Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-19T15:53:00.764Z Has data issue: false hasContentIssue false
  • English
  • Français

Fractality of metal pad instability threshold in rectangular cells

Published online by Cambridge University Press:  25 March 2021

Gerasimos Politis  and
Jānis Priede Open the ORCID record for Jānis Priede [Opens in a new window]
Gerasimos Politis
Affiliation:
Fluid and Complex Systems Research Centre, Coventry University, CV1 5FBCoventry, UK
Jānis Priede*
Affiliation:
Fluid and Complex Systems Research Centre, Coventry University, CV1 5FBCoventry, UK
*
†Email address for correspondence: J.Priede@coventry.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse linear stability of interfacial waves in an idealised model of an aluminium reduction cell consisting of two stably stratified liquid layers which carry a vertical electric current in a collinear external magnetic field. If the product of electric current and magnetic field exceeds a certain critical threshold depending on the cell design, the electromagnetic coupling of gravity wave modes can give rise to a self-amplifying rotating interfacial wave which is known as the metal pad instability. Using the eigenvalue perturbation method, we show that, in the inviscid limit, rectangular cells of horizontal aspect ratios $\alpha =\sqrt {m/n}$, where $m$ and $n$ are any two odd numbers, can be destabilised by an infinitesimally weak electromagnetic interaction while cells of other aspect ratios have finite instability thresholds. This fractal distribution of critical aspect ratios, which form an absolutely discontinuous dense set of points interspersed with aspect ratios with non-zero stability thresholds, is confirmed by accurate numerical solution of the linear stability problem. Although the fractality vanishes when viscous friction is taken into account, the instability threshold is smoothed out gradually and its principal structure, which is dominated by the major critical aspect ratios corresponding to moderate values of $m$ and $n$, is well-preserved up to relatively large dimensionless viscous friction coefficients $\gamma \sim 0.1$. With a small viscous friction, the most stable are cells with $\alpha ^{2}\approx 2.13$ which have the highest stability threshold corresponding to the electromagnetic interaction parameter $\beta \approx 4.7$.

