  We consider the motion of a fluid-immersed negatively buoyant particle in the
vicinity of a thin compressible elastic wall, a situation that arises in a
variety of technological and natural settings. We use scaling arguments to
establish different regimes of sliding, and complement these estimates using
thin-film lubrication dynamics to determine an asymptotic theory for the
sedimentation, sliding, and spinning motions of a cylinder. The resulting
theory takes the form of three coupled nonlinear singular-differential
equations. Numerical integration of the resulting equations confirms our
scaling relations and further yields a range of unexpected behaviours. Despite
the low-Reynolds feature of the flow, we demonstrate that the particle can
spontaneously oscillate when sliding, can generate lift via a Magnus-like
effect, can undergo a spin-induced reversal effect, and also shows an unusual
sedimentation singularity. Our description also allows us to address a
sedimentation-sliding transition that can lead to the particle coasting over
very long distances, similar to certain geophysical phenomena. Finally, we show
that a small modification of our theory allows to generalize the results to
account for additional effects such as wall poroelasticity.
