Solutions for models of chemically reacting mixtures

International audienceThe mathematical modeling of chemically reacting mixtures is investigated. The governing equations, that may be split between conservation equations, thermochemistry and transport fluxes, are presented as well as typical simplifications often encountered in the literature. The hyperbolic-parabolic structure of the resulting system of partial differential equations is analyzed using symmetrizing variables. The Cauchy problem is discussed for the full system derived from the kinetic theory of gases as well as relaxation towards chemical equilibrium fluids in the fast chemistry limit. The situations of traveling waves and reaction-diffusion systems is also addressed

Projected iterative algorithms with application to multicomponent transport

AbstractWe investigate projected iterative algorithms for solving constrained symmetric singular linear systems. We discuss the symmetry of generalized inverses and investigate projected standard iterative methods as well as projected conjugate-gradient algorithms. Using a generalization of Stein's theorem for singular matrices, we obtain a new proof of Keller's theorem. We also strengthen a result from Neumann and Plemmons about the spectrum of iteration matrices. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients in multicomponent gas mixtures. We obtain low-cost accurate approximate expressions for the transport coefficients that can be used in multicomponent flow models. Typical examples for the species diffusion coefficients and the volume viscosity are presented

Volume Viscosity and Internal Energy Relaxation : Error Estimates

We investigate the fast relaxation of internal energy in nonequilibrium gas models derived from the kinetic theory of gases. We establish a priori estimates and existence theorems for symmetric hyperbolic-parabolic systems of partial differential equations with small second order terms and stiff sources. We also establish the stability of the corresponding equilibrium systems. We then prove local in time error estimates between the out of equilibrium solution and the one-temperature equilibrium fluid solution for well prepared data and justify the apparition of volume viscosity terms. The situation of ill prepared data with initial layers is also addressed

Volume Viscosity and Internal Energy Relaxation : Symmetrization and Chapman-Enskog Expansion

We analyze a mathematical model for the relaxation of translational and internal temperatures in a nonequilibrium gas. The system of partial differential equations---derived from the kinetic theory of gases---is recast in its natural entropic symmetric form as well as in a convenient hyperbolic-parabolic symmetric form. We investigate the Chapman-Enskog expansion in the fast relaxation limit and establish that the temperature difference become asymptotically proportional to the divergence of the velocity field. This asymptotic behavior yields the volume viscosity term of the limiting one-temperature fluid model

Scattered wavefield in the stochastic homogenization regime

In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques, we provide asymptotic expansions of the scattered field in the background medium with respect to a scaling parameter describing the spatial random oscillations of the micro-structure. Introducing a boundary layer corrector to compensate the breakdown of stationarity assumptions at the boundary of the scattering medium, we prove quantitative $L^2$- and $H^1$- error estimates for the asymptotic first-order expansion. The theoretical results are supported by numerical experiments

A conservative model for high-throughput synthesis of nanoparticles in reacting gas flows

Considerable progress has been made over the past decades in the modeling of gas-phase synthesis of nanoparticles. However, when the nanoparticles mass fraction is large representing up to 50 % of the mixture mass fraction, some issues can be observed in the self-consistent modeling of the production process. In particular, enthalpy exchanges between gas and particle phases and differential diffusion between the two phases are usually neglected, since the particle mass fraction is generally very small. However, when high nanoparticle mass fractions are encountered, these simplifications may cause non conservation of the total enthalpy or the total mass. In the present paper, we propose a conservative model for nanoparticles production from gas-phase processes with a high throughput of nanoparticles. The model is derived in order to satisfy conservations of both enthalpy and mass and is validated on laminar one-dimensional premixed and non-premixed flames. In particular, it is shown that the enthalpy of the particle phase as well as the differential diffusion of the gas phase with respect to the particle phase cannot be generally neglected when the nanoparticles concentration is high to preserve the accuracy of the numerical results