I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the spatial $L^2$-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into traveling combustion waves.

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, $\tau $, between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on $\tau $. A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive $\tau $.

When an explosive burns, gaseous products are formed as a result. The interaction of the burning solid and gas is not well understood. More specifically, the process of the gaseous product heating the explosive is yet to be explored in detail. The present work sets out to fill some of that gap using mathematical modelling: this aims to track the temperature profile in the explosive. The work begins by modelling single-step reactions using a simple Arrhenius model. The model is then extended to include three-step reaction. An alternative asymptotic approach is also employed. There is close agreement between results from the full reaction-diffusion problem and the asymptotic problem.

We numerically study a thermal-diffusive model for smouldering combustion under microgravity with convective heat losses. In accordance with previous experimental observations, it is well known that porous materials burning against a gaseous oxidiser under microgravity exhibit various finger-like char patterns due to the destabilising effect of oxidiser transport. There is a close resemblance between the pattern-forming dynamics observed in the experiments with the mechanism of thermal-diffusive instability, similar to that occurring in low Lewis number premixtures. At large values of the Lewis number, the finger-like pattern coalesces and propagates as a stable front reminiscent of the pattern behaviour at large Péclet numbers in diffusion-limited systems. The significance of the order of the chemical kinetics for the coexistence of both upstream and downstream smoulder waves is also considered.

A diesel particulate filter (DPF) is a key technology to meet future emission standards of particulate matters (PM), mainly soot. It is generally consists of a wall-flow type filter positioned in the exhaust stream of a diesel vehicle. It is difficult to simulate the thermal flow in DPF, because we need to consider the soot deposition and combustion in the complex geometry of filter wall. In our previous study, we proposed an approach for the conjugate simulation of gas-solid flow. That is, the gas phase was simulated by the lattice Boltzmann method (LBM), coupled with the equation of heat conduction inside the solid filter substrate. However, its numerical procedure was slightly complex. In this study, to reduce numerical costs, we have tested a new boundary condition with chemical equilibrium in soot combustion at the surface of filter substrate. Based on the soot oxidation rate with catalysts evaluated in experiments, the lattice Boltzmann simulation of soot combustion in the catalyzed DPF is firstly presented to consider the process in the after-treatment of diesel exhaust gas. The heat and mass transfer is shown to discuss the effect of catalysts.

In the present paper a combined procedure for the quasi-dimensional modelling of heat transfer, combustion and knock phenomena in a “downsized” Spark Ignition two-cylinder turbocharged engine is presented. The procedure is extended to also include the effects consequent the Cyclic Variability. Heat transfer is modelled by means of a Finite Elements model. Combustion simulation is based on a fractal description of the flame front area. Cyclic Variability (CV) is characterized through the introduction of a random variation on a number of parameters controlling the rate of heat release (air/fuel ratio, initial flame kernel duration and radius, laminar flame speed, turbulence intensity). The intensity of the random variation is specified in order to realize a Coefficient Of Variation (COV) of the Indicated Mean Effective Pressure (IMEP) similar to the one measured during an experimental campaign. Moreover, the relative importance of the various concurring effects is established on the overall COV. A kinetic scheme is then solved within the unburned gas zone, characterized by different thermodynamic conditions occurring cycle-by-cycle. In this way, an optimal choice of the “knock-limited” spark advance is effected and compared with experimental data. Finally, the CV effects on the occurrence of individual knocking cycles are assessed and discussed.