  We study the motion of a solid particle immersed in a Newtonian fluid and
confined between two parallel elastic membranes possessing shear and bending
rigidity. The hydrodynamic mobility depends on the frequency of the particle
motion due to the elastic energy stored in the membrane. Unlike the
single-membrane case, a coupling between shearing and bending exists. The
commonly used approximation of superposing two single-membrane contributions is
found to give reasonable results only for motions in the parallel, but not in
the perpendicular direction. We also compute analytically the membrane
deformation resulting from the motion of the particle, showing that the
presence of the second membrane reduces deformation. Using the
fluctuation-dissipation theorem we compute the Brownian motion of the particle,
finding a long-lasting subdiffusive regime at intermediate time scales. We
finally assess the accuracy of the employed point-particle approximation via
boundary-integral simulations for a truly extended particle. They are found to
be in excellent agreement with the analytical predictions.
