Creeping motion of a solid particle inside a spherical elastic cavity

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 the 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.