In this paper we investigate the linear stability and properties of the planar travelling non-adiabatic combustion front for the cases of zero and non-zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as 'fast' and 'slow'. The Evans function approach extended by the compound-matrix method is employed to numerically solve the linear-stability problem for the travelling-wave solution. We demonstrate that the 'slow' branch of the solutions is unstable, whereas the 'fast' branch can be stable or exhibits Hopf or Bogdanov-Takens instability, depending on the parameter values.

a b s t r a c tAlthough numerical methods have been developed for diffusion through single layer materials, few have been developed for multiple layers. Diffusion processes through a multilayered material are of interest for a wide range of applications, including industrial, biological, electrical, and environmental areas. We present finite difference schemes for multilayered materials with a range of matching conditions between the layers, in particular for a jump matching condition. We show the finite difference methods are flexible, simple to implement, and help illustrate interesting behaviour in multilayered diffusion.

A numerical method is used to show that heat loss increases can lead to a perioddoubling route to the cessation of propagation of solid fuel combustion. Oscillatory combustion waves are found in certain regions of the parameter space. The behaviour of these oscillatory waves becomes more complex as the heat loss is increased until extinction of the combustion reaction occurs. Large excursions in temperature, above the adiabatic temperature, are possible in the non-adiabatic case close to this extinction point.

The traditional combustion problems of calculating flame speeds for a premixed gaseous fuel and for a premixed solid fuel are revisited using a simpler (than previously) non-dimensional temperature. It turns out to be possible to carry out asymptotic calculations for flame speed and the agreement with corresponding numerical calculations is remarkably good. In each case the uniqueness of the speed is considered using phase plane methods, with a little effort to determine the nature of the 'cold' critical point.Consideration of the stability of the travelling combustion wave fronts suggests a period doubling route to chaos for the premixed solid fuel (as the exothermicity is decreased) and corresponds with previous work using different non-dimensional temperature and parameters.

We consider travelling wave solutions of a reaction-diffusion system corresponding to a single-step homogeneous premixed combustion scheme competitively coupled with an endothermic reaction. Properties of the travelling combustion fronts, such as the wave speed and the burnt temperature are derived numerically over a range of different parameter values, such as those describing the relative enthalpies, rates and activation energies of the endothermic and exothermic reactions. Unique combustion wave solutions are shown to exist for each distinct combination of the parameter values. These solutions are linearly stable if the heat release from the exothermic reaction is sufficiently large, otherwise the combustion waves develop pulsation. In particular, using a finite element package to numerically integrate the governing partial differential equations, period-1 and period-2 type oscillatory behaviour was observed prior to wave extinction.

[1] The tidal dynamics of Darwin Harbour, Australia, are simulated using a finite volume coastal ocean model. The calibrated model agreed well with the observed sea surface elevation and current velocity. Results indicate that the harbor's hydrodynamics are driven mainly by the tides, with wind and river inputs playing only small roles. The M 2 tide is dominant, with amplitude 1.7 m and peak current speed 3.0 m s À1 . Sensitivity tests using the model indicate that the mangrove areas and tidal flats play crucial roles in modulating tidal amplitudes and phases in the embayments, especially for the shallow water tides such as M 4 . Removal of the mangrove areas and tidal flats from Darwin Harbour would dampen the M 2 amplitude due to decreased shoaling effects but generate a 75.0% greater M 4 amplitude in parts of the harbor. Mangrove areas and tidal flats also affect tidal asymmetry through the changing amplitudes and phases of mainly the M 2 and M 4 tides. In Darwin Harbour, tidal asymmetry, measured by elevation and current skewness, would increase by up to 100% if the mangrove areas were removed. If the tidal flats were removed as well, the increase would be 120%. Therefore, reclamation of the mangrove areas and tidal flats may cause sediment siltation as a result of increased flood dominance. Although this study is site-specific, the model and our findings have a wider applicability to the effects of mangrove areas and tidal flats on tides and sediment transport in harbors and estuaries.

We model the increase in temperature in compost piles due to micro-organisms undergoing exothermic reactions. The model incorporates two types of heat release: one due to biological activity; and the other due to the oxidation of cellulosic materials. In this study we also include the consumption of oxygen. We investigate the bifurcation behaviour and compare the results obtained from models that include and exclude oxygen consumption in both one-and two-dimensional geometries.

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