PhD Math Defence: Numerical solution of a single-species biofilm model on non-orthogonal grids

CANDIDATE:  MD AFSAR ALI

ABSTRACT:

Biofilms are collections of microbes attached to either a smooth or a rough surface. Within biofilms, bacteria interact with each other using a signalling communication method known as quorum sensing, which enables bacteria to execute gene-expression.

Our research focuses on studying a density-dependent, diffusion-reaction-based single-species biofilm model, in which the biomass growth equation exhibits two non-linear degeneracy effects: (i) porous medium type degeneracy as the local biomass density vanishes, (ii) a super-diffusion singularity as the local biomass density approaches its a priori known maximum value. Previously, a semi-implicit numerical method was developed to solve this model in the case of non-orthogonal grids. To solve the biofilm model, we improve and extend the existing semi-implicit numerical formulation to deal with non-orthogonal grids. In this process, governing equations are transferred to general non-orthogonal curvilinear grids, and are discretized by the control-volume-based cell-centered finite volume method. At the faces of a control volume, the diffusive flux is split into orthogonal and non-orthogonal components. The orthogonal component is handled in a conventional manner, while the non-orthogonal component is handled explicitly and treated as a part of the source term. While discretizing the non-orthogonal term at the midpoint of a control volume face, the values of a dependent variable at the corners of the control volume face are required for evaluating the non-orthogonal terms. These values are calculated using values available at the centroid locations by an area-weighted linear interpolation scheme. The extensive validation of the developed formulation shows that the semi-implicit treatment of the non-orthogonal flux component works efficiently if the maximum deviation in orthogonality in the biofilm growth region of the grid is within 15 – 20 degrees.

The developed method is applied to measure the effect of mesoscale irregularities of the substratum on the substrate diffusivity and biofilm growth activity.  Overall, we conclude that the one-dimensional biofilm model is fully able to describe the biofilm activity on the rough surfaces under the nutrient-rich condition, but under the nutrient-low condition, higher dimensional models are necessary to simulate biofilm growth accurately on a rough surface.

We further study the effect of substratum roughness irregularities on quorum sensing activity in biofilm. We find that a deeper cavity of a complex geometry is beneficial for earlier QS induction. The results also indicate that QS induction is dependent not only on the size of the bacterial population, but also on the diffusion properties of the signalling molecules.