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The Japan Society of Fluid Mechanics

Fluid Dynamics Research doi:10.1088/0169-5983/46/3/031420

Fluid Dyn. Res. 46 (2014) 031420 (7pp)

On the dynamics of point vortices in an annular region
Nadezhda N Erdakova1 and Ivan S Mamaev
1

1,2

Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia 2 Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia E-mail: enn@rcd.ru and mamaev@rcd.ru Received 1 July 2013, revised 14 April 2014 Accepted for publication 17 April 2014 Published 28 May 2014 Communicated by Y Fukumoto Abstract

This paper reviews the results of stability analysis for polygonal configurations of a point vortex system in an annular region depending on the ratio of the inner and outer radii of the annulus. Conditions are found for linear stability of Thomsons configurations for the case N < 7. The paper also shows that a system of two vortices between parallel walls is a limiting case of a two-vortex system in an annular region, as the radii of the annulus tend to infinity. (Some figures may appear in colour only in the online journal) 1. Introduction A large body of literature has been devoted to the study of the stability of polygonal vortex configurations, both in the case of a free plane, and inside and outside an annulus--see, e.g., (Thomson 1883, Havelock 1931, Kurakin 2010, 2012, Borisov and Mamaev 2005). The problem of the dynamics of two vortices in an annulus is addressed in Pashaev and Yilmaz (2011), Zueva (1996), Erdakova (2010) and Lakaniemi (2007). In Bolsinov et al (2010) and Vaskin and Erdakova (2010), a topological approach is used to search for relative equilibria and to analyze their stability. This approach has yielded new stationary configurations for the system of tree vortices in a circle and two vortices in an annulus. In this paper, we use a Hamiltonian representation for the vortex system in an annular region in terms of 1-functions and explore the stability of polygonal configurations of the system of N vortices in an annulus depending on the ratio between the inner and outer radii of the annulus. Further, we show that the system of two vortices between parallel walls is a limiting case in an annular region as the radii of the annulus tend to infinity.
0169-5983/14/031420+07$33.00 © 2014 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK 1

Fluid Dyn. Res. 46 (2014) 031420

N N Erdakova and I S Mamaev

Figure 1. Thomsons configuration of five vortices in an annular region.

2. Stability of Thomsons configurations Thomsons configurations of point vortices of equal strength in an annular region are regular N-gons rotating with constant angular velocity about their centers, which coincide with the center of the annulus. Figure 1 shows Thomsons configuration of a five-vortex system in an annular region. The positions of vortices in a polar coordinate system at time t are given by the following expressions

r = a , =

2 + t , N

(1)

where a is the radius of Thomsons orbit of the configuration, is the number of the vortex, N is the number of vortices in the configuration and w is the angular velocity of rotation of the configuration. Generalizing the expression of the Hamiltonian function for two vortices (Erdakova 2010) to the system of N vortices, we obtain
N

H = H0 +
-1

=1

2 Y ( q , ) 4

-
N

=1

4

(Y (
(

q ,

,

)

+ Y q , * - Y q , ,
-1

(

)

(

,

)

- Y q , * ,

(

))

,

( 2)

H0 =

=1

2 2 ln rout - r2 + 4

)
=1

4

ln

2 2 4 r2r - 2rout ( r , r ) + rout , 2 r2 + r - 2( r , r )

(3)

where H0 is the Hamiltonian of the system of N vortices in the cylinder and Y ( q , v ) is expressed in terms of the 1-function, represented as a quickly converging series (Abramowitz and Stegun 1972):

1 ( v 2) =2 Y ( q , v) = ln 1 2Gq 4 sin ( v 2) 1 ( q , v) = 2Gq 4 sin v
1

k=1

q

2k

(

1 - cos kv
2k

)

(

1-q

)
,

k

,

n=1

(

1 - 2q 2n cos 2v + q

4n

)

G=

n=1

(

1-q

2n

)

= const ,

(4)

2

Fluid Dyn. Res. 46 (2014) 031420

N N Erdakova and I S Mamaev

where the parameter q = rin rout is the ratio between the inner and outer radii of the annulus and , , , , , * and * are the functions of the coordinates of vortices and their complex , , conjugates
2 = iln rout r2 ,

(

)

, = - + iln r r , (5)

2 , = - + iln rr rout .

To perform a linear stability analysis of Thomsons configurations, we choose a coordinate system rotating with angular velocity and consider the deviations of the vortices from the equilibrium position (1), which are defined by the following expressions

r = a 1 + ,

= 2 N + ,

(6)

where and are small deviations of the position of the -th vortex from the equilibrium state. We expand the Hamiltonian H (2) of the system in a Taylor series near the equilibrium state (1) and then investigate the sign definiteness of the eigenvalues of its quadratic part
B = H

(

zi z

j

)

z=0

and the eigenvalues of the matrix for the linearization of the vector

field A = JB z = 0 (J is a skew-symmetric matrix of the Poisson structure), with respect to the variable z = ( 1 , 1, ... , N , N ), depending on the ratio between the inner and outer radii of the annulus q (Bolsinov et al 2010).

