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February  2020, 25(2): 749-760. doi: 10.3934/dcdsb.2019265

Bifurcation of relative equilibria generated by a circular vortex path in a circular domain

David Rojas 1,, and  Pedro J. Torres 2,

1. 

Departament d'Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, 17003 Girona, Spain
2. 

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

* Corresponding author: David Rojas (david.rojas@udg.edu)

Received  December 2018 Revised  March 2019 Published  February 2020 Early access  November 2019

Fund Project: All the authors are partially supported by the MINECO/FEDER grant MTM2017-82348-C2-1-P. The first author is also partially supported by the MINECO/FEDER grant MTM2017-86795-C3-1-P.

We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number.

Keywords: Vortex, passive transport, stirring protocol, stream function, periodic orbit, winding number.
Mathematics Subject Classification: Primary: 25C34, 37N10, 76B47.
Citation: David Rojas, Pedro J. Torres. Bifurcation of relative equilibria generated by a circular vortex path in a circular domain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 749-760. doi: 10.3934/dcdsb.2019265
References:
[1]

H. Aref, Stirring by chaotic advection, J. Fluid Mech., 143 (1984), 1-21.  doi: 10.1017/S0022112084001233.  Google Scholar

[2]

H. Aref, J. Roenby, M. A. Stremler and L. Tophøj, Nonlinear excursions of particles in ideal 2D flows, Phys. D, 240 (2011), 199-207.  doi: 10.1016/j.physd.2010.08.007.  Google Scholar

[3]

A. Boscaggin and P. J. Torres, Periodic motions of fluid particles induced by a prescribed vortex path in a circular domain, Phys. D, 261 (2013), 81-84.  doi: 10.1016/j.physd.2013.07.004.  Google Scholar

[4]

T. Carletti and A. Margheri, Measuring the mixing efficiency in a simple model of stirring: Some analytical results and a quantitative study via frequency map analysis, J. Phys. A, 39 (2006), 299-312.  doi: 10.1088/0305-4470/39/2/002.  Google Scholar

[5]

A. Fonda, M. Sabatini and F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff Theorem, Topol. Methods Nonlinear Anal., 40 (2012), 29-52.   Google Scholar

[6]

P. Franzese and L. Zannetti, Advection by a point vortex in closed domains, European J. Mech. B Fluids, 12 (1993), 43-67.   Google Scholar

[7]

R. Ortega, V. Ortega and P. J. Torres, Point-vortex stability under the influence of an external periodic flow, Nonlinearity, 31 (2018), 1849-1867.  doi: 10.1088/1361-6544/aaa5e2.  Google Scholar

[8] P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.  doi: 10.1017/CBO9780511624063.  Google Scholar
[9] P. J. Torres, Mathematical Models With Singularities, Atlantis Briefs in Differential Equations, 1, Atlantis Press, Paris, 2015.  doi: 10.2991/978-94-6239-106-2.  Google Scholar
[10]

S. Wiggins and J. M. Ottino, Foundations of chaotic mixing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362 (2004), 937-970.  doi: 10.1098/rsta.2003.1356.  Google Scholar

show all references

References:
[1]

H. Aref, Stirring by chaotic advection, J. Fluid Mech., 143 (1984), 1-21.  doi: 10.1017/S0022112084001233.  Google Scholar

[2]

H. Aref, J. Roenby, M. A. Stremler and L. Tophøj, Nonlinear excursions of particles in ideal 2D flows, Phys. D, 240 (2011), 199-207.  doi: 10.1016/j.physd.2010.08.007.  Google Scholar

[3]

A. Boscaggin and P. J. Torres, Periodic motions of fluid particles induced by a prescribed vortex path in a circular domain, Phys. D, 261 (2013), 81-84.  doi: 10.1016/j.physd.2013.07.004.  Google Scholar

[4]

T. Carletti and A. Margheri, Measuring the mixing efficiency in a simple model of stirring: Some analytical results and a quantitative study via frequency map analysis, J. Phys. A, 39 (2006), 299-312.  doi: 10.1088/0305-4470/39/2/002.  Google Scholar

[5]

A. Fonda, M. Sabatini and F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff Theorem, Topol. Methods Nonlinear Anal., 40 (2012), 29-52.   Google Scholar

[6]

P. Franzese and L. Zannetti, Advection by a point vortex in closed domains, European J. Mech. B Fluids, 12 (1993), 43-67.   Google Scholar

[7]

R. Ortega, V. Ortega and P. J. Torres, Point-vortex stability under the influence of an external periodic flow, Nonlinearity, 31 (2018), 1849-1867.  doi: 10.1088/1361-6544/aaa5e2.  Google Scholar

[8] P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.  doi: 10.1017/CBO9780511624063.  Google Scholar
[9] P. J. Torres, Mathematical Models With Singularities, Atlantis Briefs in Differential Equations, 1, Atlantis Press, Paris, 2015.  doi: 10.2991/978-94-6239-106-2.  Google Scholar
[10]

S. Wiggins and J. M. Ottino, Foundations of chaotic mixing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362 (2004), 937-970.  doi: 10.1098/rsta.2003.1356.  Google Scholar

Figure 1.  Phase portrait of system (3) depending on the parameters according to Theorem 2.1
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Figure 2.  Bifurcation diagram of the phase-portrait of system (3) on $ D_R $. The bold curve corresponds to bifurcation parameters $ \mathcal B $, whereas the remaining ones correspond to regular parameters. In Theorem 2.1 the phase portrait at each region is given
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