Front propagation in channels with spatially modulated cross section

Abstract

Propagation of traveling fronts in a three-dimensional channel with spatially varying cross section is reduced to an equivalent one-dimensional reaction-diffusion-advection equation with boundary-induced advection term. Treating the advection term as a weak perturbation, an equation of motion for the front position is derived. We analyze channels whose cross sections vary periodically with L along the propagation direction of the front. Taking the Schlögl model as a representative example, we calculate analytically the nonlinear dependence of the front velocity on the ratio L/l where l denotes the intrinsic front width. In agreement with finite-element simulations of the three-dimensional reaction-diffusion dynamics, our theoretical results predicts boundary-induced propagation failure for a finite range of L/l values. In particular, the existence of the upper bound of L/l can be completely understood based on the linear eikonal equation. Last, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants for L≪l.