  Motivated by recent experiments, we consider theoretically the compression of
droplets pinned at the bottom on a surface of finite area. We show that if the
droplet is sufficiently compressed at the top by a surface, it will always
develop a shape instability at a critical compression. When the top surface is
flat, the shape instability occurs precisely when the apparent contact angle of
the droplet at the pinned surface is pi, regardless of the contact angle of the
upper surface, reminiscent of past work on liquid bridges and sessile droplets
as first observed by Plateau. After the critical compression, the droplet
transitions from a symmetric to an asymmetric shape. The force required to
deform the droplet peaks at the critical point then progressively decreases
indicative of catastrophic buckling. We characterize the transition in droplet
shape using illustrative examples in two dimensions followed by perturbative
analysis as well as numerical simulation in three dimensions. When the upper
surface is not flat, the simple apparent contact angle criterion no longer
holds, and a detailed stability analysis is carried out to predict the critical
compression.
