Linear stability of a two-layer film flow down an inclined channel: A second-order weighted residual approach

Our interest is with the long-wavelength instability in a two-dimensional gravity-driven flow of two superposed, incompressible, immiscible, and viscous fluids bounded by an upper and a lower rigid plate. Approximate boundary-layer-like equations valid up to second order in the channel width-to-wavelength ratio are first derived. An extension of the weighted residual approach first suggested by Ruyer-Quil and Manneville [Eur. Phys. J. B 15, 357 (2000)] to model the single-layer flow is then presented. It is shown that a suitable choice of trial and weighted functions allows us to lower appreciably the dimensionality of the problem. Marginal stability results clearly indicate that unlike the Shkadov approach, which gives erroneous critical conditions, the proposed second-order model, its simplified version corresponding to parabolic velocity profiles and referred to as the one-mode Galerkin approach, and the lubrication theory (long wave expansion procedure) are all in good agreement with the Orr-Sommerfeld equations near criticality where the viscous effects are dominant. The one-mode Galerkin model is therefore sufficient to correctly predict the near-critical behavior of the long wave interfacial mode. As inertia increases, the lubrication theory provides results that diverge. The Shkadov approach gives bad results as well while the second-order weighted residual model and its simplified version continue to follow the numerical results up to moderate Reynolds numbers, the approximation being better for the former than for the latter. (C) 2007 American Institute of Physics.