TY - JOUR

T1 - Toroidal bubbles with circulation in ideal hydrodynamics: A variational approach

AU - Ruban, V.P.

AU - Juul Rasmussen, J.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - 2-E tekno

U2 - 10.1103/PhysRevE.68.056301

DO - 10.1103/PhysRevE.68.056301

M3 - Journal article

VL - 68

SP - 056301

JO - Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)

JF - Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)

SN - 2470-0045

IS - 5

ER -