Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 5, pp. 915-926. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221217

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 37J25, 76B47, 76M23

On the Stability of the System of Thomson's Vortex n-Gon and a Moving Circular Cylinder

L. G. Kurakin, I. V. Ostrovskaya

The stability problem of a moving circular cylinder of radius R and a system of n identical point vortices uniformly distributed on a circle of radius R0, with n ^ 2, is considered. The center of the vortex polygon coincides with the center of the cylinder. The circulation around the cylinder is zero. There are three parameters in the problem: the number of point vortices n, the added mass of the cylinder a and the parameter q =

The linearization matrix and the quadratic part of the Hamiltonian of the problem are studied. As a result, the parameter space of the problem is divided into the instability area and the area of linear stability where nonlinear analysis is required. In the case n = 2, 3 two domains of linear stability are found. In the case n = 4, 5, 6 there is just one domain. In the case n ^ 7 the studied solution is unstable for any value of the problem parameters. The obtained results in the limiting case as a ^^ agree with the known results for the model of point vortices outside the circular domain.

Keywords: point vortices, Hamiltonian equation, Thomson's polygon, stability

Received August 19, 2022 Accepted November 09, 2022

The work of the first author was carried out within the framework of the project no. FMWZ-2022-0001 of the State Task of the IWP RAS.

Leonid G. Kurakin lgkurakin@sfedu.ru

Water Problems Institute, RAS

ul. Gubkina 3, Moscow, 119333 Russia

Southern Mathematical Institute, VSC RAS

ul. Vatutina 53, Vladikavkaz, 362025, Russia

Southern Federal University

ul. Milchakova 8a, Rostov on Don, 344090, Russia

Irina V. Ostrovskaya ivostrovskaya@sfedu.ru

Southern Federal University

ul. Milchakova 8a, Rostov on Don, 344090, Russia

1. Introduction

Thomson's vortex polygon is a configuration of identical point vortices located at the vertices of a regular polygon. This vortex configuration owes its name to two famous scientists. W. Thomson (Lord Kelvin) posed the stability problem of such a polygon on the plane in connection with his vortex theory of the atom [1]. The linear analysis of the stability problem was begun by J.J.Thomson [2] and completed by T.H.Havelock [3]. The history of solving this problem in a linear and nonlinear setting is described in detail in [4, 5].

Numerous studies have been devoted to the dynamics of point vortices outside a circular domain (see the review [6]). The stability problem of stationary rotation of Thomson's vortex polygon outside a fixed circular cylinder with zero circulation around the cylinder had been solved by Havelock in [3]. The nonlinear analysis of this problem required the involvement of the resonance theory of equilibria of Hamiltonian systems (see the review [7]). It turned out that two resonances lead to instability, although linear stability takes place. The effect of circulation in the problem under consideration in the case of vortices outside a circle was studied in [8].

Various forms of the equations of motion for a moving rigid circular cylinder interacting with n point vortices were obtained in [9-14]. The history of the derivation of these equations is given in the introduction of [14]. These equations are invariant under the group of motions of the plane E(2). According to Noether's theorem for Hamiltonian systems, there exist three integrals of motion. Two integrals correspond to translations along the coordinate axis. These integrals are a generalization to the classical linear momentum. The third integral corresponds to the invariance under rotations about an axis perpendicular to the plane of motion. In this paper, we use the reduced system that is obtained in [12, 13] as a result of reduction of the complete equations of motion by symmetry due to the integrals corresponding to the translations.

In this paper, the stability of a system consisting of Thomson's n-gon and a moving cylinder is studied for arbitrary n ^ 2 with zero circulation around a cylinder. The centers of the vortex n-gon and the cylinder coincide. A linear stability analysis is carried out for an arbitrary number of point vortices n ^ 2. In the case n ^ 6, linear stability conditions are found under which nonlinear analysis is required to solve the stability problem. It is proved that, in the case of n ^ 7, the system considered is unstable for any value of the problem parameters. Here we also correct the erroneous results of [15] for the case n = 2.

2. Formulation of the problem

The motion of a circular cylinder interacting with n identical point vortices is considered. As a result of the reduction of the complete equations of motion, the system of equations is written in complex form in [13]

n

(" j 3

-ijzc + iloY^ (zj - zj)' j=1

R2v h i% . A/ 1 1 \ , , (2-1}

zk Zk zk zk j\ zk zj zk zj J

j=k

Here the complex variables zc = xc + iyc, zk = xk + iyk define the position of the cylinder and point vortices, v = i\ + iv2 is the cylinder's velocity zk = is the reflection of the fct.h vortex

from the boundary of the circle, R is the cylinder's radius, the constant coefficient a involves the

added mass of the cylinder, and the constants y and y0 are related to the circulation around the

r r

cylinder T and the intensity of identical point vortices T0 by the formulae 7 = ^ and 70 = 27, see Fig. 1.

