  On the basis of the linear hydrodynamic equations, we present an analytical
theory for the low-Reynolds-number motion of a solid particle moving inside a
larger spherical elastic cavity which can be seen as a model system for a fluid
vesicle. In the particular situation where the particle is concentric with the
cavity, we use the stream function technique to find exact analytical solutions
of the fluid motion equations on both sides of the elastic cavity. In this
particular situation, we find that the solution of the hydrodynamic equations
is solely determined by membrane shear properties and that bending does not
play a role. For an arbitrary position of the solid particle within the
spherical cavity, we employ the image solution technique to compute the
axisymmetric flow field induced by a point force (Stokeslet). We then obtain
analytical expressions of the leading order mobility function describing the
fluid-mediated hydrodynamic interactions between the particle and confining
elastic cavity. In the quasi-steady limit of vanishing frequency, we find that
the particle self-mobility function is higher than that predicted inside a
rigid no-slip cavity. Considering the cavity motion, we find that the
pair-mobility function is determined only by membrane shear properties. Our
analytical predictions are supplemented and validated by fully-resolved
boundary integral simulations where a very good agreement is obtained over the
whole range of applied forcing frequencies.
