Lagrangian Particle Methods for Biological Systems - Porous Media: Applications in Biological Systems and Biotechnology - page 253

et al. 2007a,b). However, the high computational cost of pore-scale models
does not allow them to completely replace Darcy-scale models in the entire
computational domain. When mineral precipitation and/or biomass growth
is highly localized, hybrid/multiphysics algorithms can provide an ecient
computational tool for combining micro- and macroscale descriptions of phys-
ical phenomena. In the following sections we present two Lagrangian particle
models, dissipative particle dynamics (DPD) and smoothed particle hydrody-
namics (SPH), and their applications to pore-scale reactive transport, biomass
growth, and mineral precipitation.
Dissipative particle dynamics, a stochastic Lagrangian approach intro-
duced by Hoogerbrugge and Koelman in 1992 is based on the idea that parti-
cles can be used to represent clusters of atoms or molecules instead of single
atoms or molecule to provide a simple and robust way of coarse graining the
molecular dynamics (MDs) of dense fluids and soft condensed matter sys-
tems. Because of the internal degrees of freedom associated with individual
DPD particles, the DPD particle-particle interactions include dissipative and
fluctuating interactions that are related by the fluctuation-dissipation theo-
rem (Kubo 1966; Espanol and Warren 1995) in addition to the conservative
particle-particle interactions, and the dissipative and fluctuating interactions
function as a thermostat for the model. The grouping of atoms or molecules
into a single DPD particle (coarse graining) leads to averaged effective con-
servative interaction potentials (soft repulsive-only potentials in the standard
DPD model) between the DPD particles. Consequently, the computational
cost is substantially lowered due to the soft potentials as well as the coarse
graining. The computational advantage of DPD over MD is about 1,000 N 5/ m ,
where N m is the number of atoms represented by a single DPD particle (Pivkin
and Karniadakis 2006). This makes DPD an effective mesoscale particle simu-
lation technique on length and timescales, that are larger than those accessible
to fully atomistic MD simulations.
Smoothed particle hydrodynamics is a fully Lagrangian particle method
(Monaghan 2005). In SPH models, fluid phases are represented by a set of
particles, and the particle positions serve as interpolation points for solving
partial differential equations, such as the Navier-Stokes (NS) and advection-
diffusion equations. The particles move with velocities given by the NS equa-
tions and the composition (concentrations) of the particles change according to
the advection-diffusion equation (diffusion equation in the Lagrangian coordi-
nate system). The size (or number density) of the particles sets the numerical
resolution of SPH simulations, and the time step is limited by the standard
Courant-Friedrichs-Lewy (CFL) conditions. Consequently, SPH models can
be regarded as continuum models with an unstructured grid, and they operate
on much larger time and length scales than mesoscale (between the atomistic
and continuum scale) DPD models. SPH can be used to model biogeochemical
systems that can be adequately described by conservation laws in the form of
deterministic partial differential equations.