2012
DOI: 10.20537/nd1201008
|View full text |Cite
|
Sign up to set email alerts
|

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

Abstract: В этой работе мы рассматриваем задачу о движении осесимметричных вихревых колец в идеальной несжимаемой жидкости. Используя топологический подход, мы указываем метод полного качественного анализа динамики в системе двух колец, и, в частности, мы полностью решаем проблему описания условий возникновения чехарды вихревых колец. Кроме того, в задаче двух вихревых колец найдены новые семейства движений, при которых взаимные расстояния остаются конечны, названные нами псевдочехардой. В задаче трех вихревых колец так… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
3
2

Citation Types

1
28
0

Year Published

2015
2015
2019
2019

Publication Types

Select...
5

Relationship

0
5

Authors

Journals

citations
Cited by 8 publications
(29 citation statements)
references
References 18 publications
1
28
0
Order By: Relevance
“…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%
See 3 more Smart Citations
“…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: {0trueṘi=1RiZijiΓjG(Ri,Zi,Rj,Zj)0truetrueZi̇=1RiRi()jinormalΓjGfalse(Ri,Zi,Rj,Zjfalse)+normalΓi4πRi()log8Riai34,where i=1,2 is the index for the two rings, Ri are the radii of the rings, Zi are the distances along the common axis of symmetry, Γi are the vorticity strengths of the rings, ai are the radii of the cross‐section of the cores, which is taken to be a constant, and G is given by Gfalse(z,r,truez,truerfalse)=false(rtruerfalse)1/22π()2kkK(k)2kE(k),k=()4rr(zz)2+…”
Section: Discussionmentioning
confidence: 99%
See 2 more Smart Citations
“…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%