Paper Info
Title
On a vorticity-based formulation for reaction-diffusion-Brinkman systems.
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.
Year
DOI
Venue
2018
10.3934/nhm.2018004
NETWORKS AND HETEROGENEOUS MEDIA
Keywords
Field
DocType
Reaction-diffusion,Brinkman flows,vorticity formulation,mixed finite elements,chemical reactions
Convergence (routing),Vorticity,Mathematical analysis,A priori and a posteriori,Flow (psychology),Weak solution,Finite element method,Reaction–diffusion system,Mathematics,Piecewise
Journal
Volume
Issue
ISSN
13
1
1556-1801
Citations 
PageRank 
References 
1
0.37
5
Authors
4
Name
Order
Citations
PageRank
Verónica Anaya162.63
Mostafa Bendahmane2359.38
David Mora3348.92
Ricardo Ruiz-Baier47713.60