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Magnetically Induced Rotating Rayleigh-Taylor Instability
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On the Stability of Rotating Drops.

A K Nurse1, S R Coriell1, G B McFadden1

  • 1National Institute of Standards and Technology, Gaithersburg, MD 20899.

Journal of Research of the National Institute of Standards and Technology
|March 10, 2016
PubMed
Summary

This study analyzes rotating fluid drop shapes and stability using a variational approach. We found that drops can form oblate, prolate, or toroidal shapes, with stability depending on rotation rate and angular momentum.

Keywords:
bifurcationlinear stabilityrotating droptoroidsvariational principle

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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.