The Dynamics of an Elastic Membrane Using the Impulse Method

A validation study of the impulse method applied to the motion of a closed elastic membrane under tension separating two incompressible, inviscid fluids in two dimensions is presented. The approach consists of a nonlinear analysis based on a small-amplitude perturbation of an exact solution. The equations of motion for the Fourier coefficients of the solution are developed to two orders beyond the leading-order problem. The nonlinear terms in the equations depict the coupling of the Fourier modes and account for the temporal variation of the tension. The last order of the expansion is used to compute frequency corrections to the driving modes. Solutions for various problems are found and compared with a numerical method based on impulse variables. The results show that the numerical periods and amplitudes of the oscillations approach the values predicted by the perturbation analysis as the numerical smoothing parameter is reduced. This validates the use of impulse methods for free-boundary motion with surface forces.