Clarke's equation

In combustion, Clarke's equation is a third-order nonlinear partial differential equation, first derived by John Frederick Clarke in 1978.^[1][2][3][4] 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.^[5] The equation reads as

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

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

See also

• Frank-Kamenetskii theory

References

1. Clarke, J. F. (1978). "A progress report on the theoretical analysis of the interaction between a shock wave and an explosive gas mixture", College of Aeronautics report. 7801, Cranfield Inst. of Tech.
2. Clarke, J. F. (1978). Small amplitude gasdynamic disturbances in an exploding atmosphere. Journal of Fluid Mechanics, 89(2), 343–355.
3. Clarke, J. F. (1981), "Propagation of Gasdynamic Disturbances in an Explosive Atmosphere", in Combustion in Reactive Systems, J.R. Bowen, R.I. Soloukhin, N. Manson, and A.K. Oppenheim (Eds), Progress in Astronautics and Aeronautics, pp. 383-402.
4. Clarke, J. F. (1982). "Non-steady Gas Dynamic Effects in the Induction Domain Behind a Strong Shock Wave", College of Aeronautics report. 8229, Cranfield Inst. of Tech. https://repository.tudelft.nl/view/aereports/uuid%3A9c064b5f-97b4-4527-a97e-a805d5e1abd7
5. Bray, K. N. C.; Riley, N. (2014). "John Frederick Clarke 1 May 1927 – 11 June 2013". Biographical Memoirs of Fellows of the Royal Society. 60: 87–106. doi:10.1098/rsbm.2014.0012.