Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 261

this approach is that it eliminates the problems associated with integration
of stochastic differential equations, and the temperature, measured via the
particle kinetic energy, deviates less from the temperature used in the DPD
algorithm (Nikunen et al. 2003).
In the current DPD biofilm model, the solution of the NS equations (7.10-
7.11) is approximated by the low Mach number flow of a slightly compressible
fluid represented by the DPD particles. The velocities and positions of the
DPD particles are found from equation (7.14) using a modified velocity Ver-
let algorithm to integrate the equation of motion (Groot and Warren 1997).
The no-slip boundary condition on the interface between the liquid and sub-
stratum/biofilm phases is implicitly implemented through the particle-particle
interactions, which are strong enough to prevent the penetration of the liquid
particles into the region occupied by the biofilm particles. This interaction can
also be tuned to allow the liquid to penetrate into the biofilm phase, which
is important for diffusion of the nutrient substrate into the biofilm phase. In
practice, the fluid velocity field will penetrate into the biofilm and there will
be slip at the fluid-solid interface. However, both of these effects are very
small (the permeability of biofilm is very small and the slip length is typically
on the order of 1 nm).
The mass of substrate and biomass carried by DPD particle i is specified
by the substrate concentration, c s,i , and the biomass density, c b,i , associated
with particle i . The changes of substrate concentration in liquid and biofilm
DPD particles is given by the advection-diffusion equation (7.12). The DPD
representation of the advection-diffusion equation is similar to the representa-
tion of the heat-conduction equation (Avalos and Mackie 1997; Espanol 1997;
Ripoll and Espanol 1998), and it has the form

=
j

dc s,i
dt

λ ij W R ( r ij )( c s,j −

c s,i )

= i

ζ c 2 α − 1 λ ij W R ( r ij ) c s,i c s,j ∆ t

+
j

k 4 c b,i c s,i
k 2 + c s,i

−

(7.18)

= i

where λ ij is the interparticle diffusion constant between particles i and j ,
which can be related to the continuum molecular diffusion coecient, and
∆ t is the time step used in the DPD simulations. In equation (7.18), α is a
material constant representing the magnitude of the concentration fluctuation,
ζ c is a random variable of the same type as ζ in equation (7.17), but it is
uncorrelated with ζ . The term ∆ t − 1/2 appears in equation (7.18) because the
average of a random force acting over the time step of length ∆ t is proportional
to ∆ t − 1/2 (Groot and Warren 1997). The last term in equation (7.18) is a
particle formulation of the rate of substrate consumption.
Because of the soft particle-particle interactions used in DPD models, the
DPD particles have a relatively large intrinsic self-diffusion coecient (the
cage effect that makes the momentum diffusion coecient much larger than
the molecular diffusion coecient in liquids is much weaker in DPD fluids) and