Related Experiment Videos
Finite time singularities in a class of hydrodynamic models
V P Ruban1, D I Podolsky, J J Rasmussen
1L.D. Landau Institute for Theoretical Physics, 2 Kosygin Street, 117334 Moscow, Russia. ruban@itp.ac.ru
Summary
This study introduces a new fluid dynamics model where vortex filaments have finite energy, enabling analysis without regularization. An analog of Crow instability and finite-time singularities were discovered in vortex filament dynamics.
Area of Science:
- Fluid Dynamics
- Mathematical Physics
Background:
- Traditional fluid dynamics models struggle with singular vortex filaments.
- Eulerian hydrodynamics (alpha=0) requires regularization for short length scales.
Purpose of the Study:
- To investigate fluid dynamics models with finite energy for vortex filaments.
- To analyze the dynamics of singular vortex filaments without regularization.
- To identify instabilities and singularities in vortex filament systems.
Main Methods:
- Consideration of inviscid incompressible fluid models with a modified kinetic energy functional (L ~ integral k(alpha)/vk/2dk).
- Linear analysis of deviations from stationary solutions for antiparallel vortex filaments.
- Derivation of a local approximate Hamiltonian for nonlinear dynamics.
- Analytical investigation of self-similar solutions.
Main Results:
- A finite value of alpha (0
- An analog of Crow instability was identified at small wave numbers for antiparallel filaments.
- Self-similar solutions indicate a finite-time singularity formation, with length scales decreasing as (t*-t)(1/(2-alpha)).
Conclusions:
- The proposed model offers a method to study vortex filament dynamics without regularization.
- The findings reveal new instabilities and singularity behaviors in fluid systems.
- The research provides analytical insights into the nonlinear dynamics and collapse of vortex structures.