In this paper we investigate the Tayler instability in an incompressible, viscous and resistive liquid metal column and in a model of a liquid metal battery (LMB). Detailed comparisons between theory and numerics, both in linear and nonlinear regimes, are performed. We identify the timescale that is well adapted to the quasi-static (QS) regime and find the range of Hartmann numbers where this approximation applies. The scaling law
for the amplitude of the Tayler destabilized flow is explained using a weakly nonlinear argument. We calculate a critical electrolyte height above which the Tayler instability is too weak to disrupt the electrolyte layer in a LMB. Applied to present day Mg-based batteries, this criterion shows that short circuits can occur only in very large batteries. Finally, preliminary results demonstrate the feasibility of direct numerical multiphase simulations of the Tayler instability in a model battery.