A combination of two semi-analytical method called “singular perturbed homotopy analysis method, (SPHAM)” applied to combustion of spray fuel droplets

In this paper we combined two well-known analytical methods: the homotopy analysis method (HAM) and the method of integral (invariant) manifold to investigate the problems of auto-ignition of a polydisperse fuel spray. We call this combination the singular perturbed homotopy analysis method (SPHAM). In many cases, combustion processes are described by mathematical models that include a set of highly nonlinear differential equations that are characterized by a different time scale (so-called multi-scale systems). For example, the temperature is a fast variable, due to the Arrhenius factor, compared to the radius evolution variable (the evaporation process). We apply the SPHAM method to problems of thermal explosion in two-phase combustible mixtures of gas with polydisperse fuel droplets. We analyze a dependence of our analytical and/or numerical results for different practical probabilistic distributions of fuel droplets, that are modeled by continuous probability distribution functions. By applying the SPHAM, we derived an analytical solution of the system for comparatively simple models of thermal explosion and compared our results with numerical simulations