dorsal/arxiv
View SchemaFinite time singularities in a class of hydrodynamic models
| Authors | V. P. Ruban, D. I. Podolsky, J. J. Rasmussen |
|---|---|
| Categories | |
| ArXiv ID | physics/0012007 |
| URL | https://arxiv.org/abs/physics/0012007 |
| DOI | 10.1103/PhysRevE.63.056306 |
| Journal | Phys. Rev. E, 63, 056306 (2001) |
Abstract
Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form ${\cal L}\sim\int k^\alpha|{\bf v_k}|^2d^3{\bf k}$ in 3D Fourier representation, where $\alpha$ is a constant, $0<\alpha< 1$. Unlike the case $\alpha=0$ (the usual Eulerian hydrodynamics), a finite value of $\alpha$ results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like $(t^*-t)^{1/(2-\alpha)}$, where $t^*$ is the singularity time.
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"abstract": "Models of inviscid incompressible fluid are considered, with the kinetic\nenergy (i.e., the Lagrangian functional) taking the form ${\\cal L}\\sim\\int\nk^\\alpha|{\\bf v_k}|^2d^3{\\bf k}$ in 3D Fourier representation, where $\\alpha$\nis a constant, $0\u003c\\alpha\u003c 1$. Unlike the case $\\alpha=0$ (the usual Eulerian\nhydrodynamics), a finite value of $\\alpha$ results in a finite energy for a\nsingular, frozen-in vortex filament. This property allows us to study the\ndynamics of such filaments without the necessity of a regularization procedure\nfor short length scales. The linear analysis of small symmetrical deviations\nfrom a stationary solution is performed for a pair of anti-parallel vortex\nfilaments and an analog of the Crow instability is found at small wave-numbers.\nA local approximate Hamiltonian is obtained for the nonlinear long-scale\ndynamics of this system. Self-similar solutions of the corresponding equations\nare found analytically. They describe the formation of a finite time\nsingularity, with all length scales decreasing like $(t^*-t)^{1/(2-\\alpha)}$,\nwhere $t^*$ is the singularity time.",
"arxiv_id": "physics/0012007",
"authors": [
"V. P. Ruban",
"D. I. Podolsky",
"J. J. Rasmussen"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1103/PhysRevE.63.056306",
"journal_ref": "Phys. Rev. E, 63, 056306 (2001)",
"title": "Finite time singularities in a class of hydrodynamic models",
"url": "https://arxiv.org/abs/physics/0012007"
},
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