  When an elastic object is dragged through a viscous fluid tangent to a rigid
boundary, it experiences a lift force perpendicular to its direction of motion.
An analogous lift mechanism occurs when a rigid symmetric object translates
parallel to an elastic interface or a soft substrate. The induced lift force is
attributed to an elastohydrodynamic coupling that arises from the breaking of
the flow reversal symmetry induced by the elastic deformation of the
translating object or the interface. Here we derive explicit analytical
expressions for the quasi-steady state lift force exerted on a rigid spherical
particle translating parallel to a finite-sized membrane exhibiting a
resistance toward both shear and bending. Our analytical approach proceeds
through the application of the Lorentz reciprocal theorem so as to obtain the
solution of the flow problem using a perturbation technique for small
deformations of the membrane. We find that the shear-related contribution to
the normal force leads to an attractive interaction between the particle and
the membrane. This emerging attractive force decreases quadratically with the
system size to eventually vanish in the limit of an infinitely-extended
membrane. In contrast, membrane bending leads to a repulsive interaction whose
effect becomes more pronounced upon increasing the system size, where the lift
force is found to diverge logarithmically for an infinitely-large membrane. The
unphysical divergence of the bending-induced lift force can be rendered finite
by regularizing the solution with a cut-off length beyond which the bending
forces become subdominant to an external body force.
