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-reactiondiffusion 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.

A 3D reaction–diffusion system describing calcium dynamics in cardiac cell

“…Remark 3.6. Note that, since the interaction terms are Lipschitz (recall that our solutions are uniformly bounded in L ∞ ) and the positivity of diffusion function, the uniqueness of weak solution can be obtained easily (for a similar proof see [1]).…”

Partitioned coupling of advection–diffusion–reaction systems and Brinkman flows

“…Here we explore very similar scenarios, but allowing the di↵usive terms to depend nonlinearly on the species concentrations, we consider the three-dimensional case as well, and we write the Brinkman equations in terms of vorticity, velocity, and pressure of the incompressible fluid. We stress that the mathematical properties of such a formulation have been addressed only recently in [4], where also an explicit finite element method was introduced for its numerical approximation. The present work essentially complements [1,4] in the sense that we define a family of four basic coupling methods to numerically solve the governing equations.…”

“…We stress that the mathematical properties of such a formulation have been addressed only recently in [4], where also an explicit finite element method was introduced for its numerical approximation. The present work essentially complements [1,4] in the sense that we define a family of four basic coupling methods to numerically solve the governing equations. The precise form of the schemes will vary depending on whether the Brinkman problem admits a pure vorticity formulation (as the one proposed in [6]), and on two main sequential substructuring techniques to decouple the advection-di↵usion from the reaction steps in the ADR system.…”

Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system

“…subject to the boundary conditions and initial data given by (4 We conclude that if 8a 11 a 21 ≥ a 2 12 and 8a 22 a 12 ≥ a 2 21 , then M(c, s) is uniformly nonnegative. Hence, the utility of assumption (45) (see, eg, the work of Bendahmane and Langlais 40 for more details).…”

“…where is the unit outward normal to Ω on Ω. Note that in the case ( a i, ) 1 ≤ i, j ≤ 2 ∶ = 0, our model can be reduced to the recent model by Anaya et al,4 in which the authors proposed a reaction-diffusion system representing the bacteria-chemical mass exchange, coupled with the Brinkman problem. To the best of our knowledge, there are few papers proposing the augmented velocity-vorticity-pressure formulation (augmented Brinkman model) without reaction-diffusion system coupling.…”