Simulation of the Tayler instability in liquid metals

Liquid metal batteries (LMBs), containing a stable density stratification of a heavy metal, a salt-electrolyte and a light-weight metal on the top are recently discussed as a possible means for large scale electricity storage.
Due to their completely liquid interior, LMBs are susceptible to fluid dynamic instabilities. One example is the current driven Tayler instability (TI), which will set the liquid interior in motion. Consequences of a sufficiently strong flow are electrode/electrolyte deformations, which may lead to a direct contact of the electrodes and – consequently – a battery failure.
While a simulation of the TI in astrophysical plasmas may be carried out by directly solving the induction equation, the typical very low magnetic Prandtl numbers of liquid metals makes it impossible to use this approach in our case.
For the simulation of the TI in one liquid metal electrode, we couple the Navier-Stokes-equation with a Poisson equation for the electric potential complemented by a magnetic field calculation using Biot-Savart’s law. This integro-differential formulation is implemented in the open source library OpenFOAM. Simulation matches fairly well the results obtained by a recent TI experiment. Further studies on the TI, as the influence of the batteries aspect ratio, helical flows as well as countermeasures to avoid the instability are discussed.