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Using the method of integral (invariant) manifolds, the intrinsic low-dimensional manifolds (ILDM) method is analysed. This is a method for identifying invariant manifolds of a system's slow dynamics and has proven to be an efficient tool in modelling of laminar and turbulent combustion. It allows treating multi-scale systems by revealing their hidden hierarchy and decomposing the system dynamics into fast and slow motions. The performed analysis shows that the original ILDM technique can be interpreted as one of the many possible realizations of the general framework, which is based on a special transformation of the original coordinates in the state space. A modification of the ILDM is proposed based on a new definition of the transformation matrix. The proposed numerical procedure is demonstrated on linear examples and highly non-linear test problems of mathematical theory of combustion and demonstrates in some cases better performance with respect to the existing one.

The problem of thermal explosion in a flammable gas mixture with addition of volatile fuel droplets is studied based on the asymptotic method of integral manifolds. The model for the radiative heating of droplets takes into account the semitransparency of droplets. A simplified model for droplet heat-up is used. The results of the analysis are applied to the modelling of thermal explosion in diesel engines. Two distinct dynamical situations have been considered, depending on the initial droplet concentration. These are 'far zone' (small initial liquid volume fraction and small droplet radii) and 'near zone' (large initial liquid volume fraction and large droplet radii). The conditions of the first zone are typical for the areas in the combustion chamber which are far from the fuel injectors, while the conditions of the second zone are typical for the areas in the combustion chamber which are relatively close to the fuel injectors. It has been pointed out that small droplets' heating and evaporation time in the far zone is smaller than the chemical ignition delay of the fuel vapor/air mixture. The total ignition delay decreases with increasing initial gas temperature. In the near zone for large droplets, the process starts with the initial gas cooling and slight heating of droplets. This is followed by a relatively slow heating of gas due to the chemical reaction, and further droplet heating. The total ignition delay in the near zone is larger than in the far zone. It is expected that before thermal explosion in the near zone takes place, the droplets break up and are removed from this zone. In optically thick gas effects of thermal radiation are negligible for small droplets but are noticeable for large droplets.