Second-order schemes for axisymmetric Navier-Stokes-Brinkman and transport equations modelling water filters
The paper concerns with the analysis and numerical approximation of the flow of a viscous fluid through a porous medium, where the fluid carries a number \(m\) of components that are absorbed by the porous medium. This kind of problem arises in designing soil-based water filtering devices. 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. Most of the filter designs in the three-dimensional domain, display rotational symmetry around their central axis, with the flow also expected to exhibit such symmetry. The axisymmetric formulation allows the reduction from three to two spatial dimensions, which reduces the computational cost associated with the numerical solution. The authors introduce an axisymmetric \(H(\operatorname{div})\)-conforming method based on two-dimensional Brezzi-Douglas-Marini (BDM) spaces combined with an implicit, second-order backward differentiation formula (BDF2) for temporal discretization. The conditional uniqueness of the solution is proved using the discrete stability properties. An optimal a priori error estimate for the numerical scheme is derived. Numerical examples with the accuracy tests, validation against experimental data, and two contaminants in the two-layer filter, illustrate the model and reconfirm the theoretical order of accuracy.
scientific article; zbMATH DE number 7317385
scientific article