Stability analysis for combustion fronts traveling in hydraulically resistant porous media

We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively stable.Comment: 21 pages, 4 figure

Selection in the Saffman-Taylor finger problem and the Taylor-Saffman bubble problem without surface tension

AbstractWe consider the Saffman-Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration, and the Taylor-Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the porous channel through which it propagates. To select the correct member of the family of solutions, the conventional approach has been to add surface tension σ and then take the limit σ → 0. We propose a selection criterion that does not rely on surface tension arguments

Upscaling of mass and thermal transports in porous media with heterogeneous combustion reactions

The present paper aims at an upscaled description of coupled heat and mass processes during solid–fluid combustion in porous media using volume-averaging theory (VAT). The fluid flows through the pores in a porous medium where a heterogeneous chemical reaction occurs at the fluid–solid interface. The chemical model is simplified into a single reaction step with Arrhenius kinetic law, but no assumption of local thermal equilibrium is made. An array of horizontal channels is chosen for the microstructure. The corresponding effective properties are obtained by solving analytically the closure problems over a representative unit cell. For a range of Péclet and Δ numbers, the results of the upscaled model are compared with microscale computations found in the literature. The results show that, under the same circumstances, the upscaled model is capable of predicting the combustion front velocity within an acceptable discrepancy, smaller than 1% when compared to the analytical solution. Furthermore, it has been found that for the Péclet and Δ numbers considered in this study, the fluid concentration and temperature profiles that stem from the present upscaled model are in accordance with those obtained using a microscale model

Pattern formation in reverse smoldering combustion : a homogenization approach

The development of fingering char pattern on the surface of porous thin materials has been investigated in the framework of reverse combustion. This macroscopic characteristic feature of combustible media has also been studied experimentally and through the use of phenomenological models. However, much attention has not been given to the behavior of the emerging patterns based on characteristic material properties. Starting from a microscopic description of the combustion process, macroscopic models of reverse combustion that are derived by the application of the homogenization technique are presented. Using proper scaling by means of a small scale parameter, e, the results of the formal asymptotic procedure are justified by qualitative multiscale numerical simulations at the microscopic and macroscopic levels. We consider two equilibrium models that are based on effective conductivity contrasts, in a simple adiabatic situation, to investigate the formation of unstable fingering patterns on the surface of a charred material. The behavior of the emerging patterns is analyzed using primarily the Péclet number as a control parameter

Travelling waves in nonlinear diffusion-convection-reaction

The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation. \u

Bistable reaction equations with doubly nonlinear diffusion

Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions $0\leq u(x,t)\leq 1$ of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r.t. the Fisher-KPP framework recently studied in \cite{AA-JLV:art}. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries. \\ Finally, as a complement of \cite{AA-JLV:art}, we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").Comment: 42 pages, 11 figures. Accepted version on Discrete Contin. Dyn. Sys