Stress tensor symmetry preserving model applied to the 2-D viscoelastic flow of a biofilm

Abstract

The symmetry of the numeric representation of the stress tensor has been shown to be important for maintaining stability, in the sense of Hadamard, of the numeric method. Also, the viscoelastic behavior of biofilms is well documented. A 2D model for the viscoelastic flow of a biofilm using a modified Navier-Stokes equation (NSE) with a novel elastic stress term are presented. The elastic stress is modeled using a numeric stress tensor symmetry preserving scheme that is based on the numeric solution to the Lie derivative and its equivalent counterpart in the form of a symmetric matrix Riccati differential equation (SMRDE). In addition, a coupled advection equation (AE) is applied to the biofilm volume fraction. Solutions to the NSE and AE are found by applying the finite element method (FEM) to the Eulerian-Lagrangian method (ELM). The ELM is solved by first determining the 'characteristic foot' for each Gaussian quadrature point and node point in the mesh. The advection equation is solved using a modified Galerkin Least Squares (GLS) method. Computations are made using the Trilinos iterative sparse matrix solver library called AztexOO which has built in matrix preconditioners and support for parallel processing. The resulting model is used to predict the deformation of a biofilm in a 2D channel. In addition, the accompanying distribution of the pressure and stresses over the evolving velocity field is presented.