Abstract
The interaction of a moving circular cylinder of radius R with the arbitrary circulation \(\gamma \) and two parallel vortex filaments of identical intensity are considered. The stability of a system of two vortex filaments and a cylinder located in the middle between them is studied. The filaments rotate around the cylinder at a constant angular velocity. The problem depends on three parameters \((q,a,\gamma )\): \(q=R^2/R_0^2,\) where \(2R_0\) is the distance between filaments, \(0<q<1,\) the added mass of the cylinder \(a>0\) and the circulation around cylinder \(\gamma \ne 0.\) The case \(\gamma =2\) was studied earlier. The purpose of this paper is to study the influence of circulation \(\gamma \) in the problem under consideration. The eigenvalues of the linearization matrix are studied for all parameter values. For fixed values of \(\gamma ,\) the parameter plane (q, a) is divided into areas of two types: the area of instability, when the linearization matrix has at least one eigenvalue in the right half-plane, and the linear stability area—all eigenvalues lie on the imaginary axis. The results of the study are consistent with the limiting case of a fixed cylinder (for \(a\rightarrow \infty \)).
Similar content being viewed by others
Data availability
Not applicable.
References
von Helmholtz, H.: Über integrale der hydrodynamischen Gleichungen, welche der Wirbelbewegung entsprechen. J. für die reine und angewandte Mathematik 55, 173–196 (1858)
Newton, P.K.: The \(n\)-Vortex Problem: Analytical Techniques. Applied Mathematical Sciences, vol. 145. Springer, New York (2001)
Borisov, A.V., Mamaev, I.S.: Mathematical Methods in the Dynamics of Vortex Structures, p. 368. Institute of Computer Sciences, Moscow–Izhevsk (2005)
Aref, H.: Point vortex dynamics: a classical mathematics playground. J. Math. Phys. 48(6), 065401 (2007)
Sokolovskiy, M.A., Verron, J.: Dynamics of Vortex Structures in a Stratified Rotating Fluid. Atmospheric and Oceanographic Sciences Library, vol. 47. Springer, Switzerland (2014)
Koshel, K.V., Ryzhov, E.A., Carton, X.J.: Vortex interactions subjected to deformation flows: a review. Fluids 4(1), 14 (2019)
Dritschel, D.G., Sokolovskiy, M.A., Stremler, M.A.: Celebrating the 200th Anniversary of the Birth of Hermann Ludwig Ferdinand von Helmholtz (31.08.1821–08.09.1894). Regul. Chaotic Dyn. 26(5), 463–466 (2021). https://doi.org/10.1134/S1560354721050014
Thomson, W.: Floating magnets (illustrating vortex systems). Nature 18, 13–14 (1878)
Thomson, J.J.: A Treatise on the Motion of Vortex Rings: An Essay to Which the Adams Prize was Adjudged in 1882, in the University of Cambridge, pp. 94–108. Macmillan, London (1883)
Havelock, T.H.: The stability of motion of rectilinear vortices in ring formation. Philos. Mag. 11(70), 617–633 (1931)
Kurakin, L.G., Yudovich, V.I.: The stability of stationary rotation of a regular vortex polygon. Chaos 12(3), 574–595 (2002)
Bogomolov, V.A.: Model of fluctuations of the centers of action of an atmosphere. Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 15, 243 (1979)
Thomson, J.J.: XXIV. On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure. Lond. Edinb. Dublin Philos. Mag. J. Sci. 7(39), 237–265 (1904)
Kurakin, L.G.: On the nonlinear stability of regular vortex polygons and polyhedrons on a sphere. Dokl. Phys. 48(2), 84–89 (2003). https://doi.org/10.1134/1.1560737
Kurakin, L.G.: Influence of annular boundaries on Thomson’s vortex polygon stability. Chaos 24(2), 023105 (2014)
Erdakova, N.N., Mamaev, I.S.: On the dynamics of point vortices in an annular region. Fluid Dyn. Res. 46(3), 031420 (2014)
Stewart, H.J.: Hydrodynamic problems arising from the investigation of the transverse circulation in the atmosphere. Bull. Am. Math. Soc. 51, 781–799 (1945)
Morikawa, G.K., Swenson, E.V.: Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14(6), 1058–1073 (1971)
Kurakin, L.G., Ostrovskaya, I.V.: On stability of Thomson’s vortex \({N}\)-gon in the geostrophic model of the point Bessel vortices. Regul. Chaotic Dyn. 22(7), 865–879 (2017). https://doi.org/10.1134/S1560354717070085
Bergmans, J., Kuvshinov, B.N., Lakhin, V.P., Schep, T.J.: Spectral stability of Alfven filament configurations. Phys. Plasmas 7(6), 2388–2403 (2000)
Kurakin, L.G., Lysenko, I.A.: On the stability of the orbit and the invariant set of Thomson’s vortex polygon in a two-fluid plasma. Russ. J. Nonlinear Dyn. 22(7), 3–11 (2020)
Artemova, E., Kilin, A.: Nonlinear stability of regular vortex polygons in a Bose–Einstein condensate. Phys. Fluids 33(12), 127105 (2021). https://doi.org/10.1063/5.0070763
Dritschel, D.G.: Ring configurations of point vortices in polar atmospheres. Regul. Chaotic Dyn. 26(5), 467–481 (2021)
