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