  We study the bendocapillary instability of a liquid droplet that part fills a
flexible walled channel. Inspired by experiments in which a `weaving' pattern
emerges as droplets of liquid are condensed slowly into deformable
microchannels, we develop a mathematical model of this instability. We describe
equilibria of the system, and use a combination of numerical methods, and
asymptotic analysis in the limit of small channel wall deflections, to
elucidate the key features of this instability. We find that configurations are
always unstable to perturbations of sufficiently small wavenumber, that the
growth rate of the instability is highly sensitive to the volume of liquid in
the channel, and that both wetting and non-wetting configurations are
susceptible to the instability in the same channel. Insight into novel
interfacial instabilities opens the possibility for their control and thus
exploitation in processes such as microfabrication.
