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• Analysed in detail a two-step nonadiabatic competitive exothermic reaction scheme. • Identified two flame regimes and showed the difference between the adiabatic and nonadiabatic combustion waves. • Investigated the effect of heat loss on the behaviour of combustion waves. • Determined the extinction conditions analytically and numerically. We study the properties of travelling combustion waves in a diffusional thermal model with a two-step competitive exothermic reaction mechanism considering the system in one spatial dimension under nonadiabatic conditions. Based on the notion of the crossover temperature, the model is first examined analytically to predict the behaviour of travelling combustion waves in the limit of large activation energies. It is then studied numerically over a wide range of parameter values, such as those describing the ratios of the enthalpies, pre-exponential factors and activation energies. It is demonstrated that the nonadiabatic flame speed as a function of these parameters is a single-valued monotonic function with two flame regimes identified, each of which represents the region of the parameter values when one reaction dominates the other, while the flame speed as a function of the activation energy of the reaction R 2 (the exothermicity parameter) is either a c- or m-shaped. The extinction conditions for these two flame regimes are investigated analytically. [ABSTRACT FROM AUTHOR]

Abstract We consider non-adiabatic combustion waves arising from a two-step exothermic system. Our previous work showed that in certain parameter regions, the combustion wave can evolve to the "fast" solution branch, the "slow" solution branch or diffuse to the ambient temperature (extinction wave). Here, we are interested to find critical initial temperature profiles which evolve to these three types of steady solutions. For a particular family of temperature profiles, we construct a weight function which can be used to predict which of these three types of waves an initial temperature profile will evolve to. [ABSTRACT FROM AUTHOR]

We consider travelling wave solutions of a reaction-diffusion system corresponding to an adiabatic two-step competitive exothermic reaction scheme. In such a scheme, a combustion process is assumed to be lumped into two different exothermic reactions. Although the rate constants of the reactions are distinct, both reactions occur simultaneously and feed on the same reactant. The travelling wave solutions are obtained via the shooting-relaxation method. The linear stability analysis is conducted using the Evans function technique and the compound matrix method. Further, threshold values of parameters corresponding to Hopf points are established. It is shown that the system exhibits pulsating behaviour when the parameter values are greater than the threshold values. The onset of instability is found for a broad range of parameter values. Two different numerical methods are then used to obtain solutions from the governing partial differential equations to validate the results. [ABSTRACT FROM AUTHOR]

In this paper we investigate the linear stability and properties, such as speed, of the planar traveling combustion front. The speed of the front is estimated both analytically, using the matched asymptotic expansion, and numerically, by means of the shooting and relaxation methods. The Evans function approach extended by the compound matrix method is employed to numerically solve the linear stability problem for the traveling wave solution. [ABSTRACT FROM AUTHOR]

We study the effects of two-dimensional chaotic advection on a chemical system characterised by competitive exothermic and endothermic reactions. In previous studies, in which advective flow and reaction processes were assumed to dominate weak diffusive effects, two distinct behaviours were observed in the system. The first, when the stirring is fast and the reaction is slow. In this case, flame quenching occurs. In contrast, when the stirring is slow and the reaction is fast, local temperature perturbations lead to a stationary flame with a complex filament structure. We show here that, when the diffusion process is more influential, as for example in many microfluidic contexts, a third type of behaviour is possible, in which an expanding swirling travelling wave develops. We explore the parameter values which separate these different behaviours. [ABSTRACT FROM AUTHOR]

Addresses the problem of determining watershed initial conditions for combustion waves using a weight-function approach. Governing partial differential equation; Pseudo-watersheds and their relationships with exact watersheds; Combustion models; Nonlinear reaction-diffusion conservation equation.

Combustion processes almost never completely exhaust all of the available fuel. In this paper, we will consider three combustion scenarios (back-to-back premixed flames in stagnation point flow, travelling combustion waves, and microgravity spherical flame balls) and show how to calculate the amount of fuel which will be left over no matter how long we allow the combustion processes to continue. Mathematical biologists will be familiar with an analogue of this in disease modelling, where there are always some susceptibles left after the passage of an epidemic. In a bushfire or forest fire context, this is seen as unburnt solid fuel. The main reason for combustion leftovers is the heat loss, which can never be completely eliminated. [Copyright &y& Elsevier]