Finite wavelength surface-tension driven instabilities in soft solids, including instability in a cylindrical channel through an elastic solid

Abstract

We deploy linear stability analysis to find the threshold wavelength ($\lambda$) and surface tension ($\gamma$) of Rayleigh-Plateau type "peristaltic" instabilities in incompressible neo-Hookean solids in a range of cylindrical geometries with radius $R_0$. First we consider a solid cylinder, and recover the well-known, infinite wavelength instability for $\gamma\ge6 \mu R_0$, where $\mu$ is the solid's shear modulus. Second, we consider a volume-conserving (e.g.\ fluid filled and sealed) cylindrical cavity through an infinite solid, and demonstrate infinite wavelength instability for $\gamma\ge 2 \mu R_0$. Third, we consider a solid cylinder embedded in a different infinite solid, and find a finite wavelength instability with $\lambda\propto R_0$, at surface tension $\gamma \propto \mu R_0$, where the constants depend on the two solids' modulus ratio. Finally, we consider an empty cylindrical channel (or filled with expellable fluid) through an infinite solid, and find an instability with finite wavelength, $\lambda \approx2 R_0$, for $\gamma\ge 2.543... \mu R_0$. Using finite-strain numerics, we show such a channel jumps at instability to a highly peristaltic state, likely precipitating it's blockage or failure. We argue that finite wavelengths are generic for elasto-capillary instabilities, with the simple cylinder's infinite wavelength being the exception rather than the rule.