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Linear stability and saddle-node bifurcation of electromagnetically driven electrolyte flow in an annular layer

Published online by Cambridge University Press:  28 January 2020

John McCloughan Open the ORCID record for John McCloughan [Opens in a new window]  and
Sergey A. Suslov Open the ORCID record for Sergey A. Suslov [Opens in a new window]
John McCloughan
Affiliation:
Department of Mathematics, H38, Swinburne University of Technology, John Street, Hawthorn, Victoria3122, Australia
Sergey A. Suslov*
Affiliation:
Department of Mathematics, H38, Swinburne University of Technology, John Street, Hawthorn, Victoria3122, Australia
*
†Email address for correspondence: ssuslov@swin.edu.au
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Abstract

Comprehensive linear stability study of flow in an annular layer of electrolyte driven by the action of the Lorentz force is conducted following the analysis of steady axisymmetric solutions of Suslov et al. (J. Fluid Mech., vol. 828, 2017, pp. 573–600). It is shown that an experimentally observed instability in the form of anticyclonic moving vortices reported in Pérez-Barrera et al. (Magnetohydrodynamics, vol. 51 (2), 2015, pp. 203–213) develops on a background of the basic flow consisting of two tori with the opposite azimuthal vorticity components. It is found that, while the background flow is driven electromagnetically, the appearance of vortices is purely due to hydrodynamic effects: shear of the flow and centrifugal inertial forcing. The current study has also revealed that the unstable two-torus basic flow has a stable single-torus counterpart, both emanating from a saddle-node bifurcation of steady states when the Lorentz force is sufficiently strong. The transition from a one-torus to two-torus flow at weaker forcing is abrupt and leads to the appearance of vortices as soon as it occurs. The ranges of layer depths and Reynolds numbers for which vortices develop on a steady background are determined. Subsequently, weakly nonlinear amplitude expansion is used to find an approximate unsteady solution beyond the saddle-node bifurcation.

