We study the motion of a fluid within a rigid spherical container subject to small-amplitude periodic rotations. The sphere is divided into two equal portions by an impermeable stretched elastic membrane whose boundary is attached to the container wall. The model aims to represent in a simplified fashion the dynamics of a vitreous membrane subject to microsaccadic movements of the human eye, assuming the vitreous to be liquefied. The vitreous is modelled as a Newtonian, incompressible fluid in irrotational motion and the problem is linearized, taking advantage of the hypothesis of small-amplitude eye rotations. Results show that, due to the presence of the fluid, the natural frequencies of oscillation of the membrane decrease significantly with respect to the case of a free membrane. Moreover, oscillations of a stretched membrane are found to be resonantly excited by rotations of the sphere with frequencies which are typical of microsaccadic eye movements. This study suggests the possibility that oscillations of vitreous membranes may induce the development of large tensile stresses capable of producing a retinal detachment. Such a conclusion will have to be further substantiated by more refined analyses accounting for further effects, such as nonlinearity and the possible viscoelastic behaviour of the vitreous located on one or both sides of the membrane.