Three, four and five point vortices which exhibit the relaxation - - PowerPoint PPT Presentation

Faculty of Mathematics, KYUSHU UNIVERSITY

Three, four and five point vortices which exhibit the relaxation

• scillation

Tatsuyuki NAKAKI

nakaki@math.kyushu-u.ac.jp

Faculty of Mathematics, Kyushu University, Japan Euromech448 (ESPCI, Paris, France, 6 September 2004)

Three, four and five point vortices which exhibit the relaxation oscillation – p.1/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Purpose of this talk

To show the assemblies of point vortices (in 2-dim) which exhibit the relaxation oscillation

Three, four and five point vortices which exhibit the relaxation oscillation – p.2/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Purpose of this talk

To show the assemblies of point vortices (in 2-dim) which exhibit the relaxation oscillation Three point vortices exhibiting the relaxation oscillation Four point vortices exhibiting the relaxation oscillation Five point vortices exhibiting the relaxation oscillation What is the relaxation oscillation?

Three, four and five point vortices which exhibit the relaxation oscillation – p.2/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Purpose of this talk

To show the assemblies of point vortices (in 2-dim) which exhibit the relaxation oscillation Three point vortices exhibiting the relaxation oscillation Four point vortices exhibiting the relaxation oscillation Five point vortices exhibiting the relaxation oscillation What is the relaxation oscillation? It is the oscillation such that steady state

• rapid motion
• steady state
• rapid motion

Click here for a numerical simulation.

Three, four and five point vortices which exhibit the relaxation oscillation – p.2/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Relaxation oscillation

Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits.

Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Relaxation oscillation

Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits.

Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Relaxation oscillation

Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits.

Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Relaxation oscillation

Mathematically, such a relaxation oscillation is induced by chain of heteroclinic orbits.

Three, four and five point vortices which exhibit the relaxation oscillation – p.3/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Relaxation oscillation (cont.)

Chain of heteroclinic orbits

• The existence of a heteroclinic orbit

(*) What we should do is to prove (*).

Three, four and five point vortices which exhibit the relaxation oscillation – p.4/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Agenda

The basic equation to be solved Known results Formulation of our problem Relaxation oscillation (five point vortices) Relaxation oscillation (four point vortices) Relaxation oscillation (three point vortices) Conclusions and Future works

Three, four and five point vortices which exhibit the relaxation oscillation – p.5/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Basic equation

Classical point vortices problem (2-dim)

• ✁

where

: complex coordinate (unknown)

: circulation (given real constant)

• f

th point vortex

Three, four and five point vortices which exhibit the relaxation oscillation – p.6/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Known results

Let

• be the number of vortices in fluid
• ✝

: Easy to solve.

• ✝

: A qualitative analysis with arbitrary strength is done by Aref (1979).

• ✂

: The ODE is not solved yet in general cases Chaotic behavior occurs (Aref and Pomphrey, 1982) Stationary configuration (O’neil, 1987) Morikawa and Swenson’s results (1971) Some five point vortices exhibit the relaxation

• scillation (N. 1999)

— another configuration shall be shown today

Three, four and five point vortices which exhibit the relaxation oscillation – p.7/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• Two parameters

and

Three, four and five point vortices which exhibit the relaxation oscillation – p.8/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• Two parameters

and

Initial configuration

• ☎

Three, four and five point vortices which exhibit the relaxation oscillation – p.8/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• Two parameters

and

Initial configuration

• ☎

Strength is

• ✝

where

is .....

Three, four and five point vortices which exhibit the relaxation oscillation – p.8/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• (cont.)

....., where

is determined so that the five point vortices is in the relative equilibrium.

Three, four and five point vortices which exhibit the relaxation oscillation – p.9/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• (cont.)

....., where

is determined so that the five point vortices is in the relative equilibrium. Definition. The solution

• ✂

is a relative equilibrium if and only if

• ✂

is an equilibrium for

.

Three, four and five point vortices which exhibit the relaxation oscillation – p.9/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• (cont.)

