Oscillations of a ring-constrained charged drop

Free drops of uncharged and charged inviscid, conducting fluids subjected to small-amplitude perturbations undergo linear oscillations (Rayleigh, Proc. R. Soc. London, vol. 29, no. 196-199, 1879, pp. 71-97; Rayleigh, Philos. Mag., vol. 14, no. 87, 1882, pp. 184-186). There exist a countably infinite number of oscillation modes, n = 2, 3, ..., each of which has a characteristic frequency and mode shape. Presence of charge (Q) lowers modal frequencies and leads to instability when Q > Q(R) (Rayleigh limit). The n = 0 and n = 1 modes are disallowed because they violate volume conservation and cause centre of mass (COM) motion. Thus, the first mode to become unstable is the n = 2 prolate-oblate mode. For free drops, there is a one-to-one correspondence between mode number and shape (Legendre polynomial P-n). Recent research has shifted to studying oscillations of spherical drops constrained by solid rings. Pinning the drop introduces a new low-frequency mode of oscillation (n = 1), one associated primarily with COM translation of the constrained drop. We analyse theoretically the effect of charge on oscillations of constrained drops. Using normal modes and solving a linear operator eigenvalue problem, we determine the frequency of each oscillation mode. Results demonstrate that for ring-constrained charged drops (RCCDs), the association between mode number and shape is lost. For certain pinning locations, oscillations exhibit eigenvalue veering as Q increases. While slightly charged RCCDs pinned at zeros of P-2 have a first mode that involves COM motion and a second mode that entails prolate-oblate oscillations, the modes flip as Q increases. Thereafter, prolate-oblate oscillations of RCCDs adopt the role of being the first mode because they exhibit the lowest vibration frequency. At the Rayleigh limit, the first eigenmode - prolate-oblate oscillations - loses stability while the second - involving COM motion - remains stable.

Automated summary

This study discusses the oscillation characteristics of free drops and charged drops constrained by solid rings under small-amplitude perturbations. It also investigates the effects of charge on oscillation modes. The research reveals that at the Rayleigh limit, oscillation frequencies of slightly charged constrained drops exhibit variation.