Finite and infinite wavelength elastocapillary instabilities with cylindrical geometry

In an elastic cylinder with shear modulus $\mu$, radius $R_0$ and surface tension $\gamma$ we can define an emergent elastocapillary length $l=\gamma/\mu$. When this length becomes comparable to $R_0$ the cylinder becomes undergoes a Rayleigh-Plateaux type instability, but surprisingly, with infinite wavelength $\lambda$ rather than with wavelength $\lambda\sim R_0\sim l$. Here we take advantage of this infinite wavelength behaviour to construct a simple 1-D model of the elastocapillary instability in a cylindrical gel which permits a high-amplitude fully non-linear treatment. In particular, we show that the instability is sub-critical and entirely dependent on the elastic cylinder being subject to tension. We also discuss elastocapillary instabilities in a range of other cylindrical geometries, such a cylindrical cavities through a bulk elastic solid, or a solid cylinder embedded in a bulk elastic solid, and show that in these cases instability has finite wavelength. Thus infinite wavelength behaviour is a curiosity of elastic cylinders rather than the generic behaviour or elasto-capiliarity.