Fractality of metal pad instability threshold in rectangular cells
| Autor: | Gerasimos Politis, Jānis Priede |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Physics
Mechanical Engineering Physics - Fluid Dynamics Mechanics Condensed Matter Physics 01 natural sciences Instability 010305 fluids & plasmas Magnetic field Mechanics of Materials Inviscid flow 0103 physical sciences Gravity wave Magnetohydrodynamics Electric current 010306 general physics Eigenvalue perturbation Linear stability |
| Zdroj: | Journal of Fluid Mechanics. 915 |
| ISSN: | 1469-7645 0022-1120 |
| DOI: | 10.1017/jfm.2021.100 |
| Popis: | 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 $\alpha^{2}\approx2.13$ which have the highest stability threshold corresponding to the electromagnetic interaction parameter $\beta\approx 4.7$. Comment: 21 pages, 6 figures (final version; to appear in J. Fluid Mech.) |
| Databáze: | OpenAIRE |
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