Front propagation in three-dimensional corrugated reaction-diffusion media

TL;DR

This paper investigates how traveling fronts propagate in three-dimensional reaction-diffusion media with spatially modulated cross-sections, combining analytical techniques and numerical simulations to understand velocity dependence and propagation failure.

Contribution

It introduces a reduction method for analyzing front propagation in corrugated media and provides analytical predictions validated by numerical simulations.

Findings

Propagation velocity depends nonlinearly on the ratio of spatial period to front width.

Propagation failure occurs at specific parameter regimes, predicted by the eikonal equation.

Front velocity is influenced by suppressed reactant diffusivity when the front width exceeds medium variation.

Abstract

Propagation of traveling fronts in three-dimensional reaction-diffusion media with spatially modulated cross-section is studied using the Schl\"ogl model as a representative example. Applying appropriate perturbation techniques leads first to a reduction of dimensionality in which the spatially dependent Neumann boundary condition translate into a boundary-induced advection term and, secondly, to an equation of motion for the traveling wave position in weakly corrugated confinements. Comparisons with numerical simulations demonstrate that our analytical results properly predicts the nonlinear dependence of the propagation velocity on ratio of the spatial period of the confinement to the intrinsic width of the front; including the peculiarity of propagation failure. Based on the eikonal equation, we obtain an analytical estimate for the finite interval of propagation failure. Lastly, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants if the intrinsic width of the front is much larger than the spatial variation of the medium.