Motion of a vortex filament in the local induction approximation: a perturbative approach

2011 ◽  
Vol 26 (1-4) ◽  
pp. 161-171 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Vortex Filament ◽  
Perturbative Approach ◽  
Local Induction Approximation

scholarly journals Helical vortex filament motion under the non-local Biot–Savart model

2014 ◽  
Vol 762 ◽  
pp. 141-155 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Logarithmic Singularity ◽  
Transverse Velocity ◽  
Rotational Velocity ◽  
Vortex Filament ◽  
Maximal Torsion ◽  
Helical Vortex ◽  
Local Contribution ◽  
Induced Motion ◽  
Non Local ◽  
Local Induction Approximation

AbstractThe thin helical vortex filament is one of the fundamental exact solutions possible under the local induction approximation (LIA). The LIA is itself an approximation to the non-local Biot–Savart dynamics governing the self-induced motion of a vortex filament, and helical filaments have also been considered for the Biot–Savart dynamics, under a variety of configurations and assumptions. We study the motion of such a helical filament in the Cartesian reference frame by determining the curve defining this filament mathematically from the Biot–Savart model. In order to do so, we consider a matched approximation to the Biot–Savart dynamics, with local effects approximated by the LIA in order to avoid the logarithmic singularity inherent in the Biot–Savart formulation. This, in turn, allows us to determine the rotational and translational velocity of the filament in terms of a local contribution (which is exactly that which is found under the LIA) and a non-local contribution, each of which depends on the wavenumber, $k$, and the helix diameter, $A$. Performing our calculations in such a way, we can easily compare our results to those of the LIA. For small $k$, the transverse velocity scales as $k^{2}$, while for large $k$, the transverse velocity scales as $k$. On the other hand, the rotational velocity attains a maximum value at some finite $k$, which corresponds to the wavenumber giving the maximal torsion.

scholarly journals Self-similar vortex filament motion under the non-local Biot–Savart model

2016 ◽  
Vol 802 ◽  
pp. 760-774 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Superfluid Helium ◽  
Small Amplitude ◽  
Similarity Solutions ◽  
Vortex Filament ◽  
Quantized Vortex ◽  
Vortex Filaments ◽  
Filament Structure ◽  
Non Local ◽  
Self Similar ◽  
Local Induction Approximation

One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.

scholarly journals Dynamics of a planar vortex filament under the quantum local induction approximation

2014 ◽  
Vol 470 (2172) ◽  
pp. 20140341 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Fluid Flow ◽  
Friction Coefficient ◽  
Exact Solutions ◽  
Vortex Filament ◽  
Mutual Friction ◽  
Zero Temperature ◽  
Filament Length ◽  
Normal Fluid ◽  
Local Induction Approximation ◽  
Over Time

The Hasimoto planar vortex filament is one of the rare exact solutions to the classical local induction approximation (LIA). This solution persists in the absence of friction or other disturbances, and it maintains its form over time. As such, the dynamics of such a filament have not been extended to more complicated physical situations. We consider the planar vortex filament under the quantum LIA, which accounts for mutual friction and the velocity of a normal fluid impinging on the filament. We show that, for most interesting situations, a filament which is planar in the absence of mutual friction at zero temperature will gradually deform owing to friction effects and the normal fluid flow corresponding to warmer temperatures. The influence of friction is to induce torsion, so the filaments bend as they rotate. Furthermore, the flow of a normal fluid along the vortex filament length will result in a growth in space of the initial planar perturbations of a line filament. For warmer temperatures, these effects increase in magnitude, since the growth in space scales with the mutual friction coefficient. A number of nice qualitative results are analytical in nature, and these results are verified numerically for physically interesting cases.

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