Solitary waves running down a vertically falling film: low-dimensional models

We consider the wavy regime of vertically falling fluid film. Derivation of LDM for this flow, based on the LW expension have a long history(see review in[1]). A crucial test of such models is the correct prediction of the properties of SW as a function of the distance from the instability threshold. The latter is usually quantified in terms of the reduced $Re$, $\delta=3^\frac{4}{3}Re^\frac{11}{9}Ka^\frac{-1}{3}$. Though most models predict similar behavior in the drag-gravity regime($\delta\ll1$), they exhibit large differences from each other in the drag-inertia regime($\delta\gg1$). Characteristics of SW solutions to available LDM are shown and contrasted with DNS results. The best agreement with DNS is found with the 4-equation model derived in[2] and the convergence rate of the wave speed to the asymptotic limit $c_\infty$ at $\delta\to\infty$ is affected by viscous diffusion terms and is governed by the $Re$ as $|c-c_\infty|\propto1/Re^2$. The asymptotic behavior of the speed, amplitude, lengths of the wave-tail and front capillary ripples are discussed. References: [1] R.V.Craster, O.K.Matar. Reviews of modern physics 81 (2009). [2] C.Ruyer-Quil, P.Manneville. Eur.Phys.J. 15 357-369 (2000).