Thermal explosion in a droplet-gas cloud

The effect of the presence of a spray of liquid fuel on thermal explosion in a combustible droplet-gas cloud is investigated. By 'thermal explosion' we refer exclusively to the initial stages of the behaviour of the combustible medium as its temperature begins to rise and various competing physical and chemical processes are called into play. A qualitative analysis of the system of governing equations is carried out using an advanced geometrical asymptotic technique (the integral manifold method). Possible types of dynamical behaviour of the system are classified and parametric regions of their existence are determined analytically. It is demonstrated that the original problem can be decomposed into two subproblems, due to the underlying hierarchical time scale structure. The first subproblem relates to the droplet heat up period, for which a relatively rapid time scale is applicable. The second subproblem begins at the saturation point. For the latter, more significant second stage, it is found that there are five main dynamical regimes: slow regimes, conventional fast explosive regimes, thermal explosion with freeze delay and two different types of thermal explosion with delay (the concentration of the combustible gas decreases or increases). Upper and lower bounds for the delay time are derived analytically and compared with results of numerical simulations, with rather satisfactory agreement.