Reinaud, J.N.: Circular vortex arrays in generalised Euler’s and quasi-geostrophic dynamics. Regul. Chaotic Dyn. 27(3), 352–368 (2022). https://doi.org/10.1134/S1560354722030066
Kilin, A.A., Borisov, A.V., Mamaev, I.S.: The dynamics of point vortices inside and outside a circular domain. In: Borisov, A.V., Mamaev, I.S., Sokolovskiy, M.A. (eds.) Fundamental and Applied Problems in the Theory of Vortices, pp. 414–440. Institute of Computer Sciences, Moscow-Izhevsk (2003)
Kurakin, L.G.: Stability, resonances, and instability of the regular vortex polygons in the circular domain. Dokl. Phys. 49(11), 658–661 (2004). https://doi.org/10.1134/1.1831532
Kurakin, L.G., Melekhov, A.P., Ostrovskaya, I.V.: A survey of the stability criteria of Thomson’s vortex polygons outside a circular domain. Bol. Soc. Mat. Mex. 22(2), 733–744 (2016)
Kurakin, L., Ostrovskaya, I.: On the effects of circulation around a circle on the stability of a Thomson vortex \({N}\)-gon. Mathematics 8(6), 1033 (2020). https://doi.org/10.3390/math8061033
Mamaev, I.S., Bizyaev, I.A.: Dynamics of an unbalanced circular foil and point vortices in an ideal fluid. Phys. Fluids 33(8), 087119 (2021)
Ramodanov, S.M., Sokolov, S.V.: Dynamics of a circular cylinder and two point vortices in a perfect fluid. Regul. Chaotic Dyn. 26(6), 675–691 (2021)
Bizyaev, I.A., Mamaev, I.S.: Qualitative analysis of the dynamics of a balanced circular foil and a vortex. Regul. Chaotic Dyn. 26(6), 658–674 (2021)
Bizyaev, I.A., Mamaev, I.S.: Dynamics of a circular foil and two pairs of point vortices: new relative equilibria and a generalization of Helmholtz leapfrogging. Symmetry 15(3), 698 (2023)
Bizyaev, I.A., Mamaev, I.S.: Dynamics of a pair of point vortices and a foil with parametric excitation in an ideal fluid. Bull. Udmurt Univ. Math. Mech. Comput. Sci. 30(4), 618–627 (2020)
Vetchanin, E.V., Mamaev, I.S.: Periodic perturbation of motion of an unbalanced circular foil in the presence of point vortices in an ideal fluid. Bull. Udmurt Univ. Math. Mech. Comput. Sci. 32(4), 630–643 (2022)
Kurakin, L.G., Ostrovskaya, I.V.: On the stability of the system of Thomson’s vortex \(n\)-gon and a moving circular cylinder. Russ. J. Nonlinear Dyn. 18(5), 915–926 (2022)
Ramodanov, S.M.: Motion of a circular cylinder and \({N}\) point vortices in a perfect fluid. Regul. Chaotic Dyn. 7(3), 291–298 (2002)
Shashikanth, B.N., Marsden, J.E., Burdick, J.W., Kelly, S.D.: The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with \({N}\) point vortices. Phys. Fluids 14(3), 1214–1227 (2002)
Borisov, A.V., Mamaev, I.S.: Integrability of the problem on motion of cylinder and vortex in the ideal fluid. Regul. Chaotic Dyn. 8(2), 163–166 (2003)
Borisov, A.V., Mamaev, I.S., Ramodanov, S.M.: Motion of a circular cylinder and \({N}\) point vortices in a perfect fluid. Regul. Chaotic Dyn. 8(4), 449–462 (2003)
Borisov, A.V., Mamaev, I.S., Ramodanov, S.M.: Dynamics of a cylinder interacting with point vortices. In: Borisov, A.V., Mamaev, I.S. (eds.) Mathematical Methods in the Dynamics of Vortex Structures, pp. 286–307. Institute of Computer Sciences, Moscow-Izhevsk (2005)
Borisov, A.V., Mamaev, I.S., Ramodanov, S.M.: Dynamic interaction of point vortices and a two-dimensional cylinder. J. Math. Phys. 48(6), 065403 (2007)
Kurakin, L.G.: On the stability of the regular \(n\)-sided polygon of vortices. Dokl. Phys. 39(4), 284–286 (1994)
Kurakin, L.G., Ostrovskaya, I.V.: Stability of the Thomson vortex polygon with evenly many vortices outside a circular domain. Sib. Math. J. 51(3), 463–474 (2010)
Kurakin, L.G., Ostrovskaya, I.V.: Nonlinear stability analysis of a regular vortex pentagon outside a circle. Regul. Chaotic Dyn. 17(5), 385–396 (2012)
Acknowledgements
We thank both reviewers for the useful comments. 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.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Explicit formulas characteristic polynomial \(\mathfrak {L}(\lambda )\)
Appendix: Explicit formulas characteristic polynomial \(\mathfrak {L}(\lambda )\)
Characteristic polynomial (18), (19) of matrix linearization (17) has the form
Using formulas (16) we write coefficients of characteristic polynomial in explicit form as functions of the parameters q, a and \(\gamma \)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kurakin, L., Ostrovskaya, I. On the influence of circulation on the linear stability of a system of a moving cylinder and two identical parallel vortex filaments. Bol. Soc. Mat. Mex. 29, 79 (2023). https://doi.org/10.1007/s40590-023-00550-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40590-023-00550-y