Weighted-residual integral boundary-layer model for the nonlinear dynamics of thin liquid films failing on an undulating vertical wall

A set of first-order weighted-residual integral boundary-layer equations describing the nonlinear dynamics of a thin liquid film falling on a corrugated periodic vertical wall is derived. The spatiotemporal dynamics of the films is analyzed analytically and numerically in the framework of this set. A steady-state flow is found in the case of an asymptotically small wall corrugation and its stability is investigated. It is found that steady flow regimes arise in the case of a relatively small wall wavelength for the Reynolds number below its critical value corresponding to the flat-wall flow and for larger amplitudes of the wall corrugation when the Reynolds number exceeds its critical value. In the case of a larger wall wavelength, the emerging flows are either genuinely nonstationary or time periodic. The temporal period of the time-periodic flows increases with the amplitude of the wall corrugation and decreases with the Reynolds number. A possibility of the emergence of reverse flows is also discussed. (C) 2008 American Institute of Physics.