O x

Fig. 1. The system of a moving cylinder and the kth vortex

Parameter r is circulation of the cylinder in the absence of point vortices. The circulation around the cylinder in the presence of vortices is the sum of the circulation r and the total intensity of the reflected vortices zk, k = 1, ..., n with intensity —rk:

n

r — £ rk.

k=l

We consider the case rk = r0, k = 1, ..., n and zero circulation around the cylinder in the presence of point vortices, that is,

Y = nYo. (2.2)

Note that in [16] a complete bifurcation analysis of the motion of the circular cylinder and two point vortices with arbitrary circulation was carried out for the case not considered here when circulation y and the total impulse of the system are equal to zero. In [14] this study was carried out for the case of two point vortices with opposite intensity. The system (2.1)-(2.2) can be written in the form

"7o 4 = ~2iHzc, Yo^fc = ~2iHzk-The Hamiltonian H = H(z, z), z = (zc, z1, ..., zn) is given by the formula

(2.3)

H = + T E (ln (N2 - - ln N2) +

k=i

+ y E (inl^-^l'-lnl^.-^l2). (2.4)

The system (2.3) has the solution (see Fig. 2)

2C = 0, zk = eiuj^uk, uk = R0ei2^{k-l)

(2.5)

n

corresponding of the stationary rotation of the n point vortices around the cylinder with constant angular velocity wn:

Yn / 2n \ R2 , N

ro '3n-l-z-r), q = (2.6)

1 - qn

Rn

The point vortices are located uniformly on a circle of radius R0, R0 > R. Then 0 < q < 1. The solution (2.5)-(2.6) exists only for n ^ 2 and therefore the results of the study are not applicable for n = 1.

Fig. 2. The stationary rotation of Thomson's vortex pentagon around a cylinder with a common center at the origin is described by solution (2.5)

Without loss of generality, we will further assume that

Yn = 1> Rn = 1.

(2.7)

The angular velocity uon increases monotonically in the interval q e (0, 1) from — to

( i \ l/n

infinity. When parameter q passes through the point qn = ( ^tj j , the angular velocity ojn changes its sign and the Thomson's vortex n-gon changes the direction of rotation from clockwise to counterclockwise. Further studies showed that the point qn lies in the instability areas for all n ^ 2.

The change of variables

2C = (qc + ipc )e

zk = V1 + 2rke>

t

2TT(fc —1)

k = 1, ..., n,

(2.8)

reduces the system (2.3)-(2.4) to the perturbation equations with the Hamiltonian E=E(p),

p = (Qc, rl,..., rn, Pc, &!,..., en):

E(p) = H(z(p), z (p)) + ^ nql + np2c -n- 2 ^ rk

k=1

The stationary solution (2.5) corresponds to a continuous family of equilibria

C = Uc = Pc = r1 = ... = rn = 0, = ... = en}.

(2.9)

(2.10)

n

The stability of the continuous family of equilibria (2.10) is equivalent to the orbital stability of the stationary solution (2.5). Let

E2(p) = (SnP, P) (2.11)

be the quadratic terms of the expansion of the Hamiltonian E(p) in a Taylor series in power of p in a neighborhood of the zero equilibrium position. Here (-, ■) is the scalar product. The symmetric matrix Sn has the form

(

naln

nblnh1 0

nblnh

lnhl

0 nb2nhn— A

F

ln

nblnh

lnhn l

G

0n

nblnhn l naln nb2nhl

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\—nb2nhn-l — G

0n

nb2nhl

F

2n

The vectors h1 and hn_ 1 are given by the formulas

'2

h1 = \/ — (1, cos v, ..., cos(n — 1 )v),

hn_1 = \l — (0, sinz/, ..., sin(??. — l)v).

Here v = ^ and the coefficients a,ln, bln and b2n are given by the formulas

n u)n V2n(l + <

= + bln=

b2n

b2n —

iU(L —

4a

(2.12)

The symmetric matrices F1n, F2n and the skew-symmetric matrix G0n are circulant mat-

rices:

F

-■- nr.

def

nl

fmO^n + ^^ fmj, G0 — ^^ g0j &

nl

*-n 1 / j J mj j=l

/o 1 0 0 0 0 10

0 0 0 0 10 0 0

j=l

0

where In is an n x n unit matrix.

