Liñán's diffusion flame theory
Liñán diffusion flame theory is a theory developed by Amable Liñán in 1974 to explain the diffusion flame structure using activation energy asymptotics and Damköhler number asymptotics. Liñán used counterflowing jets of fuel and oxidizer to study the diffusion flame structure, analyzing for the entire range of Damköhler number. His theory predicted four different types of flame structure as follows, Mathematical description. The theory is well explained in the simplest possible model. Thus, assuming a one-step irreversible Arrhenius law for the combustion chemistry with constant density and transport properties and with unity Lewis number  reactants, the governing equation for the non-dimensional temperature field formula_1 in the stagnation point flow reduces to where formula_3 is the mixture fraction, formula_4 is the Damköhler number, formula_5 is the activation temperature and the fuel mass fraction and oxidizer mass fraction are scaled with their respective feed stream values, given by with boundary conditions formula_7. Here, formula_8 is the unburnt temperature profile (frozen solution) and formula_9 is the stoichiometric parameter (mass of oxidizer stream required to burn unit mass of fuel stream). The four regime are analyzed by trying to solve above equations using activation energy asymptotics and  Damköhler number asymptotics. The solution to above problem is multi-valued. Treating mixture fraction formula_3 as independent variable reduces the equation to with boundary conditions formula_12 and formula_13. Extinction Damköhler number. The reduced Damköhler number is defined as follows where formula_15 and formula_16. The theory predicted an expression for the reduced Damköhler number at which the flame will extinguish, given by where formula_18.