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We study a traditional one-dimensional model of biofilm formation. It includes biofilm growth due to substrate consumption and biomass loss due to natural decay and detachment. The substrate diffuses from the surrounding aqueous phase into the biofilm. Its consumption by the microbes is described by the steady state diffusion-reaction model with Monod kinetics. Overall, this leads to an initial value problem for biofilm thickness that is coupled with a two-point boundary value problem for the growth limiting substrate. In a first study, we derive a simple, easy to use analytical approximation for the diffusive substrate flux across the water-biofilm interface. Starting point for this is the homotopy perturbation method, which, however, converges poorly for realistic biofilm parameters due to the particular non- linearity of the Monod kinetics. We overcome this difficulty by observing that the biofilm thickness is a small parameter, which allows us to apply a perturbation argument to the infinite series representation of the homotopy solution. The analytical approximation that we derive with this approach is verified quantitatively against a numerical solution of the full boundary value problem. A second aspect of the thesis the role of detachment for the long term behavior of the biofilm model in a narrow conduit, in particular with a view on bioclogging of the channel. We compare four heuristic different detachment rate functions, three of which have been used in the literature before and one is suggested here. Two of these detachment rates consider only material properties of the biofilm but are independent of the hydrodynamic shear forces, while the other two also take flow effects into account. Using analytical and numerical techniques, we find that persistence and the long term behavior of the system can depend critically on the flow regime as well as on the expression used to model biomass detachment. Finally, using a simple upscaling technique, we derive from our meso-scopic one-dimensional biofilm model a macroscopic model for a porous medium biofilm reactor under a convection dominated, laminar regime. The upscaled model is a stiff quasi-linear hyperbolic system. Using characteristic theory we describe the solutions qualitatively. In numerical simulations we investigate the role of meso-scopic biofilm detachment for macro-scopic system. We find that the reactor performance is largely independent of the particular choice of meso-scopic detachment function. We study a traditional one-dimensional model of biofilm formation. It includes biofilm growth due to substrate consumption and biomass loss due to natural decay and detachment. The substrate diffuses from the surrounding aqueous phase into the biofilm. Its consumption by the microbes is described by the steady state diffusion-reaction model with Monod kinetics. Overall, this leads to an initial value problem for biofilm thickness that is coupled with a two-point boundary value problem for the growth limiting substrate. |
