# American Institute of Mathematical Sciences

May  2012, 17(3): 1027-1059. doi: 10.3934/dcdsb.2012.17.1027

## Kinetic theories for biofilms

 1 Department of Mathematics and NanoCenter at USC, University of South Carolina, Columbia, SC 29208 2 Department of Mathematical Sciences, Montana State University, P.O. Box 172400, Bozeman, MT 59717-2400

Received  January 2009 Revised  September 2011 Published  January 2012

We apply the kinetic theory formulation for binary complex fluids to develop a set of hydrodynamic models for the two-phase mixture of biofilms and solvent (water). It is aimed to model nonlinear growth and transport of the biomass in the mixture and the biomass-flow interaction. In the kinetic theory formulation of binary complex fluids, the biomass consisting of EPS (Extracellular Polymeric Substance) polymer networks and bacteria is coarse-grained into an effective fluid component, termed the effective polymer solution; while the other component, termed the effective solvent, is made up of the ensemble of nutrient substrates and the solvent. The mixture is modeled as an incompressible two-phase fluid in which the presence of the effective components are quantified by their respective volume fractions. The kinetic theory framework allows the incorporation of microscopic details of the biomass and its interaction with the coexisting effective solvent. The relative motion of the biomass and the solvent relative to an average velocity is described by binary mixing kinetics along with the intrinsic molecular elasticity of the EPS network strand modeled as an elastic dumbbell. This theory is valid in both the biofilm region which consists of the mixture of the biomass and solvent and the pure solvent region, making it convenient in numerical simulations of the biomass-flow interaction. Steady states and their stability are discussed under a growth condition. Nonlinear solutions of the three models developed in this study in simple shear are calculated and compared numerically in 1-D space.
Citation: Qi Wang, Tianyu Zhang. Kinetic theories for biofilms. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1027-1059. doi: 10.3934/dcdsb.2012.17.1027
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