Constrained flows of fluids with chemically reacting pollutants through porous media with linear and quadratic drag

Rogério P. S. Gama; Douglas M. Andrade; Rogério M. Saldanha da Gama; Felipe B. de Freitas Rachid; Heraldo S. da Costa Mattos; Maria L. Martins-Costa; Rogério P. S. Gama; Douglas M. Andrade; Rogério M. Saldanha da Gama; Felipe B. de Freitas Rachid; Heraldo S. da Costa Mattos; Maria L. Martins-Costa

doi:10.3934/math.2026649

AIMS Mathematics

2026, Volume 11, Issue 6: 15765-15796. doi: 10.3934/math.2026649

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Constrained flows of fluids with chemically reacting pollutants through porous media with linear and quadratic drag

1.
Mechanical Engineering Graduate Program (FEN), Universidade do Estado do Rio de Janeiro, 524 Rua São Francisco Xavier, Rio de Janeiro, RJ 20550-013, Brazil
2.
Laboratory of Theoretical and Applied Mechanics (LMTA) Mechanical Engineering Graduate Program (TEM-PGMEC), Universidade Federal Fluminense, 156 Rua Passo da Pátria, Niterói, RJ 24210-240, Brazil

Received: 08 March 2026 Revised: 04 April 2026 Accepted: 23 April 2026 Published: 04 June 2026
MSC : 35L65, 76S99

Constrained Newtonian fluids containing reacting pollutants flowing through porous media were modeled using a continuum theory of mixtures approach, accounting for viscous linear and quadratic drag terms. The work considered very low pollutant concentrations and similar mass densities for the fluid and all the pollutant constituents. These constituents may undergo chemical reactions among themselves, but they do not react with the fluid constituent. In this case, equal velocities for the pollutant constituents and the fluid constituent may be assumed. A solid constituent (the porous matrix, rigid and at rest), an inert gas (to account for the mixture's compressibility), an incompressible fluid (the fluid constituent), and N pollutant constituents compose the mixture. The model encompasses the mass equations for the fluid constituent and the N-pollutant constituents, as well as the fluid constituent's momentum balance. The Glimm scheme, associated with an operator-splitting technique, was used to advance in time. The complete solution to the associated Riemann problem was presented. The numerical results show the evolution of the fluid constituent's velocity and saturation, and the pollutant constituents' concentrations in the mixture for selected initial data, accounting for Darcy and Forchheimer drag terms, and chemical reactions among the pollutants. The novelty lies in the combination of linear and quadratic drag terms with simple chemical reactions among the pollutants, the use of a new constitutive relation for pressure (accounting for kinematic constraints), and a new protocol for the Glimm scheme.
- constrained flows through porous media,
- chemical reactions,
- Glimm's scheme,
- operator-splitting,
- mixture theory
Citation: Rogério P. S. Gama, Douglas M. Andrade, Rogério M. Saldanha da Gama, Felipe B. de Freitas Rachid, Heraldo S. da Costa Mattos, Maria L. Martins-Costa. Constrained flows of fluids with chemically reacting pollutants through porous media with linear and quadratic drag[J]. AIMS Mathematics, 2026, 11(6): 15765-15796. doi: 10.3934/math.2026649

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Abstract

Constrained Newtonian fluids containing reacting pollutants flowing through porous media were modeled using a continuum theory of mixtures approach, accounting for viscous linear and quadratic drag terms. The work considered very low pollutant concentrations and similar mass densities for the fluid and all the pollutant constituents. These constituents may undergo chemical reactions among themselves, but they do not react with the fluid constituent. In this case, equal velocities for the pollutant constituents and the fluid constituent may be assumed. A solid constituent (the porous matrix, rigid and at rest), an inert gas (to account for the mixture's compressibility), an incompressible fluid (the fluid constituent), and N pollutant constituents compose the mixture. The model encompasses the mass equations for the fluid constituent and the N-pollutant constituents, as well as the fluid constituent's momentum balance. The Glimm scheme, associated with an operator-splitting technique, was used to advance in time. The complete solution to the associated Riemann problem was presented. The numerical results show the evolution of the fluid constituent's velocity and saturation, and the pollutant constituents' concentrations in the mixture for selected initial data, accounting for Darcy and Forchheimer drag terms, and chemical reactions among the pollutants. The novelty lies in the combination of linear and quadratic drag terms with simple chemical reactions among the pollutants, the use of a new constitutive relation for pressure (accounting for kinematic constraints), and a new protocol for the Glimm scheme.

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