Darrieus–Landau instability

Definition

Model a planar premixed flame as a surface in $\R^3$ that separates fresh gas of density $\rho_u$ from burnt gas of density $\rho_b$. Let $r = \rho_u / \rho_b$ be the expansion ratio. Assume:

1. on each side the gas is incompressible and inviscid and solves the Euler equations, and gravity is neglected;
2. the front moves at the constant normal speed $S_L$ relative to the fresh gas;
3. mass and normal momentum are conserved across the front.

Perturb the plane front by a small wrinkle of wavenumber $k > 0$ proportional to $e^{i k y + \sigma t}$. The linearised problem gives the growth rate $$\sigma = \frac{S_L k}{r + 1} \left( \sqrt{r^3 + r^2 - r} - r \right).$$ The Darrieus–Landau instability is the instability of the front expressed by $\sigma > 0$.