Stability of a rotating liquid drop immersed in a corotating fluid with different density

Abstract

The stability of a uniformly rotating, incompressible drop with density ρ i and immersed in a corotating fluid with different density ρ e is investigated. The equilibrium figure is approximated by an oblate or prolate spheroid. The linearized equations of motion are solved by means of ‘modified’ spheroidal coordinates. A dispersion relation is derived with the aid of the energy integral method. The curves of overstability are calculated for the second harmonic modes of oscillation and for the mode n = m = 3. The points of bifurcation for the n = m modes are independent of the presence of the external fluid. It appears, however, that dynamical instability may initiate before the point of bifurcation is attained. This occurs when z = ρ e /ρ i > 0.192 for the n = m = 2 mode and when z > 0.207 for the n = m = 3 mode. In several cases we observed a bending back of the curve of overstability in the ( z, e ) or ( z, h ) plane, where e and h are the oblate and prolate eccentricities, respectively. This indicates stability for low or high eccentricities (or angular momenta) and instability for intermediate eccentricities (or angular momenta).