Modelagem e simulação numérica de um escoamento através de meio poroso com restrição unilateral e com transporte de contaminantes quimicamente reagentes

Abstract

This thesis studies constrained Newtonian fluid flows containing reacting pollutants through porous media, modeled using a Mixture Theory perspective and a constitutive relation for the pressure – namely, a continuous and differentiable function of the saturation that ensures always preserving the problem hyperbolicity. The pressure equation also permits an ultra-small porous matrix (controlled) supersaturation and the transition from unsaturated to saturated flow (and vice versa). Assuming tiny pollutant constituent concentration densities that may react among themselves but never react with the fluid constituent, they all present the same order of magnitude mass densities, measured from a Continuum Mechanics viewpoint. These hypotheses lead to the assumption of equal velocities for the pollutant constituents and the fluid constituent: the mixture velocity. Since the solid constituent that represents the porous matrix is rigid and at rest, and the inert gas has a tiny mass density (included to provide some compressibility to the mixture), it suffices to solve the mass conservation equations for the fluid constituent and for the pollutant constituents and the fluid constituent’s momentum balance, giving rise to a non-homogeneous hyperbolic system. Its numerical simulation combines Glimm’s method with an operator-splitting strategy to account for the pollutants’ mass variation caused by the pollutants’ chemical reactions among themselves. Despite the Glimm method’s proven convergence, it is not adequate to approximate non-homogeneous hyperbolic systems unless combined with an operatorsplitting technique. Although other approaches have already addressed this problem, the novelty is combining Glimm’s method with operator-splitting to account for the pollutants’ mass variation caused by the pollutants’ chemical reactions among themselves. Glimm’s scheme marches in time using a formerly selected number of associated Riemann problems. The constitutive relation for the pressure – an increasing function of the saturation, with the first derivative also increasing, convex, and positive, allows obtaining explicit expressions for the Riemann invariants. The complete (exact) solution of the associated Riemann problem is presented. The results show the evolution of the fluid constituent velocity, of the saturation, and of the pollutant constituents’ concentrations in the mixture for selected initial data that suffer the influence of the Darcy and Forchheimer drag terms on the flow and of the chemical reactions among the pollutants.