Capillary instability on a hydrophilic stripe

A recent experiment showed that cylindrical segments of water filling a hydrophilic stripe on an otherwise hydrophobic surface display a capillary instability when their volume is increased beyond the critical volume at which their apparent contact angle on the surface reaches 90 degrees (Gau et al 1999 Science 283 46-9). Surprisingly, the fluid segments did not break up into droplets-as would be expected for a classical Rayleigh-Plateau instability-but instead displayed a long-wavelength instability where all excess fluid gathered in a single bulge along each stripe. We consider here the dynamics of the flow instability associated with this setup. We perform a linear stability analysis of the capillary flow problem in the inviscid limit. We first confirm previous work showing that all cylindrical segments are linearly unstable if (and only if) their apparent contact angle is larger than 90 degrees. We then demonstrate that the most unstable wavenumber for the surface perturbation decreases to zero as the apparent contact angle of the fluid on the surface approaches 90 degrees, allowing us to re-interpret the creation of bulges in the experiment as a zero-wavenumber capillary instability. A variation of the stability calculation is also considered for the case of a hydrophilic stripe located on a wedge-like geometry.