  A rigid object moving in a viscous fluid and in close proximity with an
elastic wall experiences self-generated elastohydrodynamic interactions. This
has been the subject of an intense research activity, with a recent and growing
attention given to the particular case of elastomeric and gel-like substrates.
Here, we address the situation where the elastic wall is replaced by a
capillary surface. Specifically, we analyze the lubrication flow generated by
the prescribed normal motion of a rigid infinite cylinder near the deformable
interface separating two immiscible and incompressible viscous fluids. Using a
combination of analytical and numerical treatments, we compute the emergent
capillary-lubrication force at leading order in compliance, and characterize
its dependencies with the interfacial tension, viscosities of the fluids, and
length scales of the problem. Interestingly, we identify two main
contributions: i) a velocity-dependent adhesion-like force ; ii) an
acceleration-dependant inertial-like force. Our results may have implications
for the mobility of colloids near complex interfaces and for the motility of
confined microbiological entities.
