  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 ninety degrees (Gau et al.,
Science, 283, 1999). 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 that all cylindrical segments are linearly unstable
if (and only if) their apparent contact angle is larger than ninety 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 ninety 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.
