  Pockets of viscous fluid coalescing beneath an elastic plate are encountered
in a wide range of natural phenomena and engineering processes, spanning across
scales. As the pockets merge, a bridge is formed with a height increasing as
the plate relaxes. We study the spatiotemporal dynamics of such an
elasto-hydrodynamic coalescence process by combining experiments, lubrication
theory and numerical simulations. The bridge height exhibits an exponential
growth with time, which corresponds to a self-similar solution of the
bending-driven thin-film equation. We address this unique self-similarity and
the self-similar shape of the bridge, both of which are corroborated in
numerical simulations and experiments.
