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.

scholarly journals The Biot–Savart description of Kelvin waves on a quantum vortex filament in the presence of mutual friction and a driving fluid

2015 ◽  
Vol 471 (2179) ◽  
pp. 20150149 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Stability Criterion ◽  
Fluid Velocity ◽  
Kelvin Waves ◽  
Vortex Filament ◽  
Mutual Friction ◽  
Local Models ◽  
Local Effects ◽  
Normal Fluid ◽  
Quantum Vortex ◽  
Non Local

We study the dynamics of Kelvin waves along a quantum vortex filament in the presence of mutual friction and a driving fluid while taking into account non-local effects due to Biot–Savart integrals. The Schwarz model reduces to a nonlinear and non-local dynamical system of dimension three, the solutions of which determine the translational and rotational motion of the Kelvin waves, as well as the amplification or decay of such waves. We determine the possible qualitative behaviours of the resulting Kelvin waves. It is well known from experimental and theoretical studies that the Donnelly–Glaberson instability plays a role on the amplification or decay of Kelvin waves in the presence of a driving normal fluid velocity, and we obtain the relevant stability criterion for the non-local model. While the stability criterion is the same for local and non-local models when the wavenumber is sufficiently small, we show that large differences emerge for the large wavenumber case (tightly coiled helices). The results demonstrate that non-local effects have a stabilizing effect on the Kelvin waves, and hence larger normal fluid velocities are required for amplification of large wavenumber Kelvin waves. Additional qualitative differences between the local and non-local models are explored.

scholarly journals On rolling friction

2019 ◽  
Vol 485 (3) ◽  
pp. 295-299
Author(s):  
A. P. Ivanov
Keyword(s):  
Friction Coefficient ◽  
Angular Velocity ◽  
Viscous Friction ◽  
Rolling Friction ◽  
The Body ◽  
Normal Velocity ◽  
Contact Conditions ◽  
Combined Motion ◽  
Pure Rolling ◽  
Over Time

The dependence of rolling friction on velocity for various contact conditions is discussed. The principal difference between rolling and other types of relative motion (sliding and spinning) is that the points of the body in contact with the support change over time. Due to deformations, there is a small contact area and, entering into contact, the body points have a normal velocity proportional to the diameter of this area. For describing the dependence of the friction coefficient on the angular velocity in the case of “pure” rolling, a linear dependence is proposed that admits a logical explanation and experimental verification. Under the combined motion, the rolling friction retains its properties, the sliding and spinning friction acquiring the properties of viscous friction.

Local normal-fluid helium II flow due to mutual friction interaction with the superfluid

2000 ◽  
Vol 62 (5) ◽  
pp. 3409-3415 ◽  
Author(s):  
Olusola C. Idowu ◽  
Ashley Willis ◽  
Carlo F. Barenghi ◽  
David C. Samuels
Keyword(s):  
Mutual Friction ◽  
Normal Fluid ◽  
Helium Ii ◽  
Friction Interaction

scholarly journals Some new remarks on MHD Jeffery-Hamel fluid flow problem

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 819-826
Author(s):  
Remus-Daniel Ene ◽  
Camelia Pop
Keyword(s):  
Fluid Flow ◽  
Exact Solutions ◽  
Nonlinear Stability ◽  
Flow Problem ◽  
Analytic Solutions ◽  
Equilibrium States

AbstractA Hamilton-Poisson realization of the MHD Jeffery-Hamel fluid flow problem is proposed. Tthe nonlinear stability of the equilibrium states is discussed. A comparison between the analytic solutions obtained using the OHAM method and the exact solutions provided by the Hamilton-Poisson realization are presented.

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.

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