Analysis of the dynamics of fuel spray using asymptotic methods: The method of integral invariant manifolds

The current research deals with the thermal explosion and ignition of a mixture of carbon and air. The size distribution of the carbon particles is taken to be continuous and is characterized by a probability density function. The chemical reaction term is presented in the Arrhenius form with variable pre-exponential factor. Transforming the new model to a dimensionless form enables us to rewrite the model in a singular perturbed system of ordinary differential equations. This form of the model enables us to apply the method of integral manifold (MIM). As a result of this method we can derive an explicit expression for the thermal explosion limit which depends on the initial probability density function. Comparing our numerical results to the analytical results, we observe that the effect of the thermal radiation is significant, especially at high temperatures, and cannot be ignored in the analysis of the phenomena of the explosion and ignition.