On a vorticity-based formulation for reaction-diffusion-Brinkman systems

Anaya, V; Bendahmane M.; Mora D.; Baier R.R.

Abstract

We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nedelec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

Más información

Título según WOS: ON A VORTICITY-BASED FORMULATION FOR REACTION-DIFFUSION-BRINKMAN SYSTEMS
Título según SCOPUS: On a vorticity-based formulation for reaction-diffusion-Brinkman systems
Título de la Revista: NETWORKS AND HETEROGENEOUS MEDIA
Volumen: 13
Número: 1
Editorial: AMER INST MATHEMATICAL SCIENCES-AIMS
Fecha de publicación: 2018
Página de inicio: 69
Página final: 94
Idioma: English
DOI:

10.3934/nhm.2018004

Notas: ISI, SCOPUS