The paper reports on the behaviour and dynamics of direct current electric arc in an industrial electric arc furnace. Electric arcs are intense energy sources involved in many industrial processes. The behaviour of electric arc is simulated in a 2D axisymmetric geometry. A 40 kA current flows between two electrodes with a gap of 0.25 cm. The flow of current creates a very powerful jet up to km/s. Such speeds pose the question of the importance of compressibility and what is the extent of the effects of compressibility on arc behaviour. To assess the effect of compressibility, two different simulations are performed: an incompressible and a compressible simulation. The first simulation considers a temperature-dependent density based on experimental measurement whereas the latter adopts the ideal gas law to calculate the density variation of the plasma. The incompressible results were previously validated and compared to the well-known results predicted by the literature. The numerical results of the two models are reported and compared in terms of flow, thermal fields, and voltage drop. The results show that compressibility affects several aspects of the arc. As expected, the velocity drops when compressibility is present, however, the voltage drop increase significantly. Additionally, compressibility introduces a repetitive pattern of voltage drop over time which also depicted in the arc dynamics. The pattern is divided into three distinctive dynamics: (1) High-frequency low amplitude instabilities, followed by (2) low low-frequency high amplitude instabilities region, and finally, a relatively (3) stable interval of voltage with a bell-shaped arc.