dorsal/arxiv
View SchemaToroidal bubbles with circulation in ideal hydrodynamics. A variational approach
| Authors | V. P. Ruban, J. J. Rasmussen |
|---|---|
| Categories | |
| ArXiv ID | physics/0306029 |
| URL | https://arxiv.org/abs/physics/0306029 |
| DOI | 10.1103/PhysRevE.68.056301 |
Abstract
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.
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"abstract": "Incompressible, inviscid, irrotational, and unsteady flows with circulation\n$\\Gamma$ around a distorted toroidal bubble are considered. A general\nvariational principle that determines the evolution of the bubble shape is\nformulated. For a two-dimensional (2D) cavity with a constant area $A$, exact\npseudo-differential equations of motion are derived, based on variables that\ndetermine a conformal mapping of the unit circle exterior into the region\noccupied by the fluid. A closed expression for the Hamiltonian of the 2D system\nin terms of canonical variables is obtained. Stability of a stationary drifting\n2D hollow vortex is demonstrated, when the circulation is relatively large,\n$gA^{3/2}/\\Gamma^2\\ll 1$. For a circulation-dominated regime of\nthree-dimensional flows a simplified Lagrangian is suggested, inasmuch as the\nbubble shape is well described by the center-line $\\g{R}(\\xi,t)$ and by an\napproximately circular cross-section with relatively small area, $A(\\xi,t)\\ll\n(\\oint |\\g{R}\u0027|d\\xi)^2$. In particular, a finite-dimensional dynamical system\nis derived and approximately solved for a vertically moving axisymmetric vortex\nring bubble with a compressed gas inside.",
"arxiv_id": "physics/0306029",
"authors": [
"V. P. Ruban",
"J. J. Rasmussen"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1103/PhysRevE.68.056301",
"title": "Toroidal bubbles with circulation in ideal hydrodynamics. A variational approach",
"url": "https://arxiv.org/abs/physics/0306029"
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