Diffusion of chemically reacting fluids through nonlinear elastic solids: mixture model and stabilized methods

Abstract

This paper presents a stabilized mixed finite element method for advection-diffusion-reaction phenomena that involve an anisotropic viscous fluid diffusing and chemically reacting with an anisotropic elastic solid. The reactive fluid–solid mixture theory of Hall and Rajagopal (Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory. Math Mech Solid 2012; 17: 131–164) is employed wherein energy and entropy production relations are captured via an equation describing the Lagrange multiplier that results from imposing the constraint of maximum rate of entropy production. The primary partial differential equations are thus reduced to the balance of mass and balance of linear momentum equations for the fluid and the solid, together with an equation for the Lagrange multiplier. Present implementation considers a simplification of the full system of governing equations in the context of isothermal problems, although anisothermal studies are being investigated. The method is applied to problems involving Fickian diffusion, oxidation of PMR-15 polyimide resin, and slurry infiltration, within a one-dimensional finite element context. Results of the oxidation modeling of Tandon et al. (Modeling of oxidative development in PMR-15 resin. Polym Degrad Stab 2006; 91: 1861–1869) are recovered by employing the reaction kinetics model and properties assumed there; the only additional assumed properties are two constants describing coupled chemomechanical and purely chemical dissipation, and standard values for viscosity of air and PMR-15 stiffness properties. The present model provides the individual constituent kinematic and kinetic behaviors, thus adding rich detail to the interpretation of the process in comparison to the original treatment. The last problem considered is slurry infiltration that demonstrates the applicability of the model to account for the imposed mass deposition process and consequent effects on the kinematic and kinetic behaviors of the constituents.

Variational Multiscale Method with Flexible Fine-Scale Basis for Diffusion-Reaction Equation: Built-In a Posteriori Error Estimate and Heterogenous Coefficients

“…In earlier work by Masud and collaborators [21][22][23][24], a single bubble that is zero at the element boundaries was used to represent the fine scales, including some applications to reactive phenomena [25,26]. The example of a biquadratic bubble typically used with 4-node quadrilaterals is shown in Figure 1a.…”

Diffusion of Chemically Reacting Fluids through Nonlinear Elastic Solids and 1D Stabilized Solutions

“…This solution confers stabilization properties resulting in the dampening of the oscillatory effects resulting from the boundary discontinuity. Details are given in [11].…”

“…In the present work where we employ the mixture theory, a Newtonian fluid and an elastic solid are considered. The only additional assumed properties are two constants describing coupled chemomechanical and purely chemical dissipation, and standard values for viscosity of air and PMR-15 stiffness properties [11]. Since the unknown fields in the mixture model are fluid density, fluid velocity, solid displacement and solid density, therefore kinematic and the force measures can be readily obtained from the simulations.…”