In aluminium reduction cells the intense magnetic fields and associated Lorentz forces drive steady flows in the cryolite and aluminium layers. Generally there is a discontinuity in velocity between the layers, which gives rise to a Kelvin-Helmholtz instability, and the study of this effect is the subject of the thesis. The role of the basic steady flow is focused in this analysis, while most previous MHD analyses neglect it and emphasise on the interaction between the interface wave and the electromagnetic field by the electromagnetic perturbation. The stationary magnetic field is considered important as a strong driving source for the steady velocity in our model. Steady velocity fields are studied in terms of a stream function and global stability criteria are discussed with disturbances. Non-linear evolution of flow is investigated to find a steady condition and show that steady flows are stable. A normal mode analysis is used to formulate wave evolution equations, which are integrated forwards in time. Also an improved treatment of electromagnetic damping is developed, which includes the effect of the induced electric field. Using a normal mode form of the steady-state current, magnetic field or flow, the scheme developed in this research is suitable for industrial application where these would be calculated numerically.