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Toroidal bubbles with circulation in ideal hydrodynamics: a variational approach.

V P Ruban1, J Juul Rasmussen

  • 1L. D. Landau Institute for Theoretical Physics, 2 Kosygin Street, 119334 Moscow, Russia. ruban@itp.ac.ru

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
Summary

This study models unsteady fluid flows around toroidal bubbles using a variational principle. It derives exact equations for 2D flows and a simplified model for 3D flows, demonstrating vortex stability under specific conditions.

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Area of Science:

  • Fluid dynamics
  • Theoretical physics

Background:

  • Unsteady fluid flows around toroidal bubbles present complex dynamics.
  • Understanding bubble shape evolution and stability is crucial for various applications.

Purpose of the Study:

  • To formulate a general variational principle for bubble shape evolution.
  • To derive exact equations of motion for 2D flows and a simplified model for 3D flows.
  • To analyze the stability of hollow vortices and vortex ring bubbles.

Main Methods:

  • Formulation of a general variational principle.
  • Derivation of exact pseudodifferential equations of motion using conformal mapping for 2D flows.
  • Development of a simplified Lagrangian for circulation-dominated 3D flows.
  • Analysis of stability for stationary drifting 2D hollow vortices and vertically moving axisymmetric vortex ring bubbles.

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Main Results:

  • A general variational principle for toroidal bubble shape evolution is established.
  • Exact equations of motion and a closed-form Hamiltonian are derived for 2D flows.
  • Stability of a stationary drifting 2D hollow vortex is demonstrated under small gravity conditions.
  • A finite-dimensional dynamical system is derived and solved for a moving axisymmetric vortex ring bubble.

Conclusions:

  • The study provides a robust theoretical framework for analyzing unsteady flows around toroidal bubbles.
  • The derived models offer insights into the stability and dynamics of both 2D and 3D bubble flows.
  • The findings are applicable to understanding phenomena involving vortex dynamics and bubble behavior in fluids.