ZFK equation, abbreviation for Zeldovich–Frank-Kamenetskii equation, is a reaction–diffusion equation that models premixed flame propagation. The equation is named after Yakov Zeldovich and David A. Frank-Kamenetskii who derived the equation in 1938. The equation is analogous to KPP equation except that it contains an exponential behaviour for the reaction term and it differs fundamentally from KPP equation with regards to the propagation velocity of the traveling wave. In non-dimensional form, the equation reads
∂ θ ∂ t = ∂ 2 θ ∂ x 2 + ω ( θ ) {\displaystyle {\frac {\partial \theta }{\partial t}}={\frac {\partial ^{2}\theta }{\partial x^{2}}}+\omega (\theta )}
with a typical form for ω {\displaystyle \omega } given by
ω = β 2 2 θ ( 1 − θ ) e − β ( 1 − θ ) {\displaystyle \omega ={\frac {\beta ^{2}}{2}}\theta (1-\theta )e^{-\beta (1-\theta )}}
where θ ∈ [ 0 , 1 ] {\displaystyle \theta \in [0,1]} is the non-dimensional dependent variable (typically temperature) and β {\displaystyle \beta } is the Zeldovich number. In the ZFK regime, β ≫ 1 {\displaystyle \beta \gg 1} . The equation reduces to Fisher's equation for β ≪ 1 {\displaystyle \beta \ll 1} and thus β ≪ 1 {\displaystyle \beta \ll 1} corresponds to KPP regime. The minimum propagation velocity U m i n {\displaystyle U_{min}} (which is usually the long time asymptotic speed) of a traveling wave in the ZFK regime is given by
U ZFK ∝ 2 ∫ 0 1 ω ( θ ) d θ {\displaystyle U_{\text{ZFK}}\propto {\sqrt {2\int _{0}^{1}\omega (\theta )d\theta }}}
whereas in the KPP regime, it is given by
U KPP = 2 d ω d θ | θ = 0 . {\displaystyle U_{\text{KPP}}=2{\sqrt {\left.{\frac {d\omega }{d\theta }}\right|_{\theta =0}}}.}
Traveling wave solution
Similar to Fisher's equation, a traveling wave solution can be found for this problem. Suppose the wave to be traveling from right to left with a constant velocity U {\displaystyle U} , then in the coordinate attached to the wave, i.e., z = x + U t {\displaystyle z=x+Ut} , the problem becomes steady. The ZFK equation reduces to
U d θ d z = d 2 θ d z 2 + β 2 2 θ ( 1 − θ ) e − β ( 1 − θ ) {\displaystyle U{\frac {d\theta }{dz}}={\frac {d^{2}\theta }{dz^{2}}}+{\frac {\beta ^{2}}{2}}\theta (1-\theta )e^{-\beta (1-\theta )}}
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