Behavior of a liquid jet in a static electric field is determined depending on combined effects of electric and fluid forces.In particular,instabilities and breakup have been theoretically and experimentally investigated mainly in a liquid column jet for practical use.However,they have not been sufficiently investigated in a liquid planar jet except for instabilities of periodic disturbances in a uniform sheet,in spite of the importance in applications to producing thin liquid sheets.In this work,the instabilities and breakup process of a two-dimensional incompressible viscous planar jet are analytically investigated when the jet is unsteady and nonuniform.In our analytical model,the jet is placed between two parallel less conductive walls on which an external electric field is imposed.According to the leaky dielectric model for surface charges and a membrane approximation for long wave deformations,nonlinear evolution equations for the jet thickness,mid-plane,velocities and surface electric charge densities are derived without consideration of surrounding gas motions.These equations are numerically solved under the initial-boundary condition that the flow is emanating from a slit nozzle exit.It is found that there exist two types of typical behavior: one is(i)the surface tension dominant mode where a tip of the jet grows larger as the jet flows to the downstream and the other is(ii)the electric force dominant mode where the tip of the jet is accelerated and becomes thinner and thinner to the downstream.The existing regions of these two modes are determined for only two parameters of Pe(= convective current/conductive current)and Λ(= electric pressure/fluid inertial force).Then,critical curves are found in the Pe-Λ parameter space,across which the mode is transferred from(i)to(ii)with the increase of Λ and/or the decrease of Pe.It is also found that the jet becomes unstable when a disturbance is applied to the thickness direction at the nozzle exit,where fluctuations of the jet are amplified as far as the jet is thick.However,they are suppressed as the jet becomes extremely thin,whose extent is more enhanced with the increase of Λ.