Long-wave instabilities of film flow under an electrostatic field

The free-surface behavior of a viscous liquid layer flowing down an inclined plane by gravity and interacting with an overlying uniform electrostatic field is examined in the limit of long-wave approximation. Both linear and nonlinear stability analyses are performed to address two-dimensional surface-wave evolution initiating from a flat interface. The growth of a periodic disturbance is first investigated for a linear analysis, and then to examine the nonlinear surface-wave instabilities the evolution equation for film height is solved numerically by a Fourier-spectral method. For small evolution time the calculated nonlinear modes of instability are consistent with the results obtained from the linear theory. The effect of an electrostatic field increases the wavenumbers showing a maximum linear growth rate as well as a cutoff. A significant phenomenon as Reynolds number is increasing is the appearance of the catastrophic surface waves in the long run whenever any initial wavenumber making a traveling wave linearly unstable is employed into the initial simple-harmonic disturbance.