An Asymptotic Derivation of Two Models in Flame Theory Associated with the Constant Density Approximation

Abstract

We consider the general equations of combustion theory. By expanding in appropriately chosen small parameters, we derive two simplified models for the leading term of such an expansion. Both are associated with the constant density approximation. In the first model, the equations of fluid dynamics are completely decoupled from the equations governing heat and mass transport. The resulting model is generally referred to as the constant density approximation, or as the diffusional thermal model. In the second model, there is a weak coupling between the equations of fluid dynamics and the equations for temperature and concentration. Specifically the coupling, which enters through the effect of variable density, which in turn is due to the thermal expansion of the gas in which a flame propagates, occurs only in the external forcing term, and not elsewhere in the fluid dynamical equations. Thus, our model is analogous to the Boussinesq model in hydrodynamics.