JFM classification

Geophysical and Geological Flows: Shallow water flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A101
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bojarevics, V. 1998 Nonlinear waves with electromagnetic interaction in aluminum electrolysis cells. In Progress in Fluid Flow Research: Turbulence and Applied MHD (ed. H. Branover), vol. 186, pp. 833–848. AIAA.CrossRefGoogle Scholar
Bojarevics, V. & Pericleous, K. 2006 Comparison of MHD models for aluminium reduction cells. In TMS Light Metals (ed. T.J. Galloway), pp. 347–352. TMS.Google Scholar
Bojarevics, V. & Romerio, M.V. 1994 Long wave instability of liquid metal-electrolyte interface in aluminum electrolysis cells: a generalization of Sele's criterion. Eur. J. Mech. B 13 (1), 33–56.Google Scholar
Boyd, J.P. 2013 Chebyshev and Fourier Spectral Methods: Second Revised Edition. Dover Publications.Google Scholar
Davidson, P.A. 2000 Overview overcoming instabilities in aluminium reduction cells: a route to cheaper aluminium. Mater. Sci. Technol. 16 (5), 475–479.CrossRefGoogle Scholar
Davidson, P.A. & Lindsay, R.I. 1998 Stability of interfacial waves in aluminium reduction cells. J. Fluid Mech. 362, 273–295.CrossRefGoogle Scholar
Evans, J.W. & Ziegler, D.P. 2007 The electrolytic production of aluminumn. In Encyclopedia of Electrochemistry (ed. A.J. Bard). Wiley.CrossRefGoogle Scholar
Gerbeau, J.-F., Le Bris, C. & Lelièvre, T. 2006 Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Clarendon Press.CrossRefGoogle 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.CrossRefGoogle Scholar
Herreman, W., Nore, C., Guermond, J.-L., Cappanera, L., Weber, N. & Horstmann, G.M. 2019 Perturbation theory for metal pad roll instability in cylindrical reduction cells. J. Fluid Mech. 878, 598–646.CrossRefGoogle Scholar
Hertzberg, T., Tørklep, K. & Øye, H.A. 2013 Viscosity of molten $\text {NaF}\text{--}\text {AlF}_3\text {--}\text {Al}_2\text {O}_3\text{--}\text {CaF}_2$ mixtures: selecting and fitting models in a complex system. In Essential Readings in Light Metals (ed. G. Bearne, M. Dupuis & G. Tarcy), pp. 19–24. Wiley.CrossRefGoogle Scholar
Hinch, E.J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Horstmann, G.M., Weber, N. & Weier, T. 2018 Coupling and stability of interfacial waves in liquid metal batteries. J. Fluid Mech. 845, 1–35.CrossRefGoogle Scholar
Horstmann, G.M., Wylega, M. & Weier, T. 2019 Measurement of interfacial wave dynamics in orbitally shaken cylindrical containers using ultrasound pulse-echo techniques. Exp. Fluids 60 (4), 56.CrossRefGoogle Scholar
Kelley, D.H. & Weier, T. 2018 Fluid mechanics of liquid metal batteries. Appl. Mech. Rev. 70 (2), 020801.CrossRefGoogle Scholar
Kim, H., et al. 2013 Liquid metal batteries: past, present, and future. Chem. Rev. 113 (3), 2075–2099.CrossRefGoogle ScholarPubMed
Lukyanov, A., El, G. & Molokov, S. 2001 Instability of MHD-modified interfacial gravity waves revisited. Phys. Lett. A 290 (3–4), 165–172.CrossRefGoogle Scholar
Molokov, S. 2018 The nature of interfacial instabilities in liquid metal batteries in a vertical magnetic field. Europhys. Lett. 121 (4), 44001.CrossRefGoogle Scholar
Molokov, S., El, G. & Lukyanov, A. 2011 Classification of instability modes in a model of aluminium reduction cells with a uniform magnetic field. Theor. Comput. Fluid Dyn. 25 (5), 261–279.CrossRefGoogle Scholar
Moreau, R.J. & Ziegler, D. 1988 The Moreau–Evans hydrodynamic model applied to actual Hall–Héroult cells. Metall. Trans. B 19 (5), 737–744.CrossRefGoogle Scholar
Morris, S.J.S. & Davidson, P.A. 2003 Hydromagnetic edge waves and instability in reduction cells. J. Fluid Mech. 493, 121–130.CrossRefGoogle Scholar
Pedchenko, A., Molokov, S., Priede, J., Lukyanov, A. & Thomas, P.J. 2009 Experimental model of the interfacial instability in aluminium reduction cells. Europhys. Lett. 88 (2), 24001.CrossRefGoogle Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Viscous Flow. Springer.CrossRefGoogle Scholar
Priede, J. 2017 Elementary model of internal electromagnetic pinch-type instability. J. Fluid Mech. 816, 705–718.CrossRefGoogle Scholar
Priede, J. 2020 Self-contained two-layer shallow water theory of strong internal bores. arXiv:1806.06041.Google Scholar
Roberts, P.H. 1967 An Introduction to Magnetohydrodynamics. Longmans.Google Scholar
Sele, T. 1977 Instabilities of the metal surface in electrolytic alumina reduction cells. Metall. Trans. B 8 (3), 613–618.CrossRefGoogle Scholar
Sneyd, A.D. 1985 Stability of fluid layers carrying a normal electric current. J. Fluid Mech. 156, 223–236.CrossRefGoogle Scholar
Sneyd, A.D. & Wang, A. 1994 Interfacial instability due to MHD mode coupling in aluminium reduction cells. J. Fluid Mech. 263, 343–360.CrossRefGoogle Scholar
Stefani, F., Weier, T., Gundrum, T. & 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.CrossRefGoogle Scholar
Tucs, A., Bojarevics, V. & Pericleous, K. 2018 a Magneto-hydrodynamic stability of a liquid metal battery in discharge. Europhys. Lett. 124 (2), 24001.CrossRefGoogle Scholar
Tucs, A., Bojarevics, V. & Pericleous, K. 2018 b Magnetohydrodynamic stability of large scale liquid metal batteries. J. Fluid Mech. 852, 453–483.CrossRefGoogle Scholar
Urata, N. 1985 Magnetics and metal pad instability. In Light Metals (ed. H.O. Bohner), pp. 581–589. Metallurgical Society of AIME.Google Scholar
Urata, N., Mori, K. & Ikeuchi, H. 1976 Behavior of bath and molten metal in aluminum electrolytic cell. J. Japan Inst. Light Met. 26 (11), 573–583.CrossRefGoogle 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.CrossRefGoogle Scholar
Zikanov, O. 2015 Metal pad instabilities in liquid metal batteries. Phys. Rev. E 92 (6), 063021.CrossRefGoogle ScholarPubMed
Zikanov, O., Thess, A., Davidson, P.A. & Ziegler, D.P. 2000 A new approach to numerical simulation of melt flows and interface instability in Hall–Heroult cells. Metall. Trans. B 31 (6), 1541–1550.CrossRefGoogle Scholar