Scaling, propagation, and kinetic roughening of flame fronts in random media

Abstract

We introduce a model of two coupled reaction-diffusion equations to describe the dynamics and propagation of flame fronts in random media. The model incorporates heat diffusion, its dissipation, and its production through coupling to the background reactant density. We first show analytically and numerically that there is a finite critical value of the background density, below which the front associated with the temperature field stops propagating. The critical exponents associated with this transition are shown to be consistent with mean field theory of percolation. Second, we study the kinetic roughening associated with a moving planar flame front above the critical density. By numerically calculating the time dependent width and equal time height correlation function of the front, we demonstrate that the roughening process belongs to the universality class of the Kardar-Parisi-Zhang interface equation. Finally, we show how this interface equation can be analytically derived from our model in the limit of almost uniform background density.

Untitled

“…Indeed, in the absence of nucleation it has been shown in Ref. [10] that the flame front roughens according to the KPZ interface equation, through which some of these correspondences can be made.…”

“…Such continuum reaction-diffusion equations have been used extensively in physics, chemistry, biology and engineering to describe a wide range of phenomena from pattern formation to combustion. However, the connection of reaction-diffusion equations to nucleation and interface growth has received little attention.In a recent study of slow combustion in disordered media, Provatas et al [9,10] showed that flame fronts exhibit a percolation transition, consistent with mean field theory, and that the kinetic roughening of the reaction front in slow combustion is consistent with the KardarParisi-Zhang (KPZ) [11] universality class. In this paper we make a further connection between slow combustion started by spontaneous fluctuations, and the classical theory of the nucleation and growth of droplets from a metastable phase.…”

“…In a recent study of slow combustion in disordered media, Provatas et al [9,10] showed that flame fronts exhibit a percolation transition, consistent with mean field theory, and that the kinetic roughening of the reaction front in slow combustion is consistent with the KardarParisi-Zhang (KPZ) [11] universality class. In this paper we make a further connection between slow combustion started by spontaneous fluctuations, and the classical theory of the nucleation and growth of droplets from a metastable phase.…”

“…(46) in Ref. [10] with v = (ΓΛ −λc)/σ. The constants Λ,λ and σ depend functionally on the temperature T m (x) that solves Eqs.…”

“…We choose the same values for the constants as those used in Ref. [10], which are approximately those for the combustion of wood in air: In physical units D = 1 m 2 s −1 , Γ = 0.05 s −1 , T 0 = 0.1 K, A = 500 K, and the specific heat of wood, c p = 5 Jg −1 K −1 (entering through λ 2 = 8). Length is measured in units of the reactant size, and time in units of those for the reaction to take place, λ 2 /λ 1 .…”