Liñán's equation

Near the extinction of the diffusion flame, ${\displaystyle \delta }$ is order unity. The equation has no solution for ${\displaystyle \delta <\delta _{E}}$, where ${\displaystyle \delta _{E}}$ is the extinction Damköhler number. For ${\displaystyle \delta >\delta _{E}}$ with ${\displaystyle |\gamma |<1}$, the equation possess two solutions, of which one is an unstable solution. Unique solution exist if ${\displaystyle |\gamma |>1}$ and ${\displaystyle \delta >\delta _{E}}$. The solution is unique for ${\displaystyle \delta >\delta _{I}}$, where ${\displaystyle \delta _{I}}$ is the ignition Damköhler number.

Liñán also gave a correlation formula for the extinction Damköhler number, which is increasingly accurate for ${\displaystyle 1-\gamma \ll 1}$,

${\displaystyle \delta _{E}=e[(1-\gamma )-(1-\gamma )^{2}+0.26(1-\gamma )^{3}+0.055(1-\gamma )^{4}].}$

The generalized Liñán's equation is given by

${\displaystyle {\frac {d^{2}y}{d\zeta ^{2}}}=(y-\zeta )^{m}(y+\zeta )^{n}e^{-\delta ^{-1/3}(y+\gamma \zeta )}}$

where ${\displaystyle m}$ and ${\displaystyle n}$ are constant reaction orders of fuel and oxidizer, respectively.

In the Burke–Schumann limit, ${\displaystyle \delta \rightarrow \infty }$. Then the equation reduces to

${\displaystyle {\frac {d^{2}y}{d\zeta ^{2}}}=(y-\zeta )^{m}(y+\zeta )^{n},\quad \zeta \rightarrow \pm \infty$ :\,{\frac {dy}{d\zeta }}=\pm 1.}

An approximate solution to this equation was developed by Liñán himself using integral method in 1963 for his thesis,[9]

${\displaystyle y(\zeta )=y_{m}+(\zeta -\zeta _{m})\operatorname {erf} [k(\zeta -\zeta _{m})]-{\frac {1}{{\sqrt {\pi }}k}}\left[1-e^{-k^{2}(\zeta -\zeta _{m})^{2}}\right],}$

where ${\displaystyle \mathrm {erf} }$ is the error function and

${\displaystyle {\begin{aligned}\zeta _{m}&={\frac {n-m}{n+m}}y_{m},\\y_{m}&={\frac {m+n}{2}}\left[{\frac {2}{\pi m^{2}n^{2}}}\left({\sqrt {1+{\frac {\pi (m+n)}{2mn}}}}-1\right)\right]^{\frac {1}{m+n+1}},\\k&={\frac {\sqrt {\pi }}{2}}m^{m}n^{n}\left({\frac {2y_{m}}{m+n}}\right)^{m+n}.\end{aligned}}}$

Here ${\displaystyle \zeta =\zeta _{m}}$ is the location where ${\displaystyle y(\zeta )}$ reaches its minimum value ${\displaystyle y(\zeta _{m})=y_{m}}$. When ${\displaystyle m=n=1}$, ${\displaystyle \zeta _{m}=0}$, ${\displaystyle y_{m}=0.8702}$ and ${\displaystyle k=0.6711}$.

• Liñán's diffusion flame theory