Biomedical Engineering Reference
In-Depth Information
FIGURE 7.1
A schematic representation of the DPD biofilm model: Open circles represent
liquid flowing through a channel, filled circles represent the solid substratum,
and partially filled circles represent the biomass.
coecients,
D
s
(Eberl et al. 2001). Special attention to
D
s
is needed in the
biofilm phase to account for the fact that biofilm spreading is only signifi-
cant when the biomass density,
C
b
, is close to the maximum biofilm density,
C
bm
. Grid-based cellular automata models have also been developed to solve
the coupled differential equations presented earlier (Picioreanu et al. 1998;
Picioreanu et al. 2000; Picioreanu et al. 2001). The hydrodynamic interac-
tions between flow and biofilm structure can be further considered by using
the finite-element method to solve the stress and strain in the biofilm structure
at each time step (Picioreanu et al. 2001).
In the DPD model used in this work, three types of DPD particles are used
to represent the liquid, biomass, and solid substratum phases (Figure 7.1).
In standard DPD simulations (Hoogerbrugge and Koelman 1992), a fluid
is represented by an ensemble of particles that move because of the combined
effects of conservative (nondissipative),
f
C
,
dissipative,
f
D
,
fluctuating (ran-
dom)
f
R
,
and external,
f
ext
,
forces, and the equation of motion is
+
j
=
i
dm
i
v
i
/dt
=
f
int
+
f
ext
i
=
f
i
+
f
i
+
f
i
+
f
ext
=
f
ext
i
(
f
ij
+
f
ij
+
f
ij
)
i
i
(7.14)
where
v
i
is the velocity of particle
i
and
m
i
is its mass. In models for
single-phase fluid flow, the conservative forces between particles are usually
given by a simple and purely repulsive form such as
f
ij
=
S
(1
−
r
ij
/r
0
)
r
ij
for
and
f
ij
= 0 for
r
ij
≥
r
ij
=
r
0
, where
S
is the strength of
the particle-particle interaction,
r
0
is the cut-off range of the particle-particle
interactions, and
r
ij
is the unit vector pointing from particle
j
to particle
i
|
r
ij
|
=
|
r
i
−
r
j
|
<r
0
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