Remark 1.

The matrices of the quadratic part of the Hamiltonian B and the linearization matrices of the vector field A have a block-circulant structure and are diagonalized by a discrete Fourier transform (Kurakin 2010, Borisov and Mamaev 2005). Analysis of the eigenvalues of the matrices B and A for Thomsons configurations of N vortices in neighborhoods of stationary points reveals that for N < 7 there exist stable polygonal configurations whose stability conditions lN ( a , q ) are

l 3 =-

1 10a12 + 3a10 + 6a8 + 10a 6 + 6a 4 + 3a 2 - 2 2 2 8 a4 + a2 + 1 a2 - 1

(

)(

)

+2 l4 =-

k=1

q 1-q

2k

2k

3 k 2k 2k + 1 - 1 2k - k + k cos ak + ak + cos ak - ak > 0, - cos 4 3 8 3 4 2 3 4

1 7a12 + a8 + 9a 4 - 1 +2 4 a2 + 1 2 a2 - 1 2 a4 + 1

(

)(

)(

)

k=1

q 2k 1-q

2k

k k - 1 k k 1 1 ak + cos k , ak- - ak+ - cos kak+ > 0 , в k cos + ak- + cos 2 4 2 2 4 4 4 l5 = 18a 20 + 10a16 + 15a14 + 34a10 + 15a 6 + 10a 4 - 2 1 8 4 4 4 2 2 2 a + 2 cos a + 1 - a 4 + 2 cos a - 1 1 + a 4 - 2a 5 5

(

)

(

2

)

3

Fluid Dyn. Res. 46 (2014) 031420

N N Erdakova and I S Mamaev

4 2 2 4 - a + 2 cos a - 1 1 + a 4 - 5 6k 2 2k ak+ cos cos cos - cos 5 5 5 5 k 1 4k 2k k + 1 ak - ak- - k + ak+ - ak- > + cos + cos 4 2 5 5 2 4 4 4 2 a + 2 cos a + 1 5 2k k q k + 2 + 1 - q 2k 2 4 k=1

в

(

a4 + a3 + a2 + a + 1

)(

2

a4 - a3 + a2 - a + 1

)

2

(

2a 0,

2

)

l6 = -

1 8 a2 + 1

23a18 + 13a12 + 37a 6 - 1 a4 - a2 + 1 a4 + a2 + 1 1 2k - cos ( ak - kak+ 2 3

(

)(
2k

)(

)(

2

a2 - 1

)

2

+2 +

k=1

q 2k 1-q

)
(7)

1- 1 - cos k 1 - k + k k + k + ak + cos ak + ak - ak > 0, 4 3 2 2 24

where we have introduced the notation

ak+ = a 2k + a

-2k

,

ak- = a 2k - a

-2k

.

Numerically solving the lN ( a , q ) = 0 , we specify, on the plane of parameters (a, q), the curves q ( N ) ( a ) that define the boundaries of the regions of linear stability of Thomsons configurations of the system of N vortices (see figure 2). For q = 0, the critical radii of stable Thomsons orbits are determined by the values r3* = 0.566 816, r4* = 0.574 316, r5* = 0.588 300 3, and r6* = 0.546 919 obtained earlier for Thomsons vortex configurations in a cylinder (Havelock 1931, Kurakin 2010, 2012). Figure 2 also shows the stability region for Thomsons configuration of the system of two vortices in an annulus, which was obtained previously in Erdakova (2010).

Remark 2.

We perform a limiting process in the expressions (7) under the conditions rout , rin = 1 ** and thereby determine the radii of Thomsons configurations rN of the system of N vortices outside the cylindrical region:
** r3 1.911 467 , ** r4 1.801 510 ,

r5** 1.728 780 ,

** r6 1.838 084 , (8)

which coincide with the results for Thomsons configurations of the systems of N vortices outside the circular region, described in detail in Havelock (1931). 3. Vortex system in a strip as a limiting case of the vortex system in an annulus The system of two vortices moving between parallel walls is a limiting case of the two-vortex system in an annular region. We write the stream function of the system of two point vortices with radius vectors r1, r2 and strengths , 2 in an annular region, obtained in Vaskin and 1 Erdakova (2010),
4

Fluid Dyn. Res. 46 (2014) 031420

N N Erdakova and I S Mamaev

configurations on the ratio q between the inner and outer radii for the systems of N = 2, 3, 4, 5, 6 vortices in an annular region. The region of linear stability is located * under the curves, rN correspond to critical radii of stable Thomsons orbits for the * system of vortices in the ring (for q = 0) and qN correspond to the maximum value of the ratio between the inner and outer radii of the ring for the stable Thomson configuration of N vortices.

Figure 2. Dependence of the critical values of the radii q N ( a ) of Thomsons

1 ( r, t ) = - 4

2

=1 k =-

ln

( (

r-q

-2k

r ( t )
-2k

)

2 2

2 r - rout r2q

r ( t )

)

,

q = rin rout .

