On a vorticity-based formulation for reaction-diffusion-Brinkman systems
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...
| Main Authors: | , , , |
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| Format: | Journal article |
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American Institute of Mathematical Sciences
2018
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| _version_ | 1826303620986961920 |
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| author | Anaya, V Bendahmane, M Mora, D Ruiz Baier, R |
| author_facet | Anaya, V Bendahmane, M Mora, D Ruiz Baier, R |
| author_sort | Anaya, V |
| collection | OXFORD |
| description | 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, 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. |
| first_indexed | 2024-03-07T06:05:29Z |
| format | Journal article |
| id | oxford-uuid:edb206fd-6d59-4a60-9a60-198966ad748d |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:05:29Z |
| publishDate | 2018 |
| publisher | American Institute of Mathematical Sciences |
| record_format | dspace |
| spelling | oxford-uuid:edb206fd-6d59-4a60-9a60-198966ad748d2022-03-27T11:27:02ZOn a vorticity-based formulation for reaction-diffusion-Brinkman systemsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:edb206fd-6d59-4a60-9a60-198966ad748dSymplectic Elements at OxfordAmerican Institute of Mathematical Sciences2018Anaya, VBendahmane, MMora, DRuiz Baier, RWe 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, 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. |
| spellingShingle | Anaya, V Bendahmane, M Mora, D Ruiz Baier, R On a vorticity-based formulation for reaction-diffusion-Brinkman systems |
| title | On a vorticity-based formulation for reaction-diffusion-Brinkman systems |
| title_full | On a vorticity-based formulation for reaction-diffusion-Brinkman systems |
| title_fullStr | On a vorticity-based formulation for reaction-diffusion-Brinkman systems |
| title_full_unstemmed | On a vorticity-based formulation for reaction-diffusion-Brinkman systems |
| title_short | On a vorticity-based formulation for reaction-diffusion-Brinkman systems |
| title_sort | on a vorticity based formulation for reaction diffusion brinkman systems |
| work_keys_str_mv | AT anayav onavorticitybasedformulationforreactiondiffusionbrinkmansystems AT bendahmanem onavorticitybasedformulationforreactiondiffusionbrinkmansystems AT morad onavorticitybasedformulationforreactiondiffusionbrinkmansystems AT ruizbaierr onavorticitybasedformulationforreactiondiffusionbrinkmansystems |