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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
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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