Reaction-diffusion waves in coupled isothermal autocatalytic chemical systems

The initiation and propagation of reaction-diffusion travellingwaves in two regions, coupled together by a linear diffusiveinterchange across a semipermeable membrane is considered. Twosystems are considered in detail where there are two chemicalspecies present, species A and the autocatalytic species B.The first system is governed by quadratic autocatalysis in bothregions together with a linear decay of the autocatalyst inonly one region, with the coupling taken to be via the exchangeof species A. The second system has the reaction scheme of cubicautocatalysis and linear decay of the autocatalytic speciesB in region I, while region II is taken to be chemically inertcontaining only species A, with the mode of coupling betweenthe two regions again being via A. These two systems are consideredvia an initial-value problem in which the amounts of the autocatalystare introduced locally into a uniform concentration of speciesA. A priori bounds for the initial-value problem are first obtained,and then conditions for the initiation of travelling waves forsmall initial inputs of the autocatalyst are derived. Theseresults are extended for the first system to all initial inputsof the autocatalyst. The effects of strong coupling are thenexamined. Numerical results are presented to demonstrate theeffects of varying the parameters of the two systems as predictedby our analysis. Finally, the authors examine the equationsgoverning the permanent-form travelling waves and derive generalproperties of their solution together with a solution for weakcoupling.     

The propagation of isothermal reaction-diffusion waves in coupled autocatalytic systems

The initiation of reaction-diffusion travelling waves in tworegions, each governed by simple isothermal kinetics, is consideredwhen the two regions are coupled together. The basic reactionin each region is quadratic autocatalysis with the decay ofthe autocatalyst to an inert product in just one of the regionsbeing allowed. The situation when the coupling is achieved viathe reactant is discussed in detail and compared with previousresults obtained when the coupling is via the autocatalyst.Substantial differences both in the conditions for the formationof travelling waves and in their ultimate propagation speedare seen between the two coupling mechanisms. The permanent-formtravelling wave structure is shown to depend on the initialinput of autocatalyst when the coupling is via the reactant,which is not the case when the coupling is via the autocatalyst.     

Reaction-Diffusion Model for Combustion with Fuel Consumption: I. Dirichlet Boundary Conditions

A model is derived for the reaction between a gaseous oxidantand a solid porous fuel, which allows for the consumption ofthe fuel. A single exothermic reaction, first-order in the concentrationof the oxidant and fuel with an Arrhenius temperature dependenceof the reaction rate constant is assumed. The solid reactantis taken to have an infinite slab geometry and be surroundedby a reservoir of constant oxidant concentration and fixed ambienttemperature. Numerical solutions of the governing equationsare obtained and the effects of varying the dimensionless parameters,especially the Frank-Kamenetskii parameter and the Lewis numberL, are examined. Analytic solutions are then derived for theinitial development of the solution and for large values of. This latter solution shows the development of a reaction-diffusionwave which starts at the edge and propagates across the slab,the structure of which depends on the Lewis number.     

An Existence Proof for a Problem in Singular Parabolic Partial Differential Equations

In a previous paper (Crighton & Scott, 1979) it was shownhow the method of matched asymptotic expansions applied to aproblem of non-linear acoustics leads to a parabolic equationwith asymptotic initial conditions. To ensure that the scalingsused are correct it is necessary to prove the existence of asolution to this problem. In the present paper we demonstrateexistence and in so doing we also obtain some bounds on thesolution.