(9)

Let the inner radius of the annulus rin be such that its width remains fi suppose that the geometric form of the boundaries of the annular region degenerates parallel straight lines (see figure 3). Let h denote the distance between the walls of the (which is the width of the strip as rin ). Then the outer radius of the annulus and between the inner and outer radii of the annulus are defined by the expressions

nite and into two annulus the ratio

rout = rin + h ,

q = rin rout = rin ( rin + h ).

(10)
(see figure 3) with the ~~ OX passing through the the axis OX. We express r in the system XOY in

~~~ Choose a local coordinate system XOY attached to the strip ~~ ordinate axis OY coinciding with the axis OY and the abscissa axis middle of the strip between the walls of the annulus (strip) parallel to the radius vectors of an arbitrary point of the fluid r and the vortex ~~~ terms of their local coordinates in the system XOY

r = ( x , y) = ( x , rin + h 2 + y ), ~ ~

r = ( x , y ) = x , rin + h 2 + y . ~ ~

(

)

(11)

Substituting the expressions for the radius vectors r and r (11) into the stream function of vortices in the annular region (9), and performing a limiting process under the conditions rin , we find the expressions for the stream function and the Hamiltonian of the system of two vortices in a strip with width h = 1,
5

Fluid Dyn. Res. 46 (2014) 031420

N N Erdakova and I S Mamaev

Figure 3. Point vortices with strengths and 2 in a strip with width h. 1

( x, y) = - ~~

sin2 2 y - y1 + sh2 2 ( x - x1) ~ ~ ~ ~ 1 ln 2 2 4 sin 2 y + y - 1 + sh 2 ( x - x1) ~ ~ ~ ~ 1

(

)

(

)

sin 2 y - y2 + sh2 2 ( x - x 2 ) ~ ~ ~ ~ 2 - ln 2 , 4 sin 2 y + y - 1 + sh2 2 ( x - x 2 ) ~ ~2 ~ ~

2

(

)

(

)

(12)

H=-

1 4

12 ln + 22 ln 2 cos y1 2 cos y2 ~ ~ ch ( x1 - x 2 ) - cos y1 - y2 ~ ~ ~ ~ . ch ( x1 - x 2 ) + cos y1 + y2 ~ ~ ~ ~

+ 2 ln 1

( (

) )

(13)

The resulting Hamiltonian function (12) coincides with the expression obtained in Geshev and Yezdin (1983) for the system of two vortices in the strip. In the system of two vortices in an annular region, there exists an additional integral of motion--the moment of vorticity I = r12 + 2r22 --due to the invariance of the Hamiltonian 1 under the systems rotation. Further, we show that in a system of two vortices between parallel walls, there also exists an additional integral

P=
found in Vaskin under shift along We note that the system of an References

1 lim ( I rin - ( + 2 ) ( rin + h ) ) = y1 + 2y2 , (14) ~ 1 1~ 2 r in and Erdakova (2010) and related to the invariance of the Hamiltonian (13) the strip. the resulting Hamiltonian (13) and integral (14) can be easily generalized to arbitrary number of vortices in a strip.

Havelock T H 1931 The stability of motion of rectilinear vortices in ring formation Phil. Mag. 11 61733 Thomson J J 1883 A Treatise on the Motion of Vortex Rings (London: Macmillan) Abramowitz M and Stegun I A 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover) Bolsinov A V, Borisov A V and Mamaev I S 2010 Topology and stability of integrable systems Usp. Mat. Nauk 65 71132
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Fluid Dyn. Res. 46 (2014) 031420

N N Erdakova and I S Mamaev

Borisov A V and Mamaev I S 2005 Mathematical Methods in the Dynamics of Vortex Structures (Moscow-Izhevsk: Institute of Computer Science) p 368 Bolsinov A V, Borisov A V and Mamaev I S 2012 The bifurcation analysis and the Conley index in mechanics Regul. Chaotic Dyn. 17 45778 Vaskin V V and Erdakova N N 2010 On the dynamics of two point vortices in an annular region Rus. J. Nonlin. Dyn. 6 53147 Geshev P I and Yezdin B S 1983 The motion of a vortex pair between parallel walls J. Appl. Mech. Tech. Phys. 24 6637 Erdakova N N 2010 Thomsons configurations in the dynamics of two vortices in an annular region Bulletin of Udmurt University Mathematics, Mechanics, Computer Science pp 716 Kurakin L G 2010 On the stability of Thomsons vortex configurations inside a circular domain Regul. Chaotic Dyn. 15 4058 Kurakin L G 2012 On the stability of Thomsons vortex pentagon inside a circular domain Regul. Chaotic Dyn. 17 15069 Zueva T I 1996 Helium in the rotating annulus: the equilibrium distribution of a large number of vortices Fiz. Nizk. Temp. 22 11002 Pashaev O K and Yilmaz O 2011 Hamiltonian dynamics of N vortices in concentric annular region J. Phys. A: Math. Theor. 44 185501 Lakaniemi M 2007 On the dynamics of point vortices in a quantum gas confined in an annular region arXiv:0708.1898

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