Liñán's equation


In the study of diffusion flame, Liñán's equation is a second-order nonlinear ordinary differential equation which describes the inner structure of the diffusion flame, first derived by Amable Liñán in 1974. The equation reads as
subjected to the boundary conditions
where is the reduced or rescaled Damköhler number and is the ratio of excess heat conducted to one side of the reaction sheet to the total heat generated in the reaction zone. If, more heat is transported to the oxidizer side, thereby reducing the reaction rate on the oxidizer side and consequently greater amount of fuel will be leaked into the oxidizer side. Whereas, if, more heat is transported to the fuel side of the diffusion flame, thereby reducing the reaction rate on the fuel side of the flame and increasing the oxidizer leakage into the fuel side. When , all the heat is transported to the oxidizer side and therefore the flame sustains extremely large amount of fuel leakage.
The equation is, in some aspects, universal since although Liñán derived the equation for stagnation point flow, assuming unity Lewis numbers for the reactants, the same equation is found to represent the inner structure for general laminar flamelets, having arbitrary Lewis numbers.

Existence of solutions

Near the extinction of the diffusion flame, is order unity. The equation has no solution for, where is the extinction Damköhler number. For with, the equation possess two solutions, of which one is an unstable solution. Unique solution exist if and. The solution is unique for, where is the ignition Damköhler number.
Liñán also gave a correlation formula for the extinction Damköhler number, which is increasingly accurate for,

Generalized Liñán's equation

The generalized Liñán's equation is given by
where and are constant reaction orders of fuel and oxidizer, respectively.

Large Damköhler number limit

In the Burke–Schumann limit,. Then the equation reduces to
An approximate solution to this equation was developed by Liñán himself using integral method in 1963 for his thesis,
where is the error function and
Here is the location where reaches its minimum value. When,, and.