Shallow water modeling of rolling pad instability in liquid metal batteries

Magnetohydrodynamically induced interface instability in liquid metal batteries is analyzed. The batteries are represented by a simplified system in the form of a rectangular cell, in which strong vertical electric current flows through three horizontal layers: the layer of a heavy metal at the bottom, the layer of a light metal at the top, and the layer of electrolyte in the middle. A new two-dimensional nonlinear model based on the conservative shallow water approximation is derived and utilized in a numerical study. It is found that in the case of small density difference between the electrolyte and one of the metals, the instability closely resembles the rolling pad instability observed earlier in the aluminum reduction cells. When the two electrolyte-metal density differences are comparable, the dynamics of unstable systems is more complex and characterized by interaction between two nearly symmetric or antisymmetric interfacial waves.