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
- [math]\displaystyle{ (\theta_t-\gamma e^{\theta})_{tt}=(\theta_t-e^\theta)_{xx} }[/math]
where [math]\displaystyle{ \theta }[/math] is the non-dimensional temperature perturbation and [math]\displaystyle{ \gamma }[/math] is the specific heat ratio. The term [math]\displaystyle{ \theta_t-e^\theta }[/math] describes the explosion at constant pressure and the term [math]\displaystyle{ \theta_t-\gamma e^\theta }[/math] describes the explosion at constant volume. Similarly, the term [math]\displaystyle{ (\ )_{tt}-(\ )_{xx} }[/math] describes the wave propagation at adiabatic sound speed and the term [math]\displaystyle{ \gamma(\ )_{tt}-(\ )_{xx} }[/math] describes the wave propagation at isothermal sound speed. Molecular transports are neglected in the derivation.
See also
References
- ↑ 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.
- ↑ Clarke, J. F. (1978). Small amplitude gasdynamic disturbances in an exploding atmosphere. Journal of Fluid Mechanics, 89(2), 343–355.
- ↑ 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.
- ↑ 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
- ↑ 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.
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