Numerical schemes for mixture theory models with filling constraint: application to biofilm ecosystems

Abstract

The mixture theory framework is a powerful way to describe multi-phasic systems at an intermediary scale between microscopic and macroscopic scales. In particular, mixture theory reveals a powerful approach to represent microbial biofilms where a consortium of cells is embedded in a polymeric structure. To simulate a model of microalgal biofilm, we propose an upgraded numerical scheme, consolidating the one proposed by Berthelin et al. (2016) to enforce the volume-filling constraint in mixture models including mass exchanges. The strategy consists in deducing the discrete version of the incompressibility constraint from the discretized mass balance equations. Numerical simulations show that this method constrains the total volume filling constraint, even at the discrete level. Moreover, we add viscous terms in the biofilm model to properly represent biofilms interactions with its fluidic environment. It turns out that a well-balanced numerical scheme becomes of outmost importance to capture the biofilm dynamic when including the viscosity. This modelling upgrade also involves recalibrating model parameters. In particular, the elastic tensors to recover realistic front features. With the new parameters, the numerical set-up becomes more demanding to reach convergence.