Oscillations of an inviscid ring-constrained charged drop

Dynamics of constrained liquid/gas interfaces are important in various applications e.g. capillary switches and liquid lenses. Here, linear oscillations of an inviscid conducting drop constrained by a ring of negligible thickness are studied using normal mode analysis. Similar to linear oscillations of a charged, inviscid free drop (Rayleigh 1882), theoretical analysis of the oscillations of a constrained drop yields an eigenvalue problem. The free and constrained drop problems, however, differ because of an additional boundary condition in the latter compared to the former. Specifically, vanishing of the shape perturbation at the constraint in the latter case results in a constrained optimization problem. Oscillation frequencies (eigenvalues) and drop shapes (eigenfunctions) are determined as a function of constraint location and total drop charge for the lowest modes of oscillation. Interestingly, for certain sets of parameters, the eigenvalues of certain distinct modes become nearly identical and their eigenfunctions interchange. The correctness of the analytical results are demonstrated by simulations in which the free boundary problem comprised of the Navier-Stokes, continuity, and Laplace equations is solved numerically in the limit of small viscosities.