....., where

is determined so that the five point vortices is in the relative equilibrium. Definition. The solution

• ✂

is a relative equilibrium if and only if

• ✂

is an equilibrium for

. The vortices corotate around the origin with uniform angular velocity

. Throughout this talk (including all numerical simulations), we observe the vortices in the rotating coordinates.

Three, four and five point vortices which exhibit the relaxation oscillation – p.9/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• (cont.)

The motion of point vortices on this problem has rich structure: Relaxation oscillation Stability of the configuration Unstable in the linearized sense Weakly unstable Stable in the Lyapunov sense Today’s talk: relaxation oscillation

Three, four and five point vortices which exhibit the relaxation oscillation – p.10/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• (cont.)

Conjecture. For

• ✁
• ✞

and

• ✏✞✝
• ✞

, there exists a heteroclinic orbit. Theorem. For

• ✁
• ✞

and

, there exists a heteroclinic orbit. Numerical simulations when

and

and

(Click here)

and

(Click here)

and

(Click here)

Three, four and five point vortices which exhibit the relaxation oscillation – p.11/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• In the previous problem (
• ✝

), let us consider the case when

(i.e.,

• ✁

• ✂

• ✂

• ).

The center vortex is not a vortex anymore; it is a particle. For some

, we observe the relaxation oscillation. (Click here for a numerical simulation)

Three, four and five point vortices which exhibit the relaxation oscillation – p.12/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Problem for

• (cont.)

Theorem. For

(i.e.,

), there exists a heteroclinic orbit. Conjecture. For

• ✁
• ✞
• ✁

(i.e.,

• ✄

), there exists a heteroclinic orbit.

Three, four and five point vortices which exhibit the relaxation oscillation – p.13/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Three point vortices

The qualitative analysis is already done by Aref (1979).

Three, four and five point vortices which exhibit the relaxation oscillation – p.14/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Three point vortices

The qualitative analysis is already done by Aref (1979). It seems that he does not touch the relaxation

• scillation.

Three, four and five point vortices which exhibit the relaxation oscillation – p.14/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Three point vortices

The qualitative analysis is already done by Aref (1979). It seems that he does not touch the relaxation

• scillation.

The situation where the relaxation oscillation occurs is

• ☎

and

• ☎

. (Click here for a numerical simulation)

Three, four and five point vortices which exhibit the relaxation oscillation – p.14/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Three point vortices (cont.)

Theorem. For the situation stated above, there is a heteroclinic orbit which connects two equilibria. The proof is not hard. Use the idea of the trilinear coordinates by Aref.

Three, four and five point vortices which exhibit the relaxation oscillation – p.15/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Trilinear coordinate

Configuration of three point vortices is represented by

• ☎

, where

• ✝
• ✂

• ✂

• ✁

• ✂
• ☎

• ✂

• ✂
• ☎

• ✂

and the constant

is determined by

• ✂

• ✂

• ✏

We note that

• ✄

holds.

Three, four and five point vortices which exhibit the relaxation oscillation – p.16/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Three point vortices (cont.)

Heteroclinic orbit in the trilinear coordinate:

• 10
• 8
• 6
• 4
• 2

2 4 2 4 6 8 10 E1 & E3 E2 b1 b2 b3

Three, four and five point vortices which exhibit the relaxation oscillation – p.17/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Three point vortices (cont.)

Two equilibria

• and
• ✁

(

• ✂

is not a equilibrium).

Three, four and five point vortices which exhibit the relaxation oscillation – p.18/19

Faculty of Mathematics, KYUSHU UNIVERSITY

Conclusions and Future works

We consider the classical point vortices problem in 2-dim. We show the point vortices which exhibit the relaxation

• scillation for
• ✝

, where

• is the number of

vortices in the fluid. We prove the existence of heteroclinic orbits for some cases. We have conjectures to be proved.

— Thank you for your attention —

Questions? —— Please send an email at “nakaki@math.kyushu-u.ac.jp”.

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