Capillary instability of an oscillating liquid column subjected to a periodic rigid-body rotation

Abstract

The instability of the flow resulting from the oscillations of a rigidly periodic rotating cylindrical fluid column has been investigated. A general analytical solution to the linearized problem is presented. The application of the boundary conditions leads to the appearance of the Mathieu equation with a parametric imaginary damping term. The method of multiple time scales is used to derive and analyze the necessary and sufficient conditions for stability. Transition curves which separate the stable region from the unstable region are obtained in both non-resonance and resonance cases. Two subharmonic resonances are distinguished in the linear stability analysis. It is observed that the amplitude difference of the angular velocity plays a dual role in the resonance cases: a stabilizing role in one resonance and a destabilizing role in the other resonance. The mechanism for the frequency of the periodic rotation has been changed due to the change of the behavior of the amplitude of the angular velocity. It is shown that the increase of the column radius has a stabilizing or destabilizing effect if the surrounding medium rotates, respectively, faster or slower than the fluid column. The same conclusion is observed for the azimuthal wave number which plays a dual role in the resonances.