Nonlinear Oscillations of Two-Dimensional, Rotating Inviscid Drops

Abstract

We examine the nonlinear response of a drop, rotating as a rigid body at fixed angular velocity, to two-dimensional finite-amplitude disturbances. With these restrictions, the liquid velocity becomes a superposition of the solid-body rotation and the gradient of a velocity potential. To find the drop motion, we solve an integro-differential Bernoulli's equation for the drop shape and Laplace's equation for the velocity potential field within the drop. The integral part of Bernoulli's equation couples all parts of the drop's surface and sets this problem apart from that of the oscillations of nonrotating drops. We use Galerkin's weighted residual method with finite element basis functions which are deployed on a mesh that deforms in proportion to the deformation of the free surface. To alleviate the roundoff error in the initial conditions of the drop motion, we use a Fourier filter which turns off as soon as the highest resolved oscillation mode grows above the machine noise level. The results include sequences of drop shapes, Fourier analysis of oscillation frequencies, and evolution in time of the components of total mechanical energy of the drop. The Fourier power spectral analysis of large-amplitude oscillations of the drop reveals frequency shifts similar to those of the nonrotating free drops. Constant drop volume, total energy, and angular momentum as well as vanishing mass flow across the drop surface are the standards of accuracy against which we test the nonlinear motion of the drop over tens or hundreds of oscillation periods. Finally, we demonstrate that our finite element method has superior stability, accuracy, and computational efficiency over several boundary element algorithms that have previously appeared in the literature.