On the Stability of Rotating Drops.

Area of Science:

• Fluid dynamics
• Nonlinear dynamics
• Continuum mechanics

Background:

• Understanding the behavior of rotating fluid drops is crucial in various scientific and engineering fields.
• Previous studies have explored equilibrium shapes but often lack comprehensive stability analysis under different rotational conditions.

Purpose of the Study:

• To investigate the equilibrium configurations and linear stability of rotating axisymmetric fluid drops.
• To analyze how surface energy, rotational energy, and volume constraints influence drop shapes and stability.
• To explore the transition from spheroidal to toroidal shapes and identify conditions for instabilities.

Main Methods:

• A variational principle was employed to characterize equilibrium states as stationary points of an energy functional.
• Linear stability was assessed by solving an eigenvalue problem derived from the second variation of the energy functional.
• An angle-arc length formulation was used to compute equilibrium shapes, including non-single-valued ones in spherical coordinates.

Main Results:

• Equilibria for oblate, prolate, and toroidal shapes were computed and their evolution with rotation rate was tracked.
• Instabilities were analyzed for both driven drops (constant rotation rate) and isolated drops (constant angular momentum).
• Axisymmetric and non-axisymmetric perturbations were considered, revealing bifurcation points leading to non-axisymmetric shapes and azimuthal instabilities in toroidal drops.

Conclusions:

• The study provides a comprehensive analysis of rotating fluid drop equilibrium and stability, revealing complex shape transitions and instability mechanisms.
• Results show that toroidal drops with high aspect ratios are susceptible to azimuthal instabilities, analogous to Rayleigh instability.
• Prolate spheroidal shapes, observed when a less dense drop rotates in a denser medium, appear to be linearly stable.