Front propagation in channels with spatially modulated cross section

Abstract

The problem of front propagation in a three-dimensional channel with spatially varying crosssection is reduced to an equivalent 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 representative example, we calculate analytically the nonlinear dependence of the front velocity on the ratio L/l where l denotes the intrinsic front width. Our analytical results agree well with the results obtained by numerical simulations. In particular, the peculiarity of boundary-induced propagation failure for a finite range of L/l values is predicted by analytical calculations. Lastly, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants for L l.

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

“…Several control strategies have been developed for purposeful manipulation of wave dynamics as the application of closed-loop or feedback-mediated control loops with and without delays [8][9][10][11] and open-loop control that includes external spatio-temporal forcing [10,[12][13][14], optimal control [15][16][17], and control by imposed geometric constraints and heterogeneities on the medium [18,19]. While feedback-mediated control relies on continuously monitoring of the system's state, open-loop control is based on a detailed knowledge of the system's dynamics and its parameters.…”

Wave propagation in spatially modulated tubes

“…In general, we find that the velocity c diminishes with shrinking ratio δ for a given ratio L/l. Similar to front propagation in thin corrugated channels 34 , we identify a finite interval of L/l values where propagation failure (PF) occurs, i.e. the initially traveling front becomes quenched 44 and c goes to zero.…”

“…Despite the fact that the boundary-induced advection term is proportional to Q ′ (x)/Q(x) for rotationally symmetric tubes as well as thin 3D channels with modulated rectangular cross section 34 , we emphasize that, identifying the tube's diameter with the width of a planar channel, the amplitude of the advection field is two times larger for tubes, Q(x) = πΩ 2 (x), compared to channels with rectangular cross section, Q(x) = 2Ω(x) H; here, H denotes the height of the thin 3D channel. Consequently, we expect a much stronger impact of the tube's modulation on the propagation properties of TWs.…”

“…Applying projection method 34,47 to Eq. (10), it is feasible to derive the EOM for the position ϕ(t) of TWs in response to the advection term D (v · ∇) u 0 .…”

“…These provide efficient methods to study experimentally the impact of confinement on wave propagation [29][30][31] , to construct chemical logical gates 32 , and to control or optimize the local dynamics of catalytic reactions 33 . In our recent work 34 , we have provided a first systematic treatment of how propagation of traveling waves in thin three-dimensional channels with periodically varying cross-section can be reduced to a corresponding one-dimensional reaction-diffusion-advection equation (RDAE). Using projection method, we have derived an equation of motion for the position of TWs as a function of time in the presence of the boundary-induced advection term and obtained an analytical expression for the average propagation velocity.…”