We investigate the effect of thermal expansion and gravity on the propagation and stability of flames in inhomogeneous mixtures. We focus on laminar flames in the simple configuration of an infinitely long channel with rigid porous walls in order to understand the effect of inhomogeneities on these fundamental structures. The first part of the thesis is concerned with premixed flames propagating against a prescribed parallel (Poiseuille) flow and subject to thermal expansion. We show that in a narrow channel (corresponding to a relatively thick flame), if the Peclet number is fixed and of order unity, a premixed flame propagating against a parallel flow is governed by the equation for a planar premixed flame with an effective diffusion coefficient. The enhanced diffusion is shown to correspond to Taylor dispersion, or shear-enhanced diffusion. Several important applications of the results are discussed. One of the topics of relevance is the bending effect of turbulent combustion. The results of our analysis show that, for a large flow intensity, the effective propagation speed of the premixed flame for depends only on the Peclet number (which is equal to the Reynolds number if the Prandtl number is unity). This mimics the behaviour of the turbulent premixed flame when the effective propagation speed is plotted versus the turbulence intensity for fixed values of the Reynolds number. The second part of the thesis is concerned with triple flames, subject to thermal expansion and buoyancy. A study is undertaken to investigate the stability of a diffusion flame subject to these effects, which gives rise to a problem analogous to the classical Rayleigh--B\'nard convection problem. A linear stability analysis in the Boussinesq approximation is performed, which leads to analytical results showing that the Burke-Schumann flame is unstable if the Rayleigh number is above a critical value which is determined. Numerical results confirm and complement the analytical results. A full numerical investigation of the effects of gravity and thermal expansion on triple flames propagating in a direction perpendicular to the direction of gravity is then carried out. This configuration does not seem to have received dedicated attention in the literature. It is found that the well-known monotonic relationship between the propagation speed $U$ and the flame-front thickness $\epsilon$, which exists in the constant density case when the Lewis numbers are of order unity or larger, persists for triple flames undergoing thermal expansion. Under strong enough gravitational effects, however, the relationship is no longer found to be monotonic, exhibiting hysteresis if the Rayleigh number is large enough. Finally, the initiation of triple flames from a hot two-dimensional ignition kernel is investigated. Particular attention is devoted to the energy required for ignition and the transient evolution of triple flames after initiation. Steady, non-propagating, two-dimensional solutions representing "flame tubes" are determined; their thermal energy is used to define a minimum ignition energy for the two-dimensional triple flame in the mixing layer. The transient behaviour of triple flames following "energy-increasing" or "energy-decreasing" perturbations to the flame tube solutions is described in situations where the underlying diffusion flame is either stable or unstable.

The present work is concerned with the mathematical modelling of a bed of granular material in a gravitational field vertically fluidised by a vibrating surface. The particles are in rapid motion, and lose energy by inelastic collisions. The steady state is maintained by a balance of the rate of dissipation of energy in inelastic particle collisions and the rate of transfer of energy due to particle collisions with the vibrating surface. The limit where the energy dissipation due to inelastic collisions is small compared to the mean kinetic energy of the particles is considered. This non-equilibrium steady state is similar to a dilute gas at equilibrium with a uniform temperature and an exponentially decaying density, obtained from the ideal gas equation of state. From the analysis of this state, four non-dimensional numbers are derived which uniquely specify the state of the system. A perturbative analysis about the uniform temperature state is carried out and analytical solutions to the macroscopic variables of the system are obtained using two types of approximations. The first is a hydrodynamic model using constitutive relations from the general kinetic theory of granular media, and the second is a kinetic theory formulation derived exclusively for the vibro-fluidised bed. The latter permits an anisotropy between the horizontal and vertical directions due to the anisotropic nature of the source of energy at the bottom wall. The kinetic theory is extended to incorporate the corrections due to the high density effects, which is similar to the Enskog correction to dense gases. An event driven (ED), or hard sphere molecular dynamic (MD), simulation of the vibrated bed is carried out. The quantitative predictions of the theories are validated by the simulation. A systematic probing of the parameter space within the ED simulations revealed two new phenomena in a vibro-fluidised bed which are inhomogeneous in the horizontal direction. These are convection rolls similar to the Rayleigh-Benard instability in fluids, and a clustering instability leading to a phase separation. The instabilities are characterised using a phase diagram. The homogeneous states close to these new states are adequately described by the models developed here. An analysis of the stability of this state could have implications in understanding the instabilities in driven granular materials (such as in sheared media and fluidised beds) in general, and pattern formation in vibrated beds in particular.