Interfacial instability in aluminium reduction cells in a vertical magnetic field with a transverse gradient to the sidewall

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Interfacial instability in a two-fluid system under a vertical magnetic field with a transverse gradient to the sidewall is studied with reference to aluminium reduction cells. Analytical, asymptotic and numerical solutions are obtained for the semi-bounded domain. It is shown that for the magnetic field increasing from a sidewall instability is dominated by the boundary mode, as for the uniform field. In contrast, for the magnetic field decreasing from the sidewall, the most unstable mode is determined mainly by the gradient of the magnetic field.