Matalon–Matkowsky–Clavin–Joulin theory

Matalon–Matkowsky–Clavin–Joulin theory refers to a theoretical hydrodynamic model of a premixed flame with a large-amplitude flame wrinkling, developed independently by Moshe Matalon & Bernard J. Matkowsky and Paul Clavin & Guy Joulin. The theory, for the first time, calculated the burning rate of the curved flame that differs from the burning rate of the planar flame due to flame stretch, associated with the flame curvature and the strain imposed on the flame by the flow field.

Burning rate formula
According to Matalon–Matkowsky–Clavin–Joulin theory, if $$S_L$$ and $$\delta_L$$ are the laminar burning speed and thickness of a planar flame (and $$\tau_L=D_{T,u}/S_L^2$$ be the corresponding flame residence time with $$D_{T,u}$$ being the thermal diffusivity in the unburnt gas), then the burning speed $$S_T$$ for the curved flame with respect to the unburnt gas is given by


 * $$\frac{S_T}{S_L} = 1 + \mathcal{M}_c \delta_L \nabla \cdot \mathbf{n} + \mathcal{M}_s \tau_L \mathbf{n}\cdot \nabla\mathbf v\cdot \mathbf{n}$$

where $$\mathbf{n}$$ is the unit normal to the flame surface (pointing towards the burnt gas side), $$\mathbf{v}$$ is the flow velocity field evalauted at the flame surface and $$\mathcal{M}_c$$ and $$\mathcal{M}_s$$ are the two Markstein numbers, associated with the curvature term $$\nabla \cdot \mathbf{n}$$ and the term $$\mathbf{n}\cdot \nabla\mathbf v\cdot \mathbf{n}$$ corresponding to flow strain imposed on the flame.