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Second-order schemes for axisymmetric Navier–Stokes–Brinkman and transport equations modelling water filters

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Abstract

Soil-based water filtering devices can be described by models of viscous flow in porous media coupled with an advection–diffusion–reaction system modelling the transport of distinct contaminant species within water, and being susceptible to adsorption in the medium that represents soil. Such models are analysed mathematically, and suitable numerical methods for their approximate solution are designed. The governing equations are the Navier–Stokes–Brinkman equations for the flow of the fluid through a porous medium coupled with a convection-diffusion equation for the transport of the contaminants plus a system of ordinary differential equations accounting for the degradation of the adsorption properties of each contaminant. These equations are written in meridional axisymmetric form and the corresponding weak formulation adopts a mixed-primal structure. A second-order, (axisymmetric) divergence-conforming discretisation of this problem is introduced and the solvability, stability, and spatio-temporal convergence of the numerical method are analysed. Some numerical examples illustrate the main features of the problem and the properties of the numerical scheme.

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Acknowledgements

We are thankful to Ian Griffiths (Oxford) for the stimulating discussions about water filter models and for providing the experimental data employed in Sect. 6.2. We also thank the comments of two anonymous referees, which resulted in a number of improvements to the manuscript. In addition, this work has been partially supported by Fondecyt Project 1170473; CRHIAM, Project ANID/FONDAP/15130015; project CONICYT/PIA/AFB170001; by CONICYT through the Becas-Chile program for foreign students; by the Monash Mathematics Research Fund S05802-3951284; by the Ministry of Science and Higher Education of the Russian Federation within the framework of state support for the creation and development of World-Class Research Centers “Digital biodesign and personalized healthcare” No 075-15-2020-926; and by the HPC-Europa3 Transnational Access Grant HPC175QA9K.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK

    Graham Baird

  2. Centro de Investigación en Ingeniería Matemática (CI2MA) and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Raimund Bürger & Paul E. Méndez

  3. School of Mathematical Sciences, Monash University, 9 Rainforest Walk, Clayton, VIC, 3800, Australia

    Ricardo Ruiz-Baier

  4. Laboratory of Mathematical Modelling, Institute of Personalised Medicine, Sechenov University, Moscow, Russian Federation

    Ricardo Ruiz-Baier

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  1. Graham Baird
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Baird, G., Bürger, R., Méndez, P.E. et al. Second-order schemes for axisymmetric Navier–Stokes–Brinkman and transport equations modelling water filters. Numer. Math. 147, 431–479 (2021). https://doi.org/10.1007/s00211-020-01169-1

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