Unsteady fronts in an autocatalytic system

Travelling waves in a model for autocatalytic reactions have, for some parameter regimes, been suggested to have oscillatory instabilities. This instability is confirmed by various methods, including linear stability analysis (exploiting Evan's function) and direct numerical simulations. The front instability sets in when the order of the reaction, m, exceeds some threshold, mc(;τ), that depends on the inverse of the Lewis number, τ. The stability boundary, m =mc(;τ), is found numerically for m order one. In the limit m ≫ 1 (in which the system becomes similar to combustion systems with Arrhenius kinetics), the method of matched asymptotic expansions is employed to find the asymptotic front speed and show that mc ∼ (;τ – 1)–1 as ;τ → 1. Just beyond the stability boundary, the unstable rocking of the front saturates supercritically. If the order is increased still further, period–doubling bifurcations occur, and, for small tau there is a transition to chaos through intermittency after the disappearance of a period–4 orbit.

[1]  Kapral,et al.  Biscale chaos in propagating fronts. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  Masayasu Mimura,et al.  Layer oscillations in reaction-diffusion systems , 1989 .

[3]  Paul H. Muir,et al.  Algorithm 688: EPDCOL: a more efficient PDECOL code , 1991, TOMS.

[4]  John Billingham,et al.  The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves , 1991, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[5]  David Terman,et al.  Stability of planar wave solutions to combustion model , 1990 .

[6]  Masayasu Mimura,et al.  An interfacial approach to regional segregation of two competing species mediated by a predator , 1991 .

[7]  A. Merzhanov,et al.  Propagation of a pulsating exothermic reaction front in the condensed phase , 1971 .

[8]  Michael I. Weinstein,et al.  Oscillatory instability of traveling waves for a KdV-Burgers equation , 1993 .

[9]  A. Bayliss,et al.  Period doubling gained, period doubling lost , 1989 .

[10]  Gregory I. Sivashinsky,et al.  Diffusional-Thermal Theory of Cellular Flames , 1977 .

[11]  J. Cash,et al.  High order methods for the numerical solution of two-point boundary value problems , 1982 .

[12]  I. Epstein,et al.  Stochastic behavior and stirring rate effects in the chlorite–iodide reaction , 1988 .

[13]  K. Showalter,et al.  Instabilities in propagating reaction-diffusion fronts , 1993 .

[14]  Michael I. Weinstein,et al.  Eigenvalues, and instabilities of solitary waves , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[15]  P. Fife.  Dynamics of Internal Layers and Diffusive Interfaces , 1988 .

[16]  S. Scott,et al.  Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[17]  A. Arneodo,et al.  Modeling reaction–diffusion pattern formation in the Couette flow reactor , 1991 .

[18]  A. Bayliss,et al.  Two routes to chaos in condensed phase combustion , 1990 .

[19]  S. Falle,et al.  Stability of reaction-diffusion fronts , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[20]  J. Alexander,et al.  A topological invariant arising in the stability analysis of travelling waves. , 1990 .

[21]  B. Larrouturou.  The equations of one-dimensional unsteady flame propagation: existence and uniqueness , 1988 .

[22]  J. Langer.  Instabilities and pattern formation in crystal growth , 1980 .

[23]  John A. Pojman,et al.  Numerical modeling of self-propagating polymerization fronts: The role of kinetics on front stability. , 1997, Chaos.

[24]  A. P. Aldushin,et al.  Autovibrational propagation of the combustion front in heterogeneous condensed media , 1973 .

[25]  Jonathan Swinton,et al.  Stability of travelling pulse solutions to a laser equation , 1990 .

[26]  Gregory I. Sivashinsky,et al.  PROPAGATION OF A PULSATING REACTION FRONT IN SOLID FUEL COMBUSTION , 1978 .

[27]  J. Xin,et al.  Global Solutions to a Reactive Boussinesq System with Front Data on an Infinite Domain , 1998 .

[28]  S. Margolis.  The Transition to Nonsteady Deflagration in Gasless Combustion , 1991 .

[29]  Paul A. Zegeling,et al.  Algorithm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations , 1994, TOMS.

[30]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[31]  L. Sainsaulieu,et al.  On the stability of multiple steady planar flames when the Lewis number is lees than 1 , 1993 .