Toroidal bubbles with circulation in ideal hydrodynamics. A variational approach

Incompressible, inviscid, irrotational, and unsteady flows with circulation $\Gamma$ around a distorted toroidal bubble are considered. A general variational principle that determines the evolution of the bubble shape is formulated. For a two-dimensional (2D) cavity with a constant area $A$, exact pseudo-differential 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 circulation is relatively large, $gA^{3/2}/\Gamma^2\ll 1$. For a circulation-dominated regime of three-dimensional flows a simplified Lagrangian is suggested, inasmuch as the bubble shape is well described by the center-line $\g{R}(\xi,t)$ and by an approximately circular cross-section with relatively small area, $A(\xi,t)\ll (\oint |\g{R}'|d\xi)^2$. In particular, a finite-dimensional dynamical system is derived and approximately solved for a vertically moving axisymmetric vortex ring bubble with a compressed gas inside.