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We consider a thin horizontal layer of a non-magnetic electrolyte containinga bulk solution of salt and carrying an electric current. The layer is boundedby two deformable free surfaces loaded with an insoluble surfactant and isplaced in a vertical magnetic field. The arising Lorentz force drives theelectrolyte in the plane of the layer. We employ the long-wave approximation toderive general two-dimensional hydrodynamic equations describing symmetricpinching-type deformations of the free surfaces. These equations are used tostudy the azimuthal flow in an annular film spanning the gap between twocoaxial cylindrical electrodes. In weakly deformed films, the base azimuthalflow and its linear stability with respect to azimuthally invariantperturbations are studied analytically. For relatively thick layers and weakmagnetic fields, the leading mode with the smallest decay rate is found tocorrespond to a monotonic azimuthal velocity perturbation. The Marangoni effectleads to further stabilisation of the flow while perturbations of the soluteconcentration in the bulk of the fluid have no influence on the flow stability.In strongly deformed films in the diffusion-dominated regime, the azimuthalflow becomes linearly unstable with respect to an oscillatory mixed modecharacterised by the combination of radial and azimuthal velocity perturbationswhen the voltage applied between electrodes exceeds the critical value.

Electromagnetically driven flow in unsupported electrolyte layers: lubrication theory and linear stability of annular flow