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Elementary model of internal electromagnetic pinch-type instability

Published online by Cambridge University Press:  08 March 2017

Jānis Priede Open the ORCID record for Jānis Priede [Opens in a new window]
Jānis Priede*
Affiliation:
Applied Mathematics Research Centre, Coventry University, Priory Street, Coventry CV1 5FB, UK
*
†Email address for correspondence: J.Priede@coventry.ac.uk
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Abstract

We analyse numerically a pinch-type instability in a semi-infinite planar layer of inviscid conducting liquid bounded by solid walls and carrying a uniform electric current. Our model is as simple as possible but still captures the salient features of the instability which otherwise may be obscured by the technical details of more comprehensive numerical models and laboratory experiments. Firstly, we show the instability in liquid metals, which are relatively poor conductors, differs significantly from the astrophysically relevant Tayler instability. In liquid metals, the instability develops on the magnetic response time scale, which depends on the conductivity and is much longer than the Alfvén time scale, on which the Tayler instability develops in well conducting fluids. Secondly, we show that this instability is an edge effect caused by the curvature of the magnetic field, and its growth rate is determined by the linear current density and independent of the system size. Our results suggest that this instability may affect future liquid-metal batteries when their size reaches a few metres.

JFM classification

Instability: Instability MHD and Electrohydrodynamics: MHD and Electrohydrodynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 816 , 10 April 2017 , pp. 705 - 718
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Copyright
© 2017 Cambridge University Press 

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bojarevičs, V., Freibergs, J. A., Shcherbinin, E. V. & Shilova, E. I. 1989 Electrically Induced Vortical Flows. Kluwer.CrossRefGoogle Scholar
Boyd, J. P. 2013 Chebyshev and Fourier Spectral Methods: Second Revised Edition. Dover.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics. Cambridge University Press.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.CrossRefGoogle Scholar
Haines, M. G., Lebedev, S. V., Chittenden, J. P., Beg, F. N., Bland, S. N. & Dangor, A. E. 2000 The past, present, and future of Z pinches. Phys. Plasmas 7 (5), 1672–1680.Google Scholar
Herreman, W., Nore, C., Cappanera, L. & Guermond, J.-L. 2015 Tayler instability in liquid metal columns and liquid metal batteries. J. Fluid Mech. 771, 79–114.Google Scholar
Kelley, D. H. & Sadoway, D. R. 2014 Mixing in a liquid metal electrode. Phys. Fluids 26 (5), 057102.Google Scholar
Kim, H., Boysen, D. A., Newhouse, J. M., Spatocco, B. L., Chung, B., Burke, P. J., Bradwell, D. J., Jiang, K., Tomaszowska, A. A., Wang, K. et al. 2013 Liquid metal batteries: past, present, and future. Chem. Rev. 113 (3), 2075–2099.Google Scholar
Michael, D. H. 1954 The stability of an incompressible electrically conducting fluid rotating about an axis when current flows parallel to the axis. Mathematika 1 (01), 45–50.CrossRefGoogle Scholar
Müller, U. & Bühler, L. 2001 Magnetofluiddynamics in Channels and Containers. Springer.Google Scholar
Ogilvie, G. I. & Pringle, J. E. 1996 The non-axisymmetric instability of a cylindrical shear flow containing an azimuthal magnetic field. Mon. Not. R. Astron. Soc. 279 (1), 152–164.CrossRefGoogle Scholar
Priede, J. 2011 Edge pinch instability of oblate liquid metal drops in a transverse AC magnetic field. J. Fluid Mech. 676, 218–236.Google Scholar
Priede, J. 2015 Metamorphosis of helical magnetorotational instability in the presence of axial electric current. Phys. Rev. E 91, 033014.Google Scholar
Priede, J., Etay, J. & Fautrelle, Y. 2006 Edge pinch instability of liquid metal sheet in a transverse high-frequency ac magnetic field. Phys. Rev. E 73 (6), 066303.Google Scholar
Roberts, P. H. 1967 An Introduction to Magnetohydrodynamics. Longmans.Google Scholar
Rüdiger, G., Hollerbach, R., Schultz, M. & Elstner, D. 2007 Destabilization of hydrodynamically stable rotation laws by azimuthal magnetic fields. Mon. Not. R. Astron. Soc. 377 (4), 1481–1487.Google Scholar
Rüdiger, G., Schultz, M. & Gellert, M. 2011 The Tayler instability of toroidal magnetic fields in a columnar gallium experiment. Astron. Nachr. 332 (1), 17–23.Google Scholar
Seilmayer, M., Stefani, F., Gundrum, Th., Weier, T., Gerbeth, G., Gellert, M. & Rüdiger, G. 2012 Experimental evidence for a transient Tayler instability in a cylindrical liquid–metal column. Phys. Rev. Lett. 108 (24), 244501.Google Scholar
Sommeria, J. & Moreau, R. 1982 Why, how, and when, MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507–518.Google Scholar
Spruit, H. C. 1999 Differential rotation and magnetic fields in stellar interiors. Astron. Astrophys. 349, 189–202.Google Scholar
Stefani, F., Weier, T., Gundrum, Th. & Gerbeth, G. 2011 How to circumvent the size limitation of liquid metal batteries due to the Tayler instability. Energy Convers. Manage. 52 (8), 2982–2986.Google Scholar
Tayler, R. J. 1957 Hydromagnetic instabilities of an ideally conducting fluid. Proc. Phys. Soc. B 70 (1), 31–48.Google Scholar
Tayler, R. J. 1960 Stability of twisted magnetic fields in a fluid of finite electrical conductivity. Rev. Mod. Phys. 32, 907–913.Google Scholar
Tayler, R. J. 1961 Necessary and sufficient conditions for the hydromagnetic Rayleigh–Taylor stability of a cylindrical plasma. Plasma Phys. 3 (4), 266–272.Google Scholar
Tayler, R. J. 1973 The adiabatic stability of stars containing magnetic fields – I: toroidal fields. Mon. Not. R. Astron. Soc. 161 (4), 365–380.Google Scholar
Vandakurov, Yu. V. 1972 Theory for the stability of a star with a toroidal magnetic field. Sov. Astron. 16, 265–272.Google Scholar
Velikhov, E. P. 1959 Stability of an ideally conducting liquid flowing between rotating cylinders in a magnetic field. Sov. Phys. JETP 9, 995–998.Google Scholar
Wang, K., Jiang, K., Chung, B., Ouchi, T., Burke, P. J., Boysen, D. A., Bradwell, D. J., Kim, H., Muecke, U. & Sadoway, D. R. 2014 Lithium–antimony–lead liquid metal battery for grid-level energy storage. Nature 514 (7522), 348–350.Google Scholar
Weber, N., Galindo, V., Priede, J., Stefani, F. & Weier, T. 2015 The influence of current collectors on Tayler instability and electro-vortex flows in liquid metal batteries. Phys. Fluids 27 (1), 014103.Google Scholar
Weber, N., Galindo, V., Stefani, F., Weier, T. & Wondrak, Th. 2013 Numerical simulation of the Tayler instability in liquid metals. New J. Phys. 15 (4), 043034.Google Scholar