Convective instability of dielectric liquid subjected to unipolar injection of charge trough free surface

A problem for the Marangoni instability in a liquid layer subjected to an injection of ions is studied. The eigenvalue problem obtained after the linearization of the governing equations is solved by using the Runge-Kutta-Merson´s method of fourth order. When an upper surface is deformable, we found that both stationary and oscillatory instabilities may occur. Two kinds of stationary instabilities have been detected. The corresponding critical wave number k, the Marangoni number Ma and electrical analog of the Rayleigh number Ra have been obtained numerically. Besides, we have revealed two types of wave instability: the gravity-capillar wave and thermocapillary wave with non-zero frequency ω at k=0