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

Verónica Anaya, Mostafa Bendahmane, David Mora, Ricardo Ruiz Baier

Research output: Contribution to journalArticleResearchpeer-review

11 Citations (Scopus)

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-reactiondi 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, Nédélec 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.

Original languageEnglish
Pages (from-to)69-94
Number of pages26
JournalNetworks and Heterogeneous Media
Volume13
Issue number1
DOIs
Publication statusPublished - 1 Jan 2018
Externally publishedYes

Keywords

  • Brinkman flows
  • Chemical reactions
  • Mixed finite elements
  • Reaction-diffusion
  • Vorticity formulation

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