

r j ) 2 ( r i −

·∇ i W ( r i −

r j ,h )+ g (7.27)

r j )

In the SPH model fluid and solid (e.g., soil grains) are represented by
particles. In equations (7.25) and (7.27), j indicates summation over all
particles. Particles representing soil grains are frozen in space, their velocity is
set to zero, and they enter into the calculation of the densities of fluid particles
(Equation 7.25) and forces acting on the fluid particles (equation 7.7). The
force F N− i , defined by equation (7.27), is the hydrodynamic force acting on
the fluid particles (calculated from the NS equation), v is the fluid velocity
vector, P is the pressure, g is the gravitational acceleration vector, and µ is