Oscillations of liquid captive rotating drops

Abstract

A linear analysis of the free oscillations of captive drops or bubbles is discussed. The drop is surrounded by an immiscible liquid or gas and undergoes rotation as a rigid body in the presence of gravity. Using spectral analytical methods, we provide a general formulation for both elliptic and hyperbolic oscillation regimes of the frequency spectrum, for any combination of the Weber and Bond numbers. The method uses a Green function to reduce the inviscid Navier–Stokes equations and boundary conditions to an eigenvalue problem. Both the Green function and normal velocities at the interface are expanded in the orthogonal functional space generated by the Sturm–Liouville problem associated to the interface equation. The effect on the vibration modes of the density and geometrical parameters of the captive drop and surrounding medium is analysed. We present a complete analysis of the low-frequency spectra in the elliptic regime of a set of floating liquid zones and captive drops for a continuous range of Weber and Bond numbers. It is shown that, depending on the geometrical parameters of the system, the elliptic vibration spectrum presents a sui generis modal interaction for low wavenumbers and certain ranges of Weber number.