Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 268

This model also disregards detachment and deformation of the biomass as a
result of the shear stresses.

7.3.2 Model 2
Model 2 is based on the ideas that biomass is held together by attractive forces
mediated by biopolymer, and that attractive forces between biomass and soil
grains keep biomass attached to the solid surface. In Model 2, the excess
biomass is transferred to a neighboring fluid particle with M<M 0 , and the
behavior of the biomass (deformation, attachment, detachment, and splitting)
is modeled through a combination of hydrodynamic forces and a combination
of short-range repulsive and medium-range attractive pair-wise forces that act
between all pairs of particles including particles representing soil grains and
fluid particles with and without biomass. The short-range component of the
pair-wise force prevents particles from approaching each other too closely, and
the medium-range attractive component of the force represents the attractive
forces within the biomass and between biomass and soil grains. The exact
form of the particle-particle interactions is not critical to the success of the
simulations as long as the interactions satisfy the requirements described above
and have symmetric forms to conserve linear momentum.
In this work, we used the pair-wise interaction forces:

⎧
⎨

s ij cos 1 . 5 π
h

r i −

r j

| r i − r j | , | r i − r j |≤ h

F ij = (7.31)

| r i − r j |

⎩

0 , | r i − r j | >h

where s ij is the strength of the force that depends on the properties of mate-
rials and is different for interaction between different types of particles. The
interaction strength, s ij , between pairs of particles containing biomass was
set to be larger than s ij for the interaction between pairs of particles with
and without biomass. If the interaction strength, s ij , for fluid particles con-
taining biomass and solid particles is greater than s ij for interaction between
two particles containing biomass, the model produces biomass growth in the
form of continuous biofilm. Otherwise, patchy biofilm is preferentially formed.
In this work the strength, s ij , for interactions between fluid particles con-
taining biomass and solid particles was set to be greater than s ij for interac-
tions between two particles containing biomass. The interaction strength, s ij ,
between fluid particles with zero biomass and solid particles was set to be the
same as s ij for two fluid particles containing zero biomass. Since solid particles
are immobile, interactions between pairs of solid particles were not considered,
and s ij for the forces between pairs of solid particles were not specified.
The total force acting on every fluid particle with or without biomass is a
combination of hydrodynamic forces and particle interaction forces, and the
motion of fluid particles is controlled by the momentum conservation equation
m i d v i
dt

+
j

= F N−S
i

F ij (7.32)