Long-wavelength equation for vertically falling films

An equation is derived for describing wave evolution on the surface of a vertically falling viscous film. The traditional long-wavelength scaling is replaced by a new scaling to reduce the (We) must be used instead of the Reynolds number (Re) to distinguish between viscous and inertia dominated regimes for vertically falling films. This equation includes viscous dissipation and pressure correction terms that are missing in the existing single evolution equations at the same order. Comparison of the neutral stability curves and growth rates predicted by different models to that of the Orr-Sommerfeld (OS) equation shows that our equation matches with the OS results better than the existing single evolution equations. However, our equation is not free from finite time blowup. Selective regularization leads to a two mode model in flow rate and film thickness. The regularized equation is free from finite time blowup and predicts two families of solitary waves. Numerical simulations of the derived equation and its regularized version in the traveling wave coordinate show the transition of wave structure from regular (periodic) to chaotic profiles. Model predictions on maximum wave amplitude on the low celerity branch show good agreement with experimental data.