An instability analysis of two–dimensional liquid–metal flows in a straight duct with free–slip boundary condition applied on the walls y = 0 and y = 2π is conducted. The basic flow under examination is a Kolmogorov flow and it is driven by a transverse magnetic field interacting with an electric current supplied by lines of electrodes positioned in the bottom of the duct. It is proved by a rigorous theoretical analysis that all the secondary flows transitional from the basic flow are self–oscillations and that some secondary flows only arise when the Reynolds number R passes through a critical value Rc. That is, the instabilities are analytically proved to be supercritical Hopf bifurcations. Simple numerical predictions confirm the theoretical analysis.