Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations

We study the reliability of two-dimensional models of film flows down inclined planes obtained by us [Ruyer-Quil and Manneville, Eur. Phys. J. B 15, 357 (2000)] using weighted-residual methods combined with a standard long-wavelength expansion. Such models typically involve the local thickness h of the film, the local flow rate q, and possibly other local quantities averaged over the thickness, thus eliminating the cross-stream degrees of freedom. At the linear stage, the predicted properties of the wave packets are in excellent agreement with exact results obtained by Brevdo et al. [J. Fluid Mech. 396, 37 (1999)]. The nonlinear development of waves is also satisfactorily recovered as evidenced by comparisons with laboratory experiments by Liu et al. [Phys. Fluids 7, 55 (1995)] and with numerical simulations by Ramaswamy et al. [J. Fluid Mech. 325, 163 (1996)]. Within the modeling strategy based on a polynomial expansion of the velocity field, optimal models have been shown to exist at a given order in the long-wavelength expansion. Convergence towards the optimum is studied as the order of the weighted-residual approximation is increased. Our models accurately and economically predict linear and nonlinear properties of film flows up to relatively high Reynolds numbers, thus offering valuable theoretical and applied study perspectives.