Solvers

biofilmFoam

This solver allows the resolution of density-diffusion model of biofilm growth, as proposed by Eberl et al:

\[\frac{\partial c}{\partial t} = \nabla \cdot \left( d_1 \nabla c \right) - f(c,m)\]

\[\frac{\partial m}{\partial t} = \nabla \cdot \left( d_2(m) \nabla m \right) + g(c,m)\]

where \(c(t, {\bf x})\) is the nutrient concentration, \(m(t,{\bf x})\) is the biomass density, \(d_i\) (with \(i=1,2\)) are diffusion coefficients, \(f(c,m)\) is the nutrient consumption rate and \(g(c,m)\) is the biomass production rate.

The code is written in a dimensionless form:

\[C(t, {\bf x}) = c(t, {\bf x}) / c_0\]

\[M(t, {\bf x}) = m(t, {\bf x}) / m_{max}\]

with \(c_0\) the initial nutrient concentration and \(m_{max}\) the maximal biomass density.

biofilmPimpleFoam

This solver allows to couple the previous density-diffusion biofilm model to the surrounding hydrodynamics, using a Darcy-Brinkman approach (see Soulaine et al):

\[\frac{1}{\phi} \left( \frac{\partial \rho {\bf u}}{\partial t} + \nabla \cdot \left( \frac{\rho}{\phi} {\bf u} {\bf u} \right) \right) = - \nabla p + \frac{\mu}{\phi} \nabla^2 {\bf u} - \frac{\mu}{k} {\bf u}\]

\[\nabla \cdot {\bf u} = 0\]

\[\frac{\partial c}{\partial t} + {\bf u} \cdot \nabla c = \nabla \cdot \left( d_1 \nabla c \right) - f(c,m)\]

\[\frac{\partial m}{\partial t} = \nabla \cdot \left( d_2(m) \nabla m \right) + g(c,m)\]