Nonlinear instability of a thin film flowing down a smoothly deformed surface

Here, the nonlinear flow instability of a thin film flowing down an inclined wall with smooth deformations is investigated. A nonlinear evolution equation is obtained under the small wavenumber approximation. This equation describes the film free surface deformation including the effects of inertia, viscosity, surface tension, and deformation of the wall. The equation has a forcing term that corresponds to periodic time-dependent perturbations hitting on the free surface. These time-dependent perturbations are tested against the presence of the free surface waves due to wall deformations. In particular, a wall sinusoidal profile is investigated. It is found that when the wall wavelength is small enough, the valleys of the surface response to the wall profile become very deep, showing a kind of resonance, which has an important stabilizing influence in the evolution of the superposed time-dependent perturbations. (c) 2007 American Institute of Physics.