JFM classification

Nonlinear Dynamical Systems: Bifurcation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 887 , 25 March 2020 , A23
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Antar, G., Lalti, A. & Habchi, C. 2019 The spontaneous breaking of axisymmetry in shallow rotating flows. Phys. Fluids 31, 074104.CrossRefGoogle Scholar
Bödewadt, U. T. 1940 Die Drehströmung über festem Grunde. Z. Angew. Math. Mech. 20, 241–253.CrossRefGoogle Scholar
Bondarenko, N. F., Gak, E. Z. & Gak, M. Z. 2002 Application of MHD effects in electrolytes for modeling vortex processes in natural phenomena and in solving engineering-physical problems. J. Engng Phys. Thermophys. 75 (5), 1234–1247.CrossRefGoogle Scholar
Crouch, J. D. & Herbert, T. 1993 A note on the calculation of Landau constants. Phys. Fluids A 5, 283–285.CrossRefGoogle Scholar
Davidson, P. A. 2001 An Introduction to Magneto-Hydrodynamics. Cambridge University Press.CrossRefGoogle Scholar
Dey, P. & Suslov, S. A. 2016 Thermomagnetic instabilities in a vertical layer of ferrofluid: nonlinear analysis away from a critical point. Fluid Dyn. Res. 48, 061404.CrossRefGoogle Scholar
Digilov, R. M. 2007 Making a fluid rotate: circular flow of a weakly conducting fluid induced by a Lorentz body force. Am. J. Phys. 75 (4), 361–367.CrossRefGoogle Scholar
Dolzhanskii, F. V., Krymov, V. A. & Manin, D. Y. 1990 Stability and vortex structures of quasi-two-dimensional shear flows. Sov. Phys. Uspekhi 33 (7), 495–520.CrossRefGoogle Scholar
Dovzhenko, V. A., Krymov, V. A. & Ponomarev, V. M. 1984 Experimental and theoretical study of a shear flow driven by an axisymmetric force. Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (in Russian) 20 (8), 693–704.Google Scholar
Dovzhenko, V. A., Novikov, Y. A. & Obukhov, A. M. 1979 Modelling of the process of generation of vortices in an axisymmetric azimuthal field using a magnetohydrodynamic method. Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (in Russian) 15 (11), 1199–1202.Google Scholar
Dovzhenko, V. A., Obukhov, A. M. & Ponomarev, V. M. 1981 Generation of vortices in an axisymmetric shear flow. Fluid Dyn. 16 (4), 510–518.CrossRefGoogle Scholar
Ekman, V. W. 1905 On the influence of the Earth’s rotation on ocean currents. Arkiv. Mat. Ast. Fys. 2 (11), 1–52.Google Scholar
Faller, A. J. 1991 Instability and transition of disturbed flow over a rotating disk. J. Fluid Mech. 236, 245–269.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Kenjeres, S. 2011 Electromagnetically driven dwarf tornados in turbulent convection. Phys. Fluids 23, 015103.CrossRefGoogle Scholar
Krymov, V. A. 1989 Stability and supercritical regimes of quasi-two-dimensional shear flow in the presence of external friction (experiment). Fluid Dyn. 24 (2), 170–176.CrossRefGoogle Scholar
Lilly, D. K. 1966 On the instability of Ekman boundary layer. J. Atmos. Sci. 23, 481–494.2.0.CO;2>CrossRefGoogle Scholar
MacKerrell, S. O. 2005 Stability of Bödewadt flow. Phil. Trans. R. Soc. Lond. A 363, 1181–1187.CrossRefGoogle ScholarPubMed
Manin, D. Y. 1989 Stability and supercritical regimes of quasi-two-dimensional shear flow in the presence of external friction (theory). Fluid Dyn. 24 (2), 177–183.CrossRefGoogle Scholar
Moffatt, H. K. 1991 Electromagnetic stirring. Phys. Fluids A 3, 1336–1343.CrossRefGoogle Scholar
Moisy, F., Doaré, O., Pasutto, T., Daube, O. & Rabaud, M. 2004 Experimental and numerical study of the shear layer instability between two counter-rotating disks. J. Fluid Mech. 507, 175–202.CrossRefGoogle Scholar
Pérez-Barrera, J., Ortiz, A. & Cuevas, S. 2016 Analysis of an annular MHD stirrer for microfluidic applications. In Recent Advances in Fluid Dynamcis with Environmental Applications (ed. Klapp, J., Sigalotti, L. D. G., Ovando, A. M., Villa, A. L. & Chavarría, G. R.), pp. 275–288. Springer.CrossRefGoogle Scholar
Pérez-Barrera, J., Pérez-Espinoza, J. E., Ortiz, A., Ramos, E. & Cuevas, S. 2015 Instability of electrolyte flow driven by an azimuthal Lorentz force. Magnetohydrodynamics 51 (2), 203–213.CrossRefGoogle Scholar
Pérez-Barrera, J., Ramírez-ZÚñiga, G., Grespan, E. C., Cuevas, S. & del Río, J. A. 2019 Thermographic visualization of a flow instability in an electromagnetically driven electrolyte layer. Exp. Therm. Fluid Sci. 109, 109882.CrossRefGoogle Scholar
Pham, K. G. & Suslov, S. A. 2018 On the definition of Landau constants in amplitude equations away from a critical point. R. Soc. Open Sci. 5, 180746.CrossRefGoogle ScholarPubMed
Qin, M. & Bau, H. H. 2012 Magnetohydrodynamic flow of a binary electrolyte in a concentric annulus. Phys. Fluids 24, 037101.CrossRefGoogle Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. 93, 148–154.CrossRefGoogle Scholar
Satijn, M. P., Cense, A. W., Verzicco, R., Clercx, H. J. H. & van Heijst, G. J. F. 2001 Three-dimensional structure and decay properties of vortices in shallow fluid layers. Phys. Fluids 13 (7), 1931–1945.CrossRefGoogle Scholar
Schaeffer, N. & Cardin, P. 2005 Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers. Phys. Fluids 17, 104111.CrossRefGoogle Scholar
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 17–26.CrossRefGoogle Scholar
Suslov, S. A. & Cuevas, S. 2017 Numerical modelling of axisymmetric electromagnetically driven flows in thin layers. In Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2016 (ed. Droniou, J., Page, M. & Clarke, S.), ANZIAM J., vol. 58, pp. C46–C56. Australian Mathematical Society.Google Scholar
Suslov, S. A. & Paolucci, S. 1997 Nonlinear analysis of convection flow in a tall vertical enclosure under non-Boussinesq conditions. J. Fluid Mech. 344, 1–41.CrossRefGoogle Scholar
Suslov, S. A., Pérez-Barrera, J. & Cuevas, S. 2017 Electromagnetically driven flow of electrolyte in a thin annular layer: axisymmetric solutions. J. Fluid Mech. 828, 573–600.CrossRefGoogle Scholar
Yang, W., Delbende, I., Fraigneau, Y. & Witkowski, L. M. 2019 Axisymmetric rotating flow with free surface in a cylindrical tank. J. Fluid Mech. 861, 796–814.CrossRefGoogle Scholar