Plateau–Rayleigh instability

Remarks

The linear modes satisfy Rayleigh's dispersion relation $$s^2 = \frac{\sigma}{\rho R^3} \, kR \left( 1 - k^2 R^2 \right) \frac{I_1(kR)}{I_0(kR)},$$ where $I_0$ and $I_1$ are modified Bessel functions of the first kind. So a mode is unstable exactly when $kR < 1$, that is, when its wavelength $2\pi / k$ exceeds the circumference $2\pi R$. The growth rate is largest at $kR \approx 0.697$, a wavelength of about $9.02 R$, which sets the typical spacing of the drops.

Example: for $kR = 1/2$, the perturbed column with the same volume has smaller surface area than the straight one, and $s^2 > 0$. Non-example: for $kR = 2$, the wavelength $\pi R$ is shorter than the circumference, the area increases, and $s^2 < 0$, so the mode oscillates as a capillary wave on the jet.