Toroidal bubbles with circulation in ideal hydrodynamics: A variational approach

V.P. Ruban, J. Juul Rasmussen

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    Incompressible, inviscid, irrotational, 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 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)/Gamma(2)<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 R(xi,t) and by an approximately circular cross section with relatively small area, A(xi,t)<(integralparallel toR(')parallel todxi)(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.
    Original languageEnglish
    JournalPhysical Review E
    Volume68
    Issue number5
    Pages (from-to)056301
    Number of pages11
    ISSN2470-0045
    DOIs
    Publication statusPublished - 2003

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