The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

Borisov, Alexey V.; Kilin, Alexander A.; Mamaev, Ivan S.

doi:10.20537/nd1201008

Cited by 8 publications

(29 citation statements)

References 18 publications

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“…Similar to Section 3, we give a necessary and sufficient condition for leapfrogging to occur. Finally in Section 5, we compare our results with the results obtained in Borisov et al, and also give concluding remarks.…”

Section: Introduction and Problem Settingmentioning

confidence: 61%

“…We make a comparison between the results by Borisov et al They consider the following model system:

({, \begin{matrix} 0true {\dot{R}}_{i} = - \frac{1}{R_{i}} \frac{\partial}{\partial Z_{i}} \sum_{j \neq i} Γ_{j} G (R_{i}, Z_{i}, R_{j}, Z_{j}) \\ 0true \overset{Z_{i}}{true} = \frac{1}{R_{i}} \frac{\partial}{\partial R_{i}} ((), \sum_{j \neq i}, {normalΓ}_{j}, G, false(R_{i}, Z_{i}, R_{j}, Z_{j} false)) + \frac{{normalΓ}_{i}}{4 π R_{i}} (() \log \frac{8 R_{i}}{a_{i}} - \frac{3}{4}), \end{matrix})

where

i = 1, 2

is the index for the two rings,

R_{i}

are the radii of the rings,

Z_{i}

are the distances along the common axis of symmetry,

Γ_{i}

are the vorticity strengths of the rings,

a_{i}

are the radii of the cross‐section of the cores, which is taken to be a constant, and G is given by

G false(z, r, \overset{z}{true}, \overset{r}{true} false) = \frac{{false(r \overset{r}{true} false)}^{1 / 2}}{2 π} ((() \frac{2}{k} - k) K (k) - \frac{2}{k} E (k)), k = ((), \frac{4 r \tilde{r}}{{(z - \tilde{z})}^{2} +})

…”

Section: Discussionmentioning

confidence: 99%

“…The model used in Ref. is derived as a system of ODE, in other words, they only consider thin vortex rings as opposed to filaments with general shapes. The advantages of over our model is that it can incorporate the effect of the change in shape of the core and the effect of the change in vorticity distribution inside the core.…”

Section: Discussionmentioning

confidence: 99%

“…In more recent years, detailed models for the motion of vortex rings have been obtained, for example, by Saffman and Fukumoto, in which various distributions of vorticity in the ring core, the change in shape of the core, and the effect of viscosity of the ambient fluid can be incorporated. Based on these models, Borisov et al gave a complete description of the possible motion patterns of two interacting vortex rings with a common axis of symmetry.…”

Section: Introduction and Problem Settingmentioning

confidence: 99%

See 3 more Smart Citations

On the existence of leapfrogging pair of circular vortex filaments

Aiki

2019

Stud Appl Math

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We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. As a preliminary analysis, we first consider the case when the filaments are arranged as straight and parallel lines, and explicitly solve the system to show that the motion of the lines resemble that of point vortices moving in a plane. Then, we consider the motion of a pair of coaxial circular vortex filaments. We show that in this case, the system can be reduced to a two‐dimensional Hamiltonian system. Based on this formulation, we give a condition for the initial configuration and parameters of the filaments for leapfrogging to occur, and prove that the condition is in fact necessary and sufficient.

show abstract

Section: Introduction and Problem Settingmentioning

confidence: 61%

“…We make a comparison between the results by Borisov et al They consider the following model system:

({, \begin{matrix} 0true {\dot{R}}_{i} = - \frac{1}{R_{i}} \frac{\partial}{\partial Z_{i}} \sum_{j \neq i} Γ_{j} G (R_{i}, Z_{i}, R_{j}, Z_{j}) \\ 0true \overset{Z_{i}}{true} = \frac{1}{R_{i}} \frac{\partial}{\partial R_{i}} ((), \sum_{j \neq i}, {normalΓ}_{j}, G, false(R_{i}, Z_{i}, R_{j}, Z_{j} false)) + \frac{{normalΓ}_{i}}{4 π R_{i}} (() \log \frac{8 R_{i}}{a_{i}} - \frac{3}{4}), \end{matrix})

where

i = 1, 2

is the index for the two rings,

R_{i}

are the radii of the rings,

Z_{i}

are the distances along the common axis of symmetry,

Γ_{i}

are the vorticity strengths of the rings,

a_{i}

are the radii of the cross‐section of the cores, which is taken to be a constant, and G is given by

G false(z, r, \overset{z}{true}, \overset{r}{true} false) = \frac{{false(r \overset{r}{true} false)}^{1 / 2}}{2 π} ((() \frac{2}{k} - k) K (k) - \frac{2}{k} E (k)), k = ((), \frac{4 r \tilde{r}}{{(z - \tilde{z})}^{2} +})

…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introduction and Problem Settingmentioning

confidence: 99%

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On the existence of leapfrogging pair of circular vortex filaments

Aiki

2019

Stud Appl Math

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show abstract

“…Periodic orbits in the n-vortex problem which are not relative equilibria have been studied by some authors, see [3,4,6,15]. Here we consider relative equilibria and prove the existence of families of such periodic orbits, in a rotating system, which are in the neighborhood of an equilibrium that corresponds to a ring of vortices.…”

Section: Introductionmentioning

confidence: 87%

Lyapunov orbits in the n-vortex problem on the sphere

Carvalho

Cabral

2015

Regul. Chaot. Dyn.

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Three-dimensional stability of leapfrogging quantum vortex rings

Ruban

2018

Physics of Fluids

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It is shown by numerical simulations within a regularized Biot-Savart law that dynamical systems of two or three leapfrogging coaxial quantum vortex rings having a core width ξ and initially placed near a torus of radii R0 and r0, can be three-dimensionally (quasi-)stable in some regions of parameters Λ = ln(R0/ξ) and W = r0/R0. At fixed Λ, stable bands on W are intervals between non-overlapping main parametric resonances for different (integer) azimuthal wave numbers m. The stable intervals are most wide (∆W ∼ 0.01-0.05) between m-pairs (1, 2) and (2, 3) at Λ ≈ 4-12 thus corresponding to micro/mesoscopic sizes of vortex rings in the case of superfluid 4 He. With four and more rings, at least for W > 0.1, resonances overlap for all Λ and no stable domains exist.

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The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

Cited by 8 publications

References 18 publications

On the existence of leapfrogging pair of circular vortex filaments

On the existence of leapfrogging pair of circular vortex filaments

Lyapunov orbits in the n-vortex problem on the sphere

Three-dimensional stability of leapfrogging quantum vortex rings

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