Clarke's equation

In combustion, Clarke's equation is a third-order nonlinear partial differential equation, first derived by John Frederick Clarke in 1978. The equation describes the thermal explosion process, including both effects of constant-volume and constant-pressure processes, as well as the effects of adiabatic and isothermal sound speeds. The equation reads as

$(\theta _{t}-\gamma e^{\theta })_{tt}=(\theta _{t}-e^{\theta })_{xx}$

where $\theta$ is the non-dimensional temperature perturbation and $\gamma$ is the specific heat ratio. The term $\theta _{t}-e^{\theta }$ describes the explosion at constant pressure and the term $\theta _{t}-\gamma e^{\theta }$ describes the explosion at constant volume. Similarly, the term $(\ )_{tt}-(\ )_{xx}$ describes the wave propagation at adiabatic sound speed and the term $\gamma (\ )_{tt}-(\ )_{xx}$ describes the wave propagation at isothermal sound speed. Molecular transports are neglected in the derivation.

See also

• Frank-Kamenetskii theory