Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 267

the dynamic viscosity of the solution. In general, the mass m i associated with
fluid particle i and the local viscosity µ i depend on the fluid composition,
which may change with time. In this work it was assumed that the biomass
density is equal to the fluid density and that the viscosity of fluid containing
biomass is ten times greater than the viscosity of fluid containing no biomass.
The equation of state, P i =( P eq /n eq ) n i was used to close the system
of equations (7.25-7.27). In the equation of state n eq is the average particle
density and P eq is the fluid pressure in the system at dynamic equilibrium.
Following Tartakovsky et al. (2007a) the system of diffusion/reaction equa-
tions (7.22-7.24) can be cast in the form of a system of ODEs:

dA i
dt

1
m i

D A ( m i n i + m j n j )( A i − A j )

( r i −

= r j )

r j ) 2

n i n j ( r i −

j ∈ fluid

A i
K A + A i

B i
K B + B i

·∇ i W ( r i − r j ,h ) − k s M (7.28)

m i
j∈ fluid

dB i
dt

1 D B ( m i n i + m j n j )( B i − B j )

= ( r i − r j )

n i n j ( r i − r j ) 2

A i
K A + A i

B i
K B + B i

·∇ i W ( r i − r j ,h ) − k s M (7.29)

and

dM i
dt

A i
K A + A i

B i
K B + B i −

= Yk s M i k d M i (7.30)

where ∇ i denotes the gradient with respect to the position vector r i and the

symbol j∈ fluid indicates summation over all the fluid particles. Excluding
solid particles from the summations enforces no-diffusion no-reaction condi-
tions at solid-fluid interfaces. Initially biomass was randomly distributed on
the surfaces of the porous medium by randomly selecting fluid particles near
the solid particles and assigning a biomass concentration M 0 to these parti-
cles. Zero biomass concentration was assigned to the rest of the fluid particles.
Evolution of the biomass was calculated according to equation (7.30). Once
the concentration of the biomass at any particle i exceeded M 0 , the excess
biomass was moved to the nearest fluid particle with a biomass concentration
less than M 0 . Biomass growth is also influenced by the forces exerted on the
biomass by the flowing fluid. In this chapter we investigated two models to
account for the effect of fluid shear stress on the spreading of the biofilm.

7.3.1 Model 1
Model 1 is similar to the cellular automaton model used by Knutson et al.
(2005). Particles with nonzero biomass concentration are assumed to be immo-
bile, and excess biomass is allowed to move only to the nearest particles with
a shear stress, τ = µ ( ∂v x /∂y + ∂v y /∂x ), less than the critical stress τ cr .Ifno
particles satisfy the above requirements, excess biomass is assumed to be lost.