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

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 $\gamma \ge 6\mu R_0$, 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, $\lambda \approx 2R_0$, 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 elastocapillary instabilities, with the simple cylinder's infinite wavelength being the exception rather than the rule.

Figure 1

Solid elastic cylinders undergo a peristaltic instability if surface tension, $\gamma$, is sufficient. Top: Experiment in gels from [5]. Bottom: Schematic of the instability.

Figure 2

Critical surface tension as a function of $k$ for a solid cylinder. The first unstable mode has infinite wavelength.

Figure 3

Surface-tension-driven peristaltic instability in a cylindrical cavity, radius $R_0$, through an infinite elastic solid.

Figure 4

Critical surface tension as a function of $k$ for an incompressible cylindrical cavity. The first unstable mode again has infinite wavelength.

Figure 5

Peristaltic instability in a solid cylinder of modulus ${\mu }_i$ through a bulk solid with modulus ${\mu }_o$. (a) Critical surface tension for instability for concentric cylinders with ${\mu }_i={\mu }_o$. The first unstable mode has finite wavelength with $k{R}_0=0.561992...$. (b) Surface tension for first instability as a function of modulus ratio. (c) First unstable wavelength as a function of modulus ratio.

Figure 6

(a) Critical surface tension as a function of the axial wave number for a squeezed cavity: the threshold surface tension is minimized at ${\gamma }_c\approx 2.543...\mu R_0$ by $k{R}_0=3.145...$. (b) Finite element analysis of the squeezed cavity instability. The minimum (final state) radius of the cavity, $R_{\text{min}}$, falls as $\gamma$ is increased. As shown in the insets on the right, for $\gamma$ below threshold, the radius contracts homogeneously along the whole channel as described by Eq. (28). At threshold, the instability proceeds subcritically and the system jumps to a highly peristaltic state with a zero (or almost zero) minimum radius. Inset color indicates isotropic stress (dark red, $<-1.25\mu$; dark blue, $>1.25\mu$) and the lines indicate every eighth mesh line.