Stability of a layer of dielectric fluid subjected to corona discharge: long wave approach

Experiments have shown that there exists an instability associated with corona discharge from a point to a layer of dielectric liquid. Above a certain voltage of the point electrode, the surface of the liquid deforms and convection appears in the form of large cells (rose-window instability). The typical width of the cells is much larger than the depth of the liquid layer. This justifies the analysis of the instability focusing in the behaviour of perturbations of small wave number (large wavelength). We write the linear equations for the instability of the liquid surface and solve it analytically, neglecting the liquid motion. We also study the role of the non-dimensional parameters associated with the problem.