Magnetohydrodynamic instability with bichromatic perturbations in aluminum reduction cells

The linear stability of the electrically conducting liquid-liquid interface is investigated in a finite, rectangular domain inspired by the aluminum reduction cells. The interfacial instability is known to be the effect of the Lorentz force produced by the interaction of the perturbed horizontal current with the vertical magnetic field. While much of the literature points to the analysis of monochromatic perturbations, the role of the bichromatic type is investigated, for the first time. Considering a bichromatic perturbation, with independent wavenumbers along the horizontal directions, a dispersion relation is derived that includes the effects of Lorentz force, friction, interfacial tension, and gravity. This work reveals that the horizontal component of the magnetic field can also introduce instability, particularly when friction is negligible. While the vertical magnetic field is predicted to be mainly responsible for the instability of the interface with lower modes, the horizontal magnetic field destabilizes all the modes. For a fixed value of current and current density, the inter-electrode gap at neutral stability increases with an increase in the ratio of cell length to width. The present findings are applicable to any general electrically conducting two-fluid interface, in the presence of an ambient magnetic field.