The coefficients fmj and g0j can be written as

fmo(q, a) = ^f^o(q) + fmo(a, q), fmjiq, a) = ^fmj(q) + fUa> 9)'

9oj(q, a) = g9oM) + 9oj(a, Q), j = 1, ...,n-l.

(2.13)

Here the coefficients fm j and g0j match the corresponding coefficients written out in [3, 8].

S

n

0

The coefficients fm? and gL, j = 0, n — 1 are given by the equations

2a

= <1

(2.14)

go?(a, q) =

q'2 — 1 2 a

sin Vj.

The values Amk and i\0k are eigenvalues of matrices Fmn and G0n given in the form

\nk a) = + ^mfc(<?) a),

where A0k, A^k and A0k are the same as in [3, 8] and Ajk, Al,k and A0k are defined as

(2.15)

\i = \i

A11 = A1,n-1

\1 = \1 a21 = 1

\1 = \1

A01 = —A0rn-1

n{l + q)2 4a ,

n(1 — q)2

4 a ' n(q2 - 1) 4a, :

(2.16)

^1fc = ^2fc = ^0fc = 0, k = 1,n — 1.

The symmetric matrix Sn has zero eigenvalue corresponding to the family of equilibria (2.10). The family C is Lyapunov stable in an exact nonlinear setting if the Hamiltonian E(p) has an extremum on it. To do this, it is necessary that all the eigenvalues of the matrix Sn except for a simple zero have the same sign. As shown later in Proposition 2, this does not hold.

The linearization matrix Ln of the system with Hamiltonian (2.9) about the zero solution has the form

Ln — 2K JSn, J

0

I

n+1

n+1 0

(2.17)

where K 1 is a diagonal (2n + 2) x (2n + 2) matrix:

K"1 = diag ( -, -1, n

-1, -, -1, ..., -1 n

Then the matrix Ln can be written as

/ 0 b1nhn-1

Ln = 2

1n

b1nh1 \

nb2nhn-1

—nb1nh1 —F2n 0 b2nhn-1

\ nb1nh1 F1n nb1nhn— 1 G0n J

a

1n

G0n — b1nh1

Instability of solution (2.5) occurs when the linearization matrix Ln has eigenvalues with positive real part.

In the case n = 2 the characteristic polynomial of the linearization matrix L2 is given by the equation

det(<rI6 - L2)= a2^(a2), (2.18)

where ^22(a2) is the biquadratic polynomial

^i2(a2 ) = a4 + 8pna2 + 16poi, (2.19)

Pii = a2!2 - 86162 + AnA21, (2.20)

Poi = a2i2*ii*2i - 4b2i2^2i - 4622An + 166^b^. (2.21)

If n ^ 3, the eigenvalues of the linearization matrix Ln are the roots of the following polynomial:

LfJ

det(al2n+2 - Ln) = a2 J] ^n(a2). (2.22)

j=1

Here ^2n(a2) is the bicubic polynomial

^in(a2) = a6 + 4p2i a4 + 16pna2 + 64poi, (2.23)

P21 = 2AiiA2i + 2A§i - 4n6in62n + aL

OOOO o o

Pll = 4n 6ln62n - 4nAoi6ln62n - 4nAoi A2162n - 4nAoiA1162n+

+ 4nAoiai„6i„62n. - 4nA2iA1161n62n - 2nA2iai„62n - 2nAuai,n62n+

A01 a1nb1nb2n 4nA21A11b1nb2n 2nA21a1nb1n 2nA11a1nb2n

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+ A01 - 2A21A21 An + 2A21 a2n + A21A21 + 2A21Ana2n,

P01 = (2nb1nb2nA01 + nb1nA21 + nA11b2n + a1nA01 — A11A21 a1n)

the polynomials ^jn(a2), j = 2, ..., [^^J for n > 4 are biquadratic polynomials

^jn(a2) = a4 + 8p2j a2 + 16poj, (2.24)

Plj = A1j A2j + Aoj) Poj = (A1jA2j - Aoj) . In the case of even n ^ 4 the polynomial ^n/2 n(a2) has the form

*n/2 ,n(a2 )= a2 +4Ai ,n/2 A2 n/2. (2.25)

In the case n ^ 4, the polynomials (2.24) and (2.25) do not depend on parameter a, and their roots coincide with the corresponding eigenvalues of the linearization matrix in the case of a fixed cylinder [3]. In [3] it has been shown that among them for n ^ 7 there is at least one eigenvalue with positive real part. Hence, and from the analysis of polynomials in the case of 2 ^ n ^ 6, the following statements follow.

