  It is known that an object translating parallel to a soft wall in a viscous
fluid produces hydro- dynamic stresses that deform the wall, which, in turn,
results in a lift force on the object. Recent experiments with cylinders
sliding under gravity near a soft incline, which confirmed theoretical
arguments for the lift force, also reported an unexplained steady-state
rotation of the cylinders [Saintyves et al. PNAS 113(21), 2016]. Motivated by
these observations, we show, in the lubrication limit, that an infinite
cylinder that translates in a viscous fluid parallel to a soft wall at constant
speed and separation distance must also rotate in order to remain free of
torque. Using the Lorentz reciprocal theorem, we show analytically that for
small deformations of the elastic layer, the angular velocity of the cylinder
scales with the cube of the sliding speed. These predictions are confirmed
numerically. We then apply the theory to the gravity-driven motion of a
cylinder near a soft incline and find qualitative agreement with the
experimental observations, namely that a softer elastic layer results in a
greater angular speed of the cylinder.
