Finite-amplitude surface waves on a thin film flow subject to a unipolar-charge injection

The interaction of an electric field with a dielectric liquid film is investigated as it drains under gravity down an inclined plane electrode emitting uniform positive ions into the liquid region. By applying long-wave approximation to the governing equations, the evolution equation for the free surface is derived up to the first order of a thin film parameter ξ To investigate the space charge effect on the development of a finite-amplitude surface wave, a neutral stability condition is obtained as a critical Reynolds number through a linear stability analysis, and the amplitude and velocity of a periodic disturbance are also calculated within a supercritically stable flow region. The presence of a unipolar space charge in the fluid makes a steady surface wave take on even higher amplitude and faster wave speed compared with the case of no space charge.