  A deep understanding of the physical interactions between nanoparticles and
target cell membranes is important in designing efficient nanocarrier systems
for drug delivery applications. Here, we present a theoretical framework to
describe the hydrodynamic flow field induced by a point-force singularity
(Stokeslet) directed parallel to a finite-sized elastic membrane endowed with
shear and bending rigidities. We formulate the elastohydrodynamic problem as a
mixed-boundary-value problem, which we then reduce into a well-behaved system
of integro-differential equations. It follows that shear and bending linearly
decouple so that the solution of the overall flow problem can be obtained by
linear superposition of the contributions arising from these modes of
deformation. Additionally, we probe the effect of the membrane on the
hydrodynamic drag acting on a nearby particle, finding that, in a certain range
of parameters, translational motion near an elastic membrane with only
energetic resistance toward shear can, surprisingly, be sped up compared to
bulk fluid. Our results may find applications in microrheological
characterizations of colloidal systems near elastic confinements.
