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

${\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\left({\frac {\partial \theta }{\partial t}}-\gamma \delta e^{\theta }\right)=\nabla ^{2}\left({\frac {\partial \theta }{\partial t}}-\delta e^{\theta }\right)}$

or, alternatively

${\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right){\frac {\partial \theta }{\partial t}}=\left(\gamma {\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right)\delta e^{\theta }}$

where ${\displaystyle \theta }$ is the non-dimensional temperature perturbation, ${\displaystyle \gamma >1}$ is the specific heat ratio and ${\displaystyle \delta }$ is the relevant Damköhler number. The term ${\displaystyle \partial \theta /\partial t-e^{\theta }}$ describes the thermal explosion at constant pressure and the term ${\displaystyle \partial \theta /\partial t-\gamma e^{\theta }}$ describes the thermal explosion at constant volume. Similarly, the term ${\displaystyle \partial ^{2}/\partial t^{2}-\nabla ^{2}}$ describes the wave propagation at adiabatic sound speed and the term ${\displaystyle \gamma \partial ^{2}/\partial t^{2}-\nabla ^{2}}$ describes the wave propagation at isothermal sound speed. Molecular transports are neglected in the derivation.

It may appear that the parameter ${\displaystyle \delta }$ can be removed from the equation by the transformation ${\displaystyle (x,t)\to (\delta x,\delta t)}$, it is, however, retained here since ${\displaystyle \delta }$ may also appear in the initial and boundary conditions.