Parameterisations of slow invariant manifolds: application to a spray ignition and combustion model
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A wide range of dynamic models, including those of heating, evaporation and ignition processes in fuel sprays, is characterised by large differences in the rates of change of variables. Invariant manifold theory is an effective technique for investigation of these systems. In constructing the asymptotic expansions of slow invariant manifolds, it is commonly assumed that a limiting algebraic equation allows one to find a slow surface explicitly. This is not always possible due to the fact that the degenerate equation for this surface (small parameter equal to zero) is either a high degree polynomial or transcendental. In many problems, however, the slow surface can be described in a parametric form. In this case, the slow invariant manifold can be found in parametric form using asymptotic expansions. If this is not possible, it is necessary to use an implicit presentation of the slow surface and obtain asymptotic representations for the slow invariant manifold in an implicit form. The results of development of the mathematical theory of these approaches and the applications of this theory to some examples related to modelling combustion processes, including those in sprays, are presented.
KeywordsInvariant manifold Spray ignition and combustion The system-order reduction
E. Shchepakina was supported by the Ministry of Education and Science of the Russian Federation (Project RFMEFI58716X0033). S. Sazhin was supported by EPSRC (UK) (grant EP/M002608/1).
- 6.Strygin VV, Sobolev VA (1976) Effect of geometric and kinetic parameters and energy dissipation on the orientation stability of satellites with double spin. Cosm Res 14:331–335Google Scholar
- 7.Strygin VV, Sobolev VA (1977) Asymptotic methods in the problem of stabilization of rotating bodies by using passive dampers. Mech Solids 5:19–25Google Scholar
- 8.Sobolev VA, Strygin VV (1978) Permissibility of changing over to precession equations of gyroscopic systems. Mech Solids 5:7–13Google Scholar
- 9.Gol’dshtein VM, Sobolev VA (1988) A qualitative analysis of singularly perturbed systems. Institut matematiki SO AN SSSR, Novosibirsk (in Russian)Google Scholar
- 11.Shchepakina E, Sobolev V, Mortell MR (2014) Singular perturbations: introduction to system order reduction methods with applications. Springer lecture notes in mathematics, vol 2114. Springer, BerlinGoogle Scholar
- 18.Shchepakina E, Sobolev V, Sazhin SS (2016) System order reduction methods in spray ignition problems. In: de Sercey G, Sazhin SS (eds) Proceedings of the 27th European conference on liquid atomization and spray systems. Paper P-03Google Scholar
- 25.Johnson RS (2005) Singular perturbation theory. Springer, New York, p 263Google Scholar