Proposition 1.

1. If n ^ 7, the solution (2.5)-(2.6) is unstable for any values of the problem parameters: 0 < q < 1, a > 0.

coefficients of the biquadratic polynomial ^12(ct2) satisfy the conditions

2. If n = 2, all eigenvalues of the linearization matrix L2 lie in the imaginary axes if the

i2( 2

Poi(q,a) ^ pii(q,a) ^

2 (2.26)

^i2(q> a) = Pii - 4poi ^ 0.

3. If 3 ^ n ^ 6, all eigenvalues of the linearization matrix Ln He in the imaginary axes if all the following conditions are valid :

(a) The cubic polynomial ^1n(s) is stable, that is, satisfies the Vyshnegradsky conditions :

P01(q, a) ^ 0, Pu(q, a) ^ 0, P21(q, a) ^ 0,

Aln(q, a) = P11P21— P01 ^0, (2.27)

and its discriminant D1n is not negative :

Dm(q, a) = —4p^1 P01 + p21 P21 — 4p^ + 18P01P11P21 — 27p^ ^ 0. (2.28)

(b) The coefficients and the discriminants of the quadratic polynomials ^jn(s), j =

= 2, ..., for n > 3 are not negative:

P0j (q) ^0, Pu (q) ^ 0, (2 29)

Djn(q) = p1j— 4P0j >0.

4. The linearization matrix Ln, n = 2, ..., 6 has at least one eigenvalue in the right half-plane if at least one of conditions 2) or 3) is violated.

Let us analyze the eigenvalues of the matrix Sn in the case 2 ^ n ^ 6. If n = 2, then its characteristic polynomial has the form

det(AI6 — S2) = A(A — A12)$21 (A)$22 (A), where $2m(A) are quadratic polynomials

^2m = A2 + k1mA + k0m = 0, k1m = —2a12 + , k0m = 8bm2 — 2a12K1m.

In the case n = 3, 5 the characteristic polynomials of the matrix Sn are

det(AI8 — S3) = A(A — A1a)$21(A), det(Al12 — S5) = A(A — A15)$§1(A)$22(A).

(2.30)

(2.31)

In the case n = 4, 6, we have

det(AI10 - S4) = A(A - A14)$41 (A)$42(A), det(AI14 - S6) = A(A - Ai6)$ii(A)^(A)$63(A).

The polynomials $n1(A) in (2.30), (2.31) are cubic polynomials given by the formula

$n1(A)=A3 + ^A2 + kn A + kn, (2.32)

k21 (q, a) = -na1n - A21 - A1^

a) = — + bL) n2 + a1n(K21 + K11)n + K21K11 — K

01

k01 (q, a) = (2K01b1nb2n + K11 bL + K21b1n) n2 — a1n (K21K11 — K00 n.

2

The polynomials $42(A) and $63(A) are written as

$n,n/2(A) = (A - A1 , n/2)(A - A2,n/2).

The polynomials $52(A) and $62(A) are quadratic polynomials:

$n2(A)=A2 + k^A + ko2, (2.33)

k12(q) = A12 - A22' ko2(q) = A12A22 - A02.

The calculations show that in the case 3 ^ n ^ 6 the roots of the cubic polynomials $n1(A) have different signs. A similar case takes place for the polynomial $22$22 at n = 2.

Proposition 2. The matrix Sn, 2 ^ n ^ 6, has eigenvalues with different signs for all values of the problem parameter and the quadratic form (2.11) alternates in sign. Thus, a nonlinear stability analysis is required in linear stability domains (white domains in Figs. 3-5).

3. Formulation of results

In the case n ^ 7 the solution (2.5)-(2.6) is unstable for all of the values of the problem parameter (Proposition 1).

We introduce the parameter a:

1-a

a = --. (3.1)

1 + a v 7

In the case 2 ^ n ^ 6, analysis of the eigenvalues of the matrices Ln and Sn shows that the parameter space (q, a), where 0 < q < 1 and -1 < a < 1, is divided into two types of areas shown in Figs. 3-5:

1. The white area is a linear stability area, where the eigenvalues of the linearization matrix Ln are all purely imaginary, and the eigenvalues of the matrix Sn have different signs.

In this case, a nonlinear analysis is required to conclude stability.

2. The shaded area is an instability area. The linearization matrix Ln has eigenvalues with positive real part. The solution (2.5)-(2.6) is nonlinearly unstable.

Figures 3, 4 present the cases n = 2, 3, respectively, and Fig. 5 shows the cases n = 4, 5, 6.

For n = 2, 3 two areas of linear stability are found. In Fig. 3 the curves a22 are given by the equation V22 = 0, where D22 is the discriminant of the biquadratic polynomial (2.19). The curves a22 are defined by the equation poi = 0, where poi is given by (2.21). In Fig. 4 the boundaries of linear stability domains are given by the equation D13 = 0, where D13 is given by the formula (2.28).

In the cases n = 4, 5, 6 one linear stability area is shown in Fig. 5. Its boundary consists of the curve a2n given by the equation D1n = 0 and the straight line q = qm. Here D1n is given by Eq. (2.28). The constants qm were calculated by Havelock in [3]:

q*2 & 0.148536, q„3 & 0.273695, q*4 & 0.308125, q„5 & 0.334596, q„6 & 0.295985.

1.0 0.8 0.6 0.4 0.2 a 0 -0.2 -0.4 -0.6 -0.8 -1.0

a

a

12

a

22

a

22

1*2 0.2

0.4 0.6 q

0.8 1.0

Fig. 3. Diagram of stability of stationary rotation of Thomson's vortex n-gon around a cylinder (the solution (2.5)) in the case n = 2

-0.8 -0.82 -0.84 -0.86 -0.88 a -0.9 -0.92 -0.94 -0.96 -0.98 -1

0 0.1 0.2 9*3 0.3

q

(a)

a

0.4

1

0.98 0.96 0.94 0.92 0.9

\

0 0.1 0.2 0.3 q

(b)

0.4

Fig. 4. Diagram of stability of the solution (2.5) in the case n = 3

Figures 3-5 make it possible to compare the linear analysis of this problem with related ones. There are three problem statements: 1) no cylinder (W.Thomson [1]); 2) fixed cylinder (T. H. Havelock [3]), and 3) moving cylinder (this paper).

In the case of the Thomson problem, the regime under study is linearly stable for 2 ^ n ^ 7 and unstable for n ^ 8 [2, 3].

The presence of a moving or a fixed cylinder narrows the area of linear stability. It makes the studied regime unstable for all values of the problem parameters in the case of n = 7, and leads to instability in the shaded area in Figs. 3-5 for n = 2, ..., 6.

The limiting case a = -1 in Figs. 3-5 corresponds to the case of a fixed cylinder (a = to). The linear stability of the solution takes place at 0 < q < qm, which agrees with the results of T. H. Havelock [3]. As the added mass a decreases (a increases in the interval -1 < a < an), the area of linear stability decreases and disappears when passing through the point a

n

a

-0.9 -0.92 -0.94 -0.96 -0.98 -1,

0 0.1 0.2 0.3 0.4 q

(a)

-0.9

-0.92 -0.94 -0.96 -0.98 -1

a

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a

0.5

-0.9 -0.92 -0.94 -0.96 -0.98 -1,

0

0.1 0.2 0.3 0.4 0.5

q

(b)

0 0.1 0.2 0.3 0.4 0.5 q

(c)

Fig. 5. Diagram of stability of the solution (2.5) in the case (a) n = 4, (b) n = 5, (c) n = 6

for n = 2, ..., 6:

a2 = -0.866577, a3 = -0.887762, a4 = -0.954859, a5 = -0.974687, a6 = -0.994320.

If -1 < a < an, the interval of stability of the parameter q for a fixed a lies between two intervals of instability.

In the case n = 2, 3 with further decrease in the parameter a (an increase in the parameter a), the area of linear stability reappears in Figs. 3, 4b.

In conclusion, we note possible directions for the development of studies of the problem considered. First, it is a nonlinear analysis in the white areas of the problem parameters in Figs. 3-5. It is also of interest to study the influence of an arbitrary circulation of the cylinder r on the linear stability. For the case of a fixed cylinder, such a study was carried out in [8]. It was shown that, for any fixed n, circulation can be used to make a Thomson n-gon stable. The same effect will apparently also take place in the case of a moving cylinder.

Conflict of interest

The authors declare that they have no conflict of interest.

References

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