Toroidal bubbles with circulation in ideal hydrodynamics: A variational approach
Abstract
Incompressible, inviscid, irrotational, unsteady flows with circulation Γ around a distorted toroidal bubble are considered. A general variational principle that determines the evolution of the bubble shape is formulated. For a twodimensional (2D) cavity with a constant area A, exact pseudodifferential equations of motion are derived, based on variables that determine a conformal mapping of the unit circle exterior into the region occupied by the fluid. A closed expression for the Hamiltonian of the 2D system in terms of canonical variables is obtained. Stability of a stationary drifting 2D hollow vortex is demonstrated, when the gravity is small, gA^{3/2}/Γ^{2}≪1. For a circulationdominated regime of threedimensional flows a simplified Lagrangian is suggested, inasmuch as the bubble shape is well described by the center line R(ξ,t) and by an approximately circular cross section with relatively small area, A(ξ,t)≪(∮R'dξ)^{2}. In particular, a finitedimensional dynamical system is derived and approximately solved for a vertically moving axisymmetric vortex ring bubble with a compressed gas inside.
 Publication:

Physical Review E
 Pub Date:
 November 2003
 DOI:
 10.1103/PhysRevE.68.056301
 arXiv:
 arXiv:physics/0306029
 Bibcode:
 2003PhRvE..68e6301R
 Keywords:

 47.55.Dz;
 47.15.Hg;
 47.10.+g;
 47.32.Cc;
 Physics  Fluid Dynamics
 EPrint:
 revtex4, 11 pages, 8 EPS figures, improved and extended version