Journal article · Preprint article
Toroidal bubbles with circulation in ideal hydrodynamics: A variational approach
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.
Language: | English |
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Year: | 2003 |
Pages: | 056301 |
ISSN: | 10953787 , 1063651x , 24700053 , 24700045 and 15393755 |
Types: | Journal article and Preprint article |
DOI: | 10.1103/PhysRevE.68.056301 |
ORCIDs: | Juul Rasmussen, J. |