Reaction-diffusion model for combustion with fuel consumption: II. Robin boundary conditions

A model, derived in a previous paper, for the reaction betweena gaseous oxidant and a solid porous fuel is analysed furtherfor general Robin boundary conditions. Numerical solutions areobtained and the effects of varying the dimensionless parameters,particularly the Frank-Kamenetskii parameter and the Lewisnumber L, are discussed in detail and compared with resultsobtained previously when Dirichlet boundary conditions are applied.Analytic solutions are obtained for the small-time developmentand for large values of . This latter solution shows the existenceof a propagating reaction-diffusion burning wave, and has featureswhich are qualitatively different to those derived earlier.     

The effects of coupling on pattern formation in a simple autocatalytic system

The spatiotetnporal structures that can arise in two identicalcells, each governed by cubic autocatalator kinetics and coupledvia the diffusive interchange of a reactant, are discussed.The coupling gives rise to five spatially uniform steady states,one of which exists in the uncoupled system. By studying thelinearized equations, it is found that three of these steadystates, including that of the uncoupled system, may give riseto the possibility of bifurcations to spatially nonuniform steadystates. In the case of the steady state corresponding to thatof the uncoupled system, it is seen that the coupling leadsto bifurcations not present in the uncoupled system which giverise to locally stable nonuniform steady states. A weakly nonlinearanalysis is developed for both small and large coupling strengtha, and for parameter values in a neighbourhood of the bifurcationpoints on the new steady states. This clarifies the nature ofthe nonuniform solutions close to bifurcation, which are thenfollowed numerically using a path-following technique. The couplingis found to produce extra nonuniform steady solutions whichare stable close to their bifurcation points.     

Pattern formation in a coupled cubic autocatalator system

The spatiotemporal structures that can arise in two identicalcells, each governed by cubic autocatalator kinetics and coupledvia the diffusive interchange of the autocatalyst, are discussed.The equations obtained by linearizing about the spatially uniformsolution are considered first. These are seen to give the possibilityof bifurcations to spatially nonuniform solutions at both thesame parameter values as for the uncoupled system and, for relativelyweak coupling strengths ß, at further parameter valuesnot present in the uncoupled system. A weakly nonlinear analysisis then performed to describe the solution close to the bifurcationpoints and under the assumption of small ß. This givesfurther insights into the nature of the spatially nonuniformsolutions close to bifurcation, which are then followed numericallyusing a path-following technique. AU the extra solutions whichare due to the coupling are seen to be unstable close to theirbifurcation. However, these can undergo further secondary bifurcations,to produce new stable spatially nonuniform structures.     

A reaction-diffusion model for autocatalytic polymerization: I. Permanent form travelling waves

As a simple model for radical chain polymerization, the authorsstudy a system of coupled reaction-diffusion equations in whichthe reaction terms are autocatalytic and one of the diffusivities,representative of the evolving polymer structure, vanishes ata finite and nonzero polymer concentration. The existence ofa family of travelling wave solutions to the evolution equationsis proved, and some properties of these waveforms are derived.     

Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system

The initiation and propagation of reaction-diffusion travellingwaves in two regions coupled together by the linear diffusiveinterchange of the autocatalytic species is considered via aninitial-value problem in which amounts of the autocatalyst areintroduced locally into otherwise uniform concentrations ofthe other species. The reaction in one region is given by quadraticautocatalysis, while the reaction in the other is given by quadraticautocatalysis together with the linear decay of the autocatalyst.A priori bounds for the initial-value problem are obtained first.These, together with the solution valid for small inputs ofthe autocatalyst, enable conditions to be derived under whichtravelling waves can be initiated giving a wave for all withk<2, or if k<(2–1)/(–1), where k and aredimensionless groups corresponding to the rate of chemical decayof the autocatalyst and to the strength of coupling respectively.The global asymptotic stability of the unreacted state is thendiscussed. A solution valid for strong coupling between thetwo regions is then derived. The equations governing the permanent-formtravelling waves are treated in some detail, general propertiesof their solution and a solution valid for weak coupling beingderived. Finally, the large-time solution of the initial-valueproblem is considered. This shows that, when travelling wavesare initiated, they travel with their minimum possible speed:     

A reaction-diffusion model for autocatalytic polymerization: II. The initial value problem

An initial-boundary value problem arising from a simple modelfor radical chain polymerization is discussed in detail. Generalproperties of the solution are derived first and it is shownthat a moving interface develops. This separates a region wherethe polymer is sufficiently concentrated for it to be immobilefrom one where it is still free to diffuse. An asymptotic analysisis performed in this latter region, where it is shown that apermanent-form travelling wave (treated in Part I) developsin the long time structure and that this wave travels with itsminimum possible speed. Numerical results for the full initial-boundaryvalue problem are presented which confirm the asymptotic theoryand give results in regions not accessible to this analysis.     

The propagation of travelling waves in an open cubic autocatalytic chemical system

The reaction-diffusion travelling waves that can be initiatedin an open isothermal chemical system governed by cubic autocatalytickinetics are discussed. The system is shown to be capable ofsustaining up to three spatially uniform steady states, the(trivial) unreacted state, which is always stable (a node),and two nontrivial states, one of which is always unstable (asaddle point). The third state can change its stability throughHopf bifurcation (both subcritical and supercritical). Thisallows the possibility of two sorts of travelling wave beingestablished; there are wave profiles which connect the unreactedstate ahead to the nontrivial state at the rear, and wave profiles(pulse waves) which have the unreacted state at both the frontand rear. The conditions under which a particular wave is initiatedare considered by both a discussion of the (ordinary) differentialequations governing the travelling waves and by numerical integrationsof an initial-value problem. This treatment also reveals thepossibility of a stable travelling wave propagating throughthe system, leaving behind a temporally unstable stationarystate. Under these conditions, spatiotemporal chaotic behaviouris seen to develop after the passage of the wave.     

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.