On the damped oscillations of an elastic quasi-circular membrane in a two-dimensional incompressible fluid

We propose a procedure – partly analytical and partly numerical – to find the frequency and the damping rate of the small-amplitude oscillations of a massless elastic capsule immersed in a two-dimensional viscous incompressible fluid. The unsteady Stokes equations for the stream function are decomposed into normal modes for the angular and temporal variables, leading to a fourth-order linear ordinary differential equation in the radial variable. The forcing terms are dictated by the properties of the membrane and result in jump conditions at the interface between the internal and external media. The equation can be solved numerically, and excellent agreement is found with a fully computational approach that we have developed in parallel. Comparisons are also shown with results available in the scientific literature for drops, and a model based on the concept of entrained fluid is presented, which allows for a good representation of the present results and a consistent interpretation of the underlying physics.