Nonlinear oscillations of inviscid free drops

Abstract

Inviscid oscillations of free liquid drops are analyzed by solving Bernoulli's equation for the free surface shape and Laplace's equation for the velocity potential field. The means are: (a) Galerkin's weighted residual method which converts the governing equations into a large system of nonlinear, time-dependent ordinary differential equations; (b) an implicit predictor-corrector method for time integration which automatically adjusts time steps; and (c) Newton's method which solves the large system of nonlinear algebraic equations that results from time discretization. Results presented include sequences of drop shapes, pressure distributions, particle paths, and evolution with time of kinetic and surface energies. Accuracy is attested by virtual constancy of drop volume and total energy and smallness of mass and momentum fluxes across drop surfaces. Dynamic response to small amplitude disturbances agrees with linear theory. Large-amplitude oscillations are compared to the predictions by the marker- and-cell method and second-order perturbation theory. Mode interactions and frequency shifts are analyzed by Fourier power spectra and lend further insight into the